Research ArticleThe Effect of Seasonal Weather Variation onthe Dynamics of the Plague Disease
Rigobert C Ngeleja1 Livingstone S Luboobi12 and Yaw Nkansah-Gyekye1
1Nelson Mandela African Institution of Science and Technology (NM-AIST) Arusha Tanzania2Department of Mathematics Makerere University PO Box 7062 Kampala Uganda
Correspondence should be addressed to Rigobert C Ngeleja rngelejayahoocom
Received 21 February 2017 Accepted 28 June 2017 Published 10 August 2017
Academic Editor Ram N Mohapatra
Copyright copy 2017 Rigobert C Ngeleja et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Plague is a historic disease which is also known to be the most devastating disease that ever occurred in human history causedby gram-negative bacteria known as Yersinia pestis The disease is mostly affected by variations of weather conditions as it disturbsthe normal behavior of main plague disease transmission agents namely human beings rodents fleas and pathogens in theenvironment This in turn changes the way they interact with each other and ultimately leads to a periodic transmission of plaguedisease In this paper we formulate a periodic epidemic model system by incorporating seasonal transmission rate in order tostudy the effect of seasonal weather variation on the dynamics of plague disease We compute the basic reproduction number of aproposed model We then use numerical simulation to illustrate the effect of different weather dependent parameters on the basicreproduction number We are able to deduce that infection rate progression rates from primary forms of plague disease to moresevere forms of plague disease and the infectious flea abundance affect to a large extent the number of bubonic septicemic andpneumonic plague infective agents We recommend that it is more reasonable to consider these factors that have been shown tohave a significant effect on 119877119879 for effective control strategies
1 Introduction
Plague is the ancient disease caused by the bacteriumYersiniapestis and has had significant effects on human societiesthroughout the history [1] Dynamics of plague disease arethe result of complex interactions between human beingsrodent population flea population and pathogens in theenvironment Seasonal variation particularly temperaturehumidity rainfall and precipitation greatly affects the normaltransmission capacity of plague disease by either lowering itor raising it It affects pathogen in the environment fleasrodents and even human behavior by altering their normalimmigration rate death rate survival rate and infectiouscapability [2]
11 Seasonality in Flea Development Stages and BehaviorFlearsquos survival is greatly affected by temperature and relativehumidity [3] The ectothermic characteristics of fleas make
them very sensitive to temperature fluctuations Xenopsyllacheopis is the primary vector flea for Yersinia pestis It issignificantly affected by seasonal weather variation as mostof its life stages depend on temperature humidity andprecipitation The rate of metamorphosis of this kind of fleafrom egg to adult is also regulated by temperature
Flea larvae feed on almost any organic debris but mostlythey feed on adult excreta which consist of relatively undi-gested blood [4]This adult fecalmatter when dried falls fromthe host to serve as food for the larvae Thus the availabilityof food (dried flea dirt) for larvae to feed depends on theweather condition particularly temperature and humidityThe larvae develop well in areas where the relative humidityis greater than 75 percent and the temperature is between21∘C and 32∘C [5 6] At constant temperature fleas becomemost sensitive to air saturation and are massively killedwhen the air saturation is insufficiency [7] Considering thefact that all immature flea stages occur outside the host
HindawiInternational Journal of Mathematics and Mathematical SciencesVolume 2017 Article ID 5058085 25 pageshttpsdoiorg10115520175058085
2 International Journal of Mathematics and Mathematical Sciences
development rates of flea increase with temperature until itreaches a critical value which makes flea most vulnerableHigh temperature combined with low humidity hinders flearsquossurvival at immature stages [8]
The condition where relative humidity is below 50 isunfavorable for flea growth It is at this condition that thebiting rate of flea onto the infected human and rodent or ofthe infected flea onto the susceptible human and rodent issignificantly low But when the relative humidity is 80 theflea becomes very active and as a result the biting rate andinfection increase significantly Moreover when temperatureis above 275∘C the rapid disappearance of plague bacilli fromthe flea stomach occurs resulting in reduced rates of plaguedisease transmissionThis in turn reduces the flearsquos efficiencyin its ability to transmit the plague bacillus to human beingsand rodents [9 10]
When fleas are in rodent burrows their survival ofimmature stages is affected by soil moisture that is partlycontrolled by outside precipitation [11] As a way of gettingrid of detrimental moisture losses and temperature swingsrodents normally shift to start living underground [12]On the other hand when they are attached with a highorganic load excessively wet conditions in rodent burrows(eg relative humidity 95) can stimulate the growth ofdestructive fungi that diminish flearsquos larval and egg survival[13]
Different studies justify the negative correlation betweenrainfall and plague epidemics For example Cavanaugh andMarshall Jr [3] reported that in areaswhere drains are absentorwhere drainage is insufficient as a result of soil compositionor impoundments of water flooding unquestionably causes adrop in the flea population In areas with improved drainagesuch as those with sandy soils the lessening of the fleapopulation is minimal Precipitation also influences plagueinfection for it influences the concentration of rodents fleasand humans in the same shelter
12 Seasonality in Rodents The direct effect posed on rodentpopulation due to temperature change is minor This is dueto the fact that rodents are homoeothermic and hence donot respond immediately to changes in ambient temperatures[14] Temperature indirectly affects the spread of plague inrodent population in different ways as follows at a low meantemperature of 10∘C the bacteria within host (rodent) becomevery active as a result a large number of infected rodents dyingbefore even the plague bacilli appear in their blood At thisparticular temperature rodents also lose the ability to infectother susceptible individuals
Rainfall may pose positive or negative effect on theincrease of rodent population depending on its intensity[11] A season of moderate rainfall may be considered toaffect positively the increase of rodent abundance but whenthe amount of rainfall is extremely heavy it results ina tremendous rodent population decline [15] When it ismoderate and upon a proper timing rainfall may foster theincrease of rodent population [3] This is due to the fact thatrodentrsquos reproduction period normally follows wet seasons[16ndash18] That is to say the increase of rodent population
during wet period is expected to be higher than that duringthe dry seasons This clearly concurs with the result in thestudy by Leirs et al [19] which narrates that in Tanzaniarodent population densities show clear association with theannual rainfall and its seasonal distribution However whenrainfall is of high intensity it causes flooding of rodentburrows Large number of rodents population dies and theremaining ones normally move from forest to the householdswhere they can protect themselves [3 8 20] In other casesincreased precipitation or drought stalwartly disturbs rodentpopulation dynamics as it deters food availability
13 Seasonality in Pathogens in the Environment When thebacteria are in lungs the transmission of Yersinia pestisis possible through various ways contact transmissionin which one may be infected through physical contactwith respiratory particles on the infected surface airbornetransmission which occurs through inhaling the bacte-ria causing the disease through successive contact withthe nose or mouth of an infected individual respiratoryparticles which occurs through respiratory droplets whichis through shedding of respiratory particles (ie dropletsor aerosols) from an infected human or rodent into theenvironment [21]
Extreme temperatures regularly are ruinous to the sur-vival of pathogens causing plague The changes in tempera-ture may lead to varying effects on the pathogens in the envi-ronment and vectors that live in an environment When themean temperature approaches the maximum limit that canbe endured by the pathogens a small increase in temperaturemay be very dangerous to the pathogen survival Converselywhen pathogens are in the environment characterized by lowmean temperature a small increase in temperaturemay resultin increased development incubation and replication of thepathogen in the environment [22 23]
Davis [24] compared the seasonal incidence of plaguewith usual atmospheric conditions in particular temperatureand rainfall It was depicted that human plague is morefrequent in warm moist weather between 15∘C and 27∘Cthan in hot dry (over 27∘C) or cold weather (under 15∘C)Mitscherlich and Marth [25] narrate that the solar exerts adetrimental effect on bacterial aerosol and the decay rate ofYersinia pestis is proportional to the increase of UV light
The reports by Ayyadurai et al [26] and Mollaret [27]justify the ability of the Yersinia pestis to culture the organismfromdeepwithin contaminated soil Eisen et al [28]were ableto show the great potential durability of Yersinia pestis in thesoil substrate The long duration of their survival in the soilsupports indirectly the virulence maintenance
Yersinia pestis exhibit a very slow growth at the temper-ature between 35∘C and 37∘C but they grow very fast at thetemperature 28∘CTheydie very rapid if exposed to aUV lightor temperature exceeding 40∘C or when exposed to intensivedesiccation [29ndash31] Bacteria decrease their sensitivity whenthe level of humidity drops below 76 [25]
When an infected individual coughs or sneezes thou-sands of the bacteria are released in air [32] The releasedrespiratory particles may be large and heavy that they cannot
International Journal of Mathematics and Mathematical Sciences 3
remain suspended in the air When respiratory particles arelarge the transmission can only occurwhen these particles areexpelled directly onto another close susceptible individual Insome cases the release of smaller respiratory particles mayoccur this is when the airborne transmission is possibleThe smaller released particles are easily suspended in the airrespired (ie passed to the lower respiratory tract) [33]
Relative humidity and temperature affect the transmis-sion of Yersinia pestis from one individual to the otherHumidity affects the size of the respiratory particle [34]When humidity is low the large drops partially evaporate tocreate smaller lighter drops that are more likely to remainairborne for extended periods of time [35] That is to saywhen the air is sufficiently dry the large sized particles shrinkto a size that favors long-range transport which in turn leadsto increased infection
14 Seasonality in Human Behavior Human activities andbehavior in plague-infected areas are also to be consideredas important determinants of plague transmission to andby humans [42] When occurrences of plague are due tohuman intrusions in natural plague areas it is thus importantto consider season variation as a second-order variablethat influences disease incidence through human behaviorIn Tanzania drought and famine which are the result oflack of rainfall and temperature fluctuation have a greatimpact on the farmers and pastoralists as they force themto move from one area to another searching for food forthemselves and their cattle These human intrusions fromone place to another may lead to the increase of plaguedisease transmission in rodents fleas humanpopulation andpathogens in the environment
2 Model Formulation
We describe the complex interaction that leads to plaguedisease transmission and use it to formulate a model for thedynamics of the plague disease coupled with the effect ofseasonal weather variation in its transmission The modelincludes four populations namely human beings rodentsfleas and pathogens in the environment We generallyassume that all individuals from each population are sus-ceptible to the disease the recovered individuals confertemporary immunity and return to be susceptible again andthe infectious are all individuals with either bubonic plagueor pneumonic or septicemic plague
21 Variables and Parameters Used in the Model In Nota-tions and Table 1 we present variables and parameters theirdescription and their values as used in the model We haveobtained the parameter values from the literature that relateto this study and the present information on plague diseaseand through estimation
22 Model Description The human population is dividedinto six subgroups the subgroup of people who have notcontracted the disease to be referred to as susceptible anddenoted by 119878119867 but may get it if they come into contact
with 119868119867119878 119868119867119875 119868119877119878 119868119867119875 119868119865 or 119860 people who have thedisease but have not shown any symptom and are incapableof transmitting the disease to be referred to as exposedand denoted by 119864119867 those who are infected and capable oftransmitting the disease are divided into three subgroupsthere are those who have bubonic plague denoted by 119868119867119861those with septicemic plague denoted by 119868119867119878 and thosewho have pneumonic plague disease denoted by 119868119867119875 Thefraction of population in 119868119867119861 if treated or through strong bodyimmunity may recover and move to subgroup 119877119867 otherwisethey progress either to a septicemic disease infective agent119868119867119878 or to pneumonic plague disease infective agent 119868119867119875 orelse they die The population in the subgroup 119868119867119878 throughstrong body immunity or if treated recover and progressto the subgroup 119877119867 and if not treated they progress andjoin subgroup 119868119867119875 otherwise they die The population ofthe subgroup 119868119867119875 is considered as a very dangerous stage ofplague disease it is a very fatal stage of plague disease withthe fatality rate of about 100 however if treated they recoverand join subgroup119877119867 otherwise they die So the total humanpopulation1198731 is as given by1198731 = 119878119867 + 119864119867 + 119868119867119861 + 119868119867119878 + 119868119867119875 + 119877119867 (1)
Fleas are divided into two subgroups those who have notcontracted the disease butmay get it if they get in contact withinfectious agent (rodent or human) referred to as susceptibleflea and denoted by 119878119865 and those who are infected and arecapable of transmitting the disease referred to as infectiveagents and denoted by 119868119865 The total flea population 1198732 is asgiven by 1198732 = 119878119865 + 119868119865 (2)
The rodents are divided into five subgroups those whohave not contracted the disease but may get it if they getin contact with 119868119867119878 119868119867119875 119868119877119878 119868119867119875 119868119865 or 119860 referred to assusceptible rodents and denoted by 119878119877 those who have thedisease but have not shown any symptom and are incapableof transmitting the disease referred to as exposed anddenotedby 119864119877 those who are infected and capable of transmittingthe disease are divided into three subgroups those who havebubonic plague denoted by 119868119877119861 those with septicemic plaguedenoted by 119868119877119878 and those who have pneumonic plague 119868119877119875The fraction of population in 119868119877119861 may progress either to asepticemic plague disease infective agent 119868119877119878 or to pneumonicplague disease infective agent 119868119877119875 The rodent population inthe subgroup 119868119877119878 may either progress to pneumonic plaguedisease infective agent 119868119877119875 otherwise they dieThepopulationin the subgroup 119868119877119875 is considered as a very dangerous stage ofplague disease and very fatal so the mortality due to diseasein this subgroup is approximated to be 100 Then the totalrodent population1198733 is as given by1198733 = 119878119877 + 119864119877 + 119868119877119861 + 119868119877119878 + 119868119877119875 (3)
The individuals with pneumonic plague may releasepathogens causing plague disease to the environmentdenoted by 119860 through coughing or sneezing When thecondition in soilenvironment is favorable pathogens may
4 International Journal of Mathematics and Mathematical Sciences
Table 1 Parameters and their description
Parameters Description Value ReferencesourceΓ119903119887119891(119905) Adequate contact rate between 119868119877119861 and flea 01 Eisen et al [36]Γ119903119904119891(119905) Adequate contact rate between 119868119877119878 and flea 01 Eisen et al [36]Γ119891ℎ(119905) Adequate contact rate between 119868119865 and human 00641 Eisen et al [36]Γ119891119903(119905) Adequate contact rate between 119868119865 and rodent 00641 Eisen et al [36]Γℎ119901ℎ(119905) Adequate contact rate between 119868119867119875 and 119878119867 039 EstimatedΓℎ119904ℎ(119905) Adequate contact rate between 119868119867119878 and 119878119867 012 EstimatedΓ119903119887ℎ(119905) Adequate contact rate between 119868119877119861 and 119878119867Γ119903119901ℎ(119905) Adequate contact rate between 119868119877119875 and 119878119867 019 EstimatedΓ119903119904ℎ(119905) Adequate contact rate between 119868119877119878 and 119878119867 021 Estimated1205721 Progression rate of 119878119867 to 119864119867 population 099 Estimated1205722 Progression rate out of 119864119867 to infectious state 023 Gani and Leach [37]12058811205723 Progression rate out of 119868119867119861 to 11986811986711987512058821205723 Progression rate out of 119868119867119861 to 11987711986712058831205723 Progression rate out of 119868119867119861 to 1198681198671198781205751119887 Disease induced death rate of 119868119867119861 004 Keeling and Gilligan [38]1205724 Progression rate out of 119868119867119878 to 119868119867119875 and 119877119867 006 Estimated1205751119904 Disease induced death rate of 119868119867119878 004 Estimated1205725 Progression rate out of 119868119867119875 to 119877119867 04 Gani and Leach [37]1205751119901 Disease induced death rate of 119868119867119875 063 Kugeler et al [39]1205741 Progression rate of 119878119877 to 119864119877 092 EstimatedΓℎ119887119891(119905) Adequate contact rate between 119868119867119861 and flea 01 Eisen et al [36]Γℎ119904119891(119905) Adequate contact rate between 119868119867119878 and flea 01 Eisen et al [36]Γ119903119901119903(119905) Adequate contact rate between 119868119877119875 and 119878119877 09 EstimatedΓ119903119904119903(119905) Adequate contact rate between 119868119877119878 and 119878119877 09 EstimatedΓℎ119901119903(119905) Adequate contact rate between 119868119867119875 and 119878119877 000005 EstimatedΓℎ119904119903(119905) Adequate contact rate between 119868119867119878 and 119878119877 000008 Estimated1205742 The rate at which rodent becomes infectious 098 Estimated1205743 Progression rate out of 119868119877119861 to 119868119877119878 and 119868119877119875 0194 Tollenaere et al [40]1205753119887 Disease induced death rate of 119868119877119861 01 Estimated1205744 Progression rate out of 119868119877119878 to 119868119877119875 005 Estimated1205753119904 Disease induced death rate of 119868119877119878 73 Tollenaere et al [40]1205753119901 Disease induced death rate of 119868119877119875 014 Estimated120603 Progression rate of 119877119867 to 119878119867 033 Kugeler et al [39]1205831 Natural death rate for human being 004 Keeling and Gilligan [38]1205832 Natural death rate for flea 02 Bacot and Martin [7]1205833 Natural death rate for rodent 1 Morand and Harvey [41]1205961(119905) Adequate contact rate 119860 and human being1205962(119905) Adequate contact rate 119860 and rodent1205781(119905) Recruitment rate of 119860 by 119868119867119875 02 Estimated1205782(119905) Recruitment rate of 119860 by 119868119877119875 04 Estimated1205834 Natural death rate for pathogens 01 Estimated1205951 Recruitment rate of human beings 009 Estimated120595 Recruitment rate of fleas1205953 Recruitment rate of rodents
International Journal of Mathematics and Mathematical Sciences 5
remain infectious in the environment for a long timeWhen asusceptible individual adequately interacts with the environ-ment infestedwithYersinia pestis heshe gets the disease evenin the absence of any vector
23 Description of Interactions The susceptible fleas in sub-group 119878119865 get Yersinia pestis bacteria through biting theinfected rodent 119868119877119861 or 119868119877119878 who are the primary reservoir forthe bacteria and become infected at the rates Γ119903119887119891 and Γ119903119904119891respectively Fleas may also get the disease when they bite theinfected human being with bubonic plague 119868119867119861 or septicemicplague 119868119867119878 at the rates Γℎ119887119891 and Γℎ119904119891 respectivelyThus the fleapopulation gets plague infection with the force of infectiongiven in 1198663 (119905) = Γℎ119887119891 (119905) 119868119867119861 + Γℎ119904119891 (119905) 1198681198671198781198731+ Γ119903119887119891 (119905) 119868119877119861 + Γ119903119904119891 (119905) 1198681198771198781198733 (4)
The human population may get the disease in one of thefollowing ways when the infected flea 119868119865 bites and infectsthe susceptible human being 119878119867 at a rate Γ119891ℎ when theyinteract with one another this can be with either a personwith pneumonic plague 119868119867119875 through airborne transmissionor septicemic plague 119868119867119878 through physical or sexual contactat the rates Γℎ119901ℎ and Γℎ119904ℎ respectively Another infectionis through airborne transmission through interaction withrodent infected with pneumonic plague 119868119877119875 or throughtouching or eating the infected rodent with septicemic plague119868119877119878 at rates of Γ119903119901ℎ and Γ119903119904ℎ respectively Human beings mayalso get the infection from the environment when they breathin the bacteria or physically contact the infected material atthe rate of1205961That is to say human population acquire plaguedisease following effective contact with infected humanrodent flea and the environment with force of infection 1198661given by1198661 (119905) = Γℎ119901ℎ (119905) 119868119867119875 + Γℎ119904ℎ (119905) 1198681198671198781198731 + Γ119891ℎ (119905) 1198681198651198732+ Γ119903119901ℎ (119905) 119868119877119875 + Γ119903119904ℎ (119905) 1198681198771198781198733 + 1205961 (119905) 119860 (5)
The subgroup 119878119867 after the infection progresses andbecomes latent to the disease at a rate 1205721 After 2 to 7 daysthe subgroups 119864119867 become infected into one of the threeinfectious classes 119868119867119861 119868119867119878 or 119868119867119875 (depending on the modeof transmission an individual is exposed to) and are capableof transmitting the disease The proportion of 119864119867 progressesand becomes infected by bubonic plague 119868119867119861 septicemicplague 119868119867119878 or pneumonic plague 119868119867119875 at the rate 1205722 andproportion to ]1 ]2 or ]3 respectivelyThe compartment 119868119867119861either through strong body immunity or if they get treatmentthey recover andmove to subgroup 119877119867 at a rate 1205723 otherwisethey either progress to subgroup 119868119867119875 or 119868119867119878 at a rate 1205723 ordie either naturally at a rate 1205831 or due to the disease at arate 1205751119887 The fraction of humans with septicemic plague 119868119867119878
either through strong body immunity or if treated recoverat a rate 1205724 and join 119877119867 otherwise they either progress tosubgroup 119868119867119875 at a rate 1205724 or die due to the disease at a rate1205751119904 or naturally at a rate 1205831 The compartments 119868119867119875 if treatedrecover at a rate1205725 otherwise they die either naturally at a rate1205831 or due to the disease at a rate 1205751119901 The subgroup 119877119867 attaintemporary immunity and then return andbecome susceptible119878119867 at a rate 120603
The rodent population may get a disease in one of thefollowing ways when the infected flea 119868119865 bites and infectsthe susceptible rodent 119878119877 at a rate Γ119891119903 through interactionbetween rodents themselves which may be with rodentinfected by pneumonic plague 119868119877119875 or septicemic plague 119868119877119878at the rates Γ119903119901119903 and Γ119903119904119903 respectively The other infectionmay be through interaction with human infected with eitherpneumonic plague 119868119867119875 or septicemic plague 119868119867119878 at ratesof Γℎ119901119903 and Γℎ119904119903 respectively When the susceptible rodentsufficiently interacts with the pathogens in environmentthrough breathing in the bacteria or physically touches theinfected material it gets the infections at the rate of 1205962Rodent also gets the disease through adequate interactionwith rodent human flea and pathogens in the environmentwith force of infection 1198662 given by1198662 (119905) = Γℎ119901119903 (119905) 119868119867119875 + Γℎ119904119903 (119905) 1198681198671198781198731 + Γ119891119903 (119905) 1198681198651198732+ Γ119903119901119903 (119905) 119868119877119875 + Γ119903119904119903 (119905) 1198681198771198781198733 + 1205962 (119905) 119860 (6)
The subgroup 119878119877 after the infection progress and becomelatent to the disease at a rate 1205741 After 2 to 7 days the subgroup119864119877 become infected and capable of transmitting the diseasethe fraction of it progresses and becomes infected by bubonicplague 119868119877119861 septicemic plague 119868119877119878 or pneumonic plague 119868119877119875at the rate 1205742 and proportional to 1205911 1205912 or 1205913 respectivelyTherodent in subgroup 119868119877119861 may either progress to subgroup 119868119877119875or 119868119877119878 at a rate 1205743 or die either naturally at a rate 1205833 or dueto the disease at a rate 1205753119887 The compartment 119868119877119878 may eitherprogress to 119868119877119875 at a rate 1205744 or die due to a disease at a rate 1205753119904or naturally at a rate 1205833 and the compartments 119868119877119875 die eithernaturally at a rate 1205833 or due to the disease at a rate 1205753119901
With regard to the pathogens in the environment weassume that the adequate interaction with 119878119867 and 119878119877 has anegligible effect on the dynamics of pathogens populationsize in the environment The pathogens in the environmentare populated at a constant rate 1205824 The infected human withpneumonic plague 119868119867119875 and rodent with pneumonic plague119868119877119875 also populate the environment 119860 with the bacteria atthe rates 1205781 and 1205782 respectively Thus the environment ispopulated with pathogens causing plague disease with theforce of infection 1198664 given by1198664 (119905) = 1205824 (119905) + 1205781 (119905) 1198681198671198751198731 + 1205782 (119905) 1198681198771198751198733 (7)
The pathogens within the environment suffer natural mor-tality at a rate 1205834 Human population in subgroups 119878119867 and119864119867 flea population in subgroup 119878119865 and rodent population insubgroups 119878119877 and 119864119877 suffer natural mortality at rates 1205831 1205832
6 International Journal of Mathematics and Mathematical Sciences
and 1205833 respectively The compartments 119868119867119861 119868119867119878 119868119867119875 119868119865 119868119877119861119868119877119878 and 119868119877119875 suffer both natural death at the rates12058311205832 and1205833and disease inducedmortality at rates 1205751119887 1205751119904 1205751119901 1205752 1205753119887 1205753119904and 1205753119901 respectively Human flea and rodent are recruited atthe rates 1205951 1205952 and 1205953 respectively24Model Equations for Plague Disease Nowwe assume thatthe variation of infection capability from one individual tothe other migration of individuals from one place to anotherand recruitment and death rates of individuals in differentstages due to seasonal weather variation affect only the rate atwhich the disease is transmitted from one infected individualto the other We now use the variables and parameters andtheir description given in Notations and Table 1 and thedescription of interactions to drive the system of differentialequations given as follows
Human Beings
119889119878119867119889119905 = 12059011205951 + 120603119877119867 minus 12057211198661 (119905) 119878119867 minus 1205831119878119867 (8a)119889119864119867119889119905 = (1 minus 1205901) 1205951 + 12057211198661 (119905) 119878119867 minus 1205722119864119867 minus 1205831119864119867 (8b)119889119868119867119861119889119905 = 1205722]2119864119867 minus 1205723119868119867119861 minus (1205831 + 1205751119887) 119868119867119861 (8c)119889119868119867119878119889119905 = 12057231205883119868119867119861 + 1205722]3119864119867 minus 1205724119868119867119878 minus (1205831 + 1205751119904) 119868119867119878 (8d)119889119868119867119875119889119905 = 1205722]1119864119867 + 12057231205881119868119867119861 + 1205724120585119868119867119878 minus 1205725119868119867119875minus (1205831 + 1205751119901) 119868119867119875 (8e)
119889119877119867119889119905 = 12057231205882119868119867119861 + 1205724 (1 minus 120585) 119868119867119878 + 1205725119868119867119875 minus 120603119877119867minus 1205831119877119867 (8f)
Rodents119889119878119877119889119905 = 12059021205953 minus 12057411198662 (119905) 119878119877 minus 1205833119878119877 (9a)119889119864119877119889119905 = (1 minus 1205902) 1205953 + 12057411198662 (119905) 119878119877 minus 1205742119864119877 minus 1205833119864119877 (9b)119889119868119877119861119889119905 = 12057421205913119864119877 minus 1205743119868119877119861 minus (1205833 + 1205753119887) 119868119877119861 (9c)119889119868119877119878119889119905 = 12057421205912119864119877 + 1205743 (1 minus 120601) 119868119877119861 minus 1205744119868119877119878minus (1205833 + 1205753119904) 119868119877119878 (9d)
In this section we discuss the feasible region and positivityof the plague disease model For convenience purpose andeasy presentation of the result we let 119862 denote all continuousfunctions on the real line If119891 is a periodic function in119862 thenwe use 119891 for the average value of 119891 on time interval [0 119879]defined by
for a continuous 119879-periodic function 119891(119905)31 Invariant Region Plague disease affects human rodentflea and pathogens in the environment populations For thepossible modeling process all state variables and parametersof themodel must be nonnegative for forall119905 ge 0 We thus need toverify whether the solutions of the model system ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) are in suitable feasible region whereall state variables are positive Inspired by Dumont et al [43]and Mpeshe et al [44] we first write system ((8a) (8b) (8c)(8d) (8e) (8f))ndash(11) in the following compact form
where 119883 = (119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875119878119865 119868119865 119860)119879119860(119909) is a 14times14matrix and 119865 is a column vectorWe then have
where 1198861 = (1205723 + 1205831 + 1205751119887) 1198862 = (1205724 + 1205831 + 1205751119904) 1198863 = (1205725 +1205831 + 1205751119901) 1198864 = (1205743 + 1205833 + 1205753119887) 1198865 = (1205744 + 1205833 + 1205753119904) 1198921 =(12057211198661(119905) + 1205831) 1198922 = (12057411198662(119905) + 1205833) and 1198923 = (1205731198663(119905) + 1205832)Now from submatrices 11986011 11986012 11986021 and 11986022 we can
deduce that matrix119860(119909) is aMetzler matrix such that all of itsoff-diagonal elements are nonnegative forall119909 isin R14+ and 119865 ge 0is Lipschitz continuousThus the feasible region for themodelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is the set
Thismeans that any trajectory of the system starting froman initial state in the positive orthant of R14+ remains foreverin Φ32 Positivity of the Solution We need to show that allvariables and parameters of the model are nonnegative forall119905 ge0 We now solve the equations of the system in their patchesfor testing the positivity We found that by letting the initialvalues of the systems ((8a) (8b) (8c) (8d) (8e) (8f)) ((9a)(9b) (9c) (9d) (9e)) ((10a) (10b)) and (11) be 119878119867(0) gt 0119878119877(0) gt 0 119878119865(0) gt 0 and 1198600 ge 0 119864119867(0) ge 0 119868119867119861(0) ge 0119868119867119878(0) ge 0 119868119867119875(0) ge 0 119877119867(0) ge 0 119864119877(0) ge 0 119868119877119861(0) ge 0119868119877119878(0) ge 0 119868119877119875(0) ge 0 and 119868119865(0) ge 0 in the solution
8 International Journal of Mathematics and Mathematical Sciences
set 119878119867(119905) 119878119877(119905) 119878119865(119905) 119860(119905) 119864119867(119905) 119868119867119861(119905) 119868119867119878(119905) 119868119867119875(119905)119877119867(119905) 119864119877(119905) 119868119877119861(119905) 119868119877119878(119905) 119868119877119875(119905) and 119868119865(119905) are nonnegativeforall119905 ge 04 Model Analysis
41 Disease-Free Equilibrium Solution The periodic modelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) with nonnega-tive continuous periodic functions has disease-free equilib-rium solution in which we consider the following equations119889119878119867119889119905 = 12059011205951 minus 1205831119878119867 (17)119889119878119877119889119905 = 12059021205953 minus 1205833119878119877 (18)119889119878119865119889119905 = 1205952119904 minus 1205832119878119865 (19)
Now given initial conditions 119878119867 = 1198781198670 isin R+ 119878119877 = 1198781198770 isinR+ and 119878119865 = 1198781198650 isin R+ for (17) (18) and (19) respectivelywe will have 119878119867 = 120590112059511205831 + (1198781198670 minus 120590112059511205831 ) 119890minus1205831119905
119878119877 = 120590212059531205833 + (1198781198770 minus 120590212059531205833 ) 119890minus1205833119905119878119877 = 12059521199041205832 + (1198781198650 minus 12059521199041205832 ) 119890minus1205832119905
(20)
As 119905 rarr infin (17) (18) and (19) admit unique solution 119878119867 equiv120590112059511205831 119878119877 equiv 120590212059531205833 and 119878119865 equiv 12059521199041205832 respectively whichis globally attractive in R3+
To find the disease-free equilibrium point we set thederivatives of system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11)equal to zero Then the model system has disease-freesolution which is obtained by setting 119868119867119861 = 119868119867119878 = 119868119867119875 =119864119867 = 119877119867 = 0 119868119877119861 = 119868119877119878 = 119868119877119875 = 119864119877 = 0 119868119865 = 0 and 119860 = 0for human rodent flea and pathogen system respectivelyHence system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) has adisease-free equilibrium point1198640 (1198780119867 1198640119867 1198680119867119861 1198680119867119878 1198680119867119875 1198770119867 1198780119877 1198640119877 1198680119877119861 1198680119877119878 1198680119877119875 1198780119865 11986801198651198600) = (120590112059511205831 0 0 0 0 0 120590212059531205833 0 0 0 0 12059521199041205832 0 0) (21)
5 Basic Reproduction Number
Let (R119896R119896) be the standard ordered 119896-dimensionalEuclidean space with a norm sdot For 119906 V isin R119896 we write119906 ge V provided 119906 minus V isin R119896+ 119906 gt V provided 119906 minus V isin R119896+ 0and 119906 ≫ V if 119906 minus V isin int(R119896+)
Now let 119860(119905) be the continuous cooperative irreducibleand 119879-periodic 119896 times 119896 matrix function with period 119879 gt 0
Φ119860(sdot)(119905) be the fundamental solution matrix of the linearordinary differential system119889119909119889119905 = 119860 (119905) 119909 (22)
and 120588(Φ119860(sdot)(119879)) be the spectral radius of Φ119860(sdot)(119879) By Aron-sson and Kellogg [45] it follows that Φ119860(sdot)(119905) is a matrixwith all elements positive for each 119905 gt 0 By the PerronFrobenius theorem 120588(Φ119860(sdot)(119879)) is the principal eigenvalue ofΦ119860(sdot)(119905) in the sense that it is simple and admits an eigenvectorVlowast ≫ 0 The following result is important for our subsequentcomparison argument
Proposition 1 let 120580 = (1119879) ln(120588(Φ119860(119879))) and then thereexists a positive 119879-periodic function V(119905) such that 119890120580119905V(119905) is asolution of 1199091015840 = 119860(119905)119909Proof Let Vlowast ≫ 0 be the eigenvector associated with thespectral radius 120588Φ119860(sdot)(119879)
By the change of variable119909 (119905) = 119890120583119905V (119905) (23)
system (22) becomes119889V119889119905 = 119860 (119905) V minus 120583V = (119860 (119905) minus 120583119868) V (24)
where 119868 is an identity matrixThus V(119905) = Φ(119860(sdot)minus120583119868)(119905)Vlowast is a positive solution of (24)
We can easily see that119890120583119905Φ(119860(sdot)minus120583119868) (119905) = Φ119860(sdot) (119905) (25)
Moreover
V (119879) = Φ(119860(sdot)minus120583119868) (119905) Vlowast = 119890minus120583119879Φ119860(sdot) (119879) Vlowast= 119890minus120583119879120588 (Φ119860(sdot) (119879)) Vlowast = Vlowast = V (0) (26)
Thus V(119905) is a positive 119879-periodic solution of (24) andhence 119909(119905) = 119890120583119905V(119905) is a solution of (22)
The plague disease model system ((8a) (8b) (8c) (8d)(8e) (8f))ndash(11) has unique disease-free equilibrium pointgiven in (21)
We consider a heterogeneous populationwhose individu-als are distinguishable by stage of the disease and hence iden-tifiable and put into epidemiological compartmentswhich are119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875 119878119865 119868119865 and 119860We sort the compartments so that the first 119898 compartmentscorrespond to infected individuals
We now let
F119894(119909) be the rate of appearance of new infections inthe 119894th compartmentsV+119894 (119909) be the rate of transfer of individuals into
compartment 119894 by all other means other than theepidemic onesVminus119894 (119909) be the rate of transfer of individuals out of
compartment 119894
International Journal of Mathematics and Mathematical Sciences 9
Then the plague disease transmissionmodel in ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) is governed by a periodic ordinarydifferential system given in119889119909119894119889119905 = F119894 (119905 119909) minusV119894 (119905 119909) ≜ 119891119894 (119905 119909) (27)
119894 (119909)We rearrange the system by sorting the infectious classes
(119864119867 119868119867119861 119868119867119878 119868119867119875 119864119877 119868119877119861 119868119877119878 119868119877119875 119868119865 119860) coming first Wethen have
F (119909) =((((((((((((((((
(1 minus 1205901) 1205951 + 12057211198661 (119905)119878119867000(1 minus 1205902) 1205953 + 12057411198662 (119905)1198781198770001205952119894 + 1205731198663 (119905)1198781198770
))))))))))))))))
(28)
V (119909)
=((((((((((((((((((((
1205722119864119867 + 1205831119864119867(1205723 + 1205831 + 1205751119887) 119868119867119861 minus 1205722]2119864119867(1205724 + 1205831 + 1205751119904) 119868119867119878 minus 12057231205883119868119867119861 minus 1205722]3119864119867(1205725 + 1205831 + 1205751119901) 119868119867119875 minus 1205722]1119864119867 minus 12057231205881119868119867119861 minus 12057241205851198681198671198781205742119864119877 + 1205833119864119877(1205743 + 1205833 + 1205753119887) 119868119877119861 minus 12057421205913119864119877(1205744 + 1205833 + 1205753119904) 119868119877119878 minus 12057421205912119864119877 minus 1205743 (1 minus 120579) 119868119877119861(1205833 + 1205753119901) 119868119877119875 minus 12057421205911119864119903 minus 1205743120579119868119877119861 minus 1205744119868119877119878(1205832 + 1205752) 1198681198651205834119860 minus 1205781 (119905) 1198681198671198751198731 minus 1205782 (119905) 1198681198771198751198733 + 1205824
))))))))))))))))))))
(29)
Then we have 119865 (119905) = (120597F119894120597119909119895 (1199090)) 119881 (119905) = (120597V119894120597119909119895 (1199090)) (30)
with 1 le 119894 119895 le 10Now using (30) the matrices 119865 and 119881 are as given below
Following the setting byWang and Zhao [46] and van denDriessche andWatmough [47] for epidemicmodels we checkconditions (A1)ndash(A7) for plague disease epidemic modelSystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is equivalent toperiodic ordinary differential system (27) Now consideringthis systemwe can easily see that conditions (A1)ndash(A5) statedbelow are satisfied
(A1) Since each function represents a directed transfer ofindividuals (human rodent flea and pathogens inthe environment) they are all nonnegative Thus foreach 1 le 119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) andVminus119894 (119905 119909) are nonnegative and continuous on R times R119899+
and continuously differentiable with respect to 119909(A2) There is a real number 119879 gt 0 such that for each 1 le119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) and Vminus119894 (119905 119909)
are 119879-periodic in 119905(A3) If a compartment is empty there will be no transfer
of individuals out of the compartment by any meansThat is to say if 119909119894 = 0 then Vminus
119894 = 0 In particular if119909 isin 119883119904 thenVminus119894 = 0 for 119894 = 1 119898
(A4) The incidence of infection for uninfected compart-ments is zero That is to sayF119894 = 0 for 119894 gt 119898
(A5) If the population is disease-free then the populationwill remain free of disease Thus if 119909 isin 119883119904 thenF119894 =0 andV+
119894 = 0 for 119894 = 1 119898We know that system (27) has disease-free periodicsolution given in (21) Now we define 119891(119905 119909(119905)) =F(119905 119909(119905)) minus V(119905 119909(119905)) and119872(119905) = (120597119891119894(119905119894 1198640)120597119909119895)11 le 119894 119895 le 14 where 119891119894(119905 119909(119905)) and 119909119894 are the
119894th components of 119891(119905 119909(119905)) and 119909 respectively Nowfrom (28) and (29) we obtain a 4 times 4matrix given in
A (t) = (minus1205831 120603 0 00 minus (120603 + 1205831) 0 00 0 minus1205833 00 0 0 minus1205832) (34)
We then let Φ119860(sdot)(119905) be the monodromy matrix ofthe linear 119879-periodic system 119889119911119889119905 = 119860(119905)119911 Then120588Φ119860(sdot)(119879) lt 1 which implies that 1198640 is linearlyasymptotically stable in the disease-free subspace119883119904 = 119909 ge 0 119909119894 = 0 forall119894 = 1 sdot sdot sdot 119898 where 119894 = 1 sdot sdot sdot 119898are the infected compartments Thus condition (A6)stated below holds
(A6) The disease-free periodic solution is asymptoti-cally stable in a disease-free subspace 119883119904 that is120588Φ119860(sdot)(119879) lt 1 where 120588Φ119860(sdot)(119879) is the principaleigenvalue of Φ119860(sdot)(119905)Next we set 119865(119905) and 119881(119905) as two 10 times 10 matricesdefined by (30) then using (28) and (29) we getmatri-ces 119865(119905) and 119881(119905) given in (31) and (32) respectivelyWe can further see that matrix 119865(119905) is nonnegativeand minus119881(119905) is cooperative in the sense that the off-diagonal elements are nonnegative Let 119884(119905 119904) 119905 ge 119904be the evolution operator of our 119879-periodic system119889119910119889119905 = minus119881 (119905) 119910 (35)
International Journal of Mathematics and Mathematical Sciences 11
That is for each 119904 isin R the 10 times 10 matrix 119884(119905 119904)satisfies119889119884 (119905 119904)119889119905 = minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119868119889 (36)
where 119868119889 is a 10 times 10 identity matrix Thus themonodromymatrixΦ119881(119905) of (35) equals119884(119905 0) 119905 ge 0Therefore condition (A7) stated below holds
(A7) The evolution of individuals in the infectious com-partments decays exponentially due to natural anddisease induced mortalities Thus 120588Φ119881(119879) lt 1
Now using the standard theory of linear periodic systemby Hale [48] there exist119870 gt 0 and weierp gt 0 such that119884 (119905 119904) le 119870119890minusweierp(119905minus119904) forall119905 ge 119904 119904 isin R (37)
We then have119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) le 119870 119865 (119905 minus 119886) 119890minusweierp119886forall119905 isin R 119886 isin [0infin) (38)
Considering the periodic environment we suppose thatΦ(119904) 119879-periodic in 119904 is the distribution of the new infectionat a rate 119865(119904)Φ(119904) produced by the infected individuals whowere introduced at time 119904 Given 119905 ge 119904 then 119884(119905 119904)119865(119904)Φ(119904)yields the distribution of those infected individuals who werenewly infected at time 119904 and remain in the infected class at 119905We then haveΨ (119905) = int119905
= intinfin0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886 (39)
which is the distribution of accumulative new infectionsat time 119905 produced by all those infected individual Φ(119904)introduced at previous time 119904 to 119905 (119904 le 119905)
Let 119862119879 be the ordered Banach space of all 119879-periodicfunction fromR toR119899 which is equippedwith themaximumnorm sdotinfin and the positive cone119862+119879 = Φ isin 119862119879Φ(119905) ge 0 119905 isinR Define a linear operator 119871 119862119879 rarr 119862119879 by(119871Φ) (119905) = intinfin
0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886forall119905 isin R Φ isin 119862119879 (40)
Now by Wang and Zhao [46] Diekmann et al [49] andvan denDriessche andWatmough [47] we name 119871 as the nextinfection operator then the basic reproduction number 119877119879of the periodic system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) isgiven 119877119879 = 120588 (119871) (41)
where 120588(119871) is the spectral radius of 119871
51 Characterization of 119877119879 In this subsection we investigatewhether the basic reproduction number in our periodicsystem can characterize the threshold of the disease invasionTo do this we consider the following linear 119879-periodicequation 119889119908119889119905 = [minus119881 (119905) + 119865 (119905)120582 ]119908 forall119905 isin R (42)
with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) 119905 ge 119904 119904 isin Rbe the evolution operator of system (42) on R10 We canclearly see that Φ119865minus119881(119905) = 119882(119905 0 1) forall119905 ge 0 Consideringmatrices (31) and (32)wenote that for each120582 isin (0infin) all off-diagonal elements of matrix minus119881(119905) + 119865(119905)120582 are nonnegative(cooperative matrix) It follows that the linear operator119882(119905 119904 120582) is positive in R10 for each 119905 ge 119904 119904 isin R Nowusing Perron-Frobenius theorem by Smith andWaltman [50]it entails that 120588(119882(119879 0 120582)) is an eigenvalue of 119882(119879 0 120582)with a nonnegative eigenvector Also using matrix similarityconcept by Shores [51] we can easily verify that matrix119882(119904 +119879 0 120582) is similar to the matrix119882(119879 0 120582) and hence 120590(119882(119904 +119879 0 120582)) = 120590(119882(119879 0 120582)) for any 119904 isin R where 120590(119863) is aspectrum of the matrix119863Proposition 2 (see [46]) We let (A1)ndash(A7) hold for system((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) then
(i) if 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 1205820is an eigenvalue of 119871 and hence 119877119879 gt 0
(ii) if 119877119879 gt 0 then 120582 = 119877119879 is the unique solution of120588(119882(119879 0 120582)) = 1(iii) 119877119879 = 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0This result shows that in order to find the basic reproduc-
tion number we need to find themonodromymatrixΦ119865minus119881(119905)of system (42) and evaluate itWe then find the spectral radiusof Φ119865minus119881(119879) and solve the equation 120588(Φ119865minus119881(119879)) = 1 for 120582which is the basic reproduction number
52 Computation of the Basic Reproduction Number Wecompute a time-averaged basic reproduction number 1198770using the next-generation matrix as outlined by Wesley andAllen [52] Heesterbeek [53] and Diekmann et al [49] Themethod has the advantage over the usual next-generationmethod in that the steps to reach an estimate of 1198770 and thematrix elements of the next-generation matrix have a clearbiological basis It is easy to handle complex diseases likeplague disease which has multiple transmission roots fromdifferent infectious agents
To do this we first categorize individuals by their stateat the moment they become infected (type at infection)These types at infection refer specifically to the birth ofthe infection in the individual These categories (types atinfection) differ in the way they transmit plague diseasewhich in turn differentiates their ability to produce secondarycases
In our case we categorize the individuals into eight statesand label them as follows human infected with bubonicplague (type 1) human infected with septicemic plague (type
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
2 International Journal of Mathematics and Mathematical Sciences
development rates of flea increase with temperature until itreaches a critical value which makes flea most vulnerableHigh temperature combined with low humidity hinders flearsquossurvival at immature stages [8]
The condition where relative humidity is below 50 isunfavorable for flea growth It is at this condition that thebiting rate of flea onto the infected human and rodent or ofthe infected flea onto the susceptible human and rodent issignificantly low But when the relative humidity is 80 theflea becomes very active and as a result the biting rate andinfection increase significantly Moreover when temperatureis above 275∘C the rapid disappearance of plague bacilli fromthe flea stomach occurs resulting in reduced rates of plaguedisease transmissionThis in turn reduces the flearsquos efficiencyin its ability to transmit the plague bacillus to human beingsand rodents [9 10]
When fleas are in rodent burrows their survival ofimmature stages is affected by soil moisture that is partlycontrolled by outside precipitation [11] As a way of gettingrid of detrimental moisture losses and temperature swingsrodents normally shift to start living underground [12]On the other hand when they are attached with a highorganic load excessively wet conditions in rodent burrows(eg relative humidity 95) can stimulate the growth ofdestructive fungi that diminish flearsquos larval and egg survival[13]
Different studies justify the negative correlation betweenrainfall and plague epidemics For example Cavanaugh andMarshall Jr [3] reported that in areaswhere drains are absentorwhere drainage is insufficient as a result of soil compositionor impoundments of water flooding unquestionably causes adrop in the flea population In areas with improved drainagesuch as those with sandy soils the lessening of the fleapopulation is minimal Precipitation also influences plagueinfection for it influences the concentration of rodents fleasand humans in the same shelter
12 Seasonality in Rodents The direct effect posed on rodentpopulation due to temperature change is minor This is dueto the fact that rodents are homoeothermic and hence donot respond immediately to changes in ambient temperatures[14] Temperature indirectly affects the spread of plague inrodent population in different ways as follows at a low meantemperature of 10∘C the bacteria within host (rodent) becomevery active as a result a large number of infected rodents dyingbefore even the plague bacilli appear in their blood At thisparticular temperature rodents also lose the ability to infectother susceptible individuals
Rainfall may pose positive or negative effect on theincrease of rodent population depending on its intensity[11] A season of moderate rainfall may be considered toaffect positively the increase of rodent abundance but whenthe amount of rainfall is extremely heavy it results ina tremendous rodent population decline [15] When it ismoderate and upon a proper timing rainfall may foster theincrease of rodent population [3] This is due to the fact thatrodentrsquos reproduction period normally follows wet seasons[16ndash18] That is to say the increase of rodent population
during wet period is expected to be higher than that duringthe dry seasons This clearly concurs with the result in thestudy by Leirs et al [19] which narrates that in Tanzaniarodent population densities show clear association with theannual rainfall and its seasonal distribution However whenrainfall is of high intensity it causes flooding of rodentburrows Large number of rodents population dies and theremaining ones normally move from forest to the householdswhere they can protect themselves [3 8 20] In other casesincreased precipitation or drought stalwartly disturbs rodentpopulation dynamics as it deters food availability
13 Seasonality in Pathogens in the Environment When thebacteria are in lungs the transmission of Yersinia pestisis possible through various ways contact transmissionin which one may be infected through physical contactwith respiratory particles on the infected surface airbornetransmission which occurs through inhaling the bacte-ria causing the disease through successive contact withthe nose or mouth of an infected individual respiratoryparticles which occurs through respiratory droplets whichis through shedding of respiratory particles (ie dropletsor aerosols) from an infected human or rodent into theenvironment [21]
Extreme temperatures regularly are ruinous to the sur-vival of pathogens causing plague The changes in tempera-ture may lead to varying effects on the pathogens in the envi-ronment and vectors that live in an environment When themean temperature approaches the maximum limit that canbe endured by the pathogens a small increase in temperaturemay be very dangerous to the pathogen survival Converselywhen pathogens are in the environment characterized by lowmean temperature a small increase in temperaturemay resultin increased development incubation and replication of thepathogen in the environment [22 23]
Davis [24] compared the seasonal incidence of plaguewith usual atmospheric conditions in particular temperatureand rainfall It was depicted that human plague is morefrequent in warm moist weather between 15∘C and 27∘Cthan in hot dry (over 27∘C) or cold weather (under 15∘C)Mitscherlich and Marth [25] narrate that the solar exerts adetrimental effect on bacterial aerosol and the decay rate ofYersinia pestis is proportional to the increase of UV light
The reports by Ayyadurai et al [26] and Mollaret [27]justify the ability of the Yersinia pestis to culture the organismfromdeepwithin contaminated soil Eisen et al [28]were ableto show the great potential durability of Yersinia pestis in thesoil substrate The long duration of their survival in the soilsupports indirectly the virulence maintenance
Yersinia pestis exhibit a very slow growth at the temper-ature between 35∘C and 37∘C but they grow very fast at thetemperature 28∘CTheydie very rapid if exposed to aUV lightor temperature exceeding 40∘C or when exposed to intensivedesiccation [29ndash31] Bacteria decrease their sensitivity whenthe level of humidity drops below 76 [25]
When an infected individual coughs or sneezes thou-sands of the bacteria are released in air [32] The releasedrespiratory particles may be large and heavy that they cannot
International Journal of Mathematics and Mathematical Sciences 3
remain suspended in the air When respiratory particles arelarge the transmission can only occurwhen these particles areexpelled directly onto another close susceptible individual Insome cases the release of smaller respiratory particles mayoccur this is when the airborne transmission is possibleThe smaller released particles are easily suspended in the airrespired (ie passed to the lower respiratory tract) [33]
Relative humidity and temperature affect the transmis-sion of Yersinia pestis from one individual to the otherHumidity affects the size of the respiratory particle [34]When humidity is low the large drops partially evaporate tocreate smaller lighter drops that are more likely to remainairborne for extended periods of time [35] That is to saywhen the air is sufficiently dry the large sized particles shrinkto a size that favors long-range transport which in turn leadsto increased infection
14 Seasonality in Human Behavior Human activities andbehavior in plague-infected areas are also to be consideredas important determinants of plague transmission to andby humans [42] When occurrences of plague are due tohuman intrusions in natural plague areas it is thus importantto consider season variation as a second-order variablethat influences disease incidence through human behaviorIn Tanzania drought and famine which are the result oflack of rainfall and temperature fluctuation have a greatimpact on the farmers and pastoralists as they force themto move from one area to another searching for food forthemselves and their cattle These human intrusions fromone place to another may lead to the increase of plaguedisease transmission in rodents fleas humanpopulation andpathogens in the environment
2 Model Formulation
We describe the complex interaction that leads to plaguedisease transmission and use it to formulate a model for thedynamics of the plague disease coupled with the effect ofseasonal weather variation in its transmission The modelincludes four populations namely human beings rodentsfleas and pathogens in the environment We generallyassume that all individuals from each population are sus-ceptible to the disease the recovered individuals confertemporary immunity and return to be susceptible again andthe infectious are all individuals with either bubonic plagueor pneumonic or septicemic plague
21 Variables and Parameters Used in the Model In Nota-tions and Table 1 we present variables and parameters theirdescription and their values as used in the model We haveobtained the parameter values from the literature that relateto this study and the present information on plague diseaseand through estimation
22 Model Description The human population is dividedinto six subgroups the subgroup of people who have notcontracted the disease to be referred to as susceptible anddenoted by 119878119867 but may get it if they come into contact
with 119868119867119878 119868119867119875 119868119877119878 119868119867119875 119868119865 or 119860 people who have thedisease but have not shown any symptom and are incapableof transmitting the disease to be referred to as exposedand denoted by 119864119867 those who are infected and capable oftransmitting the disease are divided into three subgroupsthere are those who have bubonic plague denoted by 119868119867119861those with septicemic plague denoted by 119868119867119878 and thosewho have pneumonic plague disease denoted by 119868119867119875 Thefraction of population in 119868119867119861 if treated or through strong bodyimmunity may recover and move to subgroup 119877119867 otherwisethey progress either to a septicemic disease infective agent119868119867119878 or to pneumonic plague disease infective agent 119868119867119875 orelse they die The population in the subgroup 119868119867119878 throughstrong body immunity or if treated recover and progressto the subgroup 119877119867 and if not treated they progress andjoin subgroup 119868119867119875 otherwise they die The population ofthe subgroup 119868119867119875 is considered as a very dangerous stage ofplague disease it is a very fatal stage of plague disease withthe fatality rate of about 100 however if treated they recoverand join subgroup119877119867 otherwise they die So the total humanpopulation1198731 is as given by1198731 = 119878119867 + 119864119867 + 119868119867119861 + 119868119867119878 + 119868119867119875 + 119877119867 (1)
Fleas are divided into two subgroups those who have notcontracted the disease butmay get it if they get in contact withinfectious agent (rodent or human) referred to as susceptibleflea and denoted by 119878119865 and those who are infected and arecapable of transmitting the disease referred to as infectiveagents and denoted by 119868119865 The total flea population 1198732 is asgiven by 1198732 = 119878119865 + 119868119865 (2)
The rodents are divided into five subgroups those whohave not contracted the disease but may get it if they getin contact with 119868119867119878 119868119867119875 119868119877119878 119868119867119875 119868119865 or 119860 referred to assusceptible rodents and denoted by 119878119877 those who have thedisease but have not shown any symptom and are incapableof transmitting the disease referred to as exposed anddenotedby 119864119877 those who are infected and capable of transmittingthe disease are divided into three subgroups those who havebubonic plague denoted by 119868119877119861 those with septicemic plaguedenoted by 119868119877119878 and those who have pneumonic plague 119868119877119875The fraction of population in 119868119877119861 may progress either to asepticemic plague disease infective agent 119868119877119878 or to pneumonicplague disease infective agent 119868119877119875 The rodent population inthe subgroup 119868119877119878 may either progress to pneumonic plaguedisease infective agent 119868119877119875 otherwise they dieThepopulationin the subgroup 119868119877119875 is considered as a very dangerous stage ofplague disease and very fatal so the mortality due to diseasein this subgroup is approximated to be 100 Then the totalrodent population1198733 is as given by1198733 = 119878119877 + 119864119877 + 119868119877119861 + 119868119877119878 + 119868119877119875 (3)
The individuals with pneumonic plague may releasepathogens causing plague disease to the environmentdenoted by 119860 through coughing or sneezing When thecondition in soilenvironment is favorable pathogens may
4 International Journal of Mathematics and Mathematical Sciences
Table 1 Parameters and their description
Parameters Description Value ReferencesourceΓ119903119887119891(119905) Adequate contact rate between 119868119877119861 and flea 01 Eisen et al [36]Γ119903119904119891(119905) Adequate contact rate between 119868119877119878 and flea 01 Eisen et al [36]Γ119891ℎ(119905) Adequate contact rate between 119868119865 and human 00641 Eisen et al [36]Γ119891119903(119905) Adequate contact rate between 119868119865 and rodent 00641 Eisen et al [36]Γℎ119901ℎ(119905) Adequate contact rate between 119868119867119875 and 119878119867 039 EstimatedΓℎ119904ℎ(119905) Adequate contact rate between 119868119867119878 and 119878119867 012 EstimatedΓ119903119887ℎ(119905) Adequate contact rate between 119868119877119861 and 119878119867Γ119903119901ℎ(119905) Adequate contact rate between 119868119877119875 and 119878119867 019 EstimatedΓ119903119904ℎ(119905) Adequate contact rate between 119868119877119878 and 119878119867 021 Estimated1205721 Progression rate of 119878119867 to 119864119867 population 099 Estimated1205722 Progression rate out of 119864119867 to infectious state 023 Gani and Leach [37]12058811205723 Progression rate out of 119868119867119861 to 11986811986711987512058821205723 Progression rate out of 119868119867119861 to 11987711986712058831205723 Progression rate out of 119868119867119861 to 1198681198671198781205751119887 Disease induced death rate of 119868119867119861 004 Keeling and Gilligan [38]1205724 Progression rate out of 119868119867119878 to 119868119867119875 and 119877119867 006 Estimated1205751119904 Disease induced death rate of 119868119867119878 004 Estimated1205725 Progression rate out of 119868119867119875 to 119877119867 04 Gani and Leach [37]1205751119901 Disease induced death rate of 119868119867119875 063 Kugeler et al [39]1205741 Progression rate of 119878119877 to 119864119877 092 EstimatedΓℎ119887119891(119905) Adequate contact rate between 119868119867119861 and flea 01 Eisen et al [36]Γℎ119904119891(119905) Adequate contact rate between 119868119867119878 and flea 01 Eisen et al [36]Γ119903119901119903(119905) Adequate contact rate between 119868119877119875 and 119878119877 09 EstimatedΓ119903119904119903(119905) Adequate contact rate between 119868119877119878 and 119878119877 09 EstimatedΓℎ119901119903(119905) Adequate contact rate between 119868119867119875 and 119878119877 000005 EstimatedΓℎ119904119903(119905) Adequate contact rate between 119868119867119878 and 119878119877 000008 Estimated1205742 The rate at which rodent becomes infectious 098 Estimated1205743 Progression rate out of 119868119877119861 to 119868119877119878 and 119868119877119875 0194 Tollenaere et al [40]1205753119887 Disease induced death rate of 119868119877119861 01 Estimated1205744 Progression rate out of 119868119877119878 to 119868119877119875 005 Estimated1205753119904 Disease induced death rate of 119868119877119878 73 Tollenaere et al [40]1205753119901 Disease induced death rate of 119868119877119875 014 Estimated120603 Progression rate of 119877119867 to 119878119867 033 Kugeler et al [39]1205831 Natural death rate for human being 004 Keeling and Gilligan [38]1205832 Natural death rate for flea 02 Bacot and Martin [7]1205833 Natural death rate for rodent 1 Morand and Harvey [41]1205961(119905) Adequate contact rate 119860 and human being1205962(119905) Adequate contact rate 119860 and rodent1205781(119905) Recruitment rate of 119860 by 119868119867119875 02 Estimated1205782(119905) Recruitment rate of 119860 by 119868119877119875 04 Estimated1205834 Natural death rate for pathogens 01 Estimated1205951 Recruitment rate of human beings 009 Estimated120595 Recruitment rate of fleas1205953 Recruitment rate of rodents
International Journal of Mathematics and Mathematical Sciences 5
remain infectious in the environment for a long timeWhen asusceptible individual adequately interacts with the environ-ment infestedwithYersinia pestis heshe gets the disease evenin the absence of any vector
23 Description of Interactions The susceptible fleas in sub-group 119878119865 get Yersinia pestis bacteria through biting theinfected rodent 119868119877119861 or 119868119877119878 who are the primary reservoir forthe bacteria and become infected at the rates Γ119903119887119891 and Γ119903119904119891respectively Fleas may also get the disease when they bite theinfected human being with bubonic plague 119868119867119861 or septicemicplague 119868119867119878 at the rates Γℎ119887119891 and Γℎ119904119891 respectivelyThus the fleapopulation gets plague infection with the force of infectiongiven in 1198663 (119905) = Γℎ119887119891 (119905) 119868119867119861 + Γℎ119904119891 (119905) 1198681198671198781198731+ Γ119903119887119891 (119905) 119868119877119861 + Γ119903119904119891 (119905) 1198681198771198781198733 (4)
The human population may get the disease in one of thefollowing ways when the infected flea 119868119865 bites and infectsthe susceptible human being 119878119867 at a rate Γ119891ℎ when theyinteract with one another this can be with either a personwith pneumonic plague 119868119867119875 through airborne transmissionor septicemic plague 119868119867119878 through physical or sexual contactat the rates Γℎ119901ℎ and Γℎ119904ℎ respectively Another infectionis through airborne transmission through interaction withrodent infected with pneumonic plague 119868119877119875 or throughtouching or eating the infected rodent with septicemic plague119868119877119878 at rates of Γ119903119901ℎ and Γ119903119904ℎ respectively Human beings mayalso get the infection from the environment when they breathin the bacteria or physically contact the infected material atthe rate of1205961That is to say human population acquire plaguedisease following effective contact with infected humanrodent flea and the environment with force of infection 1198661given by1198661 (119905) = Γℎ119901ℎ (119905) 119868119867119875 + Γℎ119904ℎ (119905) 1198681198671198781198731 + Γ119891ℎ (119905) 1198681198651198732+ Γ119903119901ℎ (119905) 119868119877119875 + Γ119903119904ℎ (119905) 1198681198771198781198733 + 1205961 (119905) 119860 (5)
The subgroup 119878119867 after the infection progresses andbecomes latent to the disease at a rate 1205721 After 2 to 7 daysthe subgroups 119864119867 become infected into one of the threeinfectious classes 119868119867119861 119868119867119878 or 119868119867119875 (depending on the modeof transmission an individual is exposed to) and are capableof transmitting the disease The proportion of 119864119867 progressesand becomes infected by bubonic plague 119868119867119861 septicemicplague 119868119867119878 or pneumonic plague 119868119867119875 at the rate 1205722 andproportion to ]1 ]2 or ]3 respectivelyThe compartment 119868119867119861either through strong body immunity or if they get treatmentthey recover andmove to subgroup 119877119867 at a rate 1205723 otherwisethey either progress to subgroup 119868119867119875 or 119868119867119878 at a rate 1205723 ordie either naturally at a rate 1205831 or due to the disease at arate 1205751119887 The fraction of humans with septicemic plague 119868119867119878
either through strong body immunity or if treated recoverat a rate 1205724 and join 119877119867 otherwise they either progress tosubgroup 119868119867119875 at a rate 1205724 or die due to the disease at a rate1205751119904 or naturally at a rate 1205831 The compartments 119868119867119875 if treatedrecover at a rate1205725 otherwise they die either naturally at a rate1205831 or due to the disease at a rate 1205751119901 The subgroup 119877119867 attaintemporary immunity and then return andbecome susceptible119878119867 at a rate 120603
The rodent population may get a disease in one of thefollowing ways when the infected flea 119868119865 bites and infectsthe susceptible rodent 119878119877 at a rate Γ119891119903 through interactionbetween rodents themselves which may be with rodentinfected by pneumonic plague 119868119877119875 or septicemic plague 119868119877119878at the rates Γ119903119901119903 and Γ119903119904119903 respectively The other infectionmay be through interaction with human infected with eitherpneumonic plague 119868119867119875 or septicemic plague 119868119867119878 at ratesof Γℎ119901119903 and Γℎ119904119903 respectively When the susceptible rodentsufficiently interacts with the pathogens in environmentthrough breathing in the bacteria or physically touches theinfected material it gets the infections at the rate of 1205962Rodent also gets the disease through adequate interactionwith rodent human flea and pathogens in the environmentwith force of infection 1198662 given by1198662 (119905) = Γℎ119901119903 (119905) 119868119867119875 + Γℎ119904119903 (119905) 1198681198671198781198731 + Γ119891119903 (119905) 1198681198651198732+ Γ119903119901119903 (119905) 119868119877119875 + Γ119903119904119903 (119905) 1198681198771198781198733 + 1205962 (119905) 119860 (6)
The subgroup 119878119877 after the infection progress and becomelatent to the disease at a rate 1205741 After 2 to 7 days the subgroup119864119877 become infected and capable of transmitting the diseasethe fraction of it progresses and becomes infected by bubonicplague 119868119877119861 septicemic plague 119868119877119878 or pneumonic plague 119868119877119875at the rate 1205742 and proportional to 1205911 1205912 or 1205913 respectivelyTherodent in subgroup 119868119877119861 may either progress to subgroup 119868119877119875or 119868119877119878 at a rate 1205743 or die either naturally at a rate 1205833 or dueto the disease at a rate 1205753119887 The compartment 119868119877119878 may eitherprogress to 119868119877119875 at a rate 1205744 or die due to a disease at a rate 1205753119904or naturally at a rate 1205833 and the compartments 119868119877119875 die eithernaturally at a rate 1205833 or due to the disease at a rate 1205753119901
With regard to the pathogens in the environment weassume that the adequate interaction with 119878119867 and 119878119877 has anegligible effect on the dynamics of pathogens populationsize in the environment The pathogens in the environmentare populated at a constant rate 1205824 The infected human withpneumonic plague 119868119867119875 and rodent with pneumonic plague119868119877119875 also populate the environment 119860 with the bacteria atthe rates 1205781 and 1205782 respectively Thus the environment ispopulated with pathogens causing plague disease with theforce of infection 1198664 given by1198664 (119905) = 1205824 (119905) + 1205781 (119905) 1198681198671198751198731 + 1205782 (119905) 1198681198771198751198733 (7)
The pathogens within the environment suffer natural mor-tality at a rate 1205834 Human population in subgroups 119878119867 and119864119867 flea population in subgroup 119878119865 and rodent population insubgroups 119878119877 and 119864119877 suffer natural mortality at rates 1205831 1205832
6 International Journal of Mathematics and Mathematical Sciences
and 1205833 respectively The compartments 119868119867119861 119868119867119878 119868119867119875 119868119865 119868119877119861119868119877119878 and 119868119877119875 suffer both natural death at the rates12058311205832 and1205833and disease inducedmortality at rates 1205751119887 1205751119904 1205751119901 1205752 1205753119887 1205753119904and 1205753119901 respectively Human flea and rodent are recruited atthe rates 1205951 1205952 and 1205953 respectively24Model Equations for Plague Disease Nowwe assume thatthe variation of infection capability from one individual tothe other migration of individuals from one place to anotherand recruitment and death rates of individuals in differentstages due to seasonal weather variation affect only the rate atwhich the disease is transmitted from one infected individualto the other We now use the variables and parameters andtheir description given in Notations and Table 1 and thedescription of interactions to drive the system of differentialequations given as follows
Human Beings
119889119878119867119889119905 = 12059011205951 + 120603119877119867 minus 12057211198661 (119905) 119878119867 minus 1205831119878119867 (8a)119889119864119867119889119905 = (1 minus 1205901) 1205951 + 12057211198661 (119905) 119878119867 minus 1205722119864119867 minus 1205831119864119867 (8b)119889119868119867119861119889119905 = 1205722]2119864119867 minus 1205723119868119867119861 minus (1205831 + 1205751119887) 119868119867119861 (8c)119889119868119867119878119889119905 = 12057231205883119868119867119861 + 1205722]3119864119867 minus 1205724119868119867119878 minus (1205831 + 1205751119904) 119868119867119878 (8d)119889119868119867119875119889119905 = 1205722]1119864119867 + 12057231205881119868119867119861 + 1205724120585119868119867119878 minus 1205725119868119867119875minus (1205831 + 1205751119901) 119868119867119875 (8e)
119889119877119867119889119905 = 12057231205882119868119867119861 + 1205724 (1 minus 120585) 119868119867119878 + 1205725119868119867119875 minus 120603119877119867minus 1205831119877119867 (8f)
Rodents119889119878119877119889119905 = 12059021205953 minus 12057411198662 (119905) 119878119877 minus 1205833119878119877 (9a)119889119864119877119889119905 = (1 minus 1205902) 1205953 + 12057411198662 (119905) 119878119877 minus 1205742119864119877 minus 1205833119864119877 (9b)119889119868119877119861119889119905 = 12057421205913119864119877 minus 1205743119868119877119861 minus (1205833 + 1205753119887) 119868119877119861 (9c)119889119868119877119878119889119905 = 12057421205912119864119877 + 1205743 (1 minus 120601) 119868119877119861 minus 1205744119868119877119878minus (1205833 + 1205753119904) 119868119877119878 (9d)
In this section we discuss the feasible region and positivityof the plague disease model For convenience purpose andeasy presentation of the result we let 119862 denote all continuousfunctions on the real line If119891 is a periodic function in119862 thenwe use 119891 for the average value of 119891 on time interval [0 119879]defined by
for a continuous 119879-periodic function 119891(119905)31 Invariant Region Plague disease affects human rodentflea and pathogens in the environment populations For thepossible modeling process all state variables and parametersof themodel must be nonnegative for forall119905 ge 0 We thus need toverify whether the solutions of the model system ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) are in suitable feasible region whereall state variables are positive Inspired by Dumont et al [43]and Mpeshe et al [44] we first write system ((8a) (8b) (8c)(8d) (8e) (8f))ndash(11) in the following compact form
where 119883 = (119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875119878119865 119868119865 119860)119879119860(119909) is a 14times14matrix and 119865 is a column vectorWe then have
where 1198861 = (1205723 + 1205831 + 1205751119887) 1198862 = (1205724 + 1205831 + 1205751119904) 1198863 = (1205725 +1205831 + 1205751119901) 1198864 = (1205743 + 1205833 + 1205753119887) 1198865 = (1205744 + 1205833 + 1205753119904) 1198921 =(12057211198661(119905) + 1205831) 1198922 = (12057411198662(119905) + 1205833) and 1198923 = (1205731198663(119905) + 1205832)Now from submatrices 11986011 11986012 11986021 and 11986022 we can
deduce that matrix119860(119909) is aMetzler matrix such that all of itsoff-diagonal elements are nonnegative forall119909 isin R14+ and 119865 ge 0is Lipschitz continuousThus the feasible region for themodelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is the set
Thismeans that any trajectory of the system starting froman initial state in the positive orthant of R14+ remains foreverin Φ32 Positivity of the Solution We need to show that allvariables and parameters of the model are nonnegative forall119905 ge0 We now solve the equations of the system in their patchesfor testing the positivity We found that by letting the initialvalues of the systems ((8a) (8b) (8c) (8d) (8e) (8f)) ((9a)(9b) (9c) (9d) (9e)) ((10a) (10b)) and (11) be 119878119867(0) gt 0119878119877(0) gt 0 119878119865(0) gt 0 and 1198600 ge 0 119864119867(0) ge 0 119868119867119861(0) ge 0119868119867119878(0) ge 0 119868119867119875(0) ge 0 119877119867(0) ge 0 119864119877(0) ge 0 119868119877119861(0) ge 0119868119877119878(0) ge 0 119868119877119875(0) ge 0 and 119868119865(0) ge 0 in the solution
8 International Journal of Mathematics and Mathematical Sciences
set 119878119867(119905) 119878119877(119905) 119878119865(119905) 119860(119905) 119864119867(119905) 119868119867119861(119905) 119868119867119878(119905) 119868119867119875(119905)119877119867(119905) 119864119877(119905) 119868119877119861(119905) 119868119877119878(119905) 119868119877119875(119905) and 119868119865(119905) are nonnegativeforall119905 ge 04 Model Analysis
41 Disease-Free Equilibrium Solution The periodic modelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) with nonnega-tive continuous periodic functions has disease-free equilib-rium solution in which we consider the following equations119889119878119867119889119905 = 12059011205951 minus 1205831119878119867 (17)119889119878119877119889119905 = 12059021205953 minus 1205833119878119877 (18)119889119878119865119889119905 = 1205952119904 minus 1205832119878119865 (19)
Now given initial conditions 119878119867 = 1198781198670 isin R+ 119878119877 = 1198781198770 isinR+ and 119878119865 = 1198781198650 isin R+ for (17) (18) and (19) respectivelywe will have 119878119867 = 120590112059511205831 + (1198781198670 minus 120590112059511205831 ) 119890minus1205831119905
119878119877 = 120590212059531205833 + (1198781198770 minus 120590212059531205833 ) 119890minus1205833119905119878119877 = 12059521199041205832 + (1198781198650 minus 12059521199041205832 ) 119890minus1205832119905
(20)
As 119905 rarr infin (17) (18) and (19) admit unique solution 119878119867 equiv120590112059511205831 119878119877 equiv 120590212059531205833 and 119878119865 equiv 12059521199041205832 respectively whichis globally attractive in R3+
To find the disease-free equilibrium point we set thederivatives of system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11)equal to zero Then the model system has disease-freesolution which is obtained by setting 119868119867119861 = 119868119867119878 = 119868119867119875 =119864119867 = 119877119867 = 0 119868119877119861 = 119868119877119878 = 119868119877119875 = 119864119877 = 0 119868119865 = 0 and 119860 = 0for human rodent flea and pathogen system respectivelyHence system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) has adisease-free equilibrium point1198640 (1198780119867 1198640119867 1198680119867119861 1198680119867119878 1198680119867119875 1198770119867 1198780119877 1198640119877 1198680119877119861 1198680119877119878 1198680119877119875 1198780119865 11986801198651198600) = (120590112059511205831 0 0 0 0 0 120590212059531205833 0 0 0 0 12059521199041205832 0 0) (21)
5 Basic Reproduction Number
Let (R119896R119896) be the standard ordered 119896-dimensionalEuclidean space with a norm sdot For 119906 V isin R119896 we write119906 ge V provided 119906 minus V isin R119896+ 119906 gt V provided 119906 minus V isin R119896+ 0and 119906 ≫ V if 119906 minus V isin int(R119896+)
Now let 119860(119905) be the continuous cooperative irreducibleand 119879-periodic 119896 times 119896 matrix function with period 119879 gt 0
Φ119860(sdot)(119905) be the fundamental solution matrix of the linearordinary differential system119889119909119889119905 = 119860 (119905) 119909 (22)
and 120588(Φ119860(sdot)(119879)) be the spectral radius of Φ119860(sdot)(119879) By Aron-sson and Kellogg [45] it follows that Φ119860(sdot)(119905) is a matrixwith all elements positive for each 119905 gt 0 By the PerronFrobenius theorem 120588(Φ119860(sdot)(119879)) is the principal eigenvalue ofΦ119860(sdot)(119905) in the sense that it is simple and admits an eigenvectorVlowast ≫ 0 The following result is important for our subsequentcomparison argument
Proposition 1 let 120580 = (1119879) ln(120588(Φ119860(119879))) and then thereexists a positive 119879-periodic function V(119905) such that 119890120580119905V(119905) is asolution of 1199091015840 = 119860(119905)119909Proof Let Vlowast ≫ 0 be the eigenvector associated with thespectral radius 120588Φ119860(sdot)(119879)
By the change of variable119909 (119905) = 119890120583119905V (119905) (23)
system (22) becomes119889V119889119905 = 119860 (119905) V minus 120583V = (119860 (119905) minus 120583119868) V (24)
where 119868 is an identity matrixThus V(119905) = Φ(119860(sdot)minus120583119868)(119905)Vlowast is a positive solution of (24)
We can easily see that119890120583119905Φ(119860(sdot)minus120583119868) (119905) = Φ119860(sdot) (119905) (25)
Moreover
V (119879) = Φ(119860(sdot)minus120583119868) (119905) Vlowast = 119890minus120583119879Φ119860(sdot) (119879) Vlowast= 119890minus120583119879120588 (Φ119860(sdot) (119879)) Vlowast = Vlowast = V (0) (26)
Thus V(119905) is a positive 119879-periodic solution of (24) andhence 119909(119905) = 119890120583119905V(119905) is a solution of (22)
The plague disease model system ((8a) (8b) (8c) (8d)(8e) (8f))ndash(11) has unique disease-free equilibrium pointgiven in (21)
We consider a heterogeneous populationwhose individu-als are distinguishable by stage of the disease and hence iden-tifiable and put into epidemiological compartmentswhich are119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875 119878119865 119868119865 and 119860We sort the compartments so that the first 119898 compartmentscorrespond to infected individuals
We now let
F119894(119909) be the rate of appearance of new infections inthe 119894th compartmentsV+119894 (119909) be the rate of transfer of individuals into
compartment 119894 by all other means other than theepidemic onesVminus119894 (119909) be the rate of transfer of individuals out of
compartment 119894
International Journal of Mathematics and Mathematical Sciences 9
Then the plague disease transmissionmodel in ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) is governed by a periodic ordinarydifferential system given in119889119909119894119889119905 = F119894 (119905 119909) minusV119894 (119905 119909) ≜ 119891119894 (119905 119909) (27)
119894 (119909)We rearrange the system by sorting the infectious classes
(119864119867 119868119867119861 119868119867119878 119868119867119875 119864119877 119868119877119861 119868119877119878 119868119877119875 119868119865 119860) coming first Wethen have
F (119909) =((((((((((((((((
(1 minus 1205901) 1205951 + 12057211198661 (119905)119878119867000(1 minus 1205902) 1205953 + 12057411198662 (119905)1198781198770001205952119894 + 1205731198663 (119905)1198781198770
))))))))))))))))
(28)
V (119909)
=((((((((((((((((((((
1205722119864119867 + 1205831119864119867(1205723 + 1205831 + 1205751119887) 119868119867119861 minus 1205722]2119864119867(1205724 + 1205831 + 1205751119904) 119868119867119878 minus 12057231205883119868119867119861 minus 1205722]3119864119867(1205725 + 1205831 + 1205751119901) 119868119867119875 minus 1205722]1119864119867 minus 12057231205881119868119867119861 minus 12057241205851198681198671198781205742119864119877 + 1205833119864119877(1205743 + 1205833 + 1205753119887) 119868119877119861 minus 12057421205913119864119877(1205744 + 1205833 + 1205753119904) 119868119877119878 minus 12057421205912119864119877 minus 1205743 (1 minus 120579) 119868119877119861(1205833 + 1205753119901) 119868119877119875 minus 12057421205911119864119903 minus 1205743120579119868119877119861 minus 1205744119868119877119878(1205832 + 1205752) 1198681198651205834119860 minus 1205781 (119905) 1198681198671198751198731 minus 1205782 (119905) 1198681198771198751198733 + 1205824
))))))))))))))))))))
(29)
Then we have 119865 (119905) = (120597F119894120597119909119895 (1199090)) 119881 (119905) = (120597V119894120597119909119895 (1199090)) (30)
with 1 le 119894 119895 le 10Now using (30) the matrices 119865 and 119881 are as given below
Following the setting byWang and Zhao [46] and van denDriessche andWatmough [47] for epidemicmodels we checkconditions (A1)ndash(A7) for plague disease epidemic modelSystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is equivalent toperiodic ordinary differential system (27) Now consideringthis systemwe can easily see that conditions (A1)ndash(A5) statedbelow are satisfied
(A1) Since each function represents a directed transfer ofindividuals (human rodent flea and pathogens inthe environment) they are all nonnegative Thus foreach 1 le 119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) andVminus119894 (119905 119909) are nonnegative and continuous on R times R119899+
and continuously differentiable with respect to 119909(A2) There is a real number 119879 gt 0 such that for each 1 le119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) and Vminus119894 (119905 119909)
are 119879-periodic in 119905(A3) If a compartment is empty there will be no transfer
of individuals out of the compartment by any meansThat is to say if 119909119894 = 0 then Vminus
119894 = 0 In particular if119909 isin 119883119904 thenVminus119894 = 0 for 119894 = 1 119898
(A4) The incidence of infection for uninfected compart-ments is zero That is to sayF119894 = 0 for 119894 gt 119898
(A5) If the population is disease-free then the populationwill remain free of disease Thus if 119909 isin 119883119904 thenF119894 =0 andV+
119894 = 0 for 119894 = 1 119898We know that system (27) has disease-free periodicsolution given in (21) Now we define 119891(119905 119909(119905)) =F(119905 119909(119905)) minus V(119905 119909(119905)) and119872(119905) = (120597119891119894(119905119894 1198640)120597119909119895)11 le 119894 119895 le 14 where 119891119894(119905 119909(119905)) and 119909119894 are the
119894th components of 119891(119905 119909(119905)) and 119909 respectively Nowfrom (28) and (29) we obtain a 4 times 4matrix given in
A (t) = (minus1205831 120603 0 00 minus (120603 + 1205831) 0 00 0 minus1205833 00 0 0 minus1205832) (34)
We then let Φ119860(sdot)(119905) be the monodromy matrix ofthe linear 119879-periodic system 119889119911119889119905 = 119860(119905)119911 Then120588Φ119860(sdot)(119879) lt 1 which implies that 1198640 is linearlyasymptotically stable in the disease-free subspace119883119904 = 119909 ge 0 119909119894 = 0 forall119894 = 1 sdot sdot sdot 119898 where 119894 = 1 sdot sdot sdot 119898are the infected compartments Thus condition (A6)stated below holds
(A6) The disease-free periodic solution is asymptoti-cally stable in a disease-free subspace 119883119904 that is120588Φ119860(sdot)(119879) lt 1 where 120588Φ119860(sdot)(119879) is the principaleigenvalue of Φ119860(sdot)(119905)Next we set 119865(119905) and 119881(119905) as two 10 times 10 matricesdefined by (30) then using (28) and (29) we getmatri-ces 119865(119905) and 119881(119905) given in (31) and (32) respectivelyWe can further see that matrix 119865(119905) is nonnegativeand minus119881(119905) is cooperative in the sense that the off-diagonal elements are nonnegative Let 119884(119905 119904) 119905 ge 119904be the evolution operator of our 119879-periodic system119889119910119889119905 = minus119881 (119905) 119910 (35)
International Journal of Mathematics and Mathematical Sciences 11
That is for each 119904 isin R the 10 times 10 matrix 119884(119905 119904)satisfies119889119884 (119905 119904)119889119905 = minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119868119889 (36)
where 119868119889 is a 10 times 10 identity matrix Thus themonodromymatrixΦ119881(119905) of (35) equals119884(119905 0) 119905 ge 0Therefore condition (A7) stated below holds
(A7) The evolution of individuals in the infectious com-partments decays exponentially due to natural anddisease induced mortalities Thus 120588Φ119881(119879) lt 1
Now using the standard theory of linear periodic systemby Hale [48] there exist119870 gt 0 and weierp gt 0 such that119884 (119905 119904) le 119870119890minusweierp(119905minus119904) forall119905 ge 119904 119904 isin R (37)
We then have119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) le 119870 119865 (119905 minus 119886) 119890minusweierp119886forall119905 isin R 119886 isin [0infin) (38)
Considering the periodic environment we suppose thatΦ(119904) 119879-periodic in 119904 is the distribution of the new infectionat a rate 119865(119904)Φ(119904) produced by the infected individuals whowere introduced at time 119904 Given 119905 ge 119904 then 119884(119905 119904)119865(119904)Φ(119904)yields the distribution of those infected individuals who werenewly infected at time 119904 and remain in the infected class at 119905We then haveΨ (119905) = int119905
= intinfin0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886 (39)
which is the distribution of accumulative new infectionsat time 119905 produced by all those infected individual Φ(119904)introduced at previous time 119904 to 119905 (119904 le 119905)
Let 119862119879 be the ordered Banach space of all 119879-periodicfunction fromR toR119899 which is equippedwith themaximumnorm sdotinfin and the positive cone119862+119879 = Φ isin 119862119879Φ(119905) ge 0 119905 isinR Define a linear operator 119871 119862119879 rarr 119862119879 by(119871Φ) (119905) = intinfin
0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886forall119905 isin R Φ isin 119862119879 (40)
Now by Wang and Zhao [46] Diekmann et al [49] andvan denDriessche andWatmough [47] we name 119871 as the nextinfection operator then the basic reproduction number 119877119879of the periodic system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) isgiven 119877119879 = 120588 (119871) (41)
where 120588(119871) is the spectral radius of 119871
51 Characterization of 119877119879 In this subsection we investigatewhether the basic reproduction number in our periodicsystem can characterize the threshold of the disease invasionTo do this we consider the following linear 119879-periodicequation 119889119908119889119905 = [minus119881 (119905) + 119865 (119905)120582 ]119908 forall119905 isin R (42)
with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) 119905 ge 119904 119904 isin Rbe the evolution operator of system (42) on R10 We canclearly see that Φ119865minus119881(119905) = 119882(119905 0 1) forall119905 ge 0 Consideringmatrices (31) and (32)wenote that for each120582 isin (0infin) all off-diagonal elements of matrix minus119881(119905) + 119865(119905)120582 are nonnegative(cooperative matrix) It follows that the linear operator119882(119905 119904 120582) is positive in R10 for each 119905 ge 119904 119904 isin R Nowusing Perron-Frobenius theorem by Smith andWaltman [50]it entails that 120588(119882(119879 0 120582)) is an eigenvalue of 119882(119879 0 120582)with a nonnegative eigenvector Also using matrix similarityconcept by Shores [51] we can easily verify that matrix119882(119904 +119879 0 120582) is similar to the matrix119882(119879 0 120582) and hence 120590(119882(119904 +119879 0 120582)) = 120590(119882(119879 0 120582)) for any 119904 isin R where 120590(119863) is aspectrum of the matrix119863Proposition 2 (see [46]) We let (A1)ndash(A7) hold for system((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) then
(i) if 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 1205820is an eigenvalue of 119871 and hence 119877119879 gt 0
(ii) if 119877119879 gt 0 then 120582 = 119877119879 is the unique solution of120588(119882(119879 0 120582)) = 1(iii) 119877119879 = 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0This result shows that in order to find the basic reproduc-
tion number we need to find themonodromymatrixΦ119865minus119881(119905)of system (42) and evaluate itWe then find the spectral radiusof Φ119865minus119881(119879) and solve the equation 120588(Φ119865minus119881(119879)) = 1 for 120582which is the basic reproduction number
52 Computation of the Basic Reproduction Number Wecompute a time-averaged basic reproduction number 1198770using the next-generation matrix as outlined by Wesley andAllen [52] Heesterbeek [53] and Diekmann et al [49] Themethod has the advantage over the usual next-generationmethod in that the steps to reach an estimate of 1198770 and thematrix elements of the next-generation matrix have a clearbiological basis It is easy to handle complex diseases likeplague disease which has multiple transmission roots fromdifferent infectious agents
To do this we first categorize individuals by their stateat the moment they become infected (type at infection)These types at infection refer specifically to the birth ofthe infection in the individual These categories (types atinfection) differ in the way they transmit plague diseasewhich in turn differentiates their ability to produce secondarycases
In our case we categorize the individuals into eight statesand label them as follows human infected with bubonicplague (type 1) human infected with septicemic plague (type
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
International Journal of Mathematics and Mathematical Sciences 3
remain suspended in the air When respiratory particles arelarge the transmission can only occurwhen these particles areexpelled directly onto another close susceptible individual Insome cases the release of smaller respiratory particles mayoccur this is when the airborne transmission is possibleThe smaller released particles are easily suspended in the airrespired (ie passed to the lower respiratory tract) [33]
Relative humidity and temperature affect the transmis-sion of Yersinia pestis from one individual to the otherHumidity affects the size of the respiratory particle [34]When humidity is low the large drops partially evaporate tocreate smaller lighter drops that are more likely to remainairborne for extended periods of time [35] That is to saywhen the air is sufficiently dry the large sized particles shrinkto a size that favors long-range transport which in turn leadsto increased infection
14 Seasonality in Human Behavior Human activities andbehavior in plague-infected areas are also to be consideredas important determinants of plague transmission to andby humans [42] When occurrences of plague are due tohuman intrusions in natural plague areas it is thus importantto consider season variation as a second-order variablethat influences disease incidence through human behaviorIn Tanzania drought and famine which are the result oflack of rainfall and temperature fluctuation have a greatimpact on the farmers and pastoralists as they force themto move from one area to another searching for food forthemselves and their cattle These human intrusions fromone place to another may lead to the increase of plaguedisease transmission in rodents fleas humanpopulation andpathogens in the environment
2 Model Formulation
We describe the complex interaction that leads to plaguedisease transmission and use it to formulate a model for thedynamics of the plague disease coupled with the effect ofseasonal weather variation in its transmission The modelincludes four populations namely human beings rodentsfleas and pathogens in the environment We generallyassume that all individuals from each population are sus-ceptible to the disease the recovered individuals confertemporary immunity and return to be susceptible again andthe infectious are all individuals with either bubonic plagueor pneumonic or septicemic plague
21 Variables and Parameters Used in the Model In Nota-tions and Table 1 we present variables and parameters theirdescription and their values as used in the model We haveobtained the parameter values from the literature that relateto this study and the present information on plague diseaseand through estimation
22 Model Description The human population is dividedinto six subgroups the subgroup of people who have notcontracted the disease to be referred to as susceptible anddenoted by 119878119867 but may get it if they come into contact
with 119868119867119878 119868119867119875 119868119877119878 119868119867119875 119868119865 or 119860 people who have thedisease but have not shown any symptom and are incapableof transmitting the disease to be referred to as exposedand denoted by 119864119867 those who are infected and capable oftransmitting the disease are divided into three subgroupsthere are those who have bubonic plague denoted by 119868119867119861those with septicemic plague denoted by 119868119867119878 and thosewho have pneumonic plague disease denoted by 119868119867119875 Thefraction of population in 119868119867119861 if treated or through strong bodyimmunity may recover and move to subgroup 119877119867 otherwisethey progress either to a septicemic disease infective agent119868119867119878 or to pneumonic plague disease infective agent 119868119867119875 orelse they die The population in the subgroup 119868119867119878 throughstrong body immunity or if treated recover and progressto the subgroup 119877119867 and if not treated they progress andjoin subgroup 119868119867119875 otherwise they die The population ofthe subgroup 119868119867119875 is considered as a very dangerous stage ofplague disease it is a very fatal stage of plague disease withthe fatality rate of about 100 however if treated they recoverand join subgroup119877119867 otherwise they die So the total humanpopulation1198731 is as given by1198731 = 119878119867 + 119864119867 + 119868119867119861 + 119868119867119878 + 119868119867119875 + 119877119867 (1)
Fleas are divided into two subgroups those who have notcontracted the disease butmay get it if they get in contact withinfectious agent (rodent or human) referred to as susceptibleflea and denoted by 119878119865 and those who are infected and arecapable of transmitting the disease referred to as infectiveagents and denoted by 119868119865 The total flea population 1198732 is asgiven by 1198732 = 119878119865 + 119868119865 (2)
The rodents are divided into five subgroups those whohave not contracted the disease but may get it if they getin contact with 119868119867119878 119868119867119875 119868119877119878 119868119867119875 119868119865 or 119860 referred to assusceptible rodents and denoted by 119878119877 those who have thedisease but have not shown any symptom and are incapableof transmitting the disease referred to as exposed anddenotedby 119864119877 those who are infected and capable of transmittingthe disease are divided into three subgroups those who havebubonic plague denoted by 119868119877119861 those with septicemic plaguedenoted by 119868119877119878 and those who have pneumonic plague 119868119877119875The fraction of population in 119868119877119861 may progress either to asepticemic plague disease infective agent 119868119877119878 or to pneumonicplague disease infective agent 119868119877119875 The rodent population inthe subgroup 119868119877119878 may either progress to pneumonic plaguedisease infective agent 119868119877119875 otherwise they dieThepopulationin the subgroup 119868119877119875 is considered as a very dangerous stage ofplague disease and very fatal so the mortality due to diseasein this subgroup is approximated to be 100 Then the totalrodent population1198733 is as given by1198733 = 119878119877 + 119864119877 + 119868119877119861 + 119868119877119878 + 119868119877119875 (3)
The individuals with pneumonic plague may releasepathogens causing plague disease to the environmentdenoted by 119860 through coughing or sneezing When thecondition in soilenvironment is favorable pathogens may
4 International Journal of Mathematics and Mathematical Sciences
Table 1 Parameters and their description
Parameters Description Value ReferencesourceΓ119903119887119891(119905) Adequate contact rate between 119868119877119861 and flea 01 Eisen et al [36]Γ119903119904119891(119905) Adequate contact rate between 119868119877119878 and flea 01 Eisen et al [36]Γ119891ℎ(119905) Adequate contact rate between 119868119865 and human 00641 Eisen et al [36]Γ119891119903(119905) Adequate contact rate between 119868119865 and rodent 00641 Eisen et al [36]Γℎ119901ℎ(119905) Adequate contact rate between 119868119867119875 and 119878119867 039 EstimatedΓℎ119904ℎ(119905) Adequate contact rate between 119868119867119878 and 119878119867 012 EstimatedΓ119903119887ℎ(119905) Adequate contact rate between 119868119877119861 and 119878119867Γ119903119901ℎ(119905) Adequate contact rate between 119868119877119875 and 119878119867 019 EstimatedΓ119903119904ℎ(119905) Adequate contact rate between 119868119877119878 and 119878119867 021 Estimated1205721 Progression rate of 119878119867 to 119864119867 population 099 Estimated1205722 Progression rate out of 119864119867 to infectious state 023 Gani and Leach [37]12058811205723 Progression rate out of 119868119867119861 to 11986811986711987512058821205723 Progression rate out of 119868119867119861 to 11987711986712058831205723 Progression rate out of 119868119867119861 to 1198681198671198781205751119887 Disease induced death rate of 119868119867119861 004 Keeling and Gilligan [38]1205724 Progression rate out of 119868119867119878 to 119868119867119875 and 119877119867 006 Estimated1205751119904 Disease induced death rate of 119868119867119878 004 Estimated1205725 Progression rate out of 119868119867119875 to 119877119867 04 Gani and Leach [37]1205751119901 Disease induced death rate of 119868119867119875 063 Kugeler et al [39]1205741 Progression rate of 119878119877 to 119864119877 092 EstimatedΓℎ119887119891(119905) Adequate contact rate between 119868119867119861 and flea 01 Eisen et al [36]Γℎ119904119891(119905) Adequate contact rate between 119868119867119878 and flea 01 Eisen et al [36]Γ119903119901119903(119905) Adequate contact rate between 119868119877119875 and 119878119877 09 EstimatedΓ119903119904119903(119905) Adequate contact rate between 119868119877119878 and 119878119877 09 EstimatedΓℎ119901119903(119905) Adequate contact rate between 119868119867119875 and 119878119877 000005 EstimatedΓℎ119904119903(119905) Adequate contact rate between 119868119867119878 and 119878119877 000008 Estimated1205742 The rate at which rodent becomes infectious 098 Estimated1205743 Progression rate out of 119868119877119861 to 119868119877119878 and 119868119877119875 0194 Tollenaere et al [40]1205753119887 Disease induced death rate of 119868119877119861 01 Estimated1205744 Progression rate out of 119868119877119878 to 119868119877119875 005 Estimated1205753119904 Disease induced death rate of 119868119877119878 73 Tollenaere et al [40]1205753119901 Disease induced death rate of 119868119877119875 014 Estimated120603 Progression rate of 119877119867 to 119878119867 033 Kugeler et al [39]1205831 Natural death rate for human being 004 Keeling and Gilligan [38]1205832 Natural death rate for flea 02 Bacot and Martin [7]1205833 Natural death rate for rodent 1 Morand and Harvey [41]1205961(119905) Adequate contact rate 119860 and human being1205962(119905) Adequate contact rate 119860 and rodent1205781(119905) Recruitment rate of 119860 by 119868119867119875 02 Estimated1205782(119905) Recruitment rate of 119860 by 119868119877119875 04 Estimated1205834 Natural death rate for pathogens 01 Estimated1205951 Recruitment rate of human beings 009 Estimated120595 Recruitment rate of fleas1205953 Recruitment rate of rodents
International Journal of Mathematics and Mathematical Sciences 5
remain infectious in the environment for a long timeWhen asusceptible individual adequately interacts with the environ-ment infestedwithYersinia pestis heshe gets the disease evenin the absence of any vector
23 Description of Interactions The susceptible fleas in sub-group 119878119865 get Yersinia pestis bacteria through biting theinfected rodent 119868119877119861 or 119868119877119878 who are the primary reservoir forthe bacteria and become infected at the rates Γ119903119887119891 and Γ119903119904119891respectively Fleas may also get the disease when they bite theinfected human being with bubonic plague 119868119867119861 or septicemicplague 119868119867119878 at the rates Γℎ119887119891 and Γℎ119904119891 respectivelyThus the fleapopulation gets plague infection with the force of infectiongiven in 1198663 (119905) = Γℎ119887119891 (119905) 119868119867119861 + Γℎ119904119891 (119905) 1198681198671198781198731+ Γ119903119887119891 (119905) 119868119877119861 + Γ119903119904119891 (119905) 1198681198771198781198733 (4)
The human population may get the disease in one of thefollowing ways when the infected flea 119868119865 bites and infectsthe susceptible human being 119878119867 at a rate Γ119891ℎ when theyinteract with one another this can be with either a personwith pneumonic plague 119868119867119875 through airborne transmissionor septicemic plague 119868119867119878 through physical or sexual contactat the rates Γℎ119901ℎ and Γℎ119904ℎ respectively Another infectionis through airborne transmission through interaction withrodent infected with pneumonic plague 119868119877119875 or throughtouching or eating the infected rodent with septicemic plague119868119877119878 at rates of Γ119903119901ℎ and Γ119903119904ℎ respectively Human beings mayalso get the infection from the environment when they breathin the bacteria or physically contact the infected material atthe rate of1205961That is to say human population acquire plaguedisease following effective contact with infected humanrodent flea and the environment with force of infection 1198661given by1198661 (119905) = Γℎ119901ℎ (119905) 119868119867119875 + Γℎ119904ℎ (119905) 1198681198671198781198731 + Γ119891ℎ (119905) 1198681198651198732+ Γ119903119901ℎ (119905) 119868119877119875 + Γ119903119904ℎ (119905) 1198681198771198781198733 + 1205961 (119905) 119860 (5)
The subgroup 119878119867 after the infection progresses andbecomes latent to the disease at a rate 1205721 After 2 to 7 daysthe subgroups 119864119867 become infected into one of the threeinfectious classes 119868119867119861 119868119867119878 or 119868119867119875 (depending on the modeof transmission an individual is exposed to) and are capableof transmitting the disease The proportion of 119864119867 progressesand becomes infected by bubonic plague 119868119867119861 septicemicplague 119868119867119878 or pneumonic plague 119868119867119875 at the rate 1205722 andproportion to ]1 ]2 or ]3 respectivelyThe compartment 119868119867119861either through strong body immunity or if they get treatmentthey recover andmove to subgroup 119877119867 at a rate 1205723 otherwisethey either progress to subgroup 119868119867119875 or 119868119867119878 at a rate 1205723 ordie either naturally at a rate 1205831 or due to the disease at arate 1205751119887 The fraction of humans with septicemic plague 119868119867119878
either through strong body immunity or if treated recoverat a rate 1205724 and join 119877119867 otherwise they either progress tosubgroup 119868119867119875 at a rate 1205724 or die due to the disease at a rate1205751119904 or naturally at a rate 1205831 The compartments 119868119867119875 if treatedrecover at a rate1205725 otherwise they die either naturally at a rate1205831 or due to the disease at a rate 1205751119901 The subgroup 119877119867 attaintemporary immunity and then return andbecome susceptible119878119867 at a rate 120603
The rodent population may get a disease in one of thefollowing ways when the infected flea 119868119865 bites and infectsthe susceptible rodent 119878119877 at a rate Γ119891119903 through interactionbetween rodents themselves which may be with rodentinfected by pneumonic plague 119868119877119875 or septicemic plague 119868119877119878at the rates Γ119903119901119903 and Γ119903119904119903 respectively The other infectionmay be through interaction with human infected with eitherpneumonic plague 119868119867119875 or septicemic plague 119868119867119878 at ratesof Γℎ119901119903 and Γℎ119904119903 respectively When the susceptible rodentsufficiently interacts with the pathogens in environmentthrough breathing in the bacteria or physically touches theinfected material it gets the infections at the rate of 1205962Rodent also gets the disease through adequate interactionwith rodent human flea and pathogens in the environmentwith force of infection 1198662 given by1198662 (119905) = Γℎ119901119903 (119905) 119868119867119875 + Γℎ119904119903 (119905) 1198681198671198781198731 + Γ119891119903 (119905) 1198681198651198732+ Γ119903119901119903 (119905) 119868119877119875 + Γ119903119904119903 (119905) 1198681198771198781198733 + 1205962 (119905) 119860 (6)
The subgroup 119878119877 after the infection progress and becomelatent to the disease at a rate 1205741 After 2 to 7 days the subgroup119864119877 become infected and capable of transmitting the diseasethe fraction of it progresses and becomes infected by bubonicplague 119868119877119861 septicemic plague 119868119877119878 or pneumonic plague 119868119877119875at the rate 1205742 and proportional to 1205911 1205912 or 1205913 respectivelyTherodent in subgroup 119868119877119861 may either progress to subgroup 119868119877119875or 119868119877119878 at a rate 1205743 or die either naturally at a rate 1205833 or dueto the disease at a rate 1205753119887 The compartment 119868119877119878 may eitherprogress to 119868119877119875 at a rate 1205744 or die due to a disease at a rate 1205753119904or naturally at a rate 1205833 and the compartments 119868119877119875 die eithernaturally at a rate 1205833 or due to the disease at a rate 1205753119901
With regard to the pathogens in the environment weassume that the adequate interaction with 119878119867 and 119878119877 has anegligible effect on the dynamics of pathogens populationsize in the environment The pathogens in the environmentare populated at a constant rate 1205824 The infected human withpneumonic plague 119868119867119875 and rodent with pneumonic plague119868119877119875 also populate the environment 119860 with the bacteria atthe rates 1205781 and 1205782 respectively Thus the environment ispopulated with pathogens causing plague disease with theforce of infection 1198664 given by1198664 (119905) = 1205824 (119905) + 1205781 (119905) 1198681198671198751198731 + 1205782 (119905) 1198681198771198751198733 (7)
The pathogens within the environment suffer natural mor-tality at a rate 1205834 Human population in subgroups 119878119867 and119864119867 flea population in subgroup 119878119865 and rodent population insubgroups 119878119877 and 119864119877 suffer natural mortality at rates 1205831 1205832
6 International Journal of Mathematics and Mathematical Sciences
and 1205833 respectively The compartments 119868119867119861 119868119867119878 119868119867119875 119868119865 119868119877119861119868119877119878 and 119868119877119875 suffer both natural death at the rates12058311205832 and1205833and disease inducedmortality at rates 1205751119887 1205751119904 1205751119901 1205752 1205753119887 1205753119904and 1205753119901 respectively Human flea and rodent are recruited atthe rates 1205951 1205952 and 1205953 respectively24Model Equations for Plague Disease Nowwe assume thatthe variation of infection capability from one individual tothe other migration of individuals from one place to anotherand recruitment and death rates of individuals in differentstages due to seasonal weather variation affect only the rate atwhich the disease is transmitted from one infected individualto the other We now use the variables and parameters andtheir description given in Notations and Table 1 and thedescription of interactions to drive the system of differentialequations given as follows
Human Beings
119889119878119867119889119905 = 12059011205951 + 120603119877119867 minus 12057211198661 (119905) 119878119867 minus 1205831119878119867 (8a)119889119864119867119889119905 = (1 minus 1205901) 1205951 + 12057211198661 (119905) 119878119867 minus 1205722119864119867 minus 1205831119864119867 (8b)119889119868119867119861119889119905 = 1205722]2119864119867 minus 1205723119868119867119861 minus (1205831 + 1205751119887) 119868119867119861 (8c)119889119868119867119878119889119905 = 12057231205883119868119867119861 + 1205722]3119864119867 minus 1205724119868119867119878 minus (1205831 + 1205751119904) 119868119867119878 (8d)119889119868119867119875119889119905 = 1205722]1119864119867 + 12057231205881119868119867119861 + 1205724120585119868119867119878 minus 1205725119868119867119875minus (1205831 + 1205751119901) 119868119867119875 (8e)
119889119877119867119889119905 = 12057231205882119868119867119861 + 1205724 (1 minus 120585) 119868119867119878 + 1205725119868119867119875 minus 120603119877119867minus 1205831119877119867 (8f)
Rodents119889119878119877119889119905 = 12059021205953 minus 12057411198662 (119905) 119878119877 minus 1205833119878119877 (9a)119889119864119877119889119905 = (1 minus 1205902) 1205953 + 12057411198662 (119905) 119878119877 minus 1205742119864119877 minus 1205833119864119877 (9b)119889119868119877119861119889119905 = 12057421205913119864119877 minus 1205743119868119877119861 minus (1205833 + 1205753119887) 119868119877119861 (9c)119889119868119877119878119889119905 = 12057421205912119864119877 + 1205743 (1 minus 120601) 119868119877119861 minus 1205744119868119877119878minus (1205833 + 1205753119904) 119868119877119878 (9d)
In this section we discuss the feasible region and positivityof the plague disease model For convenience purpose andeasy presentation of the result we let 119862 denote all continuousfunctions on the real line If119891 is a periodic function in119862 thenwe use 119891 for the average value of 119891 on time interval [0 119879]defined by
for a continuous 119879-periodic function 119891(119905)31 Invariant Region Plague disease affects human rodentflea and pathogens in the environment populations For thepossible modeling process all state variables and parametersof themodel must be nonnegative for forall119905 ge 0 We thus need toverify whether the solutions of the model system ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) are in suitable feasible region whereall state variables are positive Inspired by Dumont et al [43]and Mpeshe et al [44] we first write system ((8a) (8b) (8c)(8d) (8e) (8f))ndash(11) in the following compact form
where 119883 = (119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875119878119865 119868119865 119860)119879119860(119909) is a 14times14matrix and 119865 is a column vectorWe then have
where 1198861 = (1205723 + 1205831 + 1205751119887) 1198862 = (1205724 + 1205831 + 1205751119904) 1198863 = (1205725 +1205831 + 1205751119901) 1198864 = (1205743 + 1205833 + 1205753119887) 1198865 = (1205744 + 1205833 + 1205753119904) 1198921 =(12057211198661(119905) + 1205831) 1198922 = (12057411198662(119905) + 1205833) and 1198923 = (1205731198663(119905) + 1205832)Now from submatrices 11986011 11986012 11986021 and 11986022 we can
deduce that matrix119860(119909) is aMetzler matrix such that all of itsoff-diagonal elements are nonnegative forall119909 isin R14+ and 119865 ge 0is Lipschitz continuousThus the feasible region for themodelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is the set
Thismeans that any trajectory of the system starting froman initial state in the positive orthant of R14+ remains foreverin Φ32 Positivity of the Solution We need to show that allvariables and parameters of the model are nonnegative forall119905 ge0 We now solve the equations of the system in their patchesfor testing the positivity We found that by letting the initialvalues of the systems ((8a) (8b) (8c) (8d) (8e) (8f)) ((9a)(9b) (9c) (9d) (9e)) ((10a) (10b)) and (11) be 119878119867(0) gt 0119878119877(0) gt 0 119878119865(0) gt 0 and 1198600 ge 0 119864119867(0) ge 0 119868119867119861(0) ge 0119868119867119878(0) ge 0 119868119867119875(0) ge 0 119877119867(0) ge 0 119864119877(0) ge 0 119868119877119861(0) ge 0119868119877119878(0) ge 0 119868119877119875(0) ge 0 and 119868119865(0) ge 0 in the solution
8 International Journal of Mathematics and Mathematical Sciences
set 119878119867(119905) 119878119877(119905) 119878119865(119905) 119860(119905) 119864119867(119905) 119868119867119861(119905) 119868119867119878(119905) 119868119867119875(119905)119877119867(119905) 119864119877(119905) 119868119877119861(119905) 119868119877119878(119905) 119868119877119875(119905) and 119868119865(119905) are nonnegativeforall119905 ge 04 Model Analysis
41 Disease-Free Equilibrium Solution The periodic modelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) with nonnega-tive continuous periodic functions has disease-free equilib-rium solution in which we consider the following equations119889119878119867119889119905 = 12059011205951 minus 1205831119878119867 (17)119889119878119877119889119905 = 12059021205953 minus 1205833119878119877 (18)119889119878119865119889119905 = 1205952119904 minus 1205832119878119865 (19)
Now given initial conditions 119878119867 = 1198781198670 isin R+ 119878119877 = 1198781198770 isinR+ and 119878119865 = 1198781198650 isin R+ for (17) (18) and (19) respectivelywe will have 119878119867 = 120590112059511205831 + (1198781198670 minus 120590112059511205831 ) 119890minus1205831119905
119878119877 = 120590212059531205833 + (1198781198770 minus 120590212059531205833 ) 119890minus1205833119905119878119877 = 12059521199041205832 + (1198781198650 minus 12059521199041205832 ) 119890minus1205832119905
(20)
As 119905 rarr infin (17) (18) and (19) admit unique solution 119878119867 equiv120590112059511205831 119878119877 equiv 120590212059531205833 and 119878119865 equiv 12059521199041205832 respectively whichis globally attractive in R3+
To find the disease-free equilibrium point we set thederivatives of system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11)equal to zero Then the model system has disease-freesolution which is obtained by setting 119868119867119861 = 119868119867119878 = 119868119867119875 =119864119867 = 119877119867 = 0 119868119877119861 = 119868119877119878 = 119868119877119875 = 119864119877 = 0 119868119865 = 0 and 119860 = 0for human rodent flea and pathogen system respectivelyHence system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) has adisease-free equilibrium point1198640 (1198780119867 1198640119867 1198680119867119861 1198680119867119878 1198680119867119875 1198770119867 1198780119877 1198640119877 1198680119877119861 1198680119877119878 1198680119877119875 1198780119865 11986801198651198600) = (120590112059511205831 0 0 0 0 0 120590212059531205833 0 0 0 0 12059521199041205832 0 0) (21)
5 Basic Reproduction Number
Let (R119896R119896) be the standard ordered 119896-dimensionalEuclidean space with a norm sdot For 119906 V isin R119896 we write119906 ge V provided 119906 minus V isin R119896+ 119906 gt V provided 119906 minus V isin R119896+ 0and 119906 ≫ V if 119906 minus V isin int(R119896+)
Now let 119860(119905) be the continuous cooperative irreducibleand 119879-periodic 119896 times 119896 matrix function with period 119879 gt 0
Φ119860(sdot)(119905) be the fundamental solution matrix of the linearordinary differential system119889119909119889119905 = 119860 (119905) 119909 (22)
and 120588(Φ119860(sdot)(119879)) be the spectral radius of Φ119860(sdot)(119879) By Aron-sson and Kellogg [45] it follows that Φ119860(sdot)(119905) is a matrixwith all elements positive for each 119905 gt 0 By the PerronFrobenius theorem 120588(Φ119860(sdot)(119879)) is the principal eigenvalue ofΦ119860(sdot)(119905) in the sense that it is simple and admits an eigenvectorVlowast ≫ 0 The following result is important for our subsequentcomparison argument
Proposition 1 let 120580 = (1119879) ln(120588(Φ119860(119879))) and then thereexists a positive 119879-periodic function V(119905) such that 119890120580119905V(119905) is asolution of 1199091015840 = 119860(119905)119909Proof Let Vlowast ≫ 0 be the eigenvector associated with thespectral radius 120588Φ119860(sdot)(119879)
By the change of variable119909 (119905) = 119890120583119905V (119905) (23)
system (22) becomes119889V119889119905 = 119860 (119905) V minus 120583V = (119860 (119905) minus 120583119868) V (24)
where 119868 is an identity matrixThus V(119905) = Φ(119860(sdot)minus120583119868)(119905)Vlowast is a positive solution of (24)
We can easily see that119890120583119905Φ(119860(sdot)minus120583119868) (119905) = Φ119860(sdot) (119905) (25)
Moreover
V (119879) = Φ(119860(sdot)minus120583119868) (119905) Vlowast = 119890minus120583119879Φ119860(sdot) (119879) Vlowast= 119890minus120583119879120588 (Φ119860(sdot) (119879)) Vlowast = Vlowast = V (0) (26)
Thus V(119905) is a positive 119879-periodic solution of (24) andhence 119909(119905) = 119890120583119905V(119905) is a solution of (22)
The plague disease model system ((8a) (8b) (8c) (8d)(8e) (8f))ndash(11) has unique disease-free equilibrium pointgiven in (21)
We consider a heterogeneous populationwhose individu-als are distinguishable by stage of the disease and hence iden-tifiable and put into epidemiological compartmentswhich are119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875 119878119865 119868119865 and 119860We sort the compartments so that the first 119898 compartmentscorrespond to infected individuals
We now let
F119894(119909) be the rate of appearance of new infections inthe 119894th compartmentsV+119894 (119909) be the rate of transfer of individuals into
compartment 119894 by all other means other than theepidemic onesVminus119894 (119909) be the rate of transfer of individuals out of
compartment 119894
International Journal of Mathematics and Mathematical Sciences 9
Then the plague disease transmissionmodel in ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) is governed by a periodic ordinarydifferential system given in119889119909119894119889119905 = F119894 (119905 119909) minusV119894 (119905 119909) ≜ 119891119894 (119905 119909) (27)
119894 (119909)We rearrange the system by sorting the infectious classes
(119864119867 119868119867119861 119868119867119878 119868119867119875 119864119877 119868119877119861 119868119877119878 119868119877119875 119868119865 119860) coming first Wethen have
F (119909) =((((((((((((((((
(1 minus 1205901) 1205951 + 12057211198661 (119905)119878119867000(1 minus 1205902) 1205953 + 12057411198662 (119905)1198781198770001205952119894 + 1205731198663 (119905)1198781198770
))))))))))))))))
(28)
V (119909)
=((((((((((((((((((((
1205722119864119867 + 1205831119864119867(1205723 + 1205831 + 1205751119887) 119868119867119861 minus 1205722]2119864119867(1205724 + 1205831 + 1205751119904) 119868119867119878 minus 12057231205883119868119867119861 minus 1205722]3119864119867(1205725 + 1205831 + 1205751119901) 119868119867119875 minus 1205722]1119864119867 minus 12057231205881119868119867119861 minus 12057241205851198681198671198781205742119864119877 + 1205833119864119877(1205743 + 1205833 + 1205753119887) 119868119877119861 minus 12057421205913119864119877(1205744 + 1205833 + 1205753119904) 119868119877119878 minus 12057421205912119864119877 minus 1205743 (1 minus 120579) 119868119877119861(1205833 + 1205753119901) 119868119877119875 minus 12057421205911119864119903 minus 1205743120579119868119877119861 minus 1205744119868119877119878(1205832 + 1205752) 1198681198651205834119860 minus 1205781 (119905) 1198681198671198751198731 minus 1205782 (119905) 1198681198771198751198733 + 1205824
))))))))))))))))))))
(29)
Then we have 119865 (119905) = (120597F119894120597119909119895 (1199090)) 119881 (119905) = (120597V119894120597119909119895 (1199090)) (30)
with 1 le 119894 119895 le 10Now using (30) the matrices 119865 and 119881 are as given below
Following the setting byWang and Zhao [46] and van denDriessche andWatmough [47] for epidemicmodels we checkconditions (A1)ndash(A7) for plague disease epidemic modelSystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is equivalent toperiodic ordinary differential system (27) Now consideringthis systemwe can easily see that conditions (A1)ndash(A5) statedbelow are satisfied
(A1) Since each function represents a directed transfer ofindividuals (human rodent flea and pathogens inthe environment) they are all nonnegative Thus foreach 1 le 119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) andVminus119894 (119905 119909) are nonnegative and continuous on R times R119899+
and continuously differentiable with respect to 119909(A2) There is a real number 119879 gt 0 such that for each 1 le119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) and Vminus119894 (119905 119909)
are 119879-periodic in 119905(A3) If a compartment is empty there will be no transfer
of individuals out of the compartment by any meansThat is to say if 119909119894 = 0 then Vminus
119894 = 0 In particular if119909 isin 119883119904 thenVminus119894 = 0 for 119894 = 1 119898
(A4) The incidence of infection for uninfected compart-ments is zero That is to sayF119894 = 0 for 119894 gt 119898
(A5) If the population is disease-free then the populationwill remain free of disease Thus if 119909 isin 119883119904 thenF119894 =0 andV+
119894 = 0 for 119894 = 1 119898We know that system (27) has disease-free periodicsolution given in (21) Now we define 119891(119905 119909(119905)) =F(119905 119909(119905)) minus V(119905 119909(119905)) and119872(119905) = (120597119891119894(119905119894 1198640)120597119909119895)11 le 119894 119895 le 14 where 119891119894(119905 119909(119905)) and 119909119894 are the
119894th components of 119891(119905 119909(119905)) and 119909 respectively Nowfrom (28) and (29) we obtain a 4 times 4matrix given in
A (t) = (minus1205831 120603 0 00 minus (120603 + 1205831) 0 00 0 minus1205833 00 0 0 minus1205832) (34)
We then let Φ119860(sdot)(119905) be the monodromy matrix ofthe linear 119879-periodic system 119889119911119889119905 = 119860(119905)119911 Then120588Φ119860(sdot)(119879) lt 1 which implies that 1198640 is linearlyasymptotically stable in the disease-free subspace119883119904 = 119909 ge 0 119909119894 = 0 forall119894 = 1 sdot sdot sdot 119898 where 119894 = 1 sdot sdot sdot 119898are the infected compartments Thus condition (A6)stated below holds
(A6) The disease-free periodic solution is asymptoti-cally stable in a disease-free subspace 119883119904 that is120588Φ119860(sdot)(119879) lt 1 where 120588Φ119860(sdot)(119879) is the principaleigenvalue of Φ119860(sdot)(119905)Next we set 119865(119905) and 119881(119905) as two 10 times 10 matricesdefined by (30) then using (28) and (29) we getmatri-ces 119865(119905) and 119881(119905) given in (31) and (32) respectivelyWe can further see that matrix 119865(119905) is nonnegativeand minus119881(119905) is cooperative in the sense that the off-diagonal elements are nonnegative Let 119884(119905 119904) 119905 ge 119904be the evolution operator of our 119879-periodic system119889119910119889119905 = minus119881 (119905) 119910 (35)
International Journal of Mathematics and Mathematical Sciences 11
That is for each 119904 isin R the 10 times 10 matrix 119884(119905 119904)satisfies119889119884 (119905 119904)119889119905 = minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119868119889 (36)
where 119868119889 is a 10 times 10 identity matrix Thus themonodromymatrixΦ119881(119905) of (35) equals119884(119905 0) 119905 ge 0Therefore condition (A7) stated below holds
(A7) The evolution of individuals in the infectious com-partments decays exponentially due to natural anddisease induced mortalities Thus 120588Φ119881(119879) lt 1
Now using the standard theory of linear periodic systemby Hale [48] there exist119870 gt 0 and weierp gt 0 such that119884 (119905 119904) le 119870119890minusweierp(119905minus119904) forall119905 ge 119904 119904 isin R (37)
We then have119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) le 119870 119865 (119905 minus 119886) 119890minusweierp119886forall119905 isin R 119886 isin [0infin) (38)
Considering the periodic environment we suppose thatΦ(119904) 119879-periodic in 119904 is the distribution of the new infectionat a rate 119865(119904)Φ(119904) produced by the infected individuals whowere introduced at time 119904 Given 119905 ge 119904 then 119884(119905 119904)119865(119904)Φ(119904)yields the distribution of those infected individuals who werenewly infected at time 119904 and remain in the infected class at 119905We then haveΨ (119905) = int119905
= intinfin0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886 (39)
which is the distribution of accumulative new infectionsat time 119905 produced by all those infected individual Φ(119904)introduced at previous time 119904 to 119905 (119904 le 119905)
Let 119862119879 be the ordered Banach space of all 119879-periodicfunction fromR toR119899 which is equippedwith themaximumnorm sdotinfin and the positive cone119862+119879 = Φ isin 119862119879Φ(119905) ge 0 119905 isinR Define a linear operator 119871 119862119879 rarr 119862119879 by(119871Φ) (119905) = intinfin
0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886forall119905 isin R Φ isin 119862119879 (40)
Now by Wang and Zhao [46] Diekmann et al [49] andvan denDriessche andWatmough [47] we name 119871 as the nextinfection operator then the basic reproduction number 119877119879of the periodic system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) isgiven 119877119879 = 120588 (119871) (41)
where 120588(119871) is the spectral radius of 119871
51 Characterization of 119877119879 In this subsection we investigatewhether the basic reproduction number in our periodicsystem can characterize the threshold of the disease invasionTo do this we consider the following linear 119879-periodicequation 119889119908119889119905 = [minus119881 (119905) + 119865 (119905)120582 ]119908 forall119905 isin R (42)
with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) 119905 ge 119904 119904 isin Rbe the evolution operator of system (42) on R10 We canclearly see that Φ119865minus119881(119905) = 119882(119905 0 1) forall119905 ge 0 Consideringmatrices (31) and (32)wenote that for each120582 isin (0infin) all off-diagonal elements of matrix minus119881(119905) + 119865(119905)120582 are nonnegative(cooperative matrix) It follows that the linear operator119882(119905 119904 120582) is positive in R10 for each 119905 ge 119904 119904 isin R Nowusing Perron-Frobenius theorem by Smith andWaltman [50]it entails that 120588(119882(119879 0 120582)) is an eigenvalue of 119882(119879 0 120582)with a nonnegative eigenvector Also using matrix similarityconcept by Shores [51] we can easily verify that matrix119882(119904 +119879 0 120582) is similar to the matrix119882(119879 0 120582) and hence 120590(119882(119904 +119879 0 120582)) = 120590(119882(119879 0 120582)) for any 119904 isin R where 120590(119863) is aspectrum of the matrix119863Proposition 2 (see [46]) We let (A1)ndash(A7) hold for system((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) then
(i) if 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 1205820is an eigenvalue of 119871 and hence 119877119879 gt 0
(ii) if 119877119879 gt 0 then 120582 = 119877119879 is the unique solution of120588(119882(119879 0 120582)) = 1(iii) 119877119879 = 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0This result shows that in order to find the basic reproduc-
tion number we need to find themonodromymatrixΦ119865minus119881(119905)of system (42) and evaluate itWe then find the spectral radiusof Φ119865minus119881(119879) and solve the equation 120588(Φ119865minus119881(119879)) = 1 for 120582which is the basic reproduction number
52 Computation of the Basic Reproduction Number Wecompute a time-averaged basic reproduction number 1198770using the next-generation matrix as outlined by Wesley andAllen [52] Heesterbeek [53] and Diekmann et al [49] Themethod has the advantage over the usual next-generationmethod in that the steps to reach an estimate of 1198770 and thematrix elements of the next-generation matrix have a clearbiological basis It is easy to handle complex diseases likeplague disease which has multiple transmission roots fromdifferent infectious agents
To do this we first categorize individuals by their stateat the moment they become infected (type at infection)These types at infection refer specifically to the birth ofthe infection in the individual These categories (types atinfection) differ in the way they transmit plague diseasewhich in turn differentiates their ability to produce secondarycases
In our case we categorize the individuals into eight statesand label them as follows human infected with bubonicplague (type 1) human infected with septicemic plague (type
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
4 International Journal of Mathematics and Mathematical Sciences
Table 1 Parameters and their description
Parameters Description Value ReferencesourceΓ119903119887119891(119905) Adequate contact rate between 119868119877119861 and flea 01 Eisen et al [36]Γ119903119904119891(119905) Adequate contact rate between 119868119877119878 and flea 01 Eisen et al [36]Γ119891ℎ(119905) Adequate contact rate between 119868119865 and human 00641 Eisen et al [36]Γ119891119903(119905) Adequate contact rate between 119868119865 and rodent 00641 Eisen et al [36]Γℎ119901ℎ(119905) Adequate contact rate between 119868119867119875 and 119878119867 039 EstimatedΓℎ119904ℎ(119905) Adequate contact rate between 119868119867119878 and 119878119867 012 EstimatedΓ119903119887ℎ(119905) Adequate contact rate between 119868119877119861 and 119878119867Γ119903119901ℎ(119905) Adequate contact rate between 119868119877119875 and 119878119867 019 EstimatedΓ119903119904ℎ(119905) Adequate contact rate between 119868119877119878 and 119878119867 021 Estimated1205721 Progression rate of 119878119867 to 119864119867 population 099 Estimated1205722 Progression rate out of 119864119867 to infectious state 023 Gani and Leach [37]12058811205723 Progression rate out of 119868119867119861 to 11986811986711987512058821205723 Progression rate out of 119868119867119861 to 11987711986712058831205723 Progression rate out of 119868119867119861 to 1198681198671198781205751119887 Disease induced death rate of 119868119867119861 004 Keeling and Gilligan [38]1205724 Progression rate out of 119868119867119878 to 119868119867119875 and 119877119867 006 Estimated1205751119904 Disease induced death rate of 119868119867119878 004 Estimated1205725 Progression rate out of 119868119867119875 to 119877119867 04 Gani and Leach [37]1205751119901 Disease induced death rate of 119868119867119875 063 Kugeler et al [39]1205741 Progression rate of 119878119877 to 119864119877 092 EstimatedΓℎ119887119891(119905) Adequate contact rate between 119868119867119861 and flea 01 Eisen et al [36]Γℎ119904119891(119905) Adequate contact rate between 119868119867119878 and flea 01 Eisen et al [36]Γ119903119901119903(119905) Adequate contact rate between 119868119877119875 and 119878119877 09 EstimatedΓ119903119904119903(119905) Adequate contact rate between 119868119877119878 and 119878119877 09 EstimatedΓℎ119901119903(119905) Adequate contact rate between 119868119867119875 and 119878119877 000005 EstimatedΓℎ119904119903(119905) Adequate contact rate between 119868119867119878 and 119878119877 000008 Estimated1205742 The rate at which rodent becomes infectious 098 Estimated1205743 Progression rate out of 119868119877119861 to 119868119877119878 and 119868119877119875 0194 Tollenaere et al [40]1205753119887 Disease induced death rate of 119868119877119861 01 Estimated1205744 Progression rate out of 119868119877119878 to 119868119877119875 005 Estimated1205753119904 Disease induced death rate of 119868119877119878 73 Tollenaere et al [40]1205753119901 Disease induced death rate of 119868119877119875 014 Estimated120603 Progression rate of 119877119867 to 119878119867 033 Kugeler et al [39]1205831 Natural death rate for human being 004 Keeling and Gilligan [38]1205832 Natural death rate for flea 02 Bacot and Martin [7]1205833 Natural death rate for rodent 1 Morand and Harvey [41]1205961(119905) Adequate contact rate 119860 and human being1205962(119905) Adequate contact rate 119860 and rodent1205781(119905) Recruitment rate of 119860 by 119868119867119875 02 Estimated1205782(119905) Recruitment rate of 119860 by 119868119877119875 04 Estimated1205834 Natural death rate for pathogens 01 Estimated1205951 Recruitment rate of human beings 009 Estimated120595 Recruitment rate of fleas1205953 Recruitment rate of rodents
International Journal of Mathematics and Mathematical Sciences 5
remain infectious in the environment for a long timeWhen asusceptible individual adequately interacts with the environ-ment infestedwithYersinia pestis heshe gets the disease evenin the absence of any vector
23 Description of Interactions The susceptible fleas in sub-group 119878119865 get Yersinia pestis bacteria through biting theinfected rodent 119868119877119861 or 119868119877119878 who are the primary reservoir forthe bacteria and become infected at the rates Γ119903119887119891 and Γ119903119904119891respectively Fleas may also get the disease when they bite theinfected human being with bubonic plague 119868119867119861 or septicemicplague 119868119867119878 at the rates Γℎ119887119891 and Γℎ119904119891 respectivelyThus the fleapopulation gets plague infection with the force of infectiongiven in 1198663 (119905) = Γℎ119887119891 (119905) 119868119867119861 + Γℎ119904119891 (119905) 1198681198671198781198731+ Γ119903119887119891 (119905) 119868119877119861 + Γ119903119904119891 (119905) 1198681198771198781198733 (4)
The human population may get the disease in one of thefollowing ways when the infected flea 119868119865 bites and infectsthe susceptible human being 119878119867 at a rate Γ119891ℎ when theyinteract with one another this can be with either a personwith pneumonic plague 119868119867119875 through airborne transmissionor septicemic plague 119868119867119878 through physical or sexual contactat the rates Γℎ119901ℎ and Γℎ119904ℎ respectively Another infectionis through airborne transmission through interaction withrodent infected with pneumonic plague 119868119877119875 or throughtouching or eating the infected rodent with septicemic plague119868119877119878 at rates of Γ119903119901ℎ and Γ119903119904ℎ respectively Human beings mayalso get the infection from the environment when they breathin the bacteria or physically contact the infected material atthe rate of1205961That is to say human population acquire plaguedisease following effective contact with infected humanrodent flea and the environment with force of infection 1198661given by1198661 (119905) = Γℎ119901ℎ (119905) 119868119867119875 + Γℎ119904ℎ (119905) 1198681198671198781198731 + Γ119891ℎ (119905) 1198681198651198732+ Γ119903119901ℎ (119905) 119868119877119875 + Γ119903119904ℎ (119905) 1198681198771198781198733 + 1205961 (119905) 119860 (5)
The subgroup 119878119867 after the infection progresses andbecomes latent to the disease at a rate 1205721 After 2 to 7 daysthe subgroups 119864119867 become infected into one of the threeinfectious classes 119868119867119861 119868119867119878 or 119868119867119875 (depending on the modeof transmission an individual is exposed to) and are capableof transmitting the disease The proportion of 119864119867 progressesand becomes infected by bubonic plague 119868119867119861 septicemicplague 119868119867119878 or pneumonic plague 119868119867119875 at the rate 1205722 andproportion to ]1 ]2 or ]3 respectivelyThe compartment 119868119867119861either through strong body immunity or if they get treatmentthey recover andmove to subgroup 119877119867 at a rate 1205723 otherwisethey either progress to subgroup 119868119867119875 or 119868119867119878 at a rate 1205723 ordie either naturally at a rate 1205831 or due to the disease at arate 1205751119887 The fraction of humans with septicemic plague 119868119867119878
either through strong body immunity or if treated recoverat a rate 1205724 and join 119877119867 otherwise they either progress tosubgroup 119868119867119875 at a rate 1205724 or die due to the disease at a rate1205751119904 or naturally at a rate 1205831 The compartments 119868119867119875 if treatedrecover at a rate1205725 otherwise they die either naturally at a rate1205831 or due to the disease at a rate 1205751119901 The subgroup 119877119867 attaintemporary immunity and then return andbecome susceptible119878119867 at a rate 120603
The rodent population may get a disease in one of thefollowing ways when the infected flea 119868119865 bites and infectsthe susceptible rodent 119878119877 at a rate Γ119891119903 through interactionbetween rodents themselves which may be with rodentinfected by pneumonic plague 119868119877119875 or septicemic plague 119868119877119878at the rates Γ119903119901119903 and Γ119903119904119903 respectively The other infectionmay be through interaction with human infected with eitherpneumonic plague 119868119867119875 or septicemic plague 119868119867119878 at ratesof Γℎ119901119903 and Γℎ119904119903 respectively When the susceptible rodentsufficiently interacts with the pathogens in environmentthrough breathing in the bacteria or physically touches theinfected material it gets the infections at the rate of 1205962Rodent also gets the disease through adequate interactionwith rodent human flea and pathogens in the environmentwith force of infection 1198662 given by1198662 (119905) = Γℎ119901119903 (119905) 119868119867119875 + Γℎ119904119903 (119905) 1198681198671198781198731 + Γ119891119903 (119905) 1198681198651198732+ Γ119903119901119903 (119905) 119868119877119875 + Γ119903119904119903 (119905) 1198681198771198781198733 + 1205962 (119905) 119860 (6)
The subgroup 119878119877 after the infection progress and becomelatent to the disease at a rate 1205741 After 2 to 7 days the subgroup119864119877 become infected and capable of transmitting the diseasethe fraction of it progresses and becomes infected by bubonicplague 119868119877119861 septicemic plague 119868119877119878 or pneumonic plague 119868119877119875at the rate 1205742 and proportional to 1205911 1205912 or 1205913 respectivelyTherodent in subgroup 119868119877119861 may either progress to subgroup 119868119877119875or 119868119877119878 at a rate 1205743 or die either naturally at a rate 1205833 or dueto the disease at a rate 1205753119887 The compartment 119868119877119878 may eitherprogress to 119868119877119875 at a rate 1205744 or die due to a disease at a rate 1205753119904or naturally at a rate 1205833 and the compartments 119868119877119875 die eithernaturally at a rate 1205833 or due to the disease at a rate 1205753119901
With regard to the pathogens in the environment weassume that the adequate interaction with 119878119867 and 119878119877 has anegligible effect on the dynamics of pathogens populationsize in the environment The pathogens in the environmentare populated at a constant rate 1205824 The infected human withpneumonic plague 119868119867119875 and rodent with pneumonic plague119868119877119875 also populate the environment 119860 with the bacteria atthe rates 1205781 and 1205782 respectively Thus the environment ispopulated with pathogens causing plague disease with theforce of infection 1198664 given by1198664 (119905) = 1205824 (119905) + 1205781 (119905) 1198681198671198751198731 + 1205782 (119905) 1198681198771198751198733 (7)
The pathogens within the environment suffer natural mor-tality at a rate 1205834 Human population in subgroups 119878119867 and119864119867 flea population in subgroup 119878119865 and rodent population insubgroups 119878119877 and 119864119877 suffer natural mortality at rates 1205831 1205832
6 International Journal of Mathematics and Mathematical Sciences
and 1205833 respectively The compartments 119868119867119861 119868119867119878 119868119867119875 119868119865 119868119877119861119868119877119878 and 119868119877119875 suffer both natural death at the rates12058311205832 and1205833and disease inducedmortality at rates 1205751119887 1205751119904 1205751119901 1205752 1205753119887 1205753119904and 1205753119901 respectively Human flea and rodent are recruited atthe rates 1205951 1205952 and 1205953 respectively24Model Equations for Plague Disease Nowwe assume thatthe variation of infection capability from one individual tothe other migration of individuals from one place to anotherand recruitment and death rates of individuals in differentstages due to seasonal weather variation affect only the rate atwhich the disease is transmitted from one infected individualto the other We now use the variables and parameters andtheir description given in Notations and Table 1 and thedescription of interactions to drive the system of differentialequations given as follows
Human Beings
119889119878119867119889119905 = 12059011205951 + 120603119877119867 minus 12057211198661 (119905) 119878119867 minus 1205831119878119867 (8a)119889119864119867119889119905 = (1 minus 1205901) 1205951 + 12057211198661 (119905) 119878119867 minus 1205722119864119867 minus 1205831119864119867 (8b)119889119868119867119861119889119905 = 1205722]2119864119867 minus 1205723119868119867119861 minus (1205831 + 1205751119887) 119868119867119861 (8c)119889119868119867119878119889119905 = 12057231205883119868119867119861 + 1205722]3119864119867 minus 1205724119868119867119878 minus (1205831 + 1205751119904) 119868119867119878 (8d)119889119868119867119875119889119905 = 1205722]1119864119867 + 12057231205881119868119867119861 + 1205724120585119868119867119878 minus 1205725119868119867119875minus (1205831 + 1205751119901) 119868119867119875 (8e)
119889119877119867119889119905 = 12057231205882119868119867119861 + 1205724 (1 minus 120585) 119868119867119878 + 1205725119868119867119875 minus 120603119877119867minus 1205831119877119867 (8f)
Rodents119889119878119877119889119905 = 12059021205953 minus 12057411198662 (119905) 119878119877 minus 1205833119878119877 (9a)119889119864119877119889119905 = (1 minus 1205902) 1205953 + 12057411198662 (119905) 119878119877 minus 1205742119864119877 minus 1205833119864119877 (9b)119889119868119877119861119889119905 = 12057421205913119864119877 minus 1205743119868119877119861 minus (1205833 + 1205753119887) 119868119877119861 (9c)119889119868119877119878119889119905 = 12057421205912119864119877 + 1205743 (1 minus 120601) 119868119877119861 minus 1205744119868119877119878minus (1205833 + 1205753119904) 119868119877119878 (9d)
In this section we discuss the feasible region and positivityof the plague disease model For convenience purpose andeasy presentation of the result we let 119862 denote all continuousfunctions on the real line If119891 is a periodic function in119862 thenwe use 119891 for the average value of 119891 on time interval [0 119879]defined by
for a continuous 119879-periodic function 119891(119905)31 Invariant Region Plague disease affects human rodentflea and pathogens in the environment populations For thepossible modeling process all state variables and parametersof themodel must be nonnegative for forall119905 ge 0 We thus need toverify whether the solutions of the model system ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) are in suitable feasible region whereall state variables are positive Inspired by Dumont et al [43]and Mpeshe et al [44] we first write system ((8a) (8b) (8c)(8d) (8e) (8f))ndash(11) in the following compact form
where 119883 = (119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875119878119865 119868119865 119860)119879119860(119909) is a 14times14matrix and 119865 is a column vectorWe then have
where 1198861 = (1205723 + 1205831 + 1205751119887) 1198862 = (1205724 + 1205831 + 1205751119904) 1198863 = (1205725 +1205831 + 1205751119901) 1198864 = (1205743 + 1205833 + 1205753119887) 1198865 = (1205744 + 1205833 + 1205753119904) 1198921 =(12057211198661(119905) + 1205831) 1198922 = (12057411198662(119905) + 1205833) and 1198923 = (1205731198663(119905) + 1205832)Now from submatrices 11986011 11986012 11986021 and 11986022 we can
deduce that matrix119860(119909) is aMetzler matrix such that all of itsoff-diagonal elements are nonnegative forall119909 isin R14+ and 119865 ge 0is Lipschitz continuousThus the feasible region for themodelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is the set
Thismeans that any trajectory of the system starting froman initial state in the positive orthant of R14+ remains foreverin Φ32 Positivity of the Solution We need to show that allvariables and parameters of the model are nonnegative forall119905 ge0 We now solve the equations of the system in their patchesfor testing the positivity We found that by letting the initialvalues of the systems ((8a) (8b) (8c) (8d) (8e) (8f)) ((9a)(9b) (9c) (9d) (9e)) ((10a) (10b)) and (11) be 119878119867(0) gt 0119878119877(0) gt 0 119878119865(0) gt 0 and 1198600 ge 0 119864119867(0) ge 0 119868119867119861(0) ge 0119868119867119878(0) ge 0 119868119867119875(0) ge 0 119877119867(0) ge 0 119864119877(0) ge 0 119868119877119861(0) ge 0119868119877119878(0) ge 0 119868119877119875(0) ge 0 and 119868119865(0) ge 0 in the solution
8 International Journal of Mathematics and Mathematical Sciences
set 119878119867(119905) 119878119877(119905) 119878119865(119905) 119860(119905) 119864119867(119905) 119868119867119861(119905) 119868119867119878(119905) 119868119867119875(119905)119877119867(119905) 119864119877(119905) 119868119877119861(119905) 119868119877119878(119905) 119868119877119875(119905) and 119868119865(119905) are nonnegativeforall119905 ge 04 Model Analysis
41 Disease-Free Equilibrium Solution The periodic modelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) with nonnega-tive continuous periodic functions has disease-free equilib-rium solution in which we consider the following equations119889119878119867119889119905 = 12059011205951 minus 1205831119878119867 (17)119889119878119877119889119905 = 12059021205953 minus 1205833119878119877 (18)119889119878119865119889119905 = 1205952119904 minus 1205832119878119865 (19)
Now given initial conditions 119878119867 = 1198781198670 isin R+ 119878119877 = 1198781198770 isinR+ and 119878119865 = 1198781198650 isin R+ for (17) (18) and (19) respectivelywe will have 119878119867 = 120590112059511205831 + (1198781198670 minus 120590112059511205831 ) 119890minus1205831119905
119878119877 = 120590212059531205833 + (1198781198770 minus 120590212059531205833 ) 119890minus1205833119905119878119877 = 12059521199041205832 + (1198781198650 minus 12059521199041205832 ) 119890minus1205832119905
(20)
As 119905 rarr infin (17) (18) and (19) admit unique solution 119878119867 equiv120590112059511205831 119878119877 equiv 120590212059531205833 and 119878119865 equiv 12059521199041205832 respectively whichis globally attractive in R3+
To find the disease-free equilibrium point we set thederivatives of system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11)equal to zero Then the model system has disease-freesolution which is obtained by setting 119868119867119861 = 119868119867119878 = 119868119867119875 =119864119867 = 119877119867 = 0 119868119877119861 = 119868119877119878 = 119868119877119875 = 119864119877 = 0 119868119865 = 0 and 119860 = 0for human rodent flea and pathogen system respectivelyHence system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) has adisease-free equilibrium point1198640 (1198780119867 1198640119867 1198680119867119861 1198680119867119878 1198680119867119875 1198770119867 1198780119877 1198640119877 1198680119877119861 1198680119877119878 1198680119877119875 1198780119865 11986801198651198600) = (120590112059511205831 0 0 0 0 0 120590212059531205833 0 0 0 0 12059521199041205832 0 0) (21)
5 Basic Reproduction Number
Let (R119896R119896) be the standard ordered 119896-dimensionalEuclidean space with a norm sdot For 119906 V isin R119896 we write119906 ge V provided 119906 minus V isin R119896+ 119906 gt V provided 119906 minus V isin R119896+ 0and 119906 ≫ V if 119906 minus V isin int(R119896+)
Now let 119860(119905) be the continuous cooperative irreducibleand 119879-periodic 119896 times 119896 matrix function with period 119879 gt 0
Φ119860(sdot)(119905) be the fundamental solution matrix of the linearordinary differential system119889119909119889119905 = 119860 (119905) 119909 (22)
and 120588(Φ119860(sdot)(119879)) be the spectral radius of Φ119860(sdot)(119879) By Aron-sson and Kellogg [45] it follows that Φ119860(sdot)(119905) is a matrixwith all elements positive for each 119905 gt 0 By the PerronFrobenius theorem 120588(Φ119860(sdot)(119879)) is the principal eigenvalue ofΦ119860(sdot)(119905) in the sense that it is simple and admits an eigenvectorVlowast ≫ 0 The following result is important for our subsequentcomparison argument
Proposition 1 let 120580 = (1119879) ln(120588(Φ119860(119879))) and then thereexists a positive 119879-periodic function V(119905) such that 119890120580119905V(119905) is asolution of 1199091015840 = 119860(119905)119909Proof Let Vlowast ≫ 0 be the eigenvector associated with thespectral radius 120588Φ119860(sdot)(119879)
By the change of variable119909 (119905) = 119890120583119905V (119905) (23)
system (22) becomes119889V119889119905 = 119860 (119905) V minus 120583V = (119860 (119905) minus 120583119868) V (24)
where 119868 is an identity matrixThus V(119905) = Φ(119860(sdot)minus120583119868)(119905)Vlowast is a positive solution of (24)
We can easily see that119890120583119905Φ(119860(sdot)minus120583119868) (119905) = Φ119860(sdot) (119905) (25)
Moreover
V (119879) = Φ(119860(sdot)minus120583119868) (119905) Vlowast = 119890minus120583119879Φ119860(sdot) (119879) Vlowast= 119890minus120583119879120588 (Φ119860(sdot) (119879)) Vlowast = Vlowast = V (0) (26)
Thus V(119905) is a positive 119879-periodic solution of (24) andhence 119909(119905) = 119890120583119905V(119905) is a solution of (22)
The plague disease model system ((8a) (8b) (8c) (8d)(8e) (8f))ndash(11) has unique disease-free equilibrium pointgiven in (21)
We consider a heterogeneous populationwhose individu-als are distinguishable by stage of the disease and hence iden-tifiable and put into epidemiological compartmentswhich are119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875 119878119865 119868119865 and 119860We sort the compartments so that the first 119898 compartmentscorrespond to infected individuals
We now let
F119894(119909) be the rate of appearance of new infections inthe 119894th compartmentsV+119894 (119909) be the rate of transfer of individuals into
compartment 119894 by all other means other than theepidemic onesVminus119894 (119909) be the rate of transfer of individuals out of
compartment 119894
International Journal of Mathematics and Mathematical Sciences 9
Then the plague disease transmissionmodel in ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) is governed by a periodic ordinarydifferential system given in119889119909119894119889119905 = F119894 (119905 119909) minusV119894 (119905 119909) ≜ 119891119894 (119905 119909) (27)
119894 (119909)We rearrange the system by sorting the infectious classes
(119864119867 119868119867119861 119868119867119878 119868119867119875 119864119877 119868119877119861 119868119877119878 119868119877119875 119868119865 119860) coming first Wethen have
F (119909) =((((((((((((((((
(1 minus 1205901) 1205951 + 12057211198661 (119905)119878119867000(1 minus 1205902) 1205953 + 12057411198662 (119905)1198781198770001205952119894 + 1205731198663 (119905)1198781198770
))))))))))))))))
(28)
V (119909)
=((((((((((((((((((((
1205722119864119867 + 1205831119864119867(1205723 + 1205831 + 1205751119887) 119868119867119861 minus 1205722]2119864119867(1205724 + 1205831 + 1205751119904) 119868119867119878 minus 12057231205883119868119867119861 minus 1205722]3119864119867(1205725 + 1205831 + 1205751119901) 119868119867119875 minus 1205722]1119864119867 minus 12057231205881119868119867119861 minus 12057241205851198681198671198781205742119864119877 + 1205833119864119877(1205743 + 1205833 + 1205753119887) 119868119877119861 minus 12057421205913119864119877(1205744 + 1205833 + 1205753119904) 119868119877119878 minus 12057421205912119864119877 minus 1205743 (1 minus 120579) 119868119877119861(1205833 + 1205753119901) 119868119877119875 minus 12057421205911119864119903 minus 1205743120579119868119877119861 minus 1205744119868119877119878(1205832 + 1205752) 1198681198651205834119860 minus 1205781 (119905) 1198681198671198751198731 minus 1205782 (119905) 1198681198771198751198733 + 1205824
))))))))))))))))))))
(29)
Then we have 119865 (119905) = (120597F119894120597119909119895 (1199090)) 119881 (119905) = (120597V119894120597119909119895 (1199090)) (30)
with 1 le 119894 119895 le 10Now using (30) the matrices 119865 and 119881 are as given below
Following the setting byWang and Zhao [46] and van denDriessche andWatmough [47] for epidemicmodels we checkconditions (A1)ndash(A7) for plague disease epidemic modelSystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is equivalent toperiodic ordinary differential system (27) Now consideringthis systemwe can easily see that conditions (A1)ndash(A5) statedbelow are satisfied
(A1) Since each function represents a directed transfer ofindividuals (human rodent flea and pathogens inthe environment) they are all nonnegative Thus foreach 1 le 119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) andVminus119894 (119905 119909) are nonnegative and continuous on R times R119899+
and continuously differentiable with respect to 119909(A2) There is a real number 119879 gt 0 such that for each 1 le119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) and Vminus119894 (119905 119909)
are 119879-periodic in 119905(A3) If a compartment is empty there will be no transfer
of individuals out of the compartment by any meansThat is to say if 119909119894 = 0 then Vminus
119894 = 0 In particular if119909 isin 119883119904 thenVminus119894 = 0 for 119894 = 1 119898
(A4) The incidence of infection for uninfected compart-ments is zero That is to sayF119894 = 0 for 119894 gt 119898
(A5) If the population is disease-free then the populationwill remain free of disease Thus if 119909 isin 119883119904 thenF119894 =0 andV+
119894 = 0 for 119894 = 1 119898We know that system (27) has disease-free periodicsolution given in (21) Now we define 119891(119905 119909(119905)) =F(119905 119909(119905)) minus V(119905 119909(119905)) and119872(119905) = (120597119891119894(119905119894 1198640)120597119909119895)11 le 119894 119895 le 14 where 119891119894(119905 119909(119905)) and 119909119894 are the
119894th components of 119891(119905 119909(119905)) and 119909 respectively Nowfrom (28) and (29) we obtain a 4 times 4matrix given in
A (t) = (minus1205831 120603 0 00 minus (120603 + 1205831) 0 00 0 minus1205833 00 0 0 minus1205832) (34)
We then let Φ119860(sdot)(119905) be the monodromy matrix ofthe linear 119879-periodic system 119889119911119889119905 = 119860(119905)119911 Then120588Φ119860(sdot)(119879) lt 1 which implies that 1198640 is linearlyasymptotically stable in the disease-free subspace119883119904 = 119909 ge 0 119909119894 = 0 forall119894 = 1 sdot sdot sdot 119898 where 119894 = 1 sdot sdot sdot 119898are the infected compartments Thus condition (A6)stated below holds
(A6) The disease-free periodic solution is asymptoti-cally stable in a disease-free subspace 119883119904 that is120588Φ119860(sdot)(119879) lt 1 where 120588Φ119860(sdot)(119879) is the principaleigenvalue of Φ119860(sdot)(119905)Next we set 119865(119905) and 119881(119905) as two 10 times 10 matricesdefined by (30) then using (28) and (29) we getmatri-ces 119865(119905) and 119881(119905) given in (31) and (32) respectivelyWe can further see that matrix 119865(119905) is nonnegativeand minus119881(119905) is cooperative in the sense that the off-diagonal elements are nonnegative Let 119884(119905 119904) 119905 ge 119904be the evolution operator of our 119879-periodic system119889119910119889119905 = minus119881 (119905) 119910 (35)
International Journal of Mathematics and Mathematical Sciences 11
That is for each 119904 isin R the 10 times 10 matrix 119884(119905 119904)satisfies119889119884 (119905 119904)119889119905 = minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119868119889 (36)
where 119868119889 is a 10 times 10 identity matrix Thus themonodromymatrixΦ119881(119905) of (35) equals119884(119905 0) 119905 ge 0Therefore condition (A7) stated below holds
(A7) The evolution of individuals in the infectious com-partments decays exponentially due to natural anddisease induced mortalities Thus 120588Φ119881(119879) lt 1
Now using the standard theory of linear periodic systemby Hale [48] there exist119870 gt 0 and weierp gt 0 such that119884 (119905 119904) le 119870119890minusweierp(119905minus119904) forall119905 ge 119904 119904 isin R (37)
We then have119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) le 119870 119865 (119905 minus 119886) 119890minusweierp119886forall119905 isin R 119886 isin [0infin) (38)
Considering the periodic environment we suppose thatΦ(119904) 119879-periodic in 119904 is the distribution of the new infectionat a rate 119865(119904)Φ(119904) produced by the infected individuals whowere introduced at time 119904 Given 119905 ge 119904 then 119884(119905 119904)119865(119904)Φ(119904)yields the distribution of those infected individuals who werenewly infected at time 119904 and remain in the infected class at 119905We then haveΨ (119905) = int119905
= intinfin0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886 (39)
which is the distribution of accumulative new infectionsat time 119905 produced by all those infected individual Φ(119904)introduced at previous time 119904 to 119905 (119904 le 119905)
Let 119862119879 be the ordered Banach space of all 119879-periodicfunction fromR toR119899 which is equippedwith themaximumnorm sdotinfin and the positive cone119862+119879 = Φ isin 119862119879Φ(119905) ge 0 119905 isinR Define a linear operator 119871 119862119879 rarr 119862119879 by(119871Φ) (119905) = intinfin
0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886forall119905 isin R Φ isin 119862119879 (40)
Now by Wang and Zhao [46] Diekmann et al [49] andvan denDriessche andWatmough [47] we name 119871 as the nextinfection operator then the basic reproduction number 119877119879of the periodic system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) isgiven 119877119879 = 120588 (119871) (41)
where 120588(119871) is the spectral radius of 119871
51 Characterization of 119877119879 In this subsection we investigatewhether the basic reproduction number in our periodicsystem can characterize the threshold of the disease invasionTo do this we consider the following linear 119879-periodicequation 119889119908119889119905 = [minus119881 (119905) + 119865 (119905)120582 ]119908 forall119905 isin R (42)
with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) 119905 ge 119904 119904 isin Rbe the evolution operator of system (42) on R10 We canclearly see that Φ119865minus119881(119905) = 119882(119905 0 1) forall119905 ge 0 Consideringmatrices (31) and (32)wenote that for each120582 isin (0infin) all off-diagonal elements of matrix minus119881(119905) + 119865(119905)120582 are nonnegative(cooperative matrix) It follows that the linear operator119882(119905 119904 120582) is positive in R10 for each 119905 ge 119904 119904 isin R Nowusing Perron-Frobenius theorem by Smith andWaltman [50]it entails that 120588(119882(119879 0 120582)) is an eigenvalue of 119882(119879 0 120582)with a nonnegative eigenvector Also using matrix similarityconcept by Shores [51] we can easily verify that matrix119882(119904 +119879 0 120582) is similar to the matrix119882(119879 0 120582) and hence 120590(119882(119904 +119879 0 120582)) = 120590(119882(119879 0 120582)) for any 119904 isin R where 120590(119863) is aspectrum of the matrix119863Proposition 2 (see [46]) We let (A1)ndash(A7) hold for system((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) then
(i) if 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 1205820is an eigenvalue of 119871 and hence 119877119879 gt 0
(ii) if 119877119879 gt 0 then 120582 = 119877119879 is the unique solution of120588(119882(119879 0 120582)) = 1(iii) 119877119879 = 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0This result shows that in order to find the basic reproduc-
tion number we need to find themonodromymatrixΦ119865minus119881(119905)of system (42) and evaluate itWe then find the spectral radiusof Φ119865minus119881(119879) and solve the equation 120588(Φ119865minus119881(119879)) = 1 for 120582which is the basic reproduction number
52 Computation of the Basic Reproduction Number Wecompute a time-averaged basic reproduction number 1198770using the next-generation matrix as outlined by Wesley andAllen [52] Heesterbeek [53] and Diekmann et al [49] Themethod has the advantage over the usual next-generationmethod in that the steps to reach an estimate of 1198770 and thematrix elements of the next-generation matrix have a clearbiological basis It is easy to handle complex diseases likeplague disease which has multiple transmission roots fromdifferent infectious agents
To do this we first categorize individuals by their stateat the moment they become infected (type at infection)These types at infection refer specifically to the birth ofthe infection in the individual These categories (types atinfection) differ in the way they transmit plague diseasewhich in turn differentiates their ability to produce secondarycases
In our case we categorize the individuals into eight statesand label them as follows human infected with bubonicplague (type 1) human infected with septicemic plague (type
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
International Journal of Mathematics and Mathematical Sciences 5
remain infectious in the environment for a long timeWhen asusceptible individual adequately interacts with the environ-ment infestedwithYersinia pestis heshe gets the disease evenin the absence of any vector
23 Description of Interactions The susceptible fleas in sub-group 119878119865 get Yersinia pestis bacteria through biting theinfected rodent 119868119877119861 or 119868119877119878 who are the primary reservoir forthe bacteria and become infected at the rates Γ119903119887119891 and Γ119903119904119891respectively Fleas may also get the disease when they bite theinfected human being with bubonic plague 119868119867119861 or septicemicplague 119868119867119878 at the rates Γℎ119887119891 and Γℎ119904119891 respectivelyThus the fleapopulation gets plague infection with the force of infectiongiven in 1198663 (119905) = Γℎ119887119891 (119905) 119868119867119861 + Γℎ119904119891 (119905) 1198681198671198781198731+ Γ119903119887119891 (119905) 119868119877119861 + Γ119903119904119891 (119905) 1198681198771198781198733 (4)
The human population may get the disease in one of thefollowing ways when the infected flea 119868119865 bites and infectsthe susceptible human being 119878119867 at a rate Γ119891ℎ when theyinteract with one another this can be with either a personwith pneumonic plague 119868119867119875 through airborne transmissionor septicemic plague 119868119867119878 through physical or sexual contactat the rates Γℎ119901ℎ and Γℎ119904ℎ respectively Another infectionis through airborne transmission through interaction withrodent infected with pneumonic plague 119868119877119875 or throughtouching or eating the infected rodent with septicemic plague119868119877119878 at rates of Γ119903119901ℎ and Γ119903119904ℎ respectively Human beings mayalso get the infection from the environment when they breathin the bacteria or physically contact the infected material atthe rate of1205961That is to say human population acquire plaguedisease following effective contact with infected humanrodent flea and the environment with force of infection 1198661given by1198661 (119905) = Γℎ119901ℎ (119905) 119868119867119875 + Γℎ119904ℎ (119905) 1198681198671198781198731 + Γ119891ℎ (119905) 1198681198651198732+ Γ119903119901ℎ (119905) 119868119877119875 + Γ119903119904ℎ (119905) 1198681198771198781198733 + 1205961 (119905) 119860 (5)
The subgroup 119878119867 after the infection progresses andbecomes latent to the disease at a rate 1205721 After 2 to 7 daysthe subgroups 119864119867 become infected into one of the threeinfectious classes 119868119867119861 119868119867119878 or 119868119867119875 (depending on the modeof transmission an individual is exposed to) and are capableof transmitting the disease The proportion of 119864119867 progressesand becomes infected by bubonic plague 119868119867119861 septicemicplague 119868119867119878 or pneumonic plague 119868119867119875 at the rate 1205722 andproportion to ]1 ]2 or ]3 respectivelyThe compartment 119868119867119861either through strong body immunity or if they get treatmentthey recover andmove to subgroup 119877119867 at a rate 1205723 otherwisethey either progress to subgroup 119868119867119875 or 119868119867119878 at a rate 1205723 ordie either naturally at a rate 1205831 or due to the disease at arate 1205751119887 The fraction of humans with septicemic plague 119868119867119878
either through strong body immunity or if treated recoverat a rate 1205724 and join 119877119867 otherwise they either progress tosubgroup 119868119867119875 at a rate 1205724 or die due to the disease at a rate1205751119904 or naturally at a rate 1205831 The compartments 119868119867119875 if treatedrecover at a rate1205725 otherwise they die either naturally at a rate1205831 or due to the disease at a rate 1205751119901 The subgroup 119877119867 attaintemporary immunity and then return andbecome susceptible119878119867 at a rate 120603
The rodent population may get a disease in one of thefollowing ways when the infected flea 119868119865 bites and infectsthe susceptible rodent 119878119877 at a rate Γ119891119903 through interactionbetween rodents themselves which may be with rodentinfected by pneumonic plague 119868119877119875 or septicemic plague 119868119877119878at the rates Γ119903119901119903 and Γ119903119904119903 respectively The other infectionmay be through interaction with human infected with eitherpneumonic plague 119868119867119875 or septicemic plague 119868119867119878 at ratesof Γℎ119901119903 and Γℎ119904119903 respectively When the susceptible rodentsufficiently interacts with the pathogens in environmentthrough breathing in the bacteria or physically touches theinfected material it gets the infections at the rate of 1205962Rodent also gets the disease through adequate interactionwith rodent human flea and pathogens in the environmentwith force of infection 1198662 given by1198662 (119905) = Γℎ119901119903 (119905) 119868119867119875 + Γℎ119904119903 (119905) 1198681198671198781198731 + Γ119891119903 (119905) 1198681198651198732+ Γ119903119901119903 (119905) 119868119877119875 + Γ119903119904119903 (119905) 1198681198771198781198733 + 1205962 (119905) 119860 (6)
The subgroup 119878119877 after the infection progress and becomelatent to the disease at a rate 1205741 After 2 to 7 days the subgroup119864119877 become infected and capable of transmitting the diseasethe fraction of it progresses and becomes infected by bubonicplague 119868119877119861 septicemic plague 119868119877119878 or pneumonic plague 119868119877119875at the rate 1205742 and proportional to 1205911 1205912 or 1205913 respectivelyTherodent in subgroup 119868119877119861 may either progress to subgroup 119868119877119875or 119868119877119878 at a rate 1205743 or die either naturally at a rate 1205833 or dueto the disease at a rate 1205753119887 The compartment 119868119877119878 may eitherprogress to 119868119877119875 at a rate 1205744 or die due to a disease at a rate 1205753119904or naturally at a rate 1205833 and the compartments 119868119877119875 die eithernaturally at a rate 1205833 or due to the disease at a rate 1205753119901
With regard to the pathogens in the environment weassume that the adequate interaction with 119878119867 and 119878119877 has anegligible effect on the dynamics of pathogens populationsize in the environment The pathogens in the environmentare populated at a constant rate 1205824 The infected human withpneumonic plague 119868119867119875 and rodent with pneumonic plague119868119877119875 also populate the environment 119860 with the bacteria atthe rates 1205781 and 1205782 respectively Thus the environment ispopulated with pathogens causing plague disease with theforce of infection 1198664 given by1198664 (119905) = 1205824 (119905) + 1205781 (119905) 1198681198671198751198731 + 1205782 (119905) 1198681198771198751198733 (7)
The pathogens within the environment suffer natural mor-tality at a rate 1205834 Human population in subgroups 119878119867 and119864119867 flea population in subgroup 119878119865 and rodent population insubgroups 119878119877 and 119864119877 suffer natural mortality at rates 1205831 1205832
6 International Journal of Mathematics and Mathematical Sciences
and 1205833 respectively The compartments 119868119867119861 119868119867119878 119868119867119875 119868119865 119868119877119861119868119877119878 and 119868119877119875 suffer both natural death at the rates12058311205832 and1205833and disease inducedmortality at rates 1205751119887 1205751119904 1205751119901 1205752 1205753119887 1205753119904and 1205753119901 respectively Human flea and rodent are recruited atthe rates 1205951 1205952 and 1205953 respectively24Model Equations for Plague Disease Nowwe assume thatthe variation of infection capability from one individual tothe other migration of individuals from one place to anotherand recruitment and death rates of individuals in differentstages due to seasonal weather variation affect only the rate atwhich the disease is transmitted from one infected individualto the other We now use the variables and parameters andtheir description given in Notations and Table 1 and thedescription of interactions to drive the system of differentialequations given as follows
Human Beings
119889119878119867119889119905 = 12059011205951 + 120603119877119867 minus 12057211198661 (119905) 119878119867 minus 1205831119878119867 (8a)119889119864119867119889119905 = (1 minus 1205901) 1205951 + 12057211198661 (119905) 119878119867 minus 1205722119864119867 minus 1205831119864119867 (8b)119889119868119867119861119889119905 = 1205722]2119864119867 minus 1205723119868119867119861 minus (1205831 + 1205751119887) 119868119867119861 (8c)119889119868119867119878119889119905 = 12057231205883119868119867119861 + 1205722]3119864119867 minus 1205724119868119867119878 minus (1205831 + 1205751119904) 119868119867119878 (8d)119889119868119867119875119889119905 = 1205722]1119864119867 + 12057231205881119868119867119861 + 1205724120585119868119867119878 minus 1205725119868119867119875minus (1205831 + 1205751119901) 119868119867119875 (8e)
119889119877119867119889119905 = 12057231205882119868119867119861 + 1205724 (1 minus 120585) 119868119867119878 + 1205725119868119867119875 minus 120603119877119867minus 1205831119877119867 (8f)
Rodents119889119878119877119889119905 = 12059021205953 minus 12057411198662 (119905) 119878119877 minus 1205833119878119877 (9a)119889119864119877119889119905 = (1 minus 1205902) 1205953 + 12057411198662 (119905) 119878119877 minus 1205742119864119877 minus 1205833119864119877 (9b)119889119868119877119861119889119905 = 12057421205913119864119877 minus 1205743119868119877119861 minus (1205833 + 1205753119887) 119868119877119861 (9c)119889119868119877119878119889119905 = 12057421205912119864119877 + 1205743 (1 minus 120601) 119868119877119861 minus 1205744119868119877119878minus (1205833 + 1205753119904) 119868119877119878 (9d)
In this section we discuss the feasible region and positivityof the plague disease model For convenience purpose andeasy presentation of the result we let 119862 denote all continuousfunctions on the real line If119891 is a periodic function in119862 thenwe use 119891 for the average value of 119891 on time interval [0 119879]defined by
for a continuous 119879-periodic function 119891(119905)31 Invariant Region Plague disease affects human rodentflea and pathogens in the environment populations For thepossible modeling process all state variables and parametersof themodel must be nonnegative for forall119905 ge 0 We thus need toverify whether the solutions of the model system ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) are in suitable feasible region whereall state variables are positive Inspired by Dumont et al [43]and Mpeshe et al [44] we first write system ((8a) (8b) (8c)(8d) (8e) (8f))ndash(11) in the following compact form
where 119883 = (119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875119878119865 119868119865 119860)119879119860(119909) is a 14times14matrix and 119865 is a column vectorWe then have
where 1198861 = (1205723 + 1205831 + 1205751119887) 1198862 = (1205724 + 1205831 + 1205751119904) 1198863 = (1205725 +1205831 + 1205751119901) 1198864 = (1205743 + 1205833 + 1205753119887) 1198865 = (1205744 + 1205833 + 1205753119904) 1198921 =(12057211198661(119905) + 1205831) 1198922 = (12057411198662(119905) + 1205833) and 1198923 = (1205731198663(119905) + 1205832)Now from submatrices 11986011 11986012 11986021 and 11986022 we can
deduce that matrix119860(119909) is aMetzler matrix such that all of itsoff-diagonal elements are nonnegative forall119909 isin R14+ and 119865 ge 0is Lipschitz continuousThus the feasible region for themodelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is the set
Thismeans that any trajectory of the system starting froman initial state in the positive orthant of R14+ remains foreverin Φ32 Positivity of the Solution We need to show that allvariables and parameters of the model are nonnegative forall119905 ge0 We now solve the equations of the system in their patchesfor testing the positivity We found that by letting the initialvalues of the systems ((8a) (8b) (8c) (8d) (8e) (8f)) ((9a)(9b) (9c) (9d) (9e)) ((10a) (10b)) and (11) be 119878119867(0) gt 0119878119877(0) gt 0 119878119865(0) gt 0 and 1198600 ge 0 119864119867(0) ge 0 119868119867119861(0) ge 0119868119867119878(0) ge 0 119868119867119875(0) ge 0 119877119867(0) ge 0 119864119877(0) ge 0 119868119877119861(0) ge 0119868119877119878(0) ge 0 119868119877119875(0) ge 0 and 119868119865(0) ge 0 in the solution
8 International Journal of Mathematics and Mathematical Sciences
set 119878119867(119905) 119878119877(119905) 119878119865(119905) 119860(119905) 119864119867(119905) 119868119867119861(119905) 119868119867119878(119905) 119868119867119875(119905)119877119867(119905) 119864119877(119905) 119868119877119861(119905) 119868119877119878(119905) 119868119877119875(119905) and 119868119865(119905) are nonnegativeforall119905 ge 04 Model Analysis
41 Disease-Free Equilibrium Solution The periodic modelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) with nonnega-tive continuous periodic functions has disease-free equilib-rium solution in which we consider the following equations119889119878119867119889119905 = 12059011205951 minus 1205831119878119867 (17)119889119878119877119889119905 = 12059021205953 minus 1205833119878119877 (18)119889119878119865119889119905 = 1205952119904 minus 1205832119878119865 (19)
Now given initial conditions 119878119867 = 1198781198670 isin R+ 119878119877 = 1198781198770 isinR+ and 119878119865 = 1198781198650 isin R+ for (17) (18) and (19) respectivelywe will have 119878119867 = 120590112059511205831 + (1198781198670 minus 120590112059511205831 ) 119890minus1205831119905
119878119877 = 120590212059531205833 + (1198781198770 minus 120590212059531205833 ) 119890minus1205833119905119878119877 = 12059521199041205832 + (1198781198650 minus 12059521199041205832 ) 119890minus1205832119905
(20)
As 119905 rarr infin (17) (18) and (19) admit unique solution 119878119867 equiv120590112059511205831 119878119877 equiv 120590212059531205833 and 119878119865 equiv 12059521199041205832 respectively whichis globally attractive in R3+
To find the disease-free equilibrium point we set thederivatives of system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11)equal to zero Then the model system has disease-freesolution which is obtained by setting 119868119867119861 = 119868119867119878 = 119868119867119875 =119864119867 = 119877119867 = 0 119868119877119861 = 119868119877119878 = 119868119877119875 = 119864119877 = 0 119868119865 = 0 and 119860 = 0for human rodent flea and pathogen system respectivelyHence system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) has adisease-free equilibrium point1198640 (1198780119867 1198640119867 1198680119867119861 1198680119867119878 1198680119867119875 1198770119867 1198780119877 1198640119877 1198680119877119861 1198680119877119878 1198680119877119875 1198780119865 11986801198651198600) = (120590112059511205831 0 0 0 0 0 120590212059531205833 0 0 0 0 12059521199041205832 0 0) (21)
5 Basic Reproduction Number
Let (R119896R119896) be the standard ordered 119896-dimensionalEuclidean space with a norm sdot For 119906 V isin R119896 we write119906 ge V provided 119906 minus V isin R119896+ 119906 gt V provided 119906 minus V isin R119896+ 0and 119906 ≫ V if 119906 minus V isin int(R119896+)
Now let 119860(119905) be the continuous cooperative irreducibleand 119879-periodic 119896 times 119896 matrix function with period 119879 gt 0
Φ119860(sdot)(119905) be the fundamental solution matrix of the linearordinary differential system119889119909119889119905 = 119860 (119905) 119909 (22)
and 120588(Φ119860(sdot)(119879)) be the spectral radius of Φ119860(sdot)(119879) By Aron-sson and Kellogg [45] it follows that Φ119860(sdot)(119905) is a matrixwith all elements positive for each 119905 gt 0 By the PerronFrobenius theorem 120588(Φ119860(sdot)(119879)) is the principal eigenvalue ofΦ119860(sdot)(119905) in the sense that it is simple and admits an eigenvectorVlowast ≫ 0 The following result is important for our subsequentcomparison argument
Proposition 1 let 120580 = (1119879) ln(120588(Φ119860(119879))) and then thereexists a positive 119879-periodic function V(119905) such that 119890120580119905V(119905) is asolution of 1199091015840 = 119860(119905)119909Proof Let Vlowast ≫ 0 be the eigenvector associated with thespectral radius 120588Φ119860(sdot)(119879)
By the change of variable119909 (119905) = 119890120583119905V (119905) (23)
system (22) becomes119889V119889119905 = 119860 (119905) V minus 120583V = (119860 (119905) minus 120583119868) V (24)
where 119868 is an identity matrixThus V(119905) = Φ(119860(sdot)minus120583119868)(119905)Vlowast is a positive solution of (24)
We can easily see that119890120583119905Φ(119860(sdot)minus120583119868) (119905) = Φ119860(sdot) (119905) (25)
Moreover
V (119879) = Φ(119860(sdot)minus120583119868) (119905) Vlowast = 119890minus120583119879Φ119860(sdot) (119879) Vlowast= 119890minus120583119879120588 (Φ119860(sdot) (119879)) Vlowast = Vlowast = V (0) (26)
Thus V(119905) is a positive 119879-periodic solution of (24) andhence 119909(119905) = 119890120583119905V(119905) is a solution of (22)
The plague disease model system ((8a) (8b) (8c) (8d)(8e) (8f))ndash(11) has unique disease-free equilibrium pointgiven in (21)
We consider a heterogeneous populationwhose individu-als are distinguishable by stage of the disease and hence iden-tifiable and put into epidemiological compartmentswhich are119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875 119878119865 119868119865 and 119860We sort the compartments so that the first 119898 compartmentscorrespond to infected individuals
We now let
F119894(119909) be the rate of appearance of new infections inthe 119894th compartmentsV+119894 (119909) be the rate of transfer of individuals into
compartment 119894 by all other means other than theepidemic onesVminus119894 (119909) be the rate of transfer of individuals out of
compartment 119894
International Journal of Mathematics and Mathematical Sciences 9
Then the plague disease transmissionmodel in ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) is governed by a periodic ordinarydifferential system given in119889119909119894119889119905 = F119894 (119905 119909) minusV119894 (119905 119909) ≜ 119891119894 (119905 119909) (27)
119894 (119909)We rearrange the system by sorting the infectious classes
(119864119867 119868119867119861 119868119867119878 119868119867119875 119864119877 119868119877119861 119868119877119878 119868119877119875 119868119865 119860) coming first Wethen have
F (119909) =((((((((((((((((
(1 minus 1205901) 1205951 + 12057211198661 (119905)119878119867000(1 minus 1205902) 1205953 + 12057411198662 (119905)1198781198770001205952119894 + 1205731198663 (119905)1198781198770
))))))))))))))))
(28)
V (119909)
=((((((((((((((((((((
1205722119864119867 + 1205831119864119867(1205723 + 1205831 + 1205751119887) 119868119867119861 minus 1205722]2119864119867(1205724 + 1205831 + 1205751119904) 119868119867119878 minus 12057231205883119868119867119861 minus 1205722]3119864119867(1205725 + 1205831 + 1205751119901) 119868119867119875 minus 1205722]1119864119867 minus 12057231205881119868119867119861 minus 12057241205851198681198671198781205742119864119877 + 1205833119864119877(1205743 + 1205833 + 1205753119887) 119868119877119861 minus 12057421205913119864119877(1205744 + 1205833 + 1205753119904) 119868119877119878 minus 12057421205912119864119877 minus 1205743 (1 minus 120579) 119868119877119861(1205833 + 1205753119901) 119868119877119875 minus 12057421205911119864119903 minus 1205743120579119868119877119861 minus 1205744119868119877119878(1205832 + 1205752) 1198681198651205834119860 minus 1205781 (119905) 1198681198671198751198731 minus 1205782 (119905) 1198681198771198751198733 + 1205824
))))))))))))))))))))
(29)
Then we have 119865 (119905) = (120597F119894120597119909119895 (1199090)) 119881 (119905) = (120597V119894120597119909119895 (1199090)) (30)
with 1 le 119894 119895 le 10Now using (30) the matrices 119865 and 119881 are as given below
Following the setting byWang and Zhao [46] and van denDriessche andWatmough [47] for epidemicmodels we checkconditions (A1)ndash(A7) for plague disease epidemic modelSystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is equivalent toperiodic ordinary differential system (27) Now consideringthis systemwe can easily see that conditions (A1)ndash(A5) statedbelow are satisfied
(A1) Since each function represents a directed transfer ofindividuals (human rodent flea and pathogens inthe environment) they are all nonnegative Thus foreach 1 le 119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) andVminus119894 (119905 119909) are nonnegative and continuous on R times R119899+
and continuously differentiable with respect to 119909(A2) There is a real number 119879 gt 0 such that for each 1 le119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) and Vminus119894 (119905 119909)
are 119879-periodic in 119905(A3) If a compartment is empty there will be no transfer
of individuals out of the compartment by any meansThat is to say if 119909119894 = 0 then Vminus
119894 = 0 In particular if119909 isin 119883119904 thenVminus119894 = 0 for 119894 = 1 119898
(A4) The incidence of infection for uninfected compart-ments is zero That is to sayF119894 = 0 for 119894 gt 119898
(A5) If the population is disease-free then the populationwill remain free of disease Thus if 119909 isin 119883119904 thenF119894 =0 andV+
119894 = 0 for 119894 = 1 119898We know that system (27) has disease-free periodicsolution given in (21) Now we define 119891(119905 119909(119905)) =F(119905 119909(119905)) minus V(119905 119909(119905)) and119872(119905) = (120597119891119894(119905119894 1198640)120597119909119895)11 le 119894 119895 le 14 where 119891119894(119905 119909(119905)) and 119909119894 are the
119894th components of 119891(119905 119909(119905)) and 119909 respectively Nowfrom (28) and (29) we obtain a 4 times 4matrix given in
A (t) = (minus1205831 120603 0 00 minus (120603 + 1205831) 0 00 0 minus1205833 00 0 0 minus1205832) (34)
We then let Φ119860(sdot)(119905) be the monodromy matrix ofthe linear 119879-periodic system 119889119911119889119905 = 119860(119905)119911 Then120588Φ119860(sdot)(119879) lt 1 which implies that 1198640 is linearlyasymptotically stable in the disease-free subspace119883119904 = 119909 ge 0 119909119894 = 0 forall119894 = 1 sdot sdot sdot 119898 where 119894 = 1 sdot sdot sdot 119898are the infected compartments Thus condition (A6)stated below holds
(A6) The disease-free periodic solution is asymptoti-cally stable in a disease-free subspace 119883119904 that is120588Φ119860(sdot)(119879) lt 1 where 120588Φ119860(sdot)(119879) is the principaleigenvalue of Φ119860(sdot)(119905)Next we set 119865(119905) and 119881(119905) as two 10 times 10 matricesdefined by (30) then using (28) and (29) we getmatri-ces 119865(119905) and 119881(119905) given in (31) and (32) respectivelyWe can further see that matrix 119865(119905) is nonnegativeand minus119881(119905) is cooperative in the sense that the off-diagonal elements are nonnegative Let 119884(119905 119904) 119905 ge 119904be the evolution operator of our 119879-periodic system119889119910119889119905 = minus119881 (119905) 119910 (35)
International Journal of Mathematics and Mathematical Sciences 11
That is for each 119904 isin R the 10 times 10 matrix 119884(119905 119904)satisfies119889119884 (119905 119904)119889119905 = minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119868119889 (36)
where 119868119889 is a 10 times 10 identity matrix Thus themonodromymatrixΦ119881(119905) of (35) equals119884(119905 0) 119905 ge 0Therefore condition (A7) stated below holds
(A7) The evolution of individuals in the infectious com-partments decays exponentially due to natural anddisease induced mortalities Thus 120588Φ119881(119879) lt 1
Now using the standard theory of linear periodic systemby Hale [48] there exist119870 gt 0 and weierp gt 0 such that119884 (119905 119904) le 119870119890minusweierp(119905minus119904) forall119905 ge 119904 119904 isin R (37)
We then have119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) le 119870 119865 (119905 minus 119886) 119890minusweierp119886forall119905 isin R 119886 isin [0infin) (38)
Considering the periodic environment we suppose thatΦ(119904) 119879-periodic in 119904 is the distribution of the new infectionat a rate 119865(119904)Φ(119904) produced by the infected individuals whowere introduced at time 119904 Given 119905 ge 119904 then 119884(119905 119904)119865(119904)Φ(119904)yields the distribution of those infected individuals who werenewly infected at time 119904 and remain in the infected class at 119905We then haveΨ (119905) = int119905
= intinfin0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886 (39)
which is the distribution of accumulative new infectionsat time 119905 produced by all those infected individual Φ(119904)introduced at previous time 119904 to 119905 (119904 le 119905)
Let 119862119879 be the ordered Banach space of all 119879-periodicfunction fromR toR119899 which is equippedwith themaximumnorm sdotinfin and the positive cone119862+119879 = Φ isin 119862119879Φ(119905) ge 0 119905 isinR Define a linear operator 119871 119862119879 rarr 119862119879 by(119871Φ) (119905) = intinfin
0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886forall119905 isin R Φ isin 119862119879 (40)
Now by Wang and Zhao [46] Diekmann et al [49] andvan denDriessche andWatmough [47] we name 119871 as the nextinfection operator then the basic reproduction number 119877119879of the periodic system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) isgiven 119877119879 = 120588 (119871) (41)
where 120588(119871) is the spectral radius of 119871
51 Characterization of 119877119879 In this subsection we investigatewhether the basic reproduction number in our periodicsystem can characterize the threshold of the disease invasionTo do this we consider the following linear 119879-periodicequation 119889119908119889119905 = [minus119881 (119905) + 119865 (119905)120582 ]119908 forall119905 isin R (42)
with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) 119905 ge 119904 119904 isin Rbe the evolution operator of system (42) on R10 We canclearly see that Φ119865minus119881(119905) = 119882(119905 0 1) forall119905 ge 0 Consideringmatrices (31) and (32)wenote that for each120582 isin (0infin) all off-diagonal elements of matrix minus119881(119905) + 119865(119905)120582 are nonnegative(cooperative matrix) It follows that the linear operator119882(119905 119904 120582) is positive in R10 for each 119905 ge 119904 119904 isin R Nowusing Perron-Frobenius theorem by Smith andWaltman [50]it entails that 120588(119882(119879 0 120582)) is an eigenvalue of 119882(119879 0 120582)with a nonnegative eigenvector Also using matrix similarityconcept by Shores [51] we can easily verify that matrix119882(119904 +119879 0 120582) is similar to the matrix119882(119879 0 120582) and hence 120590(119882(119904 +119879 0 120582)) = 120590(119882(119879 0 120582)) for any 119904 isin R where 120590(119863) is aspectrum of the matrix119863Proposition 2 (see [46]) We let (A1)ndash(A7) hold for system((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) then
(i) if 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 1205820is an eigenvalue of 119871 and hence 119877119879 gt 0
(ii) if 119877119879 gt 0 then 120582 = 119877119879 is the unique solution of120588(119882(119879 0 120582)) = 1(iii) 119877119879 = 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0This result shows that in order to find the basic reproduc-
tion number we need to find themonodromymatrixΦ119865minus119881(119905)of system (42) and evaluate itWe then find the spectral radiusof Φ119865minus119881(119879) and solve the equation 120588(Φ119865minus119881(119879)) = 1 for 120582which is the basic reproduction number
52 Computation of the Basic Reproduction Number Wecompute a time-averaged basic reproduction number 1198770using the next-generation matrix as outlined by Wesley andAllen [52] Heesterbeek [53] and Diekmann et al [49] Themethod has the advantage over the usual next-generationmethod in that the steps to reach an estimate of 1198770 and thematrix elements of the next-generation matrix have a clearbiological basis It is easy to handle complex diseases likeplague disease which has multiple transmission roots fromdifferent infectious agents
To do this we first categorize individuals by their stateat the moment they become infected (type at infection)These types at infection refer specifically to the birth ofthe infection in the individual These categories (types atinfection) differ in the way they transmit plague diseasewhich in turn differentiates their ability to produce secondarycases
In our case we categorize the individuals into eight statesand label them as follows human infected with bubonicplague (type 1) human infected with septicemic plague (type
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
6 International Journal of Mathematics and Mathematical Sciences
and 1205833 respectively The compartments 119868119867119861 119868119867119878 119868119867119875 119868119865 119868119877119861119868119877119878 and 119868119877119875 suffer both natural death at the rates12058311205832 and1205833and disease inducedmortality at rates 1205751119887 1205751119904 1205751119901 1205752 1205753119887 1205753119904and 1205753119901 respectively Human flea and rodent are recruited atthe rates 1205951 1205952 and 1205953 respectively24Model Equations for Plague Disease Nowwe assume thatthe variation of infection capability from one individual tothe other migration of individuals from one place to anotherand recruitment and death rates of individuals in differentstages due to seasonal weather variation affect only the rate atwhich the disease is transmitted from one infected individualto the other We now use the variables and parameters andtheir description given in Notations and Table 1 and thedescription of interactions to drive the system of differentialequations given as follows
Human Beings
119889119878119867119889119905 = 12059011205951 + 120603119877119867 minus 12057211198661 (119905) 119878119867 minus 1205831119878119867 (8a)119889119864119867119889119905 = (1 minus 1205901) 1205951 + 12057211198661 (119905) 119878119867 minus 1205722119864119867 minus 1205831119864119867 (8b)119889119868119867119861119889119905 = 1205722]2119864119867 minus 1205723119868119867119861 minus (1205831 + 1205751119887) 119868119867119861 (8c)119889119868119867119878119889119905 = 12057231205883119868119867119861 + 1205722]3119864119867 minus 1205724119868119867119878 minus (1205831 + 1205751119904) 119868119867119878 (8d)119889119868119867119875119889119905 = 1205722]1119864119867 + 12057231205881119868119867119861 + 1205724120585119868119867119878 minus 1205725119868119867119875minus (1205831 + 1205751119901) 119868119867119875 (8e)
119889119877119867119889119905 = 12057231205882119868119867119861 + 1205724 (1 minus 120585) 119868119867119878 + 1205725119868119867119875 minus 120603119877119867minus 1205831119877119867 (8f)
Rodents119889119878119877119889119905 = 12059021205953 minus 12057411198662 (119905) 119878119877 minus 1205833119878119877 (9a)119889119864119877119889119905 = (1 minus 1205902) 1205953 + 12057411198662 (119905) 119878119877 minus 1205742119864119877 minus 1205833119864119877 (9b)119889119868119877119861119889119905 = 12057421205913119864119877 minus 1205743119868119877119861 minus (1205833 + 1205753119887) 119868119877119861 (9c)119889119868119877119878119889119905 = 12057421205912119864119877 + 1205743 (1 minus 120601) 119868119877119861 minus 1205744119868119877119878minus (1205833 + 1205753119904) 119868119877119878 (9d)
In this section we discuss the feasible region and positivityof the plague disease model For convenience purpose andeasy presentation of the result we let 119862 denote all continuousfunctions on the real line If119891 is a periodic function in119862 thenwe use 119891 for the average value of 119891 on time interval [0 119879]defined by
for a continuous 119879-periodic function 119891(119905)31 Invariant Region Plague disease affects human rodentflea and pathogens in the environment populations For thepossible modeling process all state variables and parametersof themodel must be nonnegative for forall119905 ge 0 We thus need toverify whether the solutions of the model system ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) are in suitable feasible region whereall state variables are positive Inspired by Dumont et al [43]and Mpeshe et al [44] we first write system ((8a) (8b) (8c)(8d) (8e) (8f))ndash(11) in the following compact form
where 119883 = (119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875119878119865 119868119865 119860)119879119860(119909) is a 14times14matrix and 119865 is a column vectorWe then have
where 1198861 = (1205723 + 1205831 + 1205751119887) 1198862 = (1205724 + 1205831 + 1205751119904) 1198863 = (1205725 +1205831 + 1205751119901) 1198864 = (1205743 + 1205833 + 1205753119887) 1198865 = (1205744 + 1205833 + 1205753119904) 1198921 =(12057211198661(119905) + 1205831) 1198922 = (12057411198662(119905) + 1205833) and 1198923 = (1205731198663(119905) + 1205832)Now from submatrices 11986011 11986012 11986021 and 11986022 we can
deduce that matrix119860(119909) is aMetzler matrix such that all of itsoff-diagonal elements are nonnegative forall119909 isin R14+ and 119865 ge 0is Lipschitz continuousThus the feasible region for themodelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is the set
Thismeans that any trajectory of the system starting froman initial state in the positive orthant of R14+ remains foreverin Φ32 Positivity of the Solution We need to show that allvariables and parameters of the model are nonnegative forall119905 ge0 We now solve the equations of the system in their patchesfor testing the positivity We found that by letting the initialvalues of the systems ((8a) (8b) (8c) (8d) (8e) (8f)) ((9a)(9b) (9c) (9d) (9e)) ((10a) (10b)) and (11) be 119878119867(0) gt 0119878119877(0) gt 0 119878119865(0) gt 0 and 1198600 ge 0 119864119867(0) ge 0 119868119867119861(0) ge 0119868119867119878(0) ge 0 119868119867119875(0) ge 0 119877119867(0) ge 0 119864119877(0) ge 0 119868119877119861(0) ge 0119868119877119878(0) ge 0 119868119877119875(0) ge 0 and 119868119865(0) ge 0 in the solution
8 International Journal of Mathematics and Mathematical Sciences
set 119878119867(119905) 119878119877(119905) 119878119865(119905) 119860(119905) 119864119867(119905) 119868119867119861(119905) 119868119867119878(119905) 119868119867119875(119905)119877119867(119905) 119864119877(119905) 119868119877119861(119905) 119868119877119878(119905) 119868119877119875(119905) and 119868119865(119905) are nonnegativeforall119905 ge 04 Model Analysis
41 Disease-Free Equilibrium Solution The periodic modelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) with nonnega-tive continuous periodic functions has disease-free equilib-rium solution in which we consider the following equations119889119878119867119889119905 = 12059011205951 minus 1205831119878119867 (17)119889119878119877119889119905 = 12059021205953 minus 1205833119878119877 (18)119889119878119865119889119905 = 1205952119904 minus 1205832119878119865 (19)
Now given initial conditions 119878119867 = 1198781198670 isin R+ 119878119877 = 1198781198770 isinR+ and 119878119865 = 1198781198650 isin R+ for (17) (18) and (19) respectivelywe will have 119878119867 = 120590112059511205831 + (1198781198670 minus 120590112059511205831 ) 119890minus1205831119905
119878119877 = 120590212059531205833 + (1198781198770 minus 120590212059531205833 ) 119890minus1205833119905119878119877 = 12059521199041205832 + (1198781198650 minus 12059521199041205832 ) 119890minus1205832119905
(20)
As 119905 rarr infin (17) (18) and (19) admit unique solution 119878119867 equiv120590112059511205831 119878119877 equiv 120590212059531205833 and 119878119865 equiv 12059521199041205832 respectively whichis globally attractive in R3+
To find the disease-free equilibrium point we set thederivatives of system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11)equal to zero Then the model system has disease-freesolution which is obtained by setting 119868119867119861 = 119868119867119878 = 119868119867119875 =119864119867 = 119877119867 = 0 119868119877119861 = 119868119877119878 = 119868119877119875 = 119864119877 = 0 119868119865 = 0 and 119860 = 0for human rodent flea and pathogen system respectivelyHence system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) has adisease-free equilibrium point1198640 (1198780119867 1198640119867 1198680119867119861 1198680119867119878 1198680119867119875 1198770119867 1198780119877 1198640119877 1198680119877119861 1198680119877119878 1198680119877119875 1198780119865 11986801198651198600) = (120590112059511205831 0 0 0 0 0 120590212059531205833 0 0 0 0 12059521199041205832 0 0) (21)
5 Basic Reproduction Number
Let (R119896R119896) be the standard ordered 119896-dimensionalEuclidean space with a norm sdot For 119906 V isin R119896 we write119906 ge V provided 119906 minus V isin R119896+ 119906 gt V provided 119906 minus V isin R119896+ 0and 119906 ≫ V if 119906 minus V isin int(R119896+)
Now let 119860(119905) be the continuous cooperative irreducibleand 119879-periodic 119896 times 119896 matrix function with period 119879 gt 0
Φ119860(sdot)(119905) be the fundamental solution matrix of the linearordinary differential system119889119909119889119905 = 119860 (119905) 119909 (22)
and 120588(Φ119860(sdot)(119879)) be the spectral radius of Φ119860(sdot)(119879) By Aron-sson and Kellogg [45] it follows that Φ119860(sdot)(119905) is a matrixwith all elements positive for each 119905 gt 0 By the PerronFrobenius theorem 120588(Φ119860(sdot)(119879)) is the principal eigenvalue ofΦ119860(sdot)(119905) in the sense that it is simple and admits an eigenvectorVlowast ≫ 0 The following result is important for our subsequentcomparison argument
Proposition 1 let 120580 = (1119879) ln(120588(Φ119860(119879))) and then thereexists a positive 119879-periodic function V(119905) such that 119890120580119905V(119905) is asolution of 1199091015840 = 119860(119905)119909Proof Let Vlowast ≫ 0 be the eigenvector associated with thespectral radius 120588Φ119860(sdot)(119879)
By the change of variable119909 (119905) = 119890120583119905V (119905) (23)
system (22) becomes119889V119889119905 = 119860 (119905) V minus 120583V = (119860 (119905) minus 120583119868) V (24)
where 119868 is an identity matrixThus V(119905) = Φ(119860(sdot)minus120583119868)(119905)Vlowast is a positive solution of (24)
We can easily see that119890120583119905Φ(119860(sdot)minus120583119868) (119905) = Φ119860(sdot) (119905) (25)
Moreover
V (119879) = Φ(119860(sdot)minus120583119868) (119905) Vlowast = 119890minus120583119879Φ119860(sdot) (119879) Vlowast= 119890minus120583119879120588 (Φ119860(sdot) (119879)) Vlowast = Vlowast = V (0) (26)
Thus V(119905) is a positive 119879-periodic solution of (24) andhence 119909(119905) = 119890120583119905V(119905) is a solution of (22)
The plague disease model system ((8a) (8b) (8c) (8d)(8e) (8f))ndash(11) has unique disease-free equilibrium pointgiven in (21)
We consider a heterogeneous populationwhose individu-als are distinguishable by stage of the disease and hence iden-tifiable and put into epidemiological compartmentswhich are119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875 119878119865 119868119865 and 119860We sort the compartments so that the first 119898 compartmentscorrespond to infected individuals
We now let
F119894(119909) be the rate of appearance of new infections inthe 119894th compartmentsV+119894 (119909) be the rate of transfer of individuals into
compartment 119894 by all other means other than theepidemic onesVminus119894 (119909) be the rate of transfer of individuals out of
compartment 119894
International Journal of Mathematics and Mathematical Sciences 9
Then the plague disease transmissionmodel in ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) is governed by a periodic ordinarydifferential system given in119889119909119894119889119905 = F119894 (119905 119909) minusV119894 (119905 119909) ≜ 119891119894 (119905 119909) (27)
119894 (119909)We rearrange the system by sorting the infectious classes
(119864119867 119868119867119861 119868119867119878 119868119867119875 119864119877 119868119877119861 119868119877119878 119868119877119875 119868119865 119860) coming first Wethen have
F (119909) =((((((((((((((((
(1 minus 1205901) 1205951 + 12057211198661 (119905)119878119867000(1 minus 1205902) 1205953 + 12057411198662 (119905)1198781198770001205952119894 + 1205731198663 (119905)1198781198770
))))))))))))))))
(28)
V (119909)
=((((((((((((((((((((
1205722119864119867 + 1205831119864119867(1205723 + 1205831 + 1205751119887) 119868119867119861 minus 1205722]2119864119867(1205724 + 1205831 + 1205751119904) 119868119867119878 minus 12057231205883119868119867119861 minus 1205722]3119864119867(1205725 + 1205831 + 1205751119901) 119868119867119875 minus 1205722]1119864119867 minus 12057231205881119868119867119861 minus 12057241205851198681198671198781205742119864119877 + 1205833119864119877(1205743 + 1205833 + 1205753119887) 119868119877119861 minus 12057421205913119864119877(1205744 + 1205833 + 1205753119904) 119868119877119878 minus 12057421205912119864119877 minus 1205743 (1 minus 120579) 119868119877119861(1205833 + 1205753119901) 119868119877119875 minus 12057421205911119864119903 minus 1205743120579119868119877119861 minus 1205744119868119877119878(1205832 + 1205752) 1198681198651205834119860 minus 1205781 (119905) 1198681198671198751198731 minus 1205782 (119905) 1198681198771198751198733 + 1205824
))))))))))))))))))))
(29)
Then we have 119865 (119905) = (120597F119894120597119909119895 (1199090)) 119881 (119905) = (120597V119894120597119909119895 (1199090)) (30)
with 1 le 119894 119895 le 10Now using (30) the matrices 119865 and 119881 are as given below
Following the setting byWang and Zhao [46] and van denDriessche andWatmough [47] for epidemicmodels we checkconditions (A1)ndash(A7) for plague disease epidemic modelSystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is equivalent toperiodic ordinary differential system (27) Now consideringthis systemwe can easily see that conditions (A1)ndash(A5) statedbelow are satisfied
(A1) Since each function represents a directed transfer ofindividuals (human rodent flea and pathogens inthe environment) they are all nonnegative Thus foreach 1 le 119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) andVminus119894 (119905 119909) are nonnegative and continuous on R times R119899+
and continuously differentiable with respect to 119909(A2) There is a real number 119879 gt 0 such that for each 1 le119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) and Vminus119894 (119905 119909)
are 119879-periodic in 119905(A3) If a compartment is empty there will be no transfer
of individuals out of the compartment by any meansThat is to say if 119909119894 = 0 then Vminus
119894 = 0 In particular if119909 isin 119883119904 thenVminus119894 = 0 for 119894 = 1 119898
(A4) The incidence of infection for uninfected compart-ments is zero That is to sayF119894 = 0 for 119894 gt 119898
(A5) If the population is disease-free then the populationwill remain free of disease Thus if 119909 isin 119883119904 thenF119894 =0 andV+
119894 = 0 for 119894 = 1 119898We know that system (27) has disease-free periodicsolution given in (21) Now we define 119891(119905 119909(119905)) =F(119905 119909(119905)) minus V(119905 119909(119905)) and119872(119905) = (120597119891119894(119905119894 1198640)120597119909119895)11 le 119894 119895 le 14 where 119891119894(119905 119909(119905)) and 119909119894 are the
119894th components of 119891(119905 119909(119905)) and 119909 respectively Nowfrom (28) and (29) we obtain a 4 times 4matrix given in
A (t) = (minus1205831 120603 0 00 minus (120603 + 1205831) 0 00 0 minus1205833 00 0 0 minus1205832) (34)
We then let Φ119860(sdot)(119905) be the monodromy matrix ofthe linear 119879-periodic system 119889119911119889119905 = 119860(119905)119911 Then120588Φ119860(sdot)(119879) lt 1 which implies that 1198640 is linearlyasymptotically stable in the disease-free subspace119883119904 = 119909 ge 0 119909119894 = 0 forall119894 = 1 sdot sdot sdot 119898 where 119894 = 1 sdot sdot sdot 119898are the infected compartments Thus condition (A6)stated below holds
(A6) The disease-free periodic solution is asymptoti-cally stable in a disease-free subspace 119883119904 that is120588Φ119860(sdot)(119879) lt 1 where 120588Φ119860(sdot)(119879) is the principaleigenvalue of Φ119860(sdot)(119905)Next we set 119865(119905) and 119881(119905) as two 10 times 10 matricesdefined by (30) then using (28) and (29) we getmatri-ces 119865(119905) and 119881(119905) given in (31) and (32) respectivelyWe can further see that matrix 119865(119905) is nonnegativeand minus119881(119905) is cooperative in the sense that the off-diagonal elements are nonnegative Let 119884(119905 119904) 119905 ge 119904be the evolution operator of our 119879-periodic system119889119910119889119905 = minus119881 (119905) 119910 (35)
International Journal of Mathematics and Mathematical Sciences 11
That is for each 119904 isin R the 10 times 10 matrix 119884(119905 119904)satisfies119889119884 (119905 119904)119889119905 = minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119868119889 (36)
where 119868119889 is a 10 times 10 identity matrix Thus themonodromymatrixΦ119881(119905) of (35) equals119884(119905 0) 119905 ge 0Therefore condition (A7) stated below holds
(A7) The evolution of individuals in the infectious com-partments decays exponentially due to natural anddisease induced mortalities Thus 120588Φ119881(119879) lt 1
Now using the standard theory of linear periodic systemby Hale [48] there exist119870 gt 0 and weierp gt 0 such that119884 (119905 119904) le 119870119890minusweierp(119905minus119904) forall119905 ge 119904 119904 isin R (37)
We then have119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) le 119870 119865 (119905 minus 119886) 119890minusweierp119886forall119905 isin R 119886 isin [0infin) (38)
Considering the periodic environment we suppose thatΦ(119904) 119879-periodic in 119904 is the distribution of the new infectionat a rate 119865(119904)Φ(119904) produced by the infected individuals whowere introduced at time 119904 Given 119905 ge 119904 then 119884(119905 119904)119865(119904)Φ(119904)yields the distribution of those infected individuals who werenewly infected at time 119904 and remain in the infected class at 119905We then haveΨ (119905) = int119905
= intinfin0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886 (39)
which is the distribution of accumulative new infectionsat time 119905 produced by all those infected individual Φ(119904)introduced at previous time 119904 to 119905 (119904 le 119905)
Let 119862119879 be the ordered Banach space of all 119879-periodicfunction fromR toR119899 which is equippedwith themaximumnorm sdotinfin and the positive cone119862+119879 = Φ isin 119862119879Φ(119905) ge 0 119905 isinR Define a linear operator 119871 119862119879 rarr 119862119879 by(119871Φ) (119905) = intinfin
0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886forall119905 isin R Φ isin 119862119879 (40)
Now by Wang and Zhao [46] Diekmann et al [49] andvan denDriessche andWatmough [47] we name 119871 as the nextinfection operator then the basic reproduction number 119877119879of the periodic system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) isgiven 119877119879 = 120588 (119871) (41)
where 120588(119871) is the spectral radius of 119871
51 Characterization of 119877119879 In this subsection we investigatewhether the basic reproduction number in our periodicsystem can characterize the threshold of the disease invasionTo do this we consider the following linear 119879-periodicequation 119889119908119889119905 = [minus119881 (119905) + 119865 (119905)120582 ]119908 forall119905 isin R (42)
with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) 119905 ge 119904 119904 isin Rbe the evolution operator of system (42) on R10 We canclearly see that Φ119865minus119881(119905) = 119882(119905 0 1) forall119905 ge 0 Consideringmatrices (31) and (32)wenote that for each120582 isin (0infin) all off-diagonal elements of matrix minus119881(119905) + 119865(119905)120582 are nonnegative(cooperative matrix) It follows that the linear operator119882(119905 119904 120582) is positive in R10 for each 119905 ge 119904 119904 isin R Nowusing Perron-Frobenius theorem by Smith andWaltman [50]it entails that 120588(119882(119879 0 120582)) is an eigenvalue of 119882(119879 0 120582)with a nonnegative eigenvector Also using matrix similarityconcept by Shores [51] we can easily verify that matrix119882(119904 +119879 0 120582) is similar to the matrix119882(119879 0 120582) and hence 120590(119882(119904 +119879 0 120582)) = 120590(119882(119879 0 120582)) for any 119904 isin R where 120590(119863) is aspectrum of the matrix119863Proposition 2 (see [46]) We let (A1)ndash(A7) hold for system((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) then
(i) if 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 1205820is an eigenvalue of 119871 and hence 119877119879 gt 0
(ii) if 119877119879 gt 0 then 120582 = 119877119879 is the unique solution of120588(119882(119879 0 120582)) = 1(iii) 119877119879 = 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0This result shows that in order to find the basic reproduc-
tion number we need to find themonodromymatrixΦ119865minus119881(119905)of system (42) and evaluate itWe then find the spectral radiusof Φ119865minus119881(119879) and solve the equation 120588(Φ119865minus119881(119879)) = 1 for 120582which is the basic reproduction number
52 Computation of the Basic Reproduction Number Wecompute a time-averaged basic reproduction number 1198770using the next-generation matrix as outlined by Wesley andAllen [52] Heesterbeek [53] and Diekmann et al [49] Themethod has the advantage over the usual next-generationmethod in that the steps to reach an estimate of 1198770 and thematrix elements of the next-generation matrix have a clearbiological basis It is easy to handle complex diseases likeplague disease which has multiple transmission roots fromdifferent infectious agents
To do this we first categorize individuals by their stateat the moment they become infected (type at infection)These types at infection refer specifically to the birth ofthe infection in the individual These categories (types atinfection) differ in the way they transmit plague diseasewhich in turn differentiates their ability to produce secondarycases
In our case we categorize the individuals into eight statesand label them as follows human infected with bubonicplague (type 1) human infected with septicemic plague (type
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
where 1198861 = (1205723 + 1205831 + 1205751119887) 1198862 = (1205724 + 1205831 + 1205751119904) 1198863 = (1205725 +1205831 + 1205751119901) 1198864 = (1205743 + 1205833 + 1205753119887) 1198865 = (1205744 + 1205833 + 1205753119904) 1198921 =(12057211198661(119905) + 1205831) 1198922 = (12057411198662(119905) + 1205833) and 1198923 = (1205731198663(119905) + 1205832)Now from submatrices 11986011 11986012 11986021 and 11986022 we can
deduce that matrix119860(119909) is aMetzler matrix such that all of itsoff-diagonal elements are nonnegative forall119909 isin R14+ and 119865 ge 0is Lipschitz continuousThus the feasible region for themodelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is the set
Thismeans that any trajectory of the system starting froman initial state in the positive orthant of R14+ remains foreverin Φ32 Positivity of the Solution We need to show that allvariables and parameters of the model are nonnegative forall119905 ge0 We now solve the equations of the system in their patchesfor testing the positivity We found that by letting the initialvalues of the systems ((8a) (8b) (8c) (8d) (8e) (8f)) ((9a)(9b) (9c) (9d) (9e)) ((10a) (10b)) and (11) be 119878119867(0) gt 0119878119877(0) gt 0 119878119865(0) gt 0 and 1198600 ge 0 119864119867(0) ge 0 119868119867119861(0) ge 0119868119867119878(0) ge 0 119868119867119875(0) ge 0 119877119867(0) ge 0 119864119877(0) ge 0 119868119877119861(0) ge 0119868119877119878(0) ge 0 119868119877119875(0) ge 0 and 119868119865(0) ge 0 in the solution
8 International Journal of Mathematics and Mathematical Sciences
set 119878119867(119905) 119878119877(119905) 119878119865(119905) 119860(119905) 119864119867(119905) 119868119867119861(119905) 119868119867119878(119905) 119868119867119875(119905)119877119867(119905) 119864119877(119905) 119868119877119861(119905) 119868119877119878(119905) 119868119877119875(119905) and 119868119865(119905) are nonnegativeforall119905 ge 04 Model Analysis
41 Disease-Free Equilibrium Solution The periodic modelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) with nonnega-tive continuous periodic functions has disease-free equilib-rium solution in which we consider the following equations119889119878119867119889119905 = 12059011205951 minus 1205831119878119867 (17)119889119878119877119889119905 = 12059021205953 minus 1205833119878119877 (18)119889119878119865119889119905 = 1205952119904 minus 1205832119878119865 (19)
Now given initial conditions 119878119867 = 1198781198670 isin R+ 119878119877 = 1198781198770 isinR+ and 119878119865 = 1198781198650 isin R+ for (17) (18) and (19) respectivelywe will have 119878119867 = 120590112059511205831 + (1198781198670 minus 120590112059511205831 ) 119890minus1205831119905
119878119877 = 120590212059531205833 + (1198781198770 minus 120590212059531205833 ) 119890minus1205833119905119878119877 = 12059521199041205832 + (1198781198650 minus 12059521199041205832 ) 119890minus1205832119905
(20)
As 119905 rarr infin (17) (18) and (19) admit unique solution 119878119867 equiv120590112059511205831 119878119877 equiv 120590212059531205833 and 119878119865 equiv 12059521199041205832 respectively whichis globally attractive in R3+
To find the disease-free equilibrium point we set thederivatives of system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11)equal to zero Then the model system has disease-freesolution which is obtained by setting 119868119867119861 = 119868119867119878 = 119868119867119875 =119864119867 = 119877119867 = 0 119868119877119861 = 119868119877119878 = 119868119877119875 = 119864119877 = 0 119868119865 = 0 and 119860 = 0for human rodent flea and pathogen system respectivelyHence system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) has adisease-free equilibrium point1198640 (1198780119867 1198640119867 1198680119867119861 1198680119867119878 1198680119867119875 1198770119867 1198780119877 1198640119877 1198680119877119861 1198680119877119878 1198680119877119875 1198780119865 11986801198651198600) = (120590112059511205831 0 0 0 0 0 120590212059531205833 0 0 0 0 12059521199041205832 0 0) (21)
5 Basic Reproduction Number
Let (R119896R119896) be the standard ordered 119896-dimensionalEuclidean space with a norm sdot For 119906 V isin R119896 we write119906 ge V provided 119906 minus V isin R119896+ 119906 gt V provided 119906 minus V isin R119896+ 0and 119906 ≫ V if 119906 minus V isin int(R119896+)
Now let 119860(119905) be the continuous cooperative irreducibleand 119879-periodic 119896 times 119896 matrix function with period 119879 gt 0
Φ119860(sdot)(119905) be the fundamental solution matrix of the linearordinary differential system119889119909119889119905 = 119860 (119905) 119909 (22)
and 120588(Φ119860(sdot)(119879)) be the spectral radius of Φ119860(sdot)(119879) By Aron-sson and Kellogg [45] it follows that Φ119860(sdot)(119905) is a matrixwith all elements positive for each 119905 gt 0 By the PerronFrobenius theorem 120588(Φ119860(sdot)(119879)) is the principal eigenvalue ofΦ119860(sdot)(119905) in the sense that it is simple and admits an eigenvectorVlowast ≫ 0 The following result is important for our subsequentcomparison argument
Proposition 1 let 120580 = (1119879) ln(120588(Φ119860(119879))) and then thereexists a positive 119879-periodic function V(119905) such that 119890120580119905V(119905) is asolution of 1199091015840 = 119860(119905)119909Proof Let Vlowast ≫ 0 be the eigenvector associated with thespectral radius 120588Φ119860(sdot)(119879)
By the change of variable119909 (119905) = 119890120583119905V (119905) (23)
system (22) becomes119889V119889119905 = 119860 (119905) V minus 120583V = (119860 (119905) minus 120583119868) V (24)
where 119868 is an identity matrixThus V(119905) = Φ(119860(sdot)minus120583119868)(119905)Vlowast is a positive solution of (24)
We can easily see that119890120583119905Φ(119860(sdot)minus120583119868) (119905) = Φ119860(sdot) (119905) (25)
Moreover
V (119879) = Φ(119860(sdot)minus120583119868) (119905) Vlowast = 119890minus120583119879Φ119860(sdot) (119879) Vlowast= 119890minus120583119879120588 (Φ119860(sdot) (119879)) Vlowast = Vlowast = V (0) (26)
Thus V(119905) is a positive 119879-periodic solution of (24) andhence 119909(119905) = 119890120583119905V(119905) is a solution of (22)
The plague disease model system ((8a) (8b) (8c) (8d)(8e) (8f))ndash(11) has unique disease-free equilibrium pointgiven in (21)
We consider a heterogeneous populationwhose individu-als are distinguishable by stage of the disease and hence iden-tifiable and put into epidemiological compartmentswhich are119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875 119878119865 119868119865 and 119860We sort the compartments so that the first 119898 compartmentscorrespond to infected individuals
We now let
F119894(119909) be the rate of appearance of new infections inthe 119894th compartmentsV+119894 (119909) be the rate of transfer of individuals into
compartment 119894 by all other means other than theepidemic onesVminus119894 (119909) be the rate of transfer of individuals out of
compartment 119894
International Journal of Mathematics and Mathematical Sciences 9
Then the plague disease transmissionmodel in ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) is governed by a periodic ordinarydifferential system given in119889119909119894119889119905 = F119894 (119905 119909) minusV119894 (119905 119909) ≜ 119891119894 (119905 119909) (27)
119894 (119909)We rearrange the system by sorting the infectious classes
(119864119867 119868119867119861 119868119867119878 119868119867119875 119864119877 119868119877119861 119868119877119878 119868119877119875 119868119865 119860) coming first Wethen have
F (119909) =((((((((((((((((
(1 minus 1205901) 1205951 + 12057211198661 (119905)119878119867000(1 minus 1205902) 1205953 + 12057411198662 (119905)1198781198770001205952119894 + 1205731198663 (119905)1198781198770
))))))))))))))))
(28)
V (119909)
=((((((((((((((((((((
1205722119864119867 + 1205831119864119867(1205723 + 1205831 + 1205751119887) 119868119867119861 minus 1205722]2119864119867(1205724 + 1205831 + 1205751119904) 119868119867119878 minus 12057231205883119868119867119861 minus 1205722]3119864119867(1205725 + 1205831 + 1205751119901) 119868119867119875 minus 1205722]1119864119867 minus 12057231205881119868119867119861 minus 12057241205851198681198671198781205742119864119877 + 1205833119864119877(1205743 + 1205833 + 1205753119887) 119868119877119861 minus 12057421205913119864119877(1205744 + 1205833 + 1205753119904) 119868119877119878 minus 12057421205912119864119877 minus 1205743 (1 minus 120579) 119868119877119861(1205833 + 1205753119901) 119868119877119875 minus 12057421205911119864119903 minus 1205743120579119868119877119861 minus 1205744119868119877119878(1205832 + 1205752) 1198681198651205834119860 minus 1205781 (119905) 1198681198671198751198731 minus 1205782 (119905) 1198681198771198751198733 + 1205824
))))))))))))))))))))
(29)
Then we have 119865 (119905) = (120597F119894120597119909119895 (1199090)) 119881 (119905) = (120597V119894120597119909119895 (1199090)) (30)
with 1 le 119894 119895 le 10Now using (30) the matrices 119865 and 119881 are as given below
Following the setting byWang and Zhao [46] and van denDriessche andWatmough [47] for epidemicmodels we checkconditions (A1)ndash(A7) for plague disease epidemic modelSystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is equivalent toperiodic ordinary differential system (27) Now consideringthis systemwe can easily see that conditions (A1)ndash(A5) statedbelow are satisfied
(A1) Since each function represents a directed transfer ofindividuals (human rodent flea and pathogens inthe environment) they are all nonnegative Thus foreach 1 le 119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) andVminus119894 (119905 119909) are nonnegative and continuous on R times R119899+
and continuously differentiable with respect to 119909(A2) There is a real number 119879 gt 0 such that for each 1 le119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) and Vminus119894 (119905 119909)
are 119879-periodic in 119905(A3) If a compartment is empty there will be no transfer
of individuals out of the compartment by any meansThat is to say if 119909119894 = 0 then Vminus
119894 = 0 In particular if119909 isin 119883119904 thenVminus119894 = 0 for 119894 = 1 119898
(A4) The incidence of infection for uninfected compart-ments is zero That is to sayF119894 = 0 for 119894 gt 119898
(A5) If the population is disease-free then the populationwill remain free of disease Thus if 119909 isin 119883119904 thenF119894 =0 andV+
119894 = 0 for 119894 = 1 119898We know that system (27) has disease-free periodicsolution given in (21) Now we define 119891(119905 119909(119905)) =F(119905 119909(119905)) minus V(119905 119909(119905)) and119872(119905) = (120597119891119894(119905119894 1198640)120597119909119895)11 le 119894 119895 le 14 where 119891119894(119905 119909(119905)) and 119909119894 are the
119894th components of 119891(119905 119909(119905)) and 119909 respectively Nowfrom (28) and (29) we obtain a 4 times 4matrix given in
A (t) = (minus1205831 120603 0 00 minus (120603 + 1205831) 0 00 0 minus1205833 00 0 0 minus1205832) (34)
We then let Φ119860(sdot)(119905) be the monodromy matrix ofthe linear 119879-periodic system 119889119911119889119905 = 119860(119905)119911 Then120588Φ119860(sdot)(119879) lt 1 which implies that 1198640 is linearlyasymptotically stable in the disease-free subspace119883119904 = 119909 ge 0 119909119894 = 0 forall119894 = 1 sdot sdot sdot 119898 where 119894 = 1 sdot sdot sdot 119898are the infected compartments Thus condition (A6)stated below holds
(A6) The disease-free periodic solution is asymptoti-cally stable in a disease-free subspace 119883119904 that is120588Φ119860(sdot)(119879) lt 1 where 120588Φ119860(sdot)(119879) is the principaleigenvalue of Φ119860(sdot)(119905)Next we set 119865(119905) and 119881(119905) as two 10 times 10 matricesdefined by (30) then using (28) and (29) we getmatri-ces 119865(119905) and 119881(119905) given in (31) and (32) respectivelyWe can further see that matrix 119865(119905) is nonnegativeand minus119881(119905) is cooperative in the sense that the off-diagonal elements are nonnegative Let 119884(119905 119904) 119905 ge 119904be the evolution operator of our 119879-periodic system119889119910119889119905 = minus119881 (119905) 119910 (35)
International Journal of Mathematics and Mathematical Sciences 11
That is for each 119904 isin R the 10 times 10 matrix 119884(119905 119904)satisfies119889119884 (119905 119904)119889119905 = minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119868119889 (36)
where 119868119889 is a 10 times 10 identity matrix Thus themonodromymatrixΦ119881(119905) of (35) equals119884(119905 0) 119905 ge 0Therefore condition (A7) stated below holds
(A7) The evolution of individuals in the infectious com-partments decays exponentially due to natural anddisease induced mortalities Thus 120588Φ119881(119879) lt 1
Now using the standard theory of linear periodic systemby Hale [48] there exist119870 gt 0 and weierp gt 0 such that119884 (119905 119904) le 119870119890minusweierp(119905minus119904) forall119905 ge 119904 119904 isin R (37)
We then have119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) le 119870 119865 (119905 minus 119886) 119890minusweierp119886forall119905 isin R 119886 isin [0infin) (38)
Considering the periodic environment we suppose thatΦ(119904) 119879-periodic in 119904 is the distribution of the new infectionat a rate 119865(119904)Φ(119904) produced by the infected individuals whowere introduced at time 119904 Given 119905 ge 119904 then 119884(119905 119904)119865(119904)Φ(119904)yields the distribution of those infected individuals who werenewly infected at time 119904 and remain in the infected class at 119905We then haveΨ (119905) = int119905
= intinfin0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886 (39)
which is the distribution of accumulative new infectionsat time 119905 produced by all those infected individual Φ(119904)introduced at previous time 119904 to 119905 (119904 le 119905)
Let 119862119879 be the ordered Banach space of all 119879-periodicfunction fromR toR119899 which is equippedwith themaximumnorm sdotinfin and the positive cone119862+119879 = Φ isin 119862119879Φ(119905) ge 0 119905 isinR Define a linear operator 119871 119862119879 rarr 119862119879 by(119871Φ) (119905) = intinfin
0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886forall119905 isin R Φ isin 119862119879 (40)
Now by Wang and Zhao [46] Diekmann et al [49] andvan denDriessche andWatmough [47] we name 119871 as the nextinfection operator then the basic reproduction number 119877119879of the periodic system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) isgiven 119877119879 = 120588 (119871) (41)
where 120588(119871) is the spectral radius of 119871
51 Characterization of 119877119879 In this subsection we investigatewhether the basic reproduction number in our periodicsystem can characterize the threshold of the disease invasionTo do this we consider the following linear 119879-periodicequation 119889119908119889119905 = [minus119881 (119905) + 119865 (119905)120582 ]119908 forall119905 isin R (42)
with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) 119905 ge 119904 119904 isin Rbe the evolution operator of system (42) on R10 We canclearly see that Φ119865minus119881(119905) = 119882(119905 0 1) forall119905 ge 0 Consideringmatrices (31) and (32)wenote that for each120582 isin (0infin) all off-diagonal elements of matrix minus119881(119905) + 119865(119905)120582 are nonnegative(cooperative matrix) It follows that the linear operator119882(119905 119904 120582) is positive in R10 for each 119905 ge 119904 119904 isin R Nowusing Perron-Frobenius theorem by Smith andWaltman [50]it entails that 120588(119882(119879 0 120582)) is an eigenvalue of 119882(119879 0 120582)with a nonnegative eigenvector Also using matrix similarityconcept by Shores [51] we can easily verify that matrix119882(119904 +119879 0 120582) is similar to the matrix119882(119879 0 120582) and hence 120590(119882(119904 +119879 0 120582)) = 120590(119882(119879 0 120582)) for any 119904 isin R where 120590(119863) is aspectrum of the matrix119863Proposition 2 (see [46]) We let (A1)ndash(A7) hold for system((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) then
(i) if 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 1205820is an eigenvalue of 119871 and hence 119877119879 gt 0
(ii) if 119877119879 gt 0 then 120582 = 119877119879 is the unique solution of120588(119882(119879 0 120582)) = 1(iii) 119877119879 = 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0This result shows that in order to find the basic reproduc-
tion number we need to find themonodromymatrixΦ119865minus119881(119905)of system (42) and evaluate itWe then find the spectral radiusof Φ119865minus119881(119879) and solve the equation 120588(Φ119865minus119881(119879)) = 1 for 120582which is the basic reproduction number
52 Computation of the Basic Reproduction Number Wecompute a time-averaged basic reproduction number 1198770using the next-generation matrix as outlined by Wesley andAllen [52] Heesterbeek [53] and Diekmann et al [49] Themethod has the advantage over the usual next-generationmethod in that the steps to reach an estimate of 1198770 and thematrix elements of the next-generation matrix have a clearbiological basis It is easy to handle complex diseases likeplague disease which has multiple transmission roots fromdifferent infectious agents
To do this we first categorize individuals by their stateat the moment they become infected (type at infection)These types at infection refer specifically to the birth ofthe infection in the individual These categories (types atinfection) differ in the way they transmit plague diseasewhich in turn differentiates their ability to produce secondarycases
In our case we categorize the individuals into eight statesand label them as follows human infected with bubonicplague (type 1) human infected with septicemic plague (type
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
8 International Journal of Mathematics and Mathematical Sciences
set 119878119867(119905) 119878119877(119905) 119878119865(119905) 119860(119905) 119864119867(119905) 119868119867119861(119905) 119868119867119878(119905) 119868119867119875(119905)119877119867(119905) 119864119877(119905) 119868119877119861(119905) 119868119877119878(119905) 119868119877119875(119905) and 119868119865(119905) are nonnegativeforall119905 ge 04 Model Analysis
41 Disease-Free Equilibrium Solution The periodic modelsystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) with nonnega-tive continuous periodic functions has disease-free equilib-rium solution in which we consider the following equations119889119878119867119889119905 = 12059011205951 minus 1205831119878119867 (17)119889119878119877119889119905 = 12059021205953 minus 1205833119878119877 (18)119889119878119865119889119905 = 1205952119904 minus 1205832119878119865 (19)
Now given initial conditions 119878119867 = 1198781198670 isin R+ 119878119877 = 1198781198770 isinR+ and 119878119865 = 1198781198650 isin R+ for (17) (18) and (19) respectivelywe will have 119878119867 = 120590112059511205831 + (1198781198670 minus 120590112059511205831 ) 119890minus1205831119905
119878119877 = 120590212059531205833 + (1198781198770 minus 120590212059531205833 ) 119890minus1205833119905119878119877 = 12059521199041205832 + (1198781198650 minus 12059521199041205832 ) 119890minus1205832119905
(20)
As 119905 rarr infin (17) (18) and (19) admit unique solution 119878119867 equiv120590112059511205831 119878119877 equiv 120590212059531205833 and 119878119865 equiv 12059521199041205832 respectively whichis globally attractive in R3+
To find the disease-free equilibrium point we set thederivatives of system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11)equal to zero Then the model system has disease-freesolution which is obtained by setting 119868119867119861 = 119868119867119878 = 119868119867119875 =119864119867 = 119877119867 = 0 119868119877119861 = 119868119877119878 = 119868119877119875 = 119864119877 = 0 119868119865 = 0 and 119860 = 0for human rodent flea and pathogen system respectivelyHence system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) has adisease-free equilibrium point1198640 (1198780119867 1198640119867 1198680119867119861 1198680119867119878 1198680119867119875 1198770119867 1198780119877 1198640119877 1198680119877119861 1198680119877119878 1198680119877119875 1198780119865 11986801198651198600) = (120590112059511205831 0 0 0 0 0 120590212059531205833 0 0 0 0 12059521199041205832 0 0) (21)
5 Basic Reproduction Number
Let (R119896R119896) be the standard ordered 119896-dimensionalEuclidean space with a norm sdot For 119906 V isin R119896 we write119906 ge V provided 119906 minus V isin R119896+ 119906 gt V provided 119906 minus V isin R119896+ 0and 119906 ≫ V if 119906 minus V isin int(R119896+)
Now let 119860(119905) be the continuous cooperative irreducibleand 119879-periodic 119896 times 119896 matrix function with period 119879 gt 0
Φ119860(sdot)(119905) be the fundamental solution matrix of the linearordinary differential system119889119909119889119905 = 119860 (119905) 119909 (22)
and 120588(Φ119860(sdot)(119879)) be the spectral radius of Φ119860(sdot)(119879) By Aron-sson and Kellogg [45] it follows that Φ119860(sdot)(119905) is a matrixwith all elements positive for each 119905 gt 0 By the PerronFrobenius theorem 120588(Φ119860(sdot)(119879)) is the principal eigenvalue ofΦ119860(sdot)(119905) in the sense that it is simple and admits an eigenvectorVlowast ≫ 0 The following result is important for our subsequentcomparison argument
Proposition 1 let 120580 = (1119879) ln(120588(Φ119860(119879))) and then thereexists a positive 119879-periodic function V(119905) such that 119890120580119905V(119905) is asolution of 1199091015840 = 119860(119905)119909Proof Let Vlowast ≫ 0 be the eigenvector associated with thespectral radius 120588Φ119860(sdot)(119879)
By the change of variable119909 (119905) = 119890120583119905V (119905) (23)
system (22) becomes119889V119889119905 = 119860 (119905) V minus 120583V = (119860 (119905) minus 120583119868) V (24)
where 119868 is an identity matrixThus V(119905) = Φ(119860(sdot)minus120583119868)(119905)Vlowast is a positive solution of (24)
We can easily see that119890120583119905Φ(119860(sdot)minus120583119868) (119905) = Φ119860(sdot) (119905) (25)
Moreover
V (119879) = Φ(119860(sdot)minus120583119868) (119905) Vlowast = 119890minus120583119879Φ119860(sdot) (119879) Vlowast= 119890minus120583119879120588 (Φ119860(sdot) (119879)) Vlowast = Vlowast = V (0) (26)
Thus V(119905) is a positive 119879-periodic solution of (24) andhence 119909(119905) = 119890120583119905V(119905) is a solution of (22)
The plague disease model system ((8a) (8b) (8c) (8d)(8e) (8f))ndash(11) has unique disease-free equilibrium pointgiven in (21)
We consider a heterogeneous populationwhose individu-als are distinguishable by stage of the disease and hence iden-tifiable and put into epidemiological compartmentswhich are119878119867 119864119867 119868119867119861 119868119867119878 119868119867119875 119877119867 119878119877 119864119877 119868119877119861 119868119877119878 119868119877119875 119878119865 119868119865 and 119860We sort the compartments so that the first 119898 compartmentscorrespond to infected individuals
We now let
F119894(119909) be the rate of appearance of new infections inthe 119894th compartmentsV+119894 (119909) be the rate of transfer of individuals into
compartment 119894 by all other means other than theepidemic onesVminus119894 (119909) be the rate of transfer of individuals out of
compartment 119894
International Journal of Mathematics and Mathematical Sciences 9
Then the plague disease transmissionmodel in ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) is governed by a periodic ordinarydifferential system given in119889119909119894119889119905 = F119894 (119905 119909) minusV119894 (119905 119909) ≜ 119891119894 (119905 119909) (27)
119894 (119909)We rearrange the system by sorting the infectious classes
(119864119867 119868119867119861 119868119867119878 119868119867119875 119864119877 119868119877119861 119868119877119878 119868119877119875 119868119865 119860) coming first Wethen have
F (119909) =((((((((((((((((
(1 minus 1205901) 1205951 + 12057211198661 (119905)119878119867000(1 minus 1205902) 1205953 + 12057411198662 (119905)1198781198770001205952119894 + 1205731198663 (119905)1198781198770
))))))))))))))))
(28)
V (119909)
=((((((((((((((((((((
1205722119864119867 + 1205831119864119867(1205723 + 1205831 + 1205751119887) 119868119867119861 minus 1205722]2119864119867(1205724 + 1205831 + 1205751119904) 119868119867119878 minus 12057231205883119868119867119861 minus 1205722]3119864119867(1205725 + 1205831 + 1205751119901) 119868119867119875 minus 1205722]1119864119867 minus 12057231205881119868119867119861 minus 12057241205851198681198671198781205742119864119877 + 1205833119864119877(1205743 + 1205833 + 1205753119887) 119868119877119861 minus 12057421205913119864119877(1205744 + 1205833 + 1205753119904) 119868119877119878 minus 12057421205912119864119877 minus 1205743 (1 minus 120579) 119868119877119861(1205833 + 1205753119901) 119868119877119875 minus 12057421205911119864119903 minus 1205743120579119868119877119861 minus 1205744119868119877119878(1205832 + 1205752) 1198681198651205834119860 minus 1205781 (119905) 1198681198671198751198731 minus 1205782 (119905) 1198681198771198751198733 + 1205824
))))))))))))))))))))
(29)
Then we have 119865 (119905) = (120597F119894120597119909119895 (1199090)) 119881 (119905) = (120597V119894120597119909119895 (1199090)) (30)
with 1 le 119894 119895 le 10Now using (30) the matrices 119865 and 119881 are as given below
Following the setting byWang and Zhao [46] and van denDriessche andWatmough [47] for epidemicmodels we checkconditions (A1)ndash(A7) for plague disease epidemic modelSystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is equivalent toperiodic ordinary differential system (27) Now consideringthis systemwe can easily see that conditions (A1)ndash(A5) statedbelow are satisfied
(A1) Since each function represents a directed transfer ofindividuals (human rodent flea and pathogens inthe environment) they are all nonnegative Thus foreach 1 le 119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) andVminus119894 (119905 119909) are nonnegative and continuous on R times R119899+
and continuously differentiable with respect to 119909(A2) There is a real number 119879 gt 0 such that for each 1 le119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) and Vminus119894 (119905 119909)
are 119879-periodic in 119905(A3) If a compartment is empty there will be no transfer
of individuals out of the compartment by any meansThat is to say if 119909119894 = 0 then Vminus
119894 = 0 In particular if119909 isin 119883119904 thenVminus119894 = 0 for 119894 = 1 119898
(A4) The incidence of infection for uninfected compart-ments is zero That is to sayF119894 = 0 for 119894 gt 119898
(A5) If the population is disease-free then the populationwill remain free of disease Thus if 119909 isin 119883119904 thenF119894 =0 andV+
119894 = 0 for 119894 = 1 119898We know that system (27) has disease-free periodicsolution given in (21) Now we define 119891(119905 119909(119905)) =F(119905 119909(119905)) minus V(119905 119909(119905)) and119872(119905) = (120597119891119894(119905119894 1198640)120597119909119895)11 le 119894 119895 le 14 where 119891119894(119905 119909(119905)) and 119909119894 are the
119894th components of 119891(119905 119909(119905)) and 119909 respectively Nowfrom (28) and (29) we obtain a 4 times 4matrix given in
A (t) = (minus1205831 120603 0 00 minus (120603 + 1205831) 0 00 0 minus1205833 00 0 0 minus1205832) (34)
We then let Φ119860(sdot)(119905) be the monodromy matrix ofthe linear 119879-periodic system 119889119911119889119905 = 119860(119905)119911 Then120588Φ119860(sdot)(119879) lt 1 which implies that 1198640 is linearlyasymptotically stable in the disease-free subspace119883119904 = 119909 ge 0 119909119894 = 0 forall119894 = 1 sdot sdot sdot 119898 where 119894 = 1 sdot sdot sdot 119898are the infected compartments Thus condition (A6)stated below holds
(A6) The disease-free periodic solution is asymptoti-cally stable in a disease-free subspace 119883119904 that is120588Φ119860(sdot)(119879) lt 1 where 120588Φ119860(sdot)(119879) is the principaleigenvalue of Φ119860(sdot)(119905)Next we set 119865(119905) and 119881(119905) as two 10 times 10 matricesdefined by (30) then using (28) and (29) we getmatri-ces 119865(119905) and 119881(119905) given in (31) and (32) respectivelyWe can further see that matrix 119865(119905) is nonnegativeand minus119881(119905) is cooperative in the sense that the off-diagonal elements are nonnegative Let 119884(119905 119904) 119905 ge 119904be the evolution operator of our 119879-periodic system119889119910119889119905 = minus119881 (119905) 119910 (35)
International Journal of Mathematics and Mathematical Sciences 11
That is for each 119904 isin R the 10 times 10 matrix 119884(119905 119904)satisfies119889119884 (119905 119904)119889119905 = minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119868119889 (36)
where 119868119889 is a 10 times 10 identity matrix Thus themonodromymatrixΦ119881(119905) of (35) equals119884(119905 0) 119905 ge 0Therefore condition (A7) stated below holds
(A7) The evolution of individuals in the infectious com-partments decays exponentially due to natural anddisease induced mortalities Thus 120588Φ119881(119879) lt 1
Now using the standard theory of linear periodic systemby Hale [48] there exist119870 gt 0 and weierp gt 0 such that119884 (119905 119904) le 119870119890minusweierp(119905minus119904) forall119905 ge 119904 119904 isin R (37)
We then have119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) le 119870 119865 (119905 minus 119886) 119890minusweierp119886forall119905 isin R 119886 isin [0infin) (38)
Considering the periodic environment we suppose thatΦ(119904) 119879-periodic in 119904 is the distribution of the new infectionat a rate 119865(119904)Φ(119904) produced by the infected individuals whowere introduced at time 119904 Given 119905 ge 119904 then 119884(119905 119904)119865(119904)Φ(119904)yields the distribution of those infected individuals who werenewly infected at time 119904 and remain in the infected class at 119905We then haveΨ (119905) = int119905
= intinfin0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886 (39)
which is the distribution of accumulative new infectionsat time 119905 produced by all those infected individual Φ(119904)introduced at previous time 119904 to 119905 (119904 le 119905)
Let 119862119879 be the ordered Banach space of all 119879-periodicfunction fromR toR119899 which is equippedwith themaximumnorm sdotinfin and the positive cone119862+119879 = Φ isin 119862119879Φ(119905) ge 0 119905 isinR Define a linear operator 119871 119862119879 rarr 119862119879 by(119871Φ) (119905) = intinfin
0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886forall119905 isin R Φ isin 119862119879 (40)
Now by Wang and Zhao [46] Diekmann et al [49] andvan denDriessche andWatmough [47] we name 119871 as the nextinfection operator then the basic reproduction number 119877119879of the periodic system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) isgiven 119877119879 = 120588 (119871) (41)
where 120588(119871) is the spectral radius of 119871
51 Characterization of 119877119879 In this subsection we investigatewhether the basic reproduction number in our periodicsystem can characterize the threshold of the disease invasionTo do this we consider the following linear 119879-periodicequation 119889119908119889119905 = [minus119881 (119905) + 119865 (119905)120582 ]119908 forall119905 isin R (42)
with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) 119905 ge 119904 119904 isin Rbe the evolution operator of system (42) on R10 We canclearly see that Φ119865minus119881(119905) = 119882(119905 0 1) forall119905 ge 0 Consideringmatrices (31) and (32)wenote that for each120582 isin (0infin) all off-diagonal elements of matrix minus119881(119905) + 119865(119905)120582 are nonnegative(cooperative matrix) It follows that the linear operator119882(119905 119904 120582) is positive in R10 for each 119905 ge 119904 119904 isin R Nowusing Perron-Frobenius theorem by Smith andWaltman [50]it entails that 120588(119882(119879 0 120582)) is an eigenvalue of 119882(119879 0 120582)with a nonnegative eigenvector Also using matrix similarityconcept by Shores [51] we can easily verify that matrix119882(119904 +119879 0 120582) is similar to the matrix119882(119879 0 120582) and hence 120590(119882(119904 +119879 0 120582)) = 120590(119882(119879 0 120582)) for any 119904 isin R where 120590(119863) is aspectrum of the matrix119863Proposition 2 (see [46]) We let (A1)ndash(A7) hold for system((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) then
(i) if 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 1205820is an eigenvalue of 119871 and hence 119877119879 gt 0
(ii) if 119877119879 gt 0 then 120582 = 119877119879 is the unique solution of120588(119882(119879 0 120582)) = 1(iii) 119877119879 = 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0This result shows that in order to find the basic reproduc-
tion number we need to find themonodromymatrixΦ119865minus119881(119905)of system (42) and evaluate itWe then find the spectral radiusof Φ119865minus119881(119879) and solve the equation 120588(Φ119865minus119881(119879)) = 1 for 120582which is the basic reproduction number
52 Computation of the Basic Reproduction Number Wecompute a time-averaged basic reproduction number 1198770using the next-generation matrix as outlined by Wesley andAllen [52] Heesterbeek [53] and Diekmann et al [49] Themethod has the advantage over the usual next-generationmethod in that the steps to reach an estimate of 1198770 and thematrix elements of the next-generation matrix have a clearbiological basis It is easy to handle complex diseases likeplague disease which has multiple transmission roots fromdifferent infectious agents
To do this we first categorize individuals by their stateat the moment they become infected (type at infection)These types at infection refer specifically to the birth ofthe infection in the individual These categories (types atinfection) differ in the way they transmit plague diseasewhich in turn differentiates their ability to produce secondarycases
In our case we categorize the individuals into eight statesand label them as follows human infected with bubonicplague (type 1) human infected with septicemic plague (type
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
International Journal of Mathematics and Mathematical Sciences 9
Then the plague disease transmissionmodel in ((8a) (8b)(8c) (8d) (8e) (8f))ndash(11) is governed by a periodic ordinarydifferential system given in119889119909119894119889119905 = F119894 (119905 119909) minusV119894 (119905 119909) ≜ 119891119894 (119905 119909) (27)
119894 (119909)We rearrange the system by sorting the infectious classes
(119864119867 119868119867119861 119868119867119878 119868119867119875 119864119877 119868119877119861 119868119877119878 119868119877119875 119868119865 119860) coming first Wethen have
F (119909) =((((((((((((((((
(1 minus 1205901) 1205951 + 12057211198661 (119905)119878119867000(1 minus 1205902) 1205953 + 12057411198662 (119905)1198781198770001205952119894 + 1205731198663 (119905)1198781198770
))))))))))))))))
(28)
V (119909)
=((((((((((((((((((((
1205722119864119867 + 1205831119864119867(1205723 + 1205831 + 1205751119887) 119868119867119861 minus 1205722]2119864119867(1205724 + 1205831 + 1205751119904) 119868119867119878 minus 12057231205883119868119867119861 minus 1205722]3119864119867(1205725 + 1205831 + 1205751119901) 119868119867119875 minus 1205722]1119864119867 minus 12057231205881119868119867119861 minus 12057241205851198681198671198781205742119864119877 + 1205833119864119877(1205743 + 1205833 + 1205753119887) 119868119877119861 minus 12057421205913119864119877(1205744 + 1205833 + 1205753119904) 119868119877119878 minus 12057421205912119864119877 minus 1205743 (1 minus 120579) 119868119877119861(1205833 + 1205753119901) 119868119877119875 minus 12057421205911119864119903 minus 1205743120579119868119877119861 minus 1205744119868119877119878(1205832 + 1205752) 1198681198651205834119860 minus 1205781 (119905) 1198681198671198751198731 minus 1205782 (119905) 1198681198771198751198733 + 1205824
))))))))))))))))))))
(29)
Then we have 119865 (119905) = (120597F119894120597119909119895 (1199090)) 119881 (119905) = (120597V119894120597119909119895 (1199090)) (30)
with 1 le 119894 119895 le 10Now using (30) the matrices 119865 and 119881 are as given below
Following the setting byWang and Zhao [46] and van denDriessche andWatmough [47] for epidemicmodels we checkconditions (A1)ndash(A7) for plague disease epidemic modelSystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is equivalent toperiodic ordinary differential system (27) Now consideringthis systemwe can easily see that conditions (A1)ndash(A5) statedbelow are satisfied
(A1) Since each function represents a directed transfer ofindividuals (human rodent flea and pathogens inthe environment) they are all nonnegative Thus foreach 1 le 119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) andVminus119894 (119905 119909) are nonnegative and continuous on R times R119899+
and continuously differentiable with respect to 119909(A2) There is a real number 119879 gt 0 such that for each 1 le119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) and Vminus119894 (119905 119909)
are 119879-periodic in 119905(A3) If a compartment is empty there will be no transfer
of individuals out of the compartment by any meansThat is to say if 119909119894 = 0 then Vminus
119894 = 0 In particular if119909 isin 119883119904 thenVminus119894 = 0 for 119894 = 1 119898
(A4) The incidence of infection for uninfected compart-ments is zero That is to sayF119894 = 0 for 119894 gt 119898
(A5) If the population is disease-free then the populationwill remain free of disease Thus if 119909 isin 119883119904 thenF119894 =0 andV+
119894 = 0 for 119894 = 1 119898We know that system (27) has disease-free periodicsolution given in (21) Now we define 119891(119905 119909(119905)) =F(119905 119909(119905)) minus V(119905 119909(119905)) and119872(119905) = (120597119891119894(119905119894 1198640)120597119909119895)11 le 119894 119895 le 14 where 119891119894(119905 119909(119905)) and 119909119894 are the
119894th components of 119891(119905 119909(119905)) and 119909 respectively Nowfrom (28) and (29) we obtain a 4 times 4matrix given in
A (t) = (minus1205831 120603 0 00 minus (120603 + 1205831) 0 00 0 minus1205833 00 0 0 minus1205832) (34)
We then let Φ119860(sdot)(119905) be the monodromy matrix ofthe linear 119879-periodic system 119889119911119889119905 = 119860(119905)119911 Then120588Φ119860(sdot)(119879) lt 1 which implies that 1198640 is linearlyasymptotically stable in the disease-free subspace119883119904 = 119909 ge 0 119909119894 = 0 forall119894 = 1 sdot sdot sdot 119898 where 119894 = 1 sdot sdot sdot 119898are the infected compartments Thus condition (A6)stated below holds
(A6) The disease-free periodic solution is asymptoti-cally stable in a disease-free subspace 119883119904 that is120588Φ119860(sdot)(119879) lt 1 where 120588Φ119860(sdot)(119879) is the principaleigenvalue of Φ119860(sdot)(119905)Next we set 119865(119905) and 119881(119905) as two 10 times 10 matricesdefined by (30) then using (28) and (29) we getmatri-ces 119865(119905) and 119881(119905) given in (31) and (32) respectivelyWe can further see that matrix 119865(119905) is nonnegativeand minus119881(119905) is cooperative in the sense that the off-diagonal elements are nonnegative Let 119884(119905 119904) 119905 ge 119904be the evolution operator of our 119879-periodic system119889119910119889119905 = minus119881 (119905) 119910 (35)
International Journal of Mathematics and Mathematical Sciences 11
That is for each 119904 isin R the 10 times 10 matrix 119884(119905 119904)satisfies119889119884 (119905 119904)119889119905 = minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119868119889 (36)
where 119868119889 is a 10 times 10 identity matrix Thus themonodromymatrixΦ119881(119905) of (35) equals119884(119905 0) 119905 ge 0Therefore condition (A7) stated below holds
(A7) The evolution of individuals in the infectious com-partments decays exponentially due to natural anddisease induced mortalities Thus 120588Φ119881(119879) lt 1
Now using the standard theory of linear periodic systemby Hale [48] there exist119870 gt 0 and weierp gt 0 such that119884 (119905 119904) le 119870119890minusweierp(119905minus119904) forall119905 ge 119904 119904 isin R (37)
We then have119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) le 119870 119865 (119905 minus 119886) 119890minusweierp119886forall119905 isin R 119886 isin [0infin) (38)
Considering the periodic environment we suppose thatΦ(119904) 119879-periodic in 119904 is the distribution of the new infectionat a rate 119865(119904)Φ(119904) produced by the infected individuals whowere introduced at time 119904 Given 119905 ge 119904 then 119884(119905 119904)119865(119904)Φ(119904)yields the distribution of those infected individuals who werenewly infected at time 119904 and remain in the infected class at 119905We then haveΨ (119905) = int119905
= intinfin0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886 (39)
which is the distribution of accumulative new infectionsat time 119905 produced by all those infected individual Φ(119904)introduced at previous time 119904 to 119905 (119904 le 119905)
Let 119862119879 be the ordered Banach space of all 119879-periodicfunction fromR toR119899 which is equippedwith themaximumnorm sdotinfin and the positive cone119862+119879 = Φ isin 119862119879Φ(119905) ge 0 119905 isinR Define a linear operator 119871 119862119879 rarr 119862119879 by(119871Φ) (119905) = intinfin
0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886forall119905 isin R Φ isin 119862119879 (40)
Now by Wang and Zhao [46] Diekmann et al [49] andvan denDriessche andWatmough [47] we name 119871 as the nextinfection operator then the basic reproduction number 119877119879of the periodic system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) isgiven 119877119879 = 120588 (119871) (41)
where 120588(119871) is the spectral radius of 119871
51 Characterization of 119877119879 In this subsection we investigatewhether the basic reproduction number in our periodicsystem can characterize the threshold of the disease invasionTo do this we consider the following linear 119879-periodicequation 119889119908119889119905 = [minus119881 (119905) + 119865 (119905)120582 ]119908 forall119905 isin R (42)
with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) 119905 ge 119904 119904 isin Rbe the evolution operator of system (42) on R10 We canclearly see that Φ119865minus119881(119905) = 119882(119905 0 1) forall119905 ge 0 Consideringmatrices (31) and (32)wenote that for each120582 isin (0infin) all off-diagonal elements of matrix minus119881(119905) + 119865(119905)120582 are nonnegative(cooperative matrix) It follows that the linear operator119882(119905 119904 120582) is positive in R10 for each 119905 ge 119904 119904 isin R Nowusing Perron-Frobenius theorem by Smith andWaltman [50]it entails that 120588(119882(119879 0 120582)) is an eigenvalue of 119882(119879 0 120582)with a nonnegative eigenvector Also using matrix similarityconcept by Shores [51] we can easily verify that matrix119882(119904 +119879 0 120582) is similar to the matrix119882(119879 0 120582) and hence 120590(119882(119904 +119879 0 120582)) = 120590(119882(119879 0 120582)) for any 119904 isin R where 120590(119863) is aspectrum of the matrix119863Proposition 2 (see [46]) We let (A1)ndash(A7) hold for system((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) then
(i) if 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 1205820is an eigenvalue of 119871 and hence 119877119879 gt 0
(ii) if 119877119879 gt 0 then 120582 = 119877119879 is the unique solution of120588(119882(119879 0 120582)) = 1(iii) 119877119879 = 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0This result shows that in order to find the basic reproduc-
tion number we need to find themonodromymatrixΦ119865minus119881(119905)of system (42) and evaluate itWe then find the spectral radiusof Φ119865minus119881(119879) and solve the equation 120588(Φ119865minus119881(119879)) = 1 for 120582which is the basic reproduction number
52 Computation of the Basic Reproduction Number Wecompute a time-averaged basic reproduction number 1198770using the next-generation matrix as outlined by Wesley andAllen [52] Heesterbeek [53] and Diekmann et al [49] Themethod has the advantage over the usual next-generationmethod in that the steps to reach an estimate of 1198770 and thematrix elements of the next-generation matrix have a clearbiological basis It is easy to handle complex diseases likeplague disease which has multiple transmission roots fromdifferent infectious agents
To do this we first categorize individuals by their stateat the moment they become infected (type at infection)These types at infection refer specifically to the birth ofthe infection in the individual These categories (types atinfection) differ in the way they transmit plague diseasewhich in turn differentiates their ability to produce secondarycases
In our case we categorize the individuals into eight statesand label them as follows human infected with bubonicplague (type 1) human infected with septicemic plague (type
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
Following the setting byWang and Zhao [46] and van denDriessche andWatmough [47] for epidemicmodels we checkconditions (A1)ndash(A7) for plague disease epidemic modelSystem ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) is equivalent toperiodic ordinary differential system (27) Now consideringthis systemwe can easily see that conditions (A1)ndash(A5) statedbelow are satisfied
(A1) Since each function represents a directed transfer ofindividuals (human rodent flea and pathogens inthe environment) they are all nonnegative Thus foreach 1 le 119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) andVminus119894 (119905 119909) are nonnegative and continuous on R times R119899+
and continuously differentiable with respect to 119909(A2) There is a real number 119879 gt 0 such that for each 1 le119894 le 119899 the functions F119894(119905 119909) V+
119894 (119905 119909) and Vminus119894 (119905 119909)
are 119879-periodic in 119905(A3) If a compartment is empty there will be no transfer
of individuals out of the compartment by any meansThat is to say if 119909119894 = 0 then Vminus
119894 = 0 In particular if119909 isin 119883119904 thenVminus119894 = 0 for 119894 = 1 119898
(A4) The incidence of infection for uninfected compart-ments is zero That is to sayF119894 = 0 for 119894 gt 119898
(A5) If the population is disease-free then the populationwill remain free of disease Thus if 119909 isin 119883119904 thenF119894 =0 andV+
119894 = 0 for 119894 = 1 119898We know that system (27) has disease-free periodicsolution given in (21) Now we define 119891(119905 119909(119905)) =F(119905 119909(119905)) minus V(119905 119909(119905)) and119872(119905) = (120597119891119894(119905119894 1198640)120597119909119895)11 le 119894 119895 le 14 where 119891119894(119905 119909(119905)) and 119909119894 are the
119894th components of 119891(119905 119909(119905)) and 119909 respectively Nowfrom (28) and (29) we obtain a 4 times 4matrix given in
A (t) = (minus1205831 120603 0 00 minus (120603 + 1205831) 0 00 0 minus1205833 00 0 0 minus1205832) (34)
We then let Φ119860(sdot)(119905) be the monodromy matrix ofthe linear 119879-periodic system 119889119911119889119905 = 119860(119905)119911 Then120588Φ119860(sdot)(119879) lt 1 which implies that 1198640 is linearlyasymptotically stable in the disease-free subspace119883119904 = 119909 ge 0 119909119894 = 0 forall119894 = 1 sdot sdot sdot 119898 where 119894 = 1 sdot sdot sdot 119898are the infected compartments Thus condition (A6)stated below holds
(A6) The disease-free periodic solution is asymptoti-cally stable in a disease-free subspace 119883119904 that is120588Φ119860(sdot)(119879) lt 1 where 120588Φ119860(sdot)(119879) is the principaleigenvalue of Φ119860(sdot)(119905)Next we set 119865(119905) and 119881(119905) as two 10 times 10 matricesdefined by (30) then using (28) and (29) we getmatri-ces 119865(119905) and 119881(119905) given in (31) and (32) respectivelyWe can further see that matrix 119865(119905) is nonnegativeand minus119881(119905) is cooperative in the sense that the off-diagonal elements are nonnegative Let 119884(119905 119904) 119905 ge 119904be the evolution operator of our 119879-periodic system119889119910119889119905 = minus119881 (119905) 119910 (35)
International Journal of Mathematics and Mathematical Sciences 11
That is for each 119904 isin R the 10 times 10 matrix 119884(119905 119904)satisfies119889119884 (119905 119904)119889119905 = minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119868119889 (36)
where 119868119889 is a 10 times 10 identity matrix Thus themonodromymatrixΦ119881(119905) of (35) equals119884(119905 0) 119905 ge 0Therefore condition (A7) stated below holds
(A7) The evolution of individuals in the infectious com-partments decays exponentially due to natural anddisease induced mortalities Thus 120588Φ119881(119879) lt 1
Now using the standard theory of linear periodic systemby Hale [48] there exist119870 gt 0 and weierp gt 0 such that119884 (119905 119904) le 119870119890minusweierp(119905minus119904) forall119905 ge 119904 119904 isin R (37)
We then have119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) le 119870 119865 (119905 minus 119886) 119890minusweierp119886forall119905 isin R 119886 isin [0infin) (38)
Considering the periodic environment we suppose thatΦ(119904) 119879-periodic in 119904 is the distribution of the new infectionat a rate 119865(119904)Φ(119904) produced by the infected individuals whowere introduced at time 119904 Given 119905 ge 119904 then 119884(119905 119904)119865(119904)Φ(119904)yields the distribution of those infected individuals who werenewly infected at time 119904 and remain in the infected class at 119905We then haveΨ (119905) = int119905
= intinfin0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886 (39)
which is the distribution of accumulative new infectionsat time 119905 produced by all those infected individual Φ(119904)introduced at previous time 119904 to 119905 (119904 le 119905)
Let 119862119879 be the ordered Banach space of all 119879-periodicfunction fromR toR119899 which is equippedwith themaximumnorm sdotinfin and the positive cone119862+119879 = Φ isin 119862119879Φ(119905) ge 0 119905 isinR Define a linear operator 119871 119862119879 rarr 119862119879 by(119871Φ) (119905) = intinfin
0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886forall119905 isin R Φ isin 119862119879 (40)
Now by Wang and Zhao [46] Diekmann et al [49] andvan denDriessche andWatmough [47] we name 119871 as the nextinfection operator then the basic reproduction number 119877119879of the periodic system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) isgiven 119877119879 = 120588 (119871) (41)
where 120588(119871) is the spectral radius of 119871
51 Characterization of 119877119879 In this subsection we investigatewhether the basic reproduction number in our periodicsystem can characterize the threshold of the disease invasionTo do this we consider the following linear 119879-periodicequation 119889119908119889119905 = [minus119881 (119905) + 119865 (119905)120582 ]119908 forall119905 isin R (42)
with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) 119905 ge 119904 119904 isin Rbe the evolution operator of system (42) on R10 We canclearly see that Φ119865minus119881(119905) = 119882(119905 0 1) forall119905 ge 0 Consideringmatrices (31) and (32)wenote that for each120582 isin (0infin) all off-diagonal elements of matrix minus119881(119905) + 119865(119905)120582 are nonnegative(cooperative matrix) It follows that the linear operator119882(119905 119904 120582) is positive in R10 for each 119905 ge 119904 119904 isin R Nowusing Perron-Frobenius theorem by Smith andWaltman [50]it entails that 120588(119882(119879 0 120582)) is an eigenvalue of 119882(119879 0 120582)with a nonnegative eigenvector Also using matrix similarityconcept by Shores [51] we can easily verify that matrix119882(119904 +119879 0 120582) is similar to the matrix119882(119879 0 120582) and hence 120590(119882(119904 +119879 0 120582)) = 120590(119882(119879 0 120582)) for any 119904 isin R where 120590(119863) is aspectrum of the matrix119863Proposition 2 (see [46]) We let (A1)ndash(A7) hold for system((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) then
(i) if 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 1205820is an eigenvalue of 119871 and hence 119877119879 gt 0
(ii) if 119877119879 gt 0 then 120582 = 119877119879 is the unique solution of120588(119882(119879 0 120582)) = 1(iii) 119877119879 = 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0This result shows that in order to find the basic reproduc-
tion number we need to find themonodromymatrixΦ119865minus119881(119905)of system (42) and evaluate itWe then find the spectral radiusof Φ119865minus119881(119879) and solve the equation 120588(Φ119865minus119881(119879)) = 1 for 120582which is the basic reproduction number
52 Computation of the Basic Reproduction Number Wecompute a time-averaged basic reproduction number 1198770using the next-generation matrix as outlined by Wesley andAllen [52] Heesterbeek [53] and Diekmann et al [49] Themethod has the advantage over the usual next-generationmethod in that the steps to reach an estimate of 1198770 and thematrix elements of the next-generation matrix have a clearbiological basis It is easy to handle complex diseases likeplague disease which has multiple transmission roots fromdifferent infectious agents
To do this we first categorize individuals by their stateat the moment they become infected (type at infection)These types at infection refer specifically to the birth ofthe infection in the individual These categories (types atinfection) differ in the way they transmit plague diseasewhich in turn differentiates their ability to produce secondarycases
In our case we categorize the individuals into eight statesand label them as follows human infected with bubonicplague (type 1) human infected with septicemic plague (type
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
International Journal of Mathematics and Mathematical Sciences 11
That is for each 119904 isin R the 10 times 10 matrix 119884(119905 119904)satisfies119889119884 (119905 119904)119889119905 = minus119881 (119905) 119884 (119905 119904) forall119905 ge 119904 119884 (119904 119904) = 119868119889 (36)
where 119868119889 is a 10 times 10 identity matrix Thus themonodromymatrixΦ119881(119905) of (35) equals119884(119905 0) 119905 ge 0Therefore condition (A7) stated below holds
(A7) The evolution of individuals in the infectious com-partments decays exponentially due to natural anddisease induced mortalities Thus 120588Φ119881(119879) lt 1
Now using the standard theory of linear periodic systemby Hale [48] there exist119870 gt 0 and weierp gt 0 such that119884 (119905 119904) le 119870119890minusweierp(119905minus119904) forall119905 ge 119904 119904 isin R (37)
We then have119884 (119905 119905 minus 119886) 119865 (119905 minus 119886) le 119870 119865 (119905 minus 119886) 119890minusweierp119886forall119905 isin R 119886 isin [0infin) (38)
Considering the periodic environment we suppose thatΦ(119904) 119879-periodic in 119904 is the distribution of the new infectionat a rate 119865(119904)Φ(119904) produced by the infected individuals whowere introduced at time 119904 Given 119905 ge 119904 then 119884(119905 119904)119865(119904)Φ(119904)yields the distribution of those infected individuals who werenewly infected at time 119904 and remain in the infected class at 119905We then haveΨ (119905) = int119905
= intinfin0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886 (39)
which is the distribution of accumulative new infectionsat time 119905 produced by all those infected individual Φ(119904)introduced at previous time 119904 to 119905 (119904 le 119905)
Let 119862119879 be the ordered Banach space of all 119879-periodicfunction fromR toR119899 which is equippedwith themaximumnorm sdotinfin and the positive cone119862+119879 = Φ isin 119862119879Φ(119905) ge 0 119905 isinR Define a linear operator 119871 119862119879 rarr 119862119879 by(119871Φ) (119905) = intinfin
0119884 (119905 119905 minus 119886) 119865 (119905 minus 119886)Φ (119905 minus 119886) 119889119886forall119905 isin R Φ isin 119862119879 (40)
Now by Wang and Zhao [46] Diekmann et al [49] andvan denDriessche andWatmough [47] we name 119871 as the nextinfection operator then the basic reproduction number 119877119879of the periodic system ((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) isgiven 119877119879 = 120588 (119871) (41)
where 120588(119871) is the spectral radius of 119871
51 Characterization of 119877119879 In this subsection we investigatewhether the basic reproduction number in our periodicsystem can characterize the threshold of the disease invasionTo do this we consider the following linear 119879-periodicequation 119889119908119889119905 = [minus119881 (119905) + 119865 (119905)120582 ]119908 forall119905 isin R (42)
with parameter 120582 isin (0infin) Let 119882(119905 119904 120582) 119905 ge 119904 119904 isin Rbe the evolution operator of system (42) on R10 We canclearly see that Φ119865minus119881(119905) = 119882(119905 0 1) forall119905 ge 0 Consideringmatrices (31) and (32)wenote that for each120582 isin (0infin) all off-diagonal elements of matrix minus119881(119905) + 119865(119905)120582 are nonnegative(cooperative matrix) It follows that the linear operator119882(119905 119904 120582) is positive in R10 for each 119905 ge 119904 119904 isin R Nowusing Perron-Frobenius theorem by Smith andWaltman [50]it entails that 120588(119882(119879 0 120582)) is an eigenvalue of 119882(119879 0 120582)with a nonnegative eigenvector Also using matrix similarityconcept by Shores [51] we can easily verify that matrix119882(119904 +119879 0 120582) is similar to the matrix119882(119879 0 120582) and hence 120590(119882(119904 +119879 0 120582)) = 120590(119882(119879 0 120582)) for any 119904 isin R where 120590(119863) is aspectrum of the matrix119863Proposition 2 (see [46]) We let (A1)ndash(A7) hold for system((8a) (8b) (8c) (8d) (8e) (8f))ndash(11) then
(i) if 120588(119882(119879 0 120582)) = 1 has a positive solution 1205820 then 1205820is an eigenvalue of 119871 and hence 119877119879 gt 0
(ii) if 119877119879 gt 0 then 120582 = 119877119879 is the unique solution of120588(119882(119879 0 120582)) = 1(iii) 119877119879 = 0 if and only if 120588(119882(119879 0 120582)) lt 1 forall120582 gt 0This result shows that in order to find the basic reproduc-
tion number we need to find themonodromymatrixΦ119865minus119881(119905)of system (42) and evaluate itWe then find the spectral radiusof Φ119865minus119881(119879) and solve the equation 120588(Φ119865minus119881(119879)) = 1 for 120582which is the basic reproduction number
52 Computation of the Basic Reproduction Number Wecompute a time-averaged basic reproduction number 1198770using the next-generation matrix as outlined by Wesley andAllen [52] Heesterbeek [53] and Diekmann et al [49] Themethod has the advantage over the usual next-generationmethod in that the steps to reach an estimate of 1198770 and thematrix elements of the next-generation matrix have a clearbiological basis It is easy to handle complex diseases likeplague disease which has multiple transmission roots fromdifferent infectious agents
To do this we first categorize individuals by their stateat the moment they become infected (type at infection)These types at infection refer specifically to the birth ofthe infection in the individual These categories (types atinfection) differ in the way they transmit plague diseasewhich in turn differentiates their ability to produce secondarycases
In our case we categorize the individuals into eight statesand label them as follows human infected with bubonicplague (type 1) human infected with septicemic plague (type
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
12 International Journal of Mathematics and Mathematical Sciences
2) human infected with pneumonic plague (type 3) rodentinfected with bubonic plague (type 4) rodent infected withsepticemic plague (type 5) rodent infected with pneumonicplague (type 6) flea infested with pathogens (type 7) and thepathogens in the environment (type 8)
We assume and label individual with bubonic plague asstage one of the disease septicemic plague as stage two andpneumonic plague as stage three We also assume that whenan individual in stage one graduates to stage two we onlyconsider the current stage and ignore the latter We assumethat the infection only goes in ascending direction that isfrom stages one to two or two to three but not in the reversedirection
Since the system has eight types at infection the next-generation matrix K will be an 8 times 8 matrix with elements119896119894119895rsquos Each of the elements 119896119894119895rsquos stands for expected numberof new cases of 119894 caused by one infected individual of 119895 Forexample 11989611 is the expected number of new cases of humansinfected with bubonic plague caused by one infected humanwith bubonic plague
We now define a matrix K whose entries are 119896119894119895 Theresulting next-generation matrix is as given in
Then 1198770 = 120588(119870) where 120588(119870) is spectral radius of 119870Some elements equal 0 because not all types of infections
cause all other types of infection For example humans withbubonic plague 119868119867119861 (type at infection 1) do not producetype at infections 1 (human infected with bubonic plague)4 (rodent infected with bubonic plague) 5 (rodent infectedwith septicemic plague) 6 (rodent infected with pneumonicplague) and 8 (pathogens in the environment) This meansthat 11989611 11989614 11989615 11989616 and 11989618 are 0 The type at infection
2 (human infected with septicemic plague) also does notproduce type at infections 1 (human infected with bubonicplague) 4 (rodent infected with bubonic plague) 6 (rodentinfected with pneumonic plague) and 8 (pathogens in theenvironment) This also means that 11989621 11989624 11989626 and 11989628 arezero (0) The type at infection 3 does not produce type atinfections 1 (human infected with bubonic plague) 2 (humaninfected with septicemic plague) 4 (rodent infected withbubonic plague) 5 and 7 which means that 11989631 11989632 11989634 11989635and 11989637 are zero Type at infection 4 does not produce typeat infection 1 2 3 4 or 8 which means that 11989641 11989642 11989643 11989644and 11989648 are zero Type at infection 5 does not produce typeat infections 1 3 4 and 8 then 11989651 11989653 11989654 and 11989658 are zeroThe type at infection 6 does not produce type at infections 12 4 5 and 7 thus 11989661 11989662 11989664 11989665 and 11989667 are zero Typeat infection 7 also does not produce type at infections 3 67 and 8 thus 11989673 11989676 11989677 and 11989678 are zero And the type atinfection 8 does not produce type at infections 1 2 4 5 7 and8 which means that 11989681 11989682 11989684 11989685 11989687 and 11989688 are zeroIncorporating these we modify the matrixK as shown in thefollowing matrix
We will now explain the derivation of each matrixelement in detail We employ the derivation steps by Gailand Benichou [54] to drive the expressions for 119896119894119895 Wemainlybase our derivation on the adequate contact rate betweenthe infected individual type 119895 and the susceptible individualtype 119894 the expected duration of infection of individual type119895 and the probability that the individual type 119895 survives theduration between the latent stage and the time an individualexperiences the onset of clinical disease as in
Kij = (Effective contact Rate) times (Duration of infection)times (Probability that the individual survive the incubation period) (45)
The production of 119868119867119861 depends on the probability that thetotal number of fleas that become infectious at the rate of 120573and the infected immigrants survive the incubation periodWe also consider the rate at which 119868119865 adequately bites thesusceptible human and the bite results in a human infectedwith bubonic plague 119868119867119861 at the average value of transmissionrate Γ119891ℎ The total number of humans infected with bubonic
plague caused by one flea infested with pathogens is as givenin 11989617 = ( 120573120573 + 1205832 + 12059521198941205952119894 + 1205832) ]2Γ119891ℎ1205832 + 1205752 (46)
Septicemic plague in human may be produced in var-ious ways progression of untreated human with bubonic
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
International Journal of Mathematics and Mathematical Sciences 13
plague to human with septicemic plague adequate contact(including sexual contact) between humans with septicemicplague adequate contact between rodent and human withsepticemic plague and being acquired from the flea infestedwith pathogens We consider the progression rate of infectedhuman with bubonic to septicemic 12057231205883 the adequate con-tact (it may be sexual contact) rate between humans withsepticemic plague rodent infected with septicemic plagueand the infected flea to human with septicemic plague at theaverage rates 120574ℎ119904ℎ 120574119903119904ℎ and 120574119891ℎ Then the number of humansinfected with septicemic plague from all the mentionedinfectious agents is as given in11989621 = 12057221205723]21205883(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (47a)
The proportions 1205881 and 120585 of untreated 119868119867119861 and 119868119867119878 mayprogress and become 119868119867119875 at the progression rates 1205723 and1205724 respectively We multiply the average period 119868119867119861 remaininfected by the rate at which they progress to 119868119867119875 119868119867119875 mayalso result from the airborne transmission from the humanor rodent with pneumonic plague at the average rate 120574ℎ119901ℎ or120574119903119901ℎ respectively and through the direct interaction with theenvironment at the average rate 1205961 Then the total number ofhumans infected with pneumonic plague from the stated fivesources is given in11989631 = 12057221205723]21205881(1205722]2 + 1205831) (1205723 + 1205831 + 1205751119887) (48a)
Production of number of rodents with bubonic plague119868RB depends only on the flea infested with pathogens The
infection depends on the infection period of the flea thatsurvives the incubation period and the proportion at whichthe adequate contact between infected flea and susceptiblerodent causes bubonic plague at the average rate 1205913Γ119891119903 as givenin
The septicemic plague in rodent is produced in threeways the first way is when infected rodent with bubonicplague progresses and becomes septicemic plague infectiveagent at the rate 1205743(1 minus 120601) The second way is after adequatecontact (it may also be a rodent eating or biting an infectedindividual) between the susceptible rodent and a rodentinfected with septicemic plague or human at the average rateΓ119903119904119903 or Γℎ119904119903 respectivelyThe third way is from the flea infestedwith pathogens with the proportion that the adequate contactbetween 119868119865 and the susceptible rodent results in 119868119877119878 The totalnumber of 119868119877119878 infected from these infectious agents is as givenin
119868119877119875 may be the result of airborne transmission betweenthe susceptible rodent and the human and rodent with pneu-monic plague at the average rates Γℎ119901119903 and Γ119903119901119903 respectively Itmay also occur from the progression of untreated 119868119877119861 and 119868119877119878at the rates 1205743 and 1205744 respectively The pathogens in environ-ment may also cause 119868119877119875 after the adequate interaction at theaverage rate 1205962 Now the total number of 119868119877119861 resulting fromthese interactions is in
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
Fleas are infested with pathogens from humans androdents infected with bubonic and septicemic plague at theaverage rates 120574ℎ119887119891 120574ℎ119904119891 120574119903119887119891 and 120574119903119904119891The infection is dictatedby the probability that humans and rodents with bubonicand septicemic plague survive the incubation period and theadequate rates of contact From these interactions we get thetotal number of infectious fleas given in11989671 = 1205722]2Γℎ119887119891(1205722]2 + 1205831) (1205831 + 1205723 + 1205751119887) (52a)
The pathogens are released into the environment at theaverage rates 1205781 and 1205781 from 119868119867119875 and 119868119877119875 respectively Thereleased number of pathogens at a given time depends onthe infectious period of the rodent and human infectedwith pneumonic plague and the probability that 119868119867119875 and119868119877119875 survive the incubation period The total pathogens insoilenvironment is as given in11989683 = ( 1205722]11205722]1 + 1205831 + 1205723120588112057231205881 + 1205831 + 12057241205851205724120601 + 1205831)sdot 12057811205725 + 1205831 + 1205751119901 (53a)
Each element in the matrix K represents the expectednumber of secondary cases produced by infected individual119895 during the entire infectious period of that particularindividual into a completely susceptible population 119894 [55]521 Basic Reproduction Number 1198770 We obtain the averagebasic reproduction number 1198770 by computing the maxi-mum modulus of the eigenvalues of the next-generation
matrix K [49 53] Now using Maple computing softwarepackage the basic reproduction number is1198770 = 1119879 int1198790 11989622 (119904) + 11989655 (119904)4+ 12radic1198601 + 13 3radic21198604 + 119860531198604+ 12radic1198602 minus 13 3radic21198604 minus 119860531198604+ 11986034radic1198601 + (13 3radic2)1198604 + 119860531198604 119889119904
(54)
in which1198601 = 31205993 + 8120599112 1198602 = 31205993 minus 812059916 1198603 = 412059911205993 minus 12059933 minus 812059941198604 = 13 3radic2 ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 27120599231205992)) + ((212059931 minus 7212059921205991 minus 9120599312059941205991 + 2712059924+ 271205992312059922 minus 4 (12059921 + 121205992 minus 312059931205994)3)13)12 1198605 = 3radic2 (12059921 + 121205992 minus 312059931205994)
Since the system has multiple infectious types frommultiple hosts then the next-generation matrix producesthe average value of the geometric mean of the number ofinfections per generation and the basic reproduction number
International Journal of Mathematics and Mathematical Sciences 15
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
Figure 1 Effect of progression rates from 119868119867119861 to 119868119867119878 on the periodic reproduction number
is the average number of secondary infections [56] It isshown that the basic reproduction number of plague diseasedepends on the expected number of new cases of humansinfected with bubonic plague caused by one infected flea(11989617) the expected number of new cases of humans infectedwith septicemic plague caused by one infected human withbubonic plague (11989621) the expected number of new casesof humans infected with septicemic plague caused by oneinfected human with septicemic plague (11989622) the expectednumber of new cases of rodents infected with bubonic plaguecaused by one infected flea (11989647) the expected number of newcases of rodents infected with septicemic plague caused byone infected rodent with bubonic plague (11989654) the expectednumber of new cases of rodents infected with septicemicplague caused by one infected rodent with septicemic plague(11989655) the expected number of new cases of rodents infectedwith septicemic plague caused by one infected flea (11989657) theexpected number of new cases of fleas infested with Yersiniapestis caused by one infected human with bubonic plague(11989671) the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemicplague (11989672) the expected number of new cases of fleasinfested with Yersinia pestis caused by one infected rodentwith bubonic plague (11989674) and the expected number of newcases of fleas infested with Yersinia pestis caused by oneinfected rodent with septicemic plague (11989675) The result mayalso be interpreted as follows among all elements of thematrix K 119896119894119895 which appear in 119877119874 gives more significantinvolvement in the dynamics and spread of plague disease
6 Numerical Results and Discussion
Here we use the parameters values of model system ((8a)(8b) (8c) (8d) (8e) (8f))ndash(11) given in Table 1 to study thetransmission trend of plague disease Simulation results aregiven to show the effect of different parameters on the peri-odic reproduction numberWe have also chosen temperaturedata obtained from Tanga region from January to December2013 to show the seasonal distribution in the number ofsecondary cases of plague infections It is observed thatsimulation results from time-averaged seasonal parametersand those seasonal parameters treated mathematically assinusoidal functions match the real seasonal fluctuation data(temperature)
Results shows that the increase in number of individualsinfected with bubonic plague to a large extent affects theincreases in number of individuals with septicemic and pneu-monic plague disease This is due to the fact that individualswith bubonic plague progress and become either septicemicor pneumonic plague infective agents This in turn leads tothe significant increase of plague disease transmission rateand the average number of secondary infections Figures2 and 1 show how the progression rates from individ-uals with bubonic plague to individuals with septicemicplague affect the average number of secondary infections inhuman beings and rodents respectively It is illustrated thatthe increase of human beings and rodents progressing tobecome septicemic plague infective agents affects the diseasedynamics by increasing the average number of secondary
16 International Journal of Mathematics and Mathematical Sciences
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
Figure 2 Effect of progression rates from 119868119877119861 to 119868119877119878 on the periodic reproduction number
infections We see similar result when we evaluate theperiodic reproduction number based on the temperature datafrom Tanga region (Figures 1(c) and 2(c)) and time-averagedparameters (Figures 1(b) and 2(b)) for human beings androdents respectively These findings necessitate the need forearly treatment of plague disease infective agents especiallythe primary forms (bubonic and septicemic plague) beforethey progress to highly fatal and fast spreading plague formslike pneumonic plague disease It is thus important for thegovernment and other health stakeholders to ensure theavailability of effective plague disease treatment especially inhigh risk areas
Increase of plague disease transmission through flea bitein human beings and rodents populations alters the wholedynamics of plague disease Results in Figures 3 and 4show the effect of infection from infectious flea to humanbeings with bubonic and septicemic plague on the averagenumber of secondary infections The infection from flea torodents with bubonic and septicemic plague disease alsoshows the significant effect on 119877119879 as illustrated in Figures5 and 6 respectively The results generally show that whenthe periodic infection rates from flea increase those of theinfectious human beings and rodent increase as well this inturn affects the general plague disease periodic transmissionand spread These results are in conjuncture with what isobserved when 119877119879 is evaluated using the temperature dataand time-averaged seasonal parameter as in Figures 3(c) 3(b)5(c) and 5(b) for human beings and rodents respectively
The results call attention for the need to control thenumber of infectious fleas and flea population in general asthe way of controlling the plague disease The study recom-mends that for the appropriate and most effective way to
control flea populationwe first need to study the flearsquos ecologyand its local patterns of disease transmission One of themost important and cost-effective strategies of controlling thevector flea is environmental management strategies that canreduce or eliminate vector breeding grounds For example inresidential areas people must be educated to improve theirsurrounding environment in a way that does not favor thesurvival and growth of vector flea This may be throughimproving the design of water systems improving wastedisposal and water storage discouraging deforestation andloss of biodiversity and living in well ventilated housing thatis not close to animals
There are also biological control tools like bacteriallarvicides and larvivorous fish that may be used to controlflea population [57] These control methods aim at killingvector larvae without generating the ecological impacts asthey do not use chemicals Another strategy is using chemicalmethods which mainly shorten the lifespan of vectorsThese tools include indoor residual sprays space sprayingand use of chemical larvicides and adulticides Since mostof these methods have side effects to the environmentalecology they are recommended to be used when other safestrategies fail Moreover even though these chemicals arenot environmentally friendly we advise the environmentalstakeholders to recommend the use of chemical methods ofvector control that reinforces linkages between health andenvironment
Reducing the number of flea population will reduce theinfection rates to human beings and rodents and ultimatelyreduce the number of secondary infections Figure 7(a) showshow reducing the number of infectious fleas reduces thenumber of secondary infections This is also true when
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
International Journal of Mathematics and Mathematical Sciences 17
3
4
5
6
7
8
9
10
11
RT
01 02 03 04 05 06 07 08 090Γfℎb
(a)
02 03 04 05 06 07 08 0901Γfℎb
20
30
40
50
60
70
80
90
100
RT
(b)
05
1015
RT
01 02 03 04 05 06 07 08 09 10Γfℎb
(c)
Figure 3 The effect of infection from 119868119865 to human beings with bubonic plague on the periodic reproduction number
3
4
5
6
7
8
9
10
RT
03 04 05 06 07 08 09 102Γfℎs
(a)
24
26
28
30
32
34
36
38
RT
005 010 015 0200Γfℎs
(b)
05
10152025
RT
03 04 05 06 07 08 09 102Γfℎs
(c)
Figure 4 The effect of infection from 119868119865 to human beings with septicemic plague on the periodic reproduction number
the parameters that are affected by seasonal weather vari-ation are evaluated using the temperature data in Tanga(Figure 7(c)) and using the time-averaged seasonal param-eters (Figure 7(b)) This result is in light of the fact thatthe reduction of flea population will reduce the numberof individuals with bubonic and septicemic plague and as
a result reduce the number of pneumonic plague infectiveagents that result from the progression of individual withbubonic and septicemic plague
The reduction of flea population will reduce not only theinfection from flea to other individuals but also the rate atwhich flea gets the disease from other individuals (human
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
18 International Journal of Mathematics and Mathematical Sciences
5
10
15
20RT
005 01 015 02 025 03 0350Γfrb
(a)
01 02 03 04 05 06 07 08 092030405060708090
100
RT
Γfrb
(b)
005 01 015 02 025 03 035 040
5
10
15
20
RT
Γfrb
(c)
Figure 5 The effect of infection from 119868119865 to rodents with bubonic plague on the periodic reproduction number
5
10
15
20
RT
005 01 015 02 025 03 035 040Γfrs
(a)
20
21
22
23
24
25
RT
40 50 60 70 80 90 10030Γfrs
(b)
5
10
15
20
RT
005 01 015 02 025 03 0350Γfrs
(c)
Figure 6 The effect of infection from 119868119865 to rodents with septicemic plague on the periodic reproduction number
beings and rodents) When the flea population is reducedit will as a result reduce the interaction between susceptiblefleas and other infectious individuals and vice versa Thenumber of fleas getting the disease increases with the increase
of the rate at which fleas acquire infection from infectioushuman beings with bubonic plague (see Figure 8(a)) andthose with septicemic plague (see Figure 9(a)) We fur-ther observe that the increase of infectious fleas may be
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
International Journal of Mathematics and Mathematical Sciences 19
005 01 015 02 025 03 035 0402
0
10
20
30
40
50
RT
(a)
0
100
200
300
400
RT
002 004 006 008 010 012 014 0160Fleas death rate(b)
0102030405060
RT
01 02 03 04 05 06 07 08 09 102
(c)
Figure 7 Effect of increased number of fleasrsquo death rate on the periodic reproduction number
01 02 03 04 05 06 07 08 09 10
10
20
30
40
RT
Γℎbf
(a)
04 06 08 10 12 14 16 18
RT
50
40
30
20
Γℎbf
(b)
01 02 03 04 05 06 07 08 09 10
10
20
40
30
RT
Γℎbf
(c)
Figure 8 The effect of increased infection rate to fleas from the infectious human beings (119868119867119861) on the periodic reproduction number
contributed by the infectious rodents with bubonic plague(see Figure 10(a)) and those with septicemic plague (seeFigure 11(a)) We can also observe the similar results whenthe parameters are evaluated based on the temperature data
in Tanga region and when the parameters are timely averagedas in Figures 8(c) 8(b) 10(c) and 10(b) for human beingsand rodents respectively Therefore using these results wesettle to the point that increasing transmission rate in flea
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
20 International Journal of Mathematics and Mathematical Sciences
2
4
6
8
10RT
01 02 03 04 05 06 07 080Γℎsf
(a)
24
26
28
30
32
34
36
38
RT
02 0301 04 05 06 070Γℎsf
(b)
05
10152025
RT
07 090604 05 08 103010 02Γℎsf
(c)
Figure 9 The effect of increased infection rate to fleas from the infectious human beings (119868119867119878) on the periodic reproduction number
2
4
6
8
10
12
14
RT
0802 03 04 05 06 07010Γrbf
(a)
20
22
24
26
28
30
32
RT
04 05 06 07 08 0903Γrbf
(b)
2468
10121416
RT
01 02 03 04 05 06 07 08 09 10Γrbf
(c)
Figure 10 The effect of increased infection rate to fleas from the infectious rodents (119868119877119861) on the periodic reproduction number
population fromhumanbeings and rodentswith bubonic andsepticemic plague raises the average number of secondaryplague disease infections
Physical contact that includes sexual contact between twoinfectious individuals (human beings and rodents) may lead
to septicemic plague The increase in the number of individ-uals with septicemic plague affects the general dynamics ofplague disease particularly the average number of secondaryinfections It is illustrated that increasing infection rate from ahuman being with septicemic plague to the other susceptible
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
International Journal of Mathematics and Mathematical Sciences 21
01 02 03 04 05 06 07 08 09 1031228
3123
31232
31234
31236
31238
3124
31242
31244RT
Γrsf
(a)
24
26
28
30
32
34
RT
5 10 15 200Γrsf
(b)
01 02 03 04 05 06 07 08 09 1031228
312331232312343123631238
31243124231244
RT
Γrsf
(c)
Figure 11 The effect of increased infection rate to fleas from the infectious rodents (119868119877119878) on the periodic reproduction number
02 03 04 05 06 07 08 09 1010
20
40
60
80
100
120
RT
Γℎsℎ
(a)
5
10
15
20
25
RT
01 02 03 04 05 06 07 08 090Γℎsℎ
(b)
020406080
100120
RT
02 03 04 05 06 07 08 09 101Γℎsℎ
(c)
Figure 12 Effect of infection rate (Γℎ119904ℎ) on the periodic reproduction number
human (see Figure 12(a)) and from rodent with septicemicplague to the susceptible rodents (see Figure 13(a)) increasesthe average number of secondary infections The result againshows a clear correlation when parameters are evaluated
based on the temperature data fromTanga region (see Figures12(c) and 13(c)) andwhen they are averaged (see Figures 12(b)and 13(b)) for human beings and rodents respectively Thisshows the necessity to educate human beings to practice safe
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
22 International Journal of Mathematics and Mathematical Sciences
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(a)
01 02 03 04 05 06 07 08 090
2355
2360
2365
2370
2375
RT
Γrsr
(b)
3122
31225
3123
RT
01 02 03 04 05 06 07 08 09 10Γrsr
(c)
Figure 13 Effect of infection rate (Γ119903119904119903) on the periodic reproduction number
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 14 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma
sex using protective gears and taking necessary precautionwhen handling people or animals with septicemic plagueThis also tells us that there is a necessity to quarantine peopleand animals that immigrate from areas that are infected bysepticemic plague so that they do not affect other humanbeings or animals and thus increasing the endemicity of thedisease
The distribution of the basic reproduction number isbased on the seasonal weather condition of a particular areaat a particular time This is what causes the unpredictabilityof the number of secondary cases of plague disease infection(bubonic septicemic and pneumonic plague) as it willchangewhenever theweather conditions changeWe evaluatethe distribution of the basic reproduction number basedon the data we obtained on daily temperature (∘C) andhumidity () from five regions in Tanzania from January toDecember 2013 Figures 14 15 16 17 and 18 show the seasonal
distribution of basic reproduction number when evaluatedbased on the temperature and humidity data from KigomaMbeya Mtwara Singida and Tanga regions respectively
The features displayed in these results clearly show howseasonal weather fluctuation can have significant effects onthe number of secondary cases of plague disease It canbe noted that there is a seasonal pattern in new plaguedisease infection cases We therefore vindicate the fact thatthe number of plague disease infective agents peakswheneverthe weather condition is favorable for plague disease trans-mission and it drops when the weather condition does notfavor plague disease transmission
7 Conclusion
The transmission of plague disease occurs in several pathwayswhich makes the modeling of the disease challenging and
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
International Journal of Mathematics and Mathematical Sciences 23
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 15 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(a)
50 100 150 200 250 300 350 4000Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 16 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara
02468
1012
Basic
repr
oduc
tion
num
ber
50 1000 300150 200 250 400350Time (days)
times103
(a)
02468
1012
Basic
repr
oduc
tion
num
ber
50 100 150 200 250 300 350 4000Time (days)
times103
(b)
Figure 17 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Singida
50 100 150 200 250 300 350 4000Time (days)
02468
1012
Basic
repr
oduc
tion
num
ber
times103
(a)
0 100 150 200 250 300 350 40050Time (days)
2
4
6
8
10
Basic
repr
oduc
tion
num
ber
times103
(b)
Figure 18 Distribution of 1198770 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
24 International Journal of Mathematics and Mathematical Sciences
very complex Moreover all ways that lead to plague diseasetransmission are directly or indirectly affected by seasonalweather variation which causes seasonality in plague diseaseepidemic The effect of seasonal weather variation has beena glowing concern in different epidemiological studies Thisin turn dictates that in order to study the dynamics andpropose the effective control strategies of the plague diseasewe must incorporate the effect of seasonal weather variationIn this study we have analysed the plague disease modelincorporated with the factors that are affected by the seasonalweather variation in order to study its effects on the dynamicsof the plagues disease We have computed basic reproductionnumber and depicted how it can be affected by seasonalweather variation through numerical simulation We wereable to deduce that progression rates from one primary formto secondary form of plague infection flearsquos infection rateand the vector flea abundance pose the significant effect onthe increase of the average number of secondary cases ofplague infection Therefore the effective control strategiesmust take into account these factors as they have been shownto have a significant contribution to the increase of theaverage number of secondary cases of plague infection
Notations119878119867 Susceptible human population119864119867 Exposed human population119868119867119861 Infectious human population with bubonic plague119868119867119878 Infectious human population with septicemic plague119868119867119875 Infectious human population with pneumonic plague119877119867 Recovered human population119878119877 Susceptible rodents119864119877 Exposed rodents119868119877119861 Infectious rodents with bubonic plague119868119877119878 Infectious rodents with septicemic plague119868119877119875 Infectious rodents with pneumonic plague119878119865 Susceptible fleas119868119865 Infected fleas119860 Pathogens in the soilenvironment
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D M Wagner J Klunk M Harbeck et al ldquoYersinia pestis andthe Plague of Justinian 541-543 AD A genomic analysisrdquo TheLancet Infectious Diseases vol 14 no 4 pp 319ndash326 2014
[2] S Altizer A Dobson P Hosseini P Hudson M Pascual and PRohani ldquoSeasonality and the dynamics of infectious diseasesrdquoEcology Letters vol 9 no 4 pp 467ndash484 2006
[3] D C Cavanaugh and J DMarshall Jr ldquoThe influence of climateon the seasonal prevalence of plague in the republic of vietnamrdquoJournal of Wildlife Diseases vol 8 no 1 pp 85ndash94 1972
[4] J Silverman M K Rust and D A Reierson ldquoInfluence oftemperature and humidity on survival and development of
the cat flea Ctenocephalides felis (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 18 no 1 pp 78ndash83 1981
[5] D Zentko and D Richman Cat Flea Ctenocephalides Felis Felis(Bouche) Citeseer 1997
[6] D C Cavanaugh ldquoSpecific effect of temperature upon trans-mission of the plague bacillus by the oriental rat flea xenop-sylla cheopisrdquo The American Journal of Tropical Medicine andHygiene vol 20 no 2 pp 264ndash273 1971
[7] A W Bacot and C J Martin ldquoThe respective influences oftemperature and moisture upon the survival of the rat flea(xenopsylla cheopis) away from its hostrdquo Journal of Hygiene vol23 no 1 pp 98ndash105 1924
[8] K L Gage T R Burkot R J Eisen and E B Hayes ldquoClimateand vectorborne diseasesrdquo American Journal of PreventiveMedicine vol 35 no 5 pp 436ndash450 2008
[9] R E Enscore B J Biggerstaff T L Brown et al ldquoModelingrelationships between climate and the frequency of humanplague cases in the southwestern united states 1960-1997rdquo TheAmerican Journal of Tropical Medicine and Hygiene vol 66 no2 pp 186ndash196 2002
[10] R S J Brooks ldquoThe influence of saturation deficiency and oftemperature on the course of epidemic plaguerdquo The Journal ofhygiene supplement 15 pp 881ndash889 1917
[11] R J Eisen and K L Gage ldquoAdaptive strategies of yersiniapestis to persist during inter-epizootic and epizootic periodsrdquoVeterinary Research vol 40 no 2 article no 01 2009
[12] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoEffect of Air Temperature andHumidity on the Survival of Pre-Imaginal Stages of Two Flea Species (Siphonaptera Pulicidae)rdquoJournal of Medical Entomology vol 38 no 5 pp 629ndash637 2001
[13] R R Parmenter E P Yadav C A Parmenter P Ettestad and KL Gage ldquoIncidence of plague associated with increased winter-spring precipitation in new mexicordquo The American Journal ofTropical Medicine and Hygiene vol 61 no 5 pp 814ndash821 1999
[14] L Korslund and H Steen ldquoSmall rodent winter survival Snowconditions limit access to food resourcesrdquo Journal of AnimalEcology vol 75 no 1 pp 156ndash166 2006
[15] S Roberts J Van Wagtendonk A Miles D Kelt and J LutzldquoModeling the effects of fire severity and spatial complexityon small mammals in yosemite national park californiardquo FireEcology Special Issue vol 4 pp 84ndash104 2008
[16] F M Jaksic andM Lima ldquoMyths and facts on ratadas Bambooblooms rainfall peaks and rodent outbreaks in South AmericardquoAustral Ecology vol 28 no 3 pp 237ndash251 2003
[17] P L Meserve W Bryan Milstead and J R Gutierrez ldquoResultsof a food addition experiment in a north-central Chile smallmammal assemblage Evidence for the role of ldquobottom-uprdquofactorsrdquo Oikos vol 94 no 3 pp 548ndash556 2001
[18] M Letnic B Tamayo and C R Dickman ldquoThe responses ofmammals to LaNina (El Nino SouthernOscillation)-associatedrainfall predation and wildfire in central Australiardquo Journal ofMammalogy vol 86 no 4 pp 689ndash703 2005
[19] H Leirs R VerhagenW Verheyen P Mwanjabe and T MbiseldquoForecasting rodent outbreaks in africa an ecological basis formastomys control in tanzaniardquo Journal of Applied Ecology pp937ndash943 1996
[20] C RDickman P SMahon PMasters andD FGibson ldquoLong-term dynamics of rodent populations in arid Australia Theinfluence of rainfallrdquo Wildlife Research vol 26 no 4 pp 389ndash403 1999
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
International Journal of Mathematics and Mathematical Sciences 25
[21] S L Agar J Sha S M Foltz et al ldquoCharacterization ofthe rat pneumonic plague model infection kinetics followingaerosolization of Yersinia pestis CO92rdquoMicrobes and Infectionvol 11 no 2 pp 205ndash214 2009
[22] B R Krasnov I S Khokhlova L J Fielden andNV BurdelovaldquoDevelopment rates of two Xenopsylla flea species in relationto air temperature and humidityrdquo Medical and VeterinaryEntomology vol 15 no 3 pp 249ndash258 2001
[23] B R Krasnov N V Burdelova G I Shenbrot and I SKhokhlova ldquoAnnual cycles of four flea species in the centralNegev desertrdquo Medical and Veterinary Entomology vol 16 no3 pp 266ndash276 2002
[24] D Davis ldquoPlague in Africa from 1935 to 1949rdquo WHO Bulletinvol 9 no 5 pp 665ndash669 1953
[25] E Mitscherlich and E H Marth Microbial Survival in TheEnvironment Bacteria and Rickettsiae Important in Human andAnimal Health Springer Science amp Business Media 2012
[26] S Ayyadurai L Houhamdi H Lepidi et al ldquoLong-termpersistence of virulent Yersinia pestis in soilrdquoMicrobiology vol154 no 9 pp 2865ndash2871 2008
[27] HMollaret ldquoExperimental Conservation of Plague inThe SoilrdquoDTIC Document 1964
[28] R J Eisen J M Petersen C L Higgins et al ldquoPersistenceof Yersinia pestis in soil under natural conditionsrdquo EmergingInfectious Diseases vol 14 no 6 pp 941ndash943 2008
[29] M Nozadze E Zhgenti M Meparishvili et al et al ldquoCom-parative proteomic studies of yersinia pestis strains isolatedfrom natural foci in the republic of georgiardquo Frontiers in PublicHealth vol 3 pp 145ndash239 2015
[30] J Koirala ldquoPlague Disease Management and Recognition ofAct of Terrorismrdquo Infectious Disease Clinics of North Americavol 20 no 2 pp 273ndash287 2006
[31] R R Brubaker ldquoThe genus Yersinia biochemistry and geneticsof virulencerdquo Current Topics in Microbiology and Immunologyvol 57 pp 111ndash158 1972
[32] N C Stenseth B B Atshabar M Begon et al ldquoPlague Pastpresent and futurerdquo PLoSMedicine vol 5 no 1 pp 0009ndash00132008
[33] A Bevelacqua and R Stilp TerrorismHandbook for OperationalResponders Cengage Learning 2009
[34] K F Meyer ldquoPneumonic plaguerdquo Bacteriological Reviews vol25 article 249 no 3 1961
[35] L J Rose R Donlan S N Banerjee and M J ArduinoldquoSurvival of Yersinia pestis on environmental surfacesrdquo Appliedand Environmental Microbiology vol 69 no 4 pp 2166ndash21712003
[36] R J Eisen A P Wilder S W Bearden J A Montenieri andK L Gage ldquoEarly-phase transmission of Yersinia pestis byunblocked Xenopsylla cheopis (Siphonaptera Pulicidae) is asefficient as transmission by blocked fleasrdquo Journal of MedicalEntomology vol 44 no 4 pp 678ndash682 2007
[37] R Gani and S Leach ldquoEpidemiologic Determinants for Mod-eling Pneumonic Plague Outbreaksrdquo Emerging Infectious Dis-eases vol 10 no 4 pp 608ndash614 2004
[38] M J Keeling and C A Gilligan ldquoBubonic plague A metapop-ulation model of a zoonosisrdquo Proceedings of the Royal Society BBiological Sciences vol 267 no 1458 pp 2219ndash2230 2000
[39] K J Kugeler J E Staples A F Hinckley K L Gage and PS Mead ldquoEpidemiology of human plague in the United States1900ndash2012rdquo Emerging Infectious Diseases vol 21 no 1 pp 16ndash22 2015
[40] C Tollenaere L Rahalison M Ranjalahy et al ldquoSusceptibilityto Yersinia pestis experimental infection in wild rattus rattusreservoir of plague in Madagascarrdquo EcoHealth vol 7 no 2 pp242ndash247 2010
[41] S Morand and P H Harvey ldquoMammalian metabolismlongevity and parasite species richnessrdquo Proceedings of the RoyalSociety B Biological Sciences vol 267 no 1456 pp 1999ndash20032000
[42] P Hunter ldquoClimate change and waterborne and vector-bornediseaserdquo Journal of Applied Microbiology vol 94 no s1 pp 37ndash46 2003
[43] Y Dumont F Chiroleu and C Domerg ldquoOn a temporal modelfor the Chikungunya disease modeling theory and numericsrdquoMathematical Biosciences vol 213 no 1 pp 80ndash91 2008
[44] S CMpeshe L S Luboobi andYNkansah-Gyekye ldquoModelingthe impact of climate change on the dynamics of rift valleyfeverrdquo Computational and Mathematical Methods in MedicineArticle ID 627586 Art ID 627586 12 pages 2014
[45] G Aronsson and R B Kellogg ldquoOn a differential equation aris-ing from compartmental analysisrdquo Mathematical BiosciencesAn International Journal vol 38 no 1-2 pp 113ndash122 1978
[46] W Wang and X-Q Zhao ldquoThreshold dynamics for compart-mental epidemic models in periodic environmentsrdquo Journal ofDynamics and Differential Equations vol 20 no 3 pp 699ndash7172008
[47] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002
[48] J KHaleOrdinaryDifferential Equations JohnWiley and SonsInc New York NY USA 1980
[49] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio 1198770 in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990
[50] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995
[51] T S Shores Applied Linear Algebra and Matrix AnalysisSpringer Science amp Business Media 2007
[52] C L Wesley and L J Allen ldquoThe basic reproduction numberin epidemic models with periodic demographicsrdquo Journal ofBiological Dynamics vol 3 no 2-3 pp 116ndash129 2009
[53] J Heesterbeek Mathematical Epidemiology of Infectious Dis-eases Model Building Analysis and Interpretation vol 5 JohnWiley amp Sons 2000
[54] M H Gail and J Benichou Encyclopedia of EpidemiologicMethods John Wiley amp Sons 2000
[55] N A Hartemink S E Randolph S A Davis and J A PHeesterbeek ldquoThe basic reproduction number for complex dis-ease systems Defining R 0 for tick-borne infectionsrdquo AmericanNaturalist vol 171 no 6 pp 743ndash754 2008
[56] J Li D Blakeley and R J Smith ldquoThe failure of R0rdquo Computa-tional and Mathematical Methods in Medicine vol 2011 ArticleID 527610 17 pages 2011
[57] J A Rozendaal Vector Control Methods for Use by Individualsand Communities World Health Organization 1997
