ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 16 (2020) No. 1, pp. 53-80 Modeling Pneumonic Plague Caused by Yersinia pestis in the Environment Rigobert C. Ngeleja 1 * , Livingstone S. Luboobi 2 , Verdiana Grace Masanja 3 1 General Studies Department, Dar Es Salaam Institute of Technology (DIT), P.O. Box 2958, Dar Es Salaam -Tanzania. 2 Department of Mathematics, Makerere University, P.O. Box 7062, Kampala, Uganda 3 Nelson Mandela African Institution of Science and Technology (NM-AIST), Arusha-Tanzania. (Received November 20 2019, Accepted January 10 2020) Abstract. A deterministic mathematical model to study the dynamics of pneumonic plague with causing pathogens Yersinia pestis in the environment is developed and analyzed. We compute the basic reproduc- tion number using the next generation matrix method and use it to derive and establish the condition for local and global asymptotic stability of equilibrium points. Sensitivity and elasticity analysis is used to determine the effect (positive or negative) of parameters on the basic reproduction number. The results show that R 0 is most sensitive to expected number of new cases of pathogens in the environment caused by one rodent infected with pneumonic plague and it is least sensitive to expected number of new cases of human beings infected with pneumonic plague caused by pathogens in the environment. We then use numerical simulations to show the dynamical behavior of pneumonic plague disease in the compartments. The results show clearly the vital role played by fleas, human beings agand rodents with bubonic plague in the increase of the number of individuals with pneumonic plague. The result also show that the increase of the number of individuals with pneumonic plague is greatly influenced by the pair k ij (expected number of new cases of i caused by one infected individu- al of j ) for Human being, Rodent, Flea and pathogens in the environment that constitute the basic reproduction number. These should also be considered when planning for any control strategies against the disease. Keywords: Pneumonic plague, Pathogens in the environment, Airborne transmission, Yersinia pestis. 1 Introduction Pneumonic plague arises when Yersinia Pestis infects the lungs. It is an extreme type of lung infection, exceedingly contagious and incurable unless identified within the first twenty-four hours [13]. Among all three main types of plague namely Bubonic plague, septicemic plague and pneumonic plague, it is the most serious and deadliest form of a plague epidemic. The symptoms normally come abruptly and are very severe, character- ized by rapid prostration; shallow, distressed and very rapid breathing; individual coughing watery and bloody sputum which contain bacteria (yersinia pestis); high body temperature and bleeding. Individuals with Bubonic or Septicemic plague get the disease from the bite of an infected flea that is primarily infected by the wild rats which are the primary reservoir of Yersinia pestis. If not treated, Y. pestis reaches the lungs and thus develops pneumonic plague [40]. [1] examined the progression of infection in rats that were exposed to aerosolized Yersinia Pestis in a whole body. The study was able to demonstrate direct transmission of Yersinia pestis from infected to a susceptible animals (rats) held in the same cage. The infection transpired via aerosol droplets produced when pneumonic plague infected rat coughs. These findings extend to all animals including human beings, rodents and other domestic animals with lungs. When an individual (human beings or rodents and other domestic animals) with pneumonic plague coughs, the bacteria are released into the air. Then when an individual with lungs breathes * Corresponding author. E-mail address: [email protected]. Published by World Academic Press, World Academic Union
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ISSN 1 746-7233, England, UKWorld Journal of Modelling and Simulation
Vol. 16 (2020) No. 1, pp. 53-80
Modeling Pneumonic Plague Caused by Yersinia pestis in the Environment
Rigobert C. Ngeleja1 *, Livingstone S. Luboobi2, Verdiana Grace Masanja3
1 General Studies Department, Dar Es Salaam Institute of Technology (DIT), P.O. Box 2958, Dar Es Salaam -Tanzania.2 Department of Mathematics, Makerere University, P.O. Box 7062, Kampala, Uganda
3 Nelson Mandela African Institution of Science and Technology (NM-AIST), Arusha-Tanzania.
(Received November 20 2019, Accepted January 10 2020)
Abstract. A deterministic mathematical model to study the dynamics of pneumonic plague with causingpathogens Yersinia pestis in the environment is developed and analyzed. We compute the basic reproduc-tion number using the next generation matrix method and use it to derive and establish the condition for localand global asymptotic stability of equilibrium points. Sensitivity and elasticity analysis is used to determinethe effect (positive or negative) of parameters on the basic reproduction number. The results show that R0 ismost sensitive to expected number of new cases of pathogens in the environment caused by one rodent infectedwith pneumonic plague and it is least sensitive to expected number of new cases of human beings infected withpneumonic plague caused by pathogens in the environment. We then use numerical simulations to show thedynamical behavior of pneumonic plague disease in the compartments. The results show clearly the vital roleplayed by fleas, human beings agand rodents with bubonic plague in the increase of the number of individualswith pneumonic plague. The result also show that the increase of the number of individuals with pneumonicplague is greatly influenced by the pair kij (expected number of new cases of i caused by one infected individu-al of j) for Human being, Rodent, Flea and pathogens in the environment that constitute the basic reproductionnumber. These should also be considered when planning for any control strategies against the disease.
Keywords: Pneumonic plague, Pathogens in the environment, Airborne transmission, Yersinia pestis.
1 Introduction
Pneumonic plague arises when Yersinia Pestis infects the lungs. It is an extreme type of lung infection,exceedingly contagious and incurable unless identified within the first twenty-four hours [13]. Among all threemain types of plague namely Bubonic plague, septicemic plague and pneumonic plague, it is the most seriousand deadliest form of a plague epidemic. The symptoms normally come abruptly and are very severe, character-ized by rapid prostration; shallow, distressed and very rapid breathing; individual coughing watery and bloodysputum which contain bacteria (yersinia pestis); high body temperature and bleeding. Individuals with Bubonicor Septicemic plague get the disease from the bite of an infected flea that is primarily infected by the wild ratswhich are the primary reservoir of Yersinia pestis. If not treated, Y. pestis reaches the lungs and thus developspneumonic plague [40].
[1] examined the progression of infection in rats that were exposed to aerosolized Yersinia Pestis in a wholebody. The study was able to demonstrate direct transmission of Yersinia pestis from infected to a susceptibleanimals (rats) held in the same cage. The infection transpired via aerosol droplets produced when pneumonicplague infected rat coughs. These findings extend to all animals including human beings, rodents and otherdomestic animals with lungs. When an individual (human beings or rodents and other domestic animals) withpneumonic plague coughs, the bacteria are released into the air. Then when an individual with lungs breathes
Published by World Academic Press, World Academic Union
54 R. C. Ngeleja, L. S. Luboobi, V. G. Masanja: Modeling Pneumonic Plague
in the aerosolized bacteria, there is a high possibility that this particular individual may get pneumonic plague[38].
[14] investigated the case where the patient died after contact with a dog that had captured pneumonicplague infected marmot. In this case, the infection was a result of exposure to Yersinia pestis aerosols fromsputum and throat samples. In their study, they found that all of the dogs that ate the marmot were infectedwith Y. pestis without symptoms. The disease was also detected in the serum of the doctors and in people whohad been in contact with the patient for a long period of time in which the transmission may be associatedto wearing of masks. The study justify the possibility of rodent to rodent, rodent to human and human tohuman transmission of pneumonic plague as a result of physical contact or through eating or biting the infectedindividual.
Pneumonic plague has a tremendous transmission capability from one individual to another. It is a highlycontagious disease which is the feature that makes it very dangerous when it enters in a community. It is also thereason why pneumonic plague disease appears on the top list of the diseases that can be used as a bioweapon.[3] conducted a study to investigate the communicability in a natural occurring pneumonic plague cluster.The cluster comprised of two simultaneous index patient’s caregiver pairs. The result showed that both indexpatients transmitted pneumonic plague to only one caregiver each. It justifies the possibility of the person toperson transmission of pneumonic plague through respiratory droplets in which all individuals within dropletrange became ill.
The rate of death due the disease for untreated individual with pneumonic plague increases to 100% within2 to 7 days after infection. [39] studied an outbreak of pneumonic plague occurred in Madagascar in the year2011, the disease remained in the community for the period of over twenty seven (27) days. In those twentyseven days, there were seventeen (17) human beings suspected to have gotten the disease, two (2) cases werepresumptive, and three were confirmed to have gotten the disease. The study postulates that there were fifteen(15) untreated patients and they all died due to the disease. The result shows that in this outbreak fatality ratewas 100% for all non treated patients. This is to say that the rate of fatality in Pneumonic plague is extreme,and it should therefore be given a special attention when it occurs.
Pneumonic plague is considered perilous mostly due to the fact that at present there is no effective vaccine.More over the instrument for diagnostic test especially the rapid diagnostic tests are scarce and the situationis even worse in African countries. Also it is the form of plague that its transmission capability is high as itcan be transmitted directly between individuals (human and rodent or other domestic animal) and through theinteraction with the infected environment [29, 37, 42]. It is certain highly contagious infectious disease whichis also listed as a leading critical biological agents with the high potential of being used as a bio weapon [29].
The transmission and spreading capacity that characterize pneumonic plague signify that, extreme publichealth measures should be considered if this kind of disease occur in the community. There should be sus-tainable planning to do a very rapid evaluation of the outbreak to determine the extent of exposure and helpdevelop the most effective disease containment strategies [8]. To do all of these, there is a need for public healthauthorities and all health stakeholders to conduct a thorough research on the subject.
In this study we formulate a mathematical model to enable us understand the dynamics of pneumon-ic plague. We compute the basic reproduction number, analyze the stability of equilibrium points, study thebehaviour of the model through numerical simulation, discuss the results and then make conclusions and rec-ommendations.
2 Material and methods
2.1 Model formulation
We formulate a mathematical model to study the dynamics of pneumonic plague. The developed modelrely on the following assumptions:
• Bubonic plague is the primary plague infection of pneumonic plague disease;• The primary infection of the disease is ignored when one gets the secondary infection of the same;
World Journal of Modelling and Simulation, Vol. 16 (2020) No. 1, pp. 53-80 55
• All individuals are born susceptible;• Members of the population mix homogeneously;• Age, sex, social status, do not affect the probability of being infected;• On recovery human attain a temporary immunity;• There is no recovery in non-human host populations (they remain infected until they die);• The disease transmission from the soil/environment to either susceptible human being or rodent popu-
lation at their respective adequate contact rates has a negligible effect to the dynamics of the pathogenspopulation;
• There is no vertical transmission, but only horizontal transmission is possible in all populations.
2.1.1 Description of pneumonic plague in various population groups
In the model, we have four populations namely the human population, fleas, rodents and the pathogensin the environment. The human population is divided into five sub-groups: the group of people who have notcontracted the disease but may get it if they get in contact with infectious agent to be referred to as susceptiblesand denoted as SH ; People who have the disease but have not shown any symptoms and are incapable oftransmitting the disease i.e the Exposed denoted by EH ; those who are infected and are capable of transmittingthe disease are divided into two sub-groups: there are those who have bubonic plague which, in this model, weregard as a primary stage of pneumonic plague denoted by IHA and the others who have pneumonic plaguedisease denoted by IHB . The fraction of the population in IHA if treated may recover and move to sub-groupRH and if not, they either die or progress and become pneumonic plague disease infectives IHB . The populationin the sub-group IHB then they recover and progress to the sub-group RH if treated and otherwise they die.
The flea population is divided into two sub-groups: those who have not contracted the disease but may getit if they get in contact with infectious agent i.e. susceptible flea denoted by SF and those who are infected andare capable of transmitting the disease i.e. infectious flea denoted by IF .
The rodent population is divided into four sub-groups: those who have not contracted the disease but mayget it if they get in contact with infectious agent i.e. susceptible rodent SR, those who have the disease buthave not shown any symptom and are incapable of transmitting the disease referred to as the exposed rodentdenoted by ER; those who are infected and are capable of transmitting the disease and these are divided intotwo subgroups: those who have bubonic plague denoted by IRA and others who have pneumonic plague diseasedenoted by IRB . To develop the model equations, we use the variables and parameters as described in Table 1and Table 2.
Table 1: Variables and their description for pneumonic plague.Variable DescriptionSH Susceptible human beingsEH Exposed human beingsIHA Infectious human beings with bubonic plagueIHB Infectious human beings with pneumonic plagueRH Recovered human beingsSR Number of Susceptible rodentsER Number of Exposed rodentsIRA Number of Infectious rodents with bubonic plagueIRB Number of Infectious rodents with pneumonic plagueSF Number of susceptible fleasIF Number of infected fleasA Number of pathogens in the environment
56 R. C. Ngeleja, L. S. Luboobi, V. G. Masanja: Modeling Pneumonic Plague
2.1.2 Description of interactions
Fleas in sub-group SF get Yersinia pestis bacteria through biting infected rodents who are the primaryreservoir for the bacteria and/or infected human being at the rates Γrf and Γhf respectively, and become theinfected flea IF .
The groups SH and SR may get the disease in various ways; one is through the bites by the infected flea(IF ) at the rates Γfh and Γfr respectively and then become latently infected and thus progress to be exposedhuman population EH and exposed rodent population ER at the rates α1 and γ1 respectively. They may aswell get the disease when they adequately contact the subgroups IHB and IRB who are infected by pneumonicplague. The transmission may be through airborne and/or physical contact (bloody sputum). The interactionmay be in such a way that IHB may come into contact and infect the subgroups SH and SR at the rate Γhh andΓhr respectively. Similarly, IRB may come into contact and infect the sub-groups SH and SR at the rates Γrhand Γrr respectively.
After 2 to 7 days the sub-groups EH and ER become infectious and capable of transmitting the disease.The proportion τ1 of exposed human beings (EH ) progress to subgroup IHA and the other proportion (1− τ1)to sub-group IHB at the rate α2, a proportion τ2 of the exposed rodents (ER) progress to the sub-group IRAand other proportion (1− τ1) to sub-group IRB at the rate γ2.
If treated the fraction of compartment IHA recover and attain temporary immunity at a rate α3 and thusprogress to a subgroup RH which then return to a sub-group SH at a rate $. Other human beings with bubonicplague (IHA) progress to sub-group IHB at a rate α3 and the rest die either naturally or due to the disease atrates µ1 and κ1, respectively.
Individuals in compartment IHB if treated they recover and progress to RH at the rate α4 which thenreturn to a sub-group SH at a rate $. Otherwise they die either from the disease at a rate δ1 or naturally at arate of µ1. After 2 to 7 days of infection the compartment IRA may progress to subgroup IRB at a rate γ3 andthe rest die either naturally or due to a disease at a rates µ3 and κ2 respectively. Subgroup IRB die either fromthe disease at a rate δ3 or naturally at a rate of µ3.
The pathogen may survive in the environment if the conditions are favorable for their survival. Throughairborne transmission or touching the contaminated soil/environment may cause infections to SH and SR at therates of ω1 and ω2 respectively. Pathogens are constantly recruited into the environment at a rate λ4. Howeverthe human beings and rodents infected with pneumonic plague (IHB and IRB) also shad yersinia pestis bacteriain the environment A at rates η1 and η2 respectively.
Pathogens in the environment suffer natural mortality at a rate µ4. The human population in subgroupsSH and EH , flea population in sub-group SF and rodent population in sub-groups SR and ER suffer naturalmortality at rates µ1, µ2 and µ3 respectively. The compartments IHA, IHB , IF , IRA and IRB suffer both naturaldeath at rates µ1, µ2 and µ3 and disease induced mortality at rates κ1, δ1, δ2, κ2 and δ3, respectively. Humanbeings, fleas and rodents are recruited through immigration at rates ψ1, ψ2 and ψ3 respectively.
Basing on the assumptions and the description of interactions stated above, the dynamics of pneumonicplague is as in Fig. 1.
2.1.3 Model equations for pneumonic plague
Using the assumptions stated above, variables and parameters and their description in Tables 1 and 2,description of the dynamics and compartmental diagram in Fig. 1, the SEIR model for pneumonic plague isthe following set of ordinary differential equations:
58 R. C. Ngeleja, L. S. Luboobi, V. G. Masanja: Modeling Pneumonic Plague
Theorem 1. Let the initial values of the system (1), (2), (3) and (4) be: (SH(0)SR(0), SF (0), A0) >0 and (EH(0), IHA(0), IHB(0), RH(0), ER(0), IRA(0), IRB(0), IF (0)) ≥ 0. Then the solution setSH(t), SR(t), SF (t), A(t), EH(t), IHA(t), IHB(t), RH(t), ER(t), IRA(t), IRB(t) and IF (t) are positive ∀t ≥0.
Proof. Using the first equation in the system we have,
dSHdt
= ψ1 +$RH − α1(ΓhhIHBN1
+ ΓfhIFN2
+ ΓrhIRBN3
+ ω1A)SH − µ1SH
≥ −α1(ΓhhIHBN1
+ ΓfhIFN2
+ ΓrhIRBN3
+ ω1A)SH − µ1SH .
dSHdt≥ −(α1(Γhh
IHBN1
+ ΓfhIFN2
+ ΓrhIRBN3
+ ω1A) + µ1)SH .
Integration yields
SH ≥ SH0e−∫ t0 (α1(Γhh
IHBN1
+ΓfhIFN2
+ΓrhIRBN3
+ω1A)+µ1)dτ > 0
sincee−∫ t0 (α1(Γhh
IHBN1
+ΓfhIFN2
+ΓrhIRBN3
+ω1A)+µ1)dτ > 0.
From the second equation we have
dEHdt
= α1(ΓhhIHBN1
+ ΓfhIFN2
+ ΓrhIRBN3
+ ω1A)SH − τ1α2EH − (1− τ1)α2EH − µ1EH .
ThusdEHdt≥ −(α2 + µ1)EH .
Integration yieldsEH ≥ EH0e
−(α2+µ1)t > 0
sincee−(α2+µ1) > 0.
Following the same procedure we can show that all the variables in the pneumonic plague disease model arepositive.
3.2 Invariant region
Since pneumonic plague involves human being, rodent, vector and pathogens populations, then, in themodeling process, we assume that all state variables and parameters of the model are non-negative for ∀t ≥ 0.The model system has four subgroups which are analyzed separately. The model system is analyzed in a suitablefeasible region where all state variables are positive. This region will be obtained under the following theorem;
Theorem 2. All forward solutions in R12+ of the system are feasible ∀t ≥ 0 if they enter the invariant region Φ
for Φ = ΩH ×ΩR ×ΩF ×ΩA
whereΩH = (SH , EH , IHA, IHB, RH) ∈ R5
+ : SH + EH + IHA + IHB +RH ≤ N1
ΩR = (SR, ER, IRA, IRB) ∈ R4+ : SR + ER + IRA + IRB ≤ N3
ΩF = (SF , IF ) ∈ R2+ : SF + IF ≤ N2
ΩA = A ∈ R1+
and Φ is the positive invariant region of the pneumonic plague system.
World Journal of Modelling and Simulation, Vol. 16 (2020) No. 1, pp. 53-80 59
Proof. For human population:We need to prove that the solution of the system (1) are feasible ∀t > 0 as they enter invariant region ΩH .We now letΩH = (SH , EH , IHA, IHB, RH) ∈ R5 be solution space of the system (1) with non-negative initialconditions.The total human population is
N1 = SH + EH + IHA + IHB +RH .
Then,dN1
dt=dSHdt
+dEHdt
+dIHAdt
+dIHBdt
+dRHdt
(5)
Adding up the system (1) we get,
dN1
dt= ψ1 − µ1N1 − δ1IHB − κ1IHA
We will then havedN1
dt≤ ψ1 − µ1N1
We then getdN1
dt+ µ1N1 ≤ ψ1
Finding the integrating factor IF = eµ1t and multiplying it through out we get
eµ1tdN1
dt+ eµ1tN1µ1 ≤ ψ1e
µ1t
which givesd(N1e
µ1t)
dt≤ ψ1e
µ1t
Integrating on both sides yields
N1eµ1t ≤ ψ1
µ1eµ1t + C
Multiplying the equation by e−µ1t we get
N1 ≤ψ1
µ1+ Ce−µ1t
Using the initial condition t = 0, N1(t = 0) = N10
then we will get
N10 −ψ1
µ1≤ C
Substituting for the constant C we get
N1 ≤ψ1
µ1+ (N10 −
ψ1
µ1)e−µ1t
When N10 >ψ1
µ1, the population decreases asymptotically to ψ1
µ1and when N10 <
ψ1
µ1the human population
increases asymptotically to ψ1
µ1as in Fig. 2. Hence all the feasible solutions of the system enter the region
ΩH =
(SH , EH , IHA, IHB, RH) : N1 ≤Max
N10,
ψ1
µ1
Following the same procedure we can also prove that the feasible solutions of the system for roden-
t, flea and pathogens in the environment given in (2), (2) and (2) respectively enter the regions ΩR =(SR, ER, IRA, IRB) : N3 ≤Max
60 R. C. Ngeleja, L. S. Luboobi, V. G. Masanja: Modeling Pneumonic Plague
Fig. 2: Feasible region for human system
4 Model analysis
In this section, we examine the existence of equilibrium states, reproduction number and stability of theequilibrium states.
4.1 Disease free equilibrium
The model has a disease free equilibrium which is obtained by setting IHA = IHB = EH = RH = 0,IRA = IRB = ER = 0, IF = 0 and A = 0 for human beings, rodents, fleas and pathogens systemsrespectively. We substitute the above into the system (1) - (4) which are the systems for human beings, rodents,fleas and pathogens in the environment respectively. Then we have the disease free-equilibrium point given asE0H =
(ψ1
µ1, 0, 0, 0, 0
), E0
R =(ψ3
µ3, 0, 0, 0
), E0
F =(ψ2s
µ2, 0)
and E0A = 0 for human being, rodent, flea and
pathogen, respectively.
Then the disease free equilibrium of the entire system
E0(S0H , E
0H , I
0HA, I
0HB , R
0H , S
0R, E
0R, I
0RA, I
0RB , S
0F , I
0F , A
0) =
(ψ1
µ1, 0, 0, 0, 0,
ψ3
µ3, 0, 0, 0,
ψ2s
µ2, 0, 0
).
4.2 The next-generation matrix
We define the basic reproduction number as the expected number of secondary cases produced by a s-ingle infectious individual during the entire infectious period of that particular individual into a completelysusceptible population. The value of this dimensionless quantity (R0 ) dictate different epidemiological criteriasuch that: If R0 < 1 then an infected individual in entirely susceptible population can produce less than onesecondary cases of infection. This indicates that the disease cannot develop and may be eradicated from thepopulation, which means that the disease-free equilibrium point is asymptotically stable. On the other hand, IfR0 > 1 it means that an infected individual in entirely susceptible population produce more than one secondarycases of infection. This indicates the persistence of the disease in the population for a long time and that thedisease free equilibrium point is unstable [2].
We compute the basic reproduction number R0 using the next generation matrix as outlined by [17] and[33]. We first categorize individuals by their state at the moment they become infected (type at infection). Thesetypes-at-infection refers specifically to the birth of the infection in the individual. These categories (types atinfection) differ in the way they transmit disease and their ability to produce secondary cases.
World Journal of Modelling and Simulation, Vol. 16 (2020) No. 1, pp. 53-80 61
For our case, we have six categories and we label them as follows: Human infected with bubonic plague(type 1), human infected with pneumonic plague (type 2), rodent infected with bubonic plague (type 3), rodentinfected with pneumonic plague (type 4), flea infested with pathogens (type 5) and the pathogens in the envi-ronment (type 6). Since the system has six types-at-infection, the next-generation matrix, K, will be a 6 × 6matrix with elements kij ,s. Each of the elements kij stands for expected number of new cases of i caused byone infected individual of j. We now define the next-generation matrix K whose entries are kij . This matrix isgiven as;
Then, R0 = ρ(K) where ρ(K) is spectral radius of K.The element k11 of the matrix 6 is the expected number of new cases of human beings infected with
bubonic plague caused by one infected human beings with bubonic plague, k12 is the expected number of newcases of human beings infected with bubonic plague caused by one infected human beings with pneumonicplague, k13 is the expected number of new cases of human infected with bubonic plague caused by one infectedrodent with bubonic plague, k14 is the expected number of new cases of human beings infected with bubonicplague caused by one infected rodent with pneumonic plague, k15 is the expected number of new cases ofhuman beings infected with bubonic plague caused by one infected flea, k16 is the expected number of newcases of human beings infected with bubonic plague caused by infected environment.
k21 is the expected number of new cases of human beings infected with pneumonic plague caused by oneinfected human beings with bubonic plague, k22 is the expected number of new cases of human beings infectedwith pneumonic plague caused by one infected human beings with pneumonic plague, k23 is the expectednumber of new cases of human beings infected with pneumonic plague caused by one infected rodent withbubonic plague, k24 is the expected number of new cases of human beings infected with pneumonic plaguecaused by one infected rodent with pneumonic plague, k25 is the expected number of new cases of humanbeings infected with pneumonic plague caused by one infected flea, k26 is the expected number of new casesof human beings infected with pneumonic plague caused by infected environment.
k31 is the expected number of new cases of rodent infected with bubonic plague caused by one infectedhuman beings with bubonic plague, k32 is the expected number of new cases of rodent infected with bubonicplague caused by one infected human beings with pneumonic plague, k33 is the expected number of newcases of rodent infected with bubonic plague caused by one infected rodent with bubonic plague, k34 is theexpected number of new cases of rodent infected with bubonic plague caused by one infected rodent withpneumonic plague, k35 is the expected number of new cases of rodent infected with bubonic plague caused byone infected flea, k36 is the expected number of new cases of rodent infected with bubonic plague caused byinfected environment.
k41 is the expected number of new cases of rodent infected with pneumonic plague caused by one infectedhuman beings with bubonic plague, k42 is the expected number of new cases of rodent infected with pneumonicplague caused by one infected human beings with pneumonic plague, k43 is the expected number of newcases of rodent infected with pneumonic plague caused by one infected rodent with bubonic plague, k44 is theexpected number of new cases of rodent infected with pneumonic plague caused by one infected rodent withpneumonic plague, k45 is the expected number of new cases of rodent infected with pneumonic plague causedby one infected flea, k46 is the expected number of new cases of rodent infected with pneumonic plague causedby infected environment.
k51 is the expected number of new cases of infected flea caused by one infected human beings with bubonicplague, k52 is the expected number of new cases of infected flea caused by one infected human beings withpneumonic plague, k53 is the expected number of new cases of infected flea caused by one infected rodent withbubonic plague, k54 is the expected number of new cases of infected flea caused by one infected rodent with
62 R. C. Ngeleja, L. S. Luboobi, V. G. Masanja: Modeling Pneumonic Plague
pneumonic plague, k55 is the expected number of new cases of infected flea caused by one infected flea, k56 isthe expected number of new cases of infected flea caused by infected environment.
k61 is the expected number of new cases of infected environment caused by one infected human beingswith bubonic plague, k62 is the expected number of new cases of infected environment caused by one infectedhuman beings with pneumonic plague, k63 is the expected number of new cases of infected environment causedby one infected rodent with bubonic plague, k64 is the expected number of new cases of infected environmentcaused by one infected rodent with pneumonic plague, k65 is the expected number of new cases of infectedenvironment caused by one infected flea and k66 is the expected number of new cases of infected environmentcaused by infected environment.
Some elements are equal to zero since not all type at infection individuals infect others. For example,human beings and rodent infected with bubonic and pneumonic do not cause new cases of infected humanbeings and rodent with bubonic plague, this means that k11, k12, k13 and k14 are equal to zero. There are nonew cases of human beings infected with pneumonic plague caused by rodent infected with pneumonic plagueand from the infected fleas, thus k23 and k25 are equal to zero. Human and rodent infected with bubonic andpneumonic do not cause new cases of infected rodent with bubonic plague, this means that k31, k32, k33 andk34 are equal to zero.
Also no single case of rodent with pnemonic plague is caused by a human beings or rodent infected withbubonic plague and from the infected fleas which again means k41, k43 and k45 are equal to zero. A flea canneither infect itself nor by the environment and no new cases of the infected environment (pathogens in theenvironment) is caused by human beings and rodent infected with bubonic plague, the infected flea or by itselfwhich means k55, k56, k61, k63, k65 and k66 are equal to zero. There are no new cases of human beings androdent infected with bubonic plague caused by infected environment (pathogens in the environment), and alsono disease transmission from human beings and rodent with pneumonic plague to flea. This means k16, k36,k52 and k54 are equal to zero [18, 31].
Now replacing in matrix K the kij elements with value zero, the matrix K becomes
The expected number of new cases of i caused by one infectious individual of j generally depends on theinfectious period of individual of type j, the progression rate from one infective class to another within theindividual type j, the probability that the individual of type j survives the incubation and the adequate contactrate: individual type j to individual type i depending on the particular type of infected individual j underconsideration. For example, k15 depends on the infectious period of flea, probability that fleas survives theincubation period and the adequate contact rate: infected flea to human being. Using the method outlined by[11], we now derive the expressions for kij basing on the adequate contact rate between the infected individualtype j and the susceptible individual type i, the expected duration of infection of individual type j and theprobability that the individual type j survive the duration between the latent stage to the time an individualexperience the onset clinical disease as in (8)
World Journal of Modelling and Simulation, Vol. 16 (2020) No. 1, pp. 53-80 63
k15 =
(β
β + µ2
)Γfh
µ2 + δ2
k26 =ω1λ4
µ4(µ4 + λ4)
k22 =(1− τ1)α2Γhh
((1− τ1)α2 + µ1)(µ1 + δ1 + α4)
k24 =(1− τ2)γ2Γrh
((1− τ2)γ2 + µ3)(µ3 + δ3)
k21 =
(τ1α2
τ1α2 + µ1
)ρα3
µ1 + κ1 + α3
k46 =ω2λ4
µ4(µ4 + λ4)
k35 =
(β
β + µ2
)Γfr
µ2 + δ2
k44 =(1− τ2)γ2Γrr
((1− τ2)γ2 + µ3)(µ3 + δ3)
k43 =
(τ2γ2
τ2γ2 + µ3
)γ3
γ3 + µ3 + κ2
k42 =(1− τ1)α2Γhr
((1− τ1)α2 + µ1)(µ1 + δ1 + α4)
k53 =
(τ2γ2
τ2γ2 + µ3
)ρ3Γrf
γ3 + µ3 + κ2
k51 =
(τ1α2
τ1α2 + µ1
)ρ1Γhf
µ1 + κ1 + α3
k62 =
((1− τ1)α2
(1− τ1)α2 + µ1+
ρα3
ρα3 + µ1 + κ1
)η1
µ1 + δ1 + α4
k64 =
((1− τ2)γ2
(1− τ2)γ2 + µ3+
γ3γ3 + µ3 + κ2
)η2
µ3 + δ3
Each element of the matrix K is the reproduction number for pairs of considered types [15]. The generalinterpretation of the matrix elements kij is that; the elements k11,k12, k21, k22 and k33, k34, k43, k44 arisewithin human beings and rodents respectively as there are two groups of infectious classes which are those withbubonic plague IHA and IRA and those with pneumonic plague IHB and IRB . These two groups differ in theway they transmit Yersinia pestis. The bubonic plague infectious cases occur when bacteria infect the lymphaticsystem and it is mainly transmitted through flea bite. In very rare cases the disease may be transmitted throughthe interaction with the environment, and in almost negligible cases the disease can be transmitted betweenhuman - human, human - rodent and rodent - rodent. These are the reasons why the value of k11, k12, k13, k14and k16 are zero. While the group of human beings and rodents infected with pneumonic plague occur whenthe bacteria infect the lungs, it is transmitted through airborne transmission.
4.2.1 Basic reproduction number r0
[9] and [17] postulates that we obtain the basic reproduction number R0 by computing the maximummodulus of the eigenvalues of the next-generation matrix. Using mapple computing software package, thebasic reproduction number is
Since pneumonic plague has multiple transmission cycles, the next-generation matrix method gives the geo-metric mean of the number of infections per generation [27]. It depends on the expected number of new casesof human beings infected with pneumonic plague caused by one infected human beings with pneumonic plague(k22), the expected number of new cases of human beings infected with pneumonic plague caused by one in-fected rodent with pneumonic plague (k24), the expected number of new cases of human beings infected withpneumonic plague caused by infected environment (k26), the expected number of new cases of rodent infect-ed with pneumonic plague caused by one infected human beings with pneumonic plague (k42), the expectednumber of new cases of rodent infected with pneumonic plague caused by one infected rodent with pneumonicplague (k44), the expected number of new cases of rodent infected with pneumonic plague caused by infectedenvironment (k46), the expected number of new cases of infected environment caused by one infected humanbeings with pneumonic plague (k62) and the expected number of new cases of infected environment caused byone infected rodent with pneumonic plague (k64).
4.3 Local stability of the disease free equilibrium point
In this section, we assess the local stability of the Disease Free Equilibrium (DFE) point of the pneumonicplague disease system, in which we prove that the trajectories start arbitrary close to the equilibrium point butdo not precisely reach it. We do this by evaluating the Jacobian matrix of system (1) - (4) at DFE point:Then we have
J(E0) =
(J11 J12J21 J22
)(9)
where J11, J12, J21 and J22 are (6× 6) matrices given by;
From the combined matrix J(E0), the diagonal entries from the first, fifth, sixth and tenth column makes thefour eigenvalues of the matrix (9). These are−µ1,−($+µ1),−µ3 and−µ2, now canceling their correspondingrows and columns we modify (9) and remain with an (8× 8) matrix with the modified J11, J12, J21 and J22 asgiven in (14), (15), (16) and (17) respectively;
Making further computation we find the other negative eigenvalues of the matrix 9 as−µ4,−(µ2+δ2),−(µ3+δ3), −(γ3 + µ3 + κ2) and −(γ2 + µ2). Also there are complex eigenvalues with very long expressions andnegative real part, we name them as −p1 + q1i and −p2 + q2i where p1, p2 and q1, q2 are real and imaginarypart respectively. The computation show that the last eigenvalue is negative if and only ifR0 < 1. By [32] theseresults prove that the equilibrium point E0 is locally asymptotically stable. It then leads to Theorem 3.
Theorem 3. The Disease Free Equilibrium E0 of pneumonic plague is locally asymptotically stable if R0 <1and unstable if R0 > 1.
4.4 Global stability of the disease-free equilibrium point
We employ the Metzler matrix method as described by [5]. We divide the general pneumonic plaguesystem (1) - (4) into transmitting and non-transmitting components as stated below.Let Yn be the vector for non-transmitting compartments, Yi be the vector for transmitting compartments andYE0,n be the vector of disease free point.
dYndt
= A1(Yn − YE0,n) +A2Yi
dYidt
= A3Yi
(18)
We will then have
Yn = (SH , RH , SR, SF )T Yi = (EH , IHA, IHB, ER, IRA, IRB, IF , A)
YE0,n = (ψ1
µ1, 0,
ψ3
µ3,ψ2s
µ2)
Yn −YE0,n =
SH − ψ1
µ1RH
SR − ψ3
µ3
SF − ψ2s
µ2
In order to prove that the DFE point is globally and asymptotically stable, we are required to show that
Matrix A1 has real negative eigenvalues and A3 is a Metzler matrix in which all off diagonal element must benon-negative. Referring to (26) we write the general model as given below;
Computing the eigenvalues of matrixA1, we find that the eigenvalues are−µ1,−µ2,−µ3 and−($+µ1).The result now confirms that the system
dYndt
= A1(Yn − YE0,n) +A2Yi
is globally and asymptotically stable at YE0 . Also we find that all its off-diagonal elements of the matrix A3 arenon-negative and thus A3 is a Metzler stable matrix.Therefore Disease Free Equilibrium point for pneumonicplague system is globally asymptotically stable and as a result we have the following theorem:
68 R. C. Ngeleja, L. S. Luboobi, V. G. Masanja: Modeling Pneumonic Plague
Theorem 4. The disease-free equilibrium point is globally asymptotically stable in E0 if R0 < 1and unstableif R0 > 1.
4.5 Existence of endemic equilibrium
Now, we investigate conditions for existence of the endemic equilibrium points of pneumonic plaguedisease. The equilibrium point E∗(S∗H , E
∗H , I
∗HA, I
∗HB, R
∗H , S
∗R, E
∗R, I
∗RA, I∗RB , S∗F , I∗F , A∗) is obtained by
solving the equations obtained by setting the derivatives of (1)-(4) equal to zero.If we let λ∗H , λ∗R, λ∗F and λ∗A be the force of infection for human beings, rodents, fleas and the environment
respectively as given in (22) - (25).
λ∗H = ΓhhI∗HBN∗1
+ ΓfhI∗FN∗2
+ ΓrhI∗RBN∗3
+ ω1A∗ (22)
λ∗R = ΓrrI∗RBN∗3
+ ΓfrI∗FN∗2
+ ΓhrI∗HBN∗1
+ ω2A∗ (23)
λ∗R = ΓhfρI∗HAN∗1
+ Γrf (1− ρ)I∗RAN∗3
(24)
λ∗A = λ4 + η1I∗HBN∗1
+ η2I∗RBN∗3
(25)
It is clear that λ∗H is an increasing function of IHB , IRB , IF and A. When the the force of infection ishigh, the rate at which human beings progress from susceptible to exposed will increase, thus the number ofhuman beings becoming exposed to the disease will as well increase. The increase of number of exposed humanbeings will lead to the increases of the number of human beings progressing and become bubonic or pneumonicplague infectives. However when the force of infection is low, the rate at which human beings progress fromsusceptible to exposed decreases and consequently it decreases the infection rate. That is to say if we assumethat when the force of infection is high then IHB = IHB1, IRB = IRB1, IF = IF1 and A = A1 and when theforce of infection is low then IHB = IHB2, IRB = IRB2, IF = IF2 and A = A1, since λ∗H is an increasingfunction then λ∗H(IHB1, IRB1, IF1, A1) > λ∗H(IHB2, IRB2, IF2, A2).
If the force of infection for human being is assumed to be very high, that is λ∗H → ∞, gradually thesusceptible human will approach zero SH ≈ 0 and recovered human beings will approach a non-zero endemicpoint. Moreover the number of exposed human beings (EH ), human beings infected with bubonic plague (IHA)and human beings infected with pneumonic plague (IHB) will rise approaching a non-zero endemic point E∗H ,I∗HA and I∗HB .
The force of infection in rodent λ∗R is an increasing function of IRB , IHB , IF and A. As we increasethe force of infection for rodent, that is λ∗R → ∞, gradually the number of susceptible rodents will approachzero SR ≈ 0. As the progression rate of the susceptible rodent to infected increases, the rodent exposed tothe disease ER, the number rodent infected with bubonic plague IRA and the number of rodent infected withpneumonic plague IRB will rise and approach a non-negative endemic point, E∗R, I∗RA and I∗RB .
The force of infection in flea (λ∗F ) is an increasing function of IRA and IHA. Now assuming the enormousincreasing force of infection for flea, that is as λ∗F → ∞, the number of susceptible fleas, SF , will graduallyapproach zero. As the rate at which a flea gets infection increases, IF will approach non-zero endemic level I∗F .
Force of infection in the environment in our case is an overall rate at which pathogens are populated in theenvironment. λ∗A is the increasing function of IHB and IRB . Now if we assume the mammoth increase of theforce of infection to the environment, that is as λ∗A →∞ will lead to the proportional increase the the numberof pathogens shad in the environment.
Using the study by [7], we study the existence of endemic equilibrium through numerical simulation. Weuse parameter values in which some are obtained from the literatures that relate to this study, and others areestimated using sensitivity analysis and simulation refer to Table 2.
In order to examine the existence of endemic equilibrium point we show that the exposed, infected andrecovered classes in human beings, rodents and fleas and the number of pathogens in the environment are
World Journal of Modelling and Simulation, Vol. 16 (2020) No. 1, pp. 53-80 69
Fig. 3: The solution trajectories showing the endemic equilibrium point.
different from zero. Fig. 3b is the zoomed view of Fig. 3a which shows that; the susceptible population inhuman being, rodent and flea approaches zero while on the other hand the exposed, infected and the recoveryclasses in human being, rodent flea and the pathogens in environment reaches maximum and then convergingto non-zero endemic equilibrium point.
We then derive the conditions under which the endemic equilibrium points are stable or unstable. That is,we show whether the solution starting sufficiently close to the equilibrium remains close to the equilibrium andapproaches the equilibrium as t→∞ , or if there are solutions starting arbitrary close to the equilibrium whichdo not approach it respectively.
4.6 Global stability of endemic equilibrium point
[41] postulate that the local stability of the Disease Free Equilibrium advocates for local stability of theEndemic Equilibrium for the reverse condition. We therefore focus on finding the global stability of Endemicequilibrium. We use Korobeinikov approach in which we formulate a suitable Lyapunov function for pneumonicplague model [22, 23, 41].The Lyapunov function is as given in the form below;
V =∑
ai(yi − y∗i ln yi)
where ai is defined as a properly selected positive constant, yi defines the population of the ith compartment,and y∗i is the equilibrium point.We will have the following Lyapunov function:
The constants Wi are non-negative in Φ for i = 1, 2, 3...12. The function V together with its constantsW1,W2...W12 are chosen such that V is continuous and differentiable in Φ.
70 R. C. Ngeleja, L. S. Luboobi, V. G. Masanja: Modeling Pneumonic Plague
dVdt = W1(1−
S∗HSH
)[ψ1 +$RH − α1(ΓhhIHBN1
+ ΓfhIFN2
+ ΓrhIRBN3
+ ω1A)SH − µ1SH , ]+W2(1−
E∗H
EH)[α1(Γhh
IHBN1
+ ΓfhIFN2
+ ΓrhIRBN3
+ ω1A)SH − (α2 + µ1)EH ]
+W3(1−I∗HAIHA
)[τ1α2EH − ρα3IHA − (1− ρ)α3IHA − (µ1 + κ1)IHA]
+W4(1−I∗HBIHB
)[(1− τ1)α2EH + ρα3IHA − α4IHB − (µ1 + δ1)IHB]
+W5(1−R∗
HRH
)[α4IHB + (1− ρ)α3IHA −$RH − µ1RH ]+W6(1−
S∗RSR
)[ψ3 − γ1(Γrr IRBN3
+ ΓfrIFN2
+ ΓhrIHBN1
+ ω2A)SR − µ3SR]+W7(1−
E∗R
ER)[γ1(Γrr
IRBN3
+ ΓfrIFN2
+ ΓhrIHBN1
+ ω2A)SR − (γ2 + µ3)ER]
+W8(1−I∗RAIRA
)[τ2γ2ER − γ3IRA − (µ3 + κ2)IRA]
+W9(1−I∗RBIRB
)[(1− τ2)γ2ER + γ3IRA − (µ3 + δ3)IRB]
+W10(1−S∗FSF
)[ψ2s − β(Γhf ρ1IHA+ρ2IHBN1
+ Γrfρ3IRA+ρ4IRB
N3)SF − µ2SF ]
+W11(1−I∗FIF
)[β(Γhfρ1IHA+ρ2IHB
N1+ Γrf
ρ3IRA+ρ4IRBN3
)SF − (µ2 + δ2)IF ]
+W12(1− A∗
A )[λ4 + η1IHBN1
+ η2IRBN3− µ4A]
Using system (1) - (4) at endemic equilibrium we derive the following;
dVdt = −W1(1−
S∗HSH
)2 −W2(1−E∗
HEH
)2 −W3(1−I∗HAIHA
)2 −W4(1−I∗HBIHB
)2
−W5(1−R∗
HRH
)2 −W6(1−S∗RSR
)2 −W7(1−E∗
RER
)2 −W8(1−I∗RAIRA
)2
−W9(1−I∗RBIRB
)2 −W10(1−S∗FSF
)2 −W11(1−I∗FIF
)2
−W12(1− A∗
A )2 + F (SH , EH , IHA, IHB, RH , SR, ER, IRA, IRB, SF , IF , A)
where the function F (SH , EH , IHA, IHB, RH , SR, ER, IRA, IRB, SF , IF , A) is non-positive, Now followingthe procedures by [30] and [24]. We take that
F (SH , EH , IHA, IHB, RH , SR, ER, IRA, IRB, SF , IF , A) ≤ 0
for all values ofSH , EH , IHA, IHB, RH , SR, ER, IRA, IRB, SF , IF , A.
Then dVdt ≤ 0 for all values of SH , EH , IHA, IHB, RH , SR, ER, IRA, IRB, SF , IF , A and it is zero when
SH = S∗H , EH = E∗H , IHA = I∗HA, IHB = I∗HB, RH = R∗H , SR = S∗R, ER = E∗R, IRA =I∗RA, IRB = I∗RB, SF = S∗F , IF = I∗F , A = A∗. Hence the largest compact invariant set inSH , EH , IHA, IHB, RH , SR, ER, IRA, IRB, SF , IF , A such that dVdt = 0 is the singleton E∗ which is the En-demic Equilibrium point of the pneumonic plague system (1) - (4). Now using LaSalles’s invariant principleby [25], it implies that E∗ is globally asymptotically stable in the interior of the region of SH , EH , IHA, IHB ,RH , SR, ER, IRA, IRB , SF , IF , A and thus leads to the theorem below:
Theorem 5. If R0 > 1 then the model system (1) - (4) of pneumonic plague has a unique endemic equilibriumpoint E∗ which is globally asymptotically stable in SH , EH , IHA, IHB , RH , SR, ER, IRA, IRB , SF , IF , A.
5 Sensitivity and elasticity analysis and numerical simulation
In this section, we determine the behavior and strength of model predictions with respect to parametervalues. We use sensitivity and elasticity analysis to determine the impact of kij on the basic reproductionnumber R0 in order to set the required control strategies for pneumonic plague.
5.1 Parameter estimation
The parameters are taken from the literature that relate to this study, the present information on pneumon-ic plague and through estimation using sensitivity analysis and simulations. Table 2 shows the values of theparameters and their description as used in the model.
World Journal of Modelling and Simulation, Vol. 16 (2020) No. 1, pp. 53-80 71
Table 2: Parameter values and Description for plague disease model.Parameters Description Value Reference/SourceΓrf Adequate contact rate: rodent to flea 0.6 [34]Γfh Adequate contact rate: flea to human being 0.09 [4]Γfr Adequate contact rate: flea to rodent 4.7 [28]ρ The probability that IHA progresses to either IHB or RH 0.7 Estimatedα1 Progression rate of susceptible human being to exposed 0.9 [34]α4 Rate of recovery of IHB 0.006 Estimatedγ1 Progression rate of susceptible rodent to exposed 0.9 [34]Γhf Adequate contact rate: human being to flea 0.28 [4]λ4 Pathogens multiplication rate 0.89 [34]α2 Progression rate of exposed human being to infected 0.95 Estimatedγ2 Progression rate of exposed rodent to infected 0.91 Estimatedα3 Progression rate of human being infected by bubonic plague 0.6 estimated$ Progression rate of recovered human being to susceptible 0.1 [20]µ1 Natural death rate for human being 0.04 [20]η1 Recruitment rate of pathogens in the environment by IHB 0.37 Estimatedδ1 Pneumonic plague disease induced death rate for human 0.04 [20]η2 Recruitment rate of pathogens in the environment by IRB 0.89 Estimatedδ3 Pneumonic plague disease induced death rate 0.05 [21]µ3 Natural death rate for rodent 0.2 [12]ω1 Adequate contact rate: Pathogens in the environment to human being 0.8 Estimatedω2 Adequate contact rate: Pathogens in the environment to rodent 0.04 Estimatedµ4 Natural death rate for pathogens 0.1 [34]µ2 Natural death rate for flea 0.07 [4]τ1 The probability that EH progresses to either IHA or IHB 0.6 Estimatedτ2 The probability that ER progresses to either IRA or IRB 0.4 Estimatedδ2 Bubonic plague disease induced death rate for flea 0.03 [4]γ3 Progression rate from IRA to IRB 0.015 Estimatedψ1 Immigration rate of human being 0.09 [34]ψ2S Immigration rate of susceptible flea 0.008 [21]ψ3 Immigration rate of rodent 0.03 [20]β The rate at which flea become infected 0.99 [34]Γhh Adequate contact rate: IHB to SH 0.019 EstimatedΓrr Adequate contact rate: IRB to SR 0.029 EstimatedΓhr Adequate contact rate: IHB to SR 0.005 EstimatedΓrh Adequate contact rate: IRB to SH 0.09 Estimated
Fig. 4 shows the dynamics of the disease in human beings, rodents, fleas and pathogens in the environmentrespectively. In human beings, we see that the the exposed EH , bubonic and pneumonic plague infectious IHAand IHB , and recovery RH classes slightly increase before it settle at its equilibrium points. The susceptibleclass SH experience a fast decrease within the first year and then it gradually decrease to its endemic point.In rodent population, all compartments SR, ER , IRA and IRB show a marginal increase before they all attainthe endemic equilibrium point. The compartment in fleas and pathogens in the environment also experience themarginal increase before they reach the endemic point.
Bubonic plague serves as the primary stage of pneumonic plague in this study, it is mainly transferredwhen the infected flea bites the susceptible human being or rodent. The fraction of human being and rodentinfected with bubonic plague if not treated may progress and become the pneumonic plague infectives. Thismeans that the increase of the number of human beings and rodents with bubonic plague will also increase thenumber of human beings and rodents with pneumonic plague [10]. Fig. 4 shows the influence of number ofinfectious individuals due to the increased number of infected flea to the human beings and rodents infectedwith bubonic plague. Fig. 4 shows the influence of number of infectious individuals due to human beings androdents infected with bubonic plague to the human beings and rodents with pneumonic plague. This output is
72 R. C. Ngeleja, L. S. Luboobi, V. G. Masanja: Modeling Pneumonic Plague
Fig. 4: The dynamics of human beings, rodents, fleas and pathogens in the environment with baseline parametervalues given in Table 2.
because the increase of the number of individual with bubonic plague consequently increases the progressionrate of individuals (human being and rodent) with Bubonic Plague to individuals with Pneumonic Plague.
Fig. 4 shows that when the number of infected flea increases the number of human beings and rodentsinfected with Bubonic plague also increase. It implies that the increased number of infected fleas will increasethe probability of a human being or a rodent to be bitten by the infected flea. As a result it increases the numberof human beings and rodents infected with Bubonic plague.
The environment infested with pathogens also plays a great role in transmitting the pneumonic plague bac-teria to the human being or rodent populations. The bacteria may be transmitted through airborne transmission(droplet contact). When a human being or rodent with pneumonic plague coughs or sneezes the bacteria aremoved to the environment and upon adequate contact it may lead to infection to human beings or rodents [38].
Fig. 7 shows the influence of pathogens in the environment to the number of human beings and rodentswith pneumonic plague. The figure portrays that the increase in the number of pathogens in the environmentproportionally increases the number of human beings and rodents with Pneumonic Plague, this is due to thefact that the increase the pathogens in the environment will also increase the rate/probability that air that one(human being or rodents) breathes in contain yersinia pestis which may leads to infection.
Environment also is greatly affected when the number of human and rodent with Pneumonic Plague isincreased [19]. In Fig. 8, we see that as the number of human and rodent with pneumonic plague increases thepathogens in the environment increase as well. This result is realistic due to the fact that human and rodentinfected with Pneumonic Plague do release yersinia pestis bacteria into the environment through coughing orsneezing [36].
World Journal of Modelling and Simulation, Vol. 16 (2020) No. 1, pp. 53-80 73
Fig. 5: The effect of increased infected fleas on the number of human beings and rodents with Bubonic Plague
Fig. 6: The effect of increased number of human beings and rodents with Bubonic Plague on the number ofhuman beings and rodents with Pneumonic Plague
5.2 Sensitivity and elasticity analysis of r0
In this section, we determine the effect of parameters in the variation of the basic reproduction numberusing sensitivity analysis. We also perform the elasticity analysis to quantify the relative change in R0 inresponse to the change in a parameter. [16] analyzed the steps to study the sensitivity and elasticity of the basicreproduction number R0 to the changes in elements kij or to the parameters that describe them. We employ thesteps in our model as given below.
5.2.1 Sensitivity
The sensitivity sij of a matrix K is defined as the change in the basic reproduction number (R0) which isthe the maximum modulus of the eigenvalues of the matrix K due to change in elements kij given by
74 R. C. Ngeleja, L. S. Luboobi, V. G. Masanja: Modeling Pneumonic Plague
Fig. 7: The effect of increased number pathogens in the environment on the number of human beings androdents with Pneumonic Plague
Fig. 8: The effect of increased number of human beings and rodents with Pneumonic Plague on the number ofpathogens in the environment
The values sij form a sensitivity matrix Sij which is computed from the left and right eigenvectors of the nextgeneration matrix corresponding to its dominant eigenvalue [6].
For individual parameters the sensitivity s(λ) is given by
s(λ) =∑ij
∂R0
∂kij
∂kij∂λ
. (27)
5.2.2 Elasticity
Elasticity is defined as the proportional change in R0 due to a proportional change in the matrix element.Now the elasticity eij of a matrix element kij is defined as
eij =kijR0
∂R0
∂kij. (28)
For individual parameters the elasticity e(λ) is given by
World Journal of Modelling and Simulation, Vol. 16 (2020) No. 1, pp. 53-80 75
Table 3 shows the sensitivity and elasticity of the basic reproduction number R0 for the given parametervalues. From the table we see that R0 is most sensitive to expected number of new cases of the contaminatedenvironment caused by one rodent infected with pneumonic plague k64. R0 is also sensitive to other ksij likethe the expected number of new cases of rodent infected with pneumonic plague caused by one infected rodentwith pneumonic plague (k44), the expected number of new cases of contaminated environment caused by oneinfected human beings with pneumonic plague (k62), the expected number of new cases of human beingsinfected with pneumonic plague caused by one infected rodent with pneumonic plague (k24), the expectednumber of new cases of rodent infected with pneumonic plague caused by one infected human beings withpneumonic plague (k42) and the expected number of new cases of rodent infected with pneumonic plaguecaused by contaminated environment (k46). The basic reproduction number (R0) is least sensitive to (k26)which is the expected number of new cases of human beings infected with pneumonic plague caused by infectedenvironment. From Table 3, we also see that R0 is more elastic to the expected number of new cases of infectedenvironment caused by one infected rodent with pneumonic plague (k64). It is also least elastic to the expectednumber of new cases of human beings infected with pneumonic plague caused by one infected human beingswith pneumonic plague (k22).
Table 3: Sensitivity and elasticity of R0 for pneumonic plague
From the Table 3 it can be seen that the sensitivity of all kijs are positive. The positive sign impliesthat increasing (decreasing) any kij will consequently increase (decrease) the basic reproduction number. Forexample, the sensitivity of k64 = 0.408 implies that increasing the expected number of new cases of infectedenvironment caused by one infected rodent with pneumonic plague by 10% will increase the value of the basicreproduction number by 4%. Fig. 9, shows the effect of k22, k24, k26, k42, k44, k46, k62 and k64 on the basicreproduction number .
The marginal increase of each kij brings about a significant increase in the basic reproduction number,which means that to effectively control the disease an effort should be made to reduce the magnitude of eachkij . We then need to reduce expected number of new cases of human beings infected with pneumonic plaguecaused by one infected human beings with pneumonic plague k22.
This may be done through reducing the probability that human being survive the incubation period, hu-man’s incubation period, and adequate contact rate between people infected with pneumonic plague. k24 maybe reduced through reducing adequate contact rate between rodent infected with Pneumonic Plague to hu-man being, The probability that a rodent infected with pneumonic plague survives the incubation period andthe infectious period of rodent with pneumonic plague, k26 may be reduced by reducing the period that theenvironment remains contaminated with pathogens causing the disease, The probability that the environmen-t survive the period taken by pathogens to reach the threshold necessary to infect the environment and thustransmit pneumonic plague disease to human beings and Adequate contact rate: Pathogens in the environmentto human beings. We can reduce the value of k42 by reducing the probability that human beings with Pneu-monic Plague survive the incubation period, Infectious period of human beings with pneumonic plague and theadequate contact rate between the human beings infected with Pneumonic Plague and rodent. Reducing k44may be by reducing the infectious period of rodent infected with pneumonic plague, the adequate contact rate
World Journal of Modelling and Simulation, Vol. 16 (2020) No. 1, pp. 53-80 77
between the rodent with pnemonic plague and a susceptible rodent and the probability that a rodent survivesthe period between exposure and onset of symptoms of pneumonic plague. k46 can be controlled by reducingthe adequate contact rate from pathogens in the environment to rodent, the period that the environment remaincontaminated with pathogens causing the disease and the probability that pathogens survive the period to reachthe threshold necessary to contaminate the environment and thus transmit pneumonic plague disease to rodent.k62 may be reduced by reducing the probability that a human beings with pneumonic plague and capable oftransmitting the pathogens to the environment survive the incubation period, the infectious period of a humanbeing infected with pneumonic plague, and the shading rate of pathogens in the soil/environment from a humanbeing infected with pneumonic plague and we can as well reduce k64 by reducing the probability that a rodentwith pneumonic plague and capable of transmitting the pathogens to the environment survive the incubationperiod, the infective period of a rodent infected with pneumonic plague, and the shading rate of pathogens inthe soil/environment from a rodent infected with pneumonic plague.
6 Discussion and conclusion
The magnificent transmission capacity displayed in the numerical results, show that, Pneumonic plague isvery fatal and threaten the life of human beings, rodents (including the domestic animals) and the fleas. Theresults demonstrate the vital role played by human beings and rodents with bubonic plague as agents in thetransmission and spread of the pneumonic plague disease. It is a fact that if an individual (human being orrodent) with bubonic plague is not treated, the probability of progressing and becoming the pneumonic plagueinfective is very high [26]. Thus the result justifies the reason why an infected flea plays the vital role in thetransmission of pneumonic plague, for it is the main agent for bubonic plague transmission to both humanbeings and rodents.
Pneumonic plague is on top of list of diseases that could be used as a bio-weapon [35]. The results in thisstudy show a significant relationship between the increase of the number of human beings and rodents withpneumonic plague and the pathogens in the environment. This implies that when the environment is favorablefor the pathogens to spread the disease becoming extremely dangerous with increased prevalence and deaths.
Results in Fig. 9 show a positive relationship between the basic reproduction number and the expectednumber of new cases of each pair kij . This means that as the number of new cases of i caused by one infectedindividual j increases. It consequently increases the basic reproduction number. Now since all individuals actas the potential agents for transmission of the disease, this indicates to us that when the disease occur it willspread to a large community very fast.
The strategy that may have a great and positive impact on the control of pneumonic plague, is the one thatwill reduce the effect of kij on the basic reproduction number. This may generally be done through the followingstrategies: one is reducing the individual’s infectious period; two is reducing the probability that the individualsurvives the incubation period, and three is reducing the adequate contact rate between one infective agent andthe other susceptible individuals. These three strategies will reduce the number of infections an individual canproduce by reducing the values of kij and as a result reduce the value of the basic reproduction number.
6.1 Conclusion
A deterministic SEIR model with modification was developed and analysed to study the dynamics of p-neumonic plague. The analytical results show that the disease free equilibrium point (DFE) and the endemicequilibrium point (EE) exist and were found to be locally and globally asymptotically stable whenever theyexist. In order to determine the number of infections an individual can produce we computed the basic repro-duction number using the next generation Matrix method. This dimensionless quantity shows that the numberof secondary cases depends on the four agents of the disease (human being, rodent, flea and pathogens in theenvironment). The R0 also shows the contribution of each agent through pair Kij in which the effect posed byeach individual to the transmission and spread of the disease is quantified.
From this study it is clear, from the analytical results and simulations, that pneumonic plague can be verydangerous to a point of being fatal. There must be plans that will effectively analyze the control strategies of the
78 R. C. Ngeleja, L. S. Luboobi, V. G. Masanja: Modeling Pneumonic Plague
disease when it occurs. With the support of the numerical analyses in this study we recommend that any controlstrategy for pneumonic plague should concentrate on reducing the effect of the expected number of new casesof each pair kij has to the basic reproduction number.
Acknowledgments
The author appreciate the support from two supervisors for their helpful comments that improved thepaper.
World Journal of Modelling and Simulation, Vol. 16 (2020) No. 1, pp. 53-80 79
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