Applied Mathematics and Computational Intelligence Volume 6, 2017 [29-40]

DeterministicandStochasticSISModelofCommonColdinUniversiti

MalaysiaPerlis

NurFarhanaHazwaniAbdulShamad1,AmranAhmed1,aandMohammadIqbalOmar2

1InstituteofEngineeringMathematics,UniversitiMalaysiaPerlis,KampusPauhPutra,

02600Arau,Perlis,Malaysia.2SchoolofMechatronicEngineering,UniversitiMalaysiaPerlis,KampusPauhPutra,

02600Arau,Perlis,Malaysia.

ABSTRACT

TheepidemiologicalofcommoncoldwithSusceptible‐Infected‐Susceptible(SIS)modelisthe description of the dynamics of a disease that is contact transmittedwith no longlasting immunity. This is the first attempt to develop SISmodel on common cold. Thepurposeof this study is tocomparebetween thedeterministicand stochasticSISmodelwithdemographyandwithoutdemography(presenceofbirthsanddeaths),toderivethereproductivenumber, betweenthemodelsandtocomparetheSISmodelsdemographywithoutpharmacologicaltreatmentandwithpharmacologicaltreatment.TherearetwogroupstestedinSISmodelwhichisUniMAP’sstudentsandUniMAP’sstaffsandthesedataweretakenfromUniMAP’suniversityhealthcentreonSeptember2015.Inthisstudy,SISmodelswere implementedas set ofdeterministic ordinarydifferential equations (ODE)thatcanbesolvedbyusingdifferentnumericalmethodsandadiscretetimeMarkovchain(DTMC)process instochasticsimulations.Gillespiealgorithmhadbeenusedtogeneratestochastic simulations efficiently in R program. Then, differential equations will beconstructedwhichdescribedthemeanstatisticsofeachprocess.Hence,thederivationofreproductive number, had been obtained by using the next generation operatormethod. In these cases, the number of infected persons in SIS demography willcontinuously decrease as there are presence of births and deaths in the population.Pharmacological treatment had been used to improve and control the infection ofcommon cold from spread to population. This controlmeasures help tominimize thenumbers of infected individuals in the population. Therefore, the pharmacologicaltreatment increases the recovery rate and helps them to recovermore quickly. Basicreproductive number, for everymodelswithout demography andwith demographywere derived for determiningwhether a disease persist in the population or not. Thedisease will continuously spread out into population if as all the models aregreaterthan1.

Keywords:SISmodel,reproductionnumber,deterministic,stochasticity

1. INTRODUCTION Susceptible‐ Infected‐ Susceptible (SIS) model is a model of the behaviour of an infectiousdisease in a large population. The SIS model consist of susceptible state and infected state.These states are generally called compartments, and the corresponding models are calledcompartmentmodels. Individuals inpopulationareappoints intodifferentcompartmentsandlettersareusedtoshowthedifferentstages incompartmentalmodels.Compartmentalmodelbrieflyexplainswhathappensatthepopulationscale(Vynncycky&White,2010).Deterministicmodellingisdescribedbyordinarydifferentialequations(ODEs)thatcanbedealwithbyusingdifferentnumericalmethods.It isapplicableforlargepopulationanddeterministicsimulationthatcontainnorandomvariablesandnodegreeofrandomness.Theoutputofthemodelisfullydeterminedbytheparametervaluesandtheinitialconditions.

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Stochastic model is developed as a stochastic process with a collection of random variablesevolvingtime(Allen,2008).ThebehaviourofdynamicsstochasticmodellingcanbeinterpretedbydiscretetimeMarkovChain(DTMC)(Vynnycky&White,2010).ThediscretetimestochasticSISmodelisaMarkovchainwithfinitestatespaceandassumedthatatmostoneeventoccursin the time period ∆t (Allen & Burgin, 2000). The same set of parameter values and initialconditionswill lead to an ensemble of different outputs. Event‐ driven approach is amethodthatrequiresexplicitconsiderationofevents.Populationformsthatemergefromtheirregularwayofoccasionsatleveloftheindividualorotherwisecalledastransitionratesisportrayedasdemographystochastic(Keeling&Rohani,2008).Thetransitionprobabilitiescanbeperformedwithindividualsexperiencedifferentratesduetoeachevent(Bloomfield,2014).Infectious diseases are defined as a disease caused by an infectious agent and have beencharacterized by their biological properties. It can be difficult to distinguish between thecommoncoldand influenza.Thedifferencesbetween thesediseasesare the typeofpathogeninvolvedinthedisease.Forcommoncold,pathogeninvolvedwererhinoviruseswhileinfluenzawereinfluenzavirusesAorB.Thispathogencanbetransmittedbyaninfectedindividualandthisepidemiology isdealwithpopulations (Krämer&Krickberg,2010).CommoncoldcanbecategorizedasSISmodel.CommoncoldisasicknessbroughtaboutbyvirusinfectionsituatedinthenoseandFigure1showsananatomyofthenose.Itisbecauseofinfectionbyanextensivevariety of respiratory infections, of which the rhinoviruses are the most widely recognized(Kumar et al., 2007). Rhinoviruses are spread effortlessly through individual‐to‐individualcontactbecause thereareno less than100differentantigenicstrainsof rhinovirus,making ittroublesomefortheimmunesystemtoconferprotection(Kumar&Clark,2009).

Figure1.Anillustrationanatomyofthenose.2. METHODS2.1 Method1:Withoutdemography

Thesimplestmodelwithoutdemography,nobirth,nodeathandnomigrationwhichisalsoknownasclosedpopulationwasused.LetNbethesizeofthepopulation.Schematicofthemodel in Figure 2 showed that the transition rates, indicated by arrow, between eachcompartment shown. Parameters β and γ represent the transition rates betweencompartmental model of susceptible and infected. Let S be the number of susceptibleindividualsandIbethenumberofinfectedindividuals.Thetransmissionrateisdenotedbyβwhilerecoveryrateisγ.Theinfectedindividualsreturntobesusceptibleclassonrecoverybecausethediseaseconfersnoimmunityagainstinfection.

β

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Figure2.AschematicSISgeneralmodeldiagram(Chitnis,2011)DeterministicofSISmodelTheSISmodelisgiveninthefollowingsetofequations:

where,

susceptibleindividualsmakescontactwithinfectedindividualstotransmitdisease, inunittime,t.

infectedindividualswhorecoverthenenteragaininsusceptibleclassinunittime,t.ThediscretetimedeterministicSISmodelwiththetotalpopulationheldconstanthastheform(Allen&Burgin,2000).

where is a fixed timed interval (days), and LetS(t)bethenumberofsusceptibleindividualsinthepopulationattimetandI(t)bethenumberofinfectedindividualsinthepopulationattimet. isdefinedby the increase in the number of susceptible individuals from time t to time while

isdefinedbythenumberofinfectedindividualsfromtimettotime .Thetotalpopulation S(t) + I(t) = N is constant. The model is set to disease free state to obtainreproductivenumber.Fordiseasefreestate,considerinfectedcompartment, .StochasticSISmodelGillespiealgorithmwillbeusedinRprogramtogeneratestochasticsimulation.TheideaoftheGillespiealgorithmistodeterminewhenthenexteventwilloccur.Thediscretetimestochasticversion of the Markovian model SIS (without demography) model with finite state space isdefinedbythefollowingeventsandratesinTable1(Keeling&Rohani,2008).Table1Transitionratesandchangeinstatespaceoftheprocess tothenexteventinsmall

time,

Event Transition TransitionrateInfectionofsusceptible & Recoveryofinfection &

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2.2 Method2:Withdemography(Withoutpharmacologicaltreatment) Thenatural birth anddeath rates are included in thismodel and assumed that all births aresusceptibleanddeathrateisequalforallcompartments.Letthebirthanddeathratesareequalsothatthetotalpopulationremainsconstantwithrespecttotime.Birthrateisdefinedasthebirth rate per capita and death rate indicates the measure of deaths per capita.

Figure3.AschematicSISmodelwithdemographydiagram(Hethcote,1989).DeterministicSISmodelwithdemographyThissystemisdescribedbythefollowingequations:

Newparameter,birthanddeathrate, wasaddedintoEq.(5)and(6).Byaddingtheequations,itisshownthatthetotalpopulationisconstantatanyinstant isS(t)+I(t)=N.Then,

Hence,thediscretetimedeterministicSISmodelhastheform:

where is a fixed timed interval (in days), and Where the total population S(t) + I(t) = N is constant. Let all theparameters are positive, and . Let S(t) be the number of susceptibleindividuals in the population at time t and I(t) be the number of infected individuals in thepopulation at time t. is the number of births or deaths per individuals during the timeinterval. is the number of individuals that recover in the time interval, . Infectedindividuals immediately become susceptible once they have recovered because they do notdevelopimmunitytowardscommoncold.StochasticSISmodelwithdemographyIn the stochastic DTMC model, Gillespie algorithm will be used in this study to generatestochasticsimulationandtheideaoftheGillespiealgorithmistodeterminewhenthenexteventwilloccur.ThediscretetimestochasticversionoftheMarkovianmodelSIS(withdemography)modelwithfinitestatespaceisdefinedbythefollowingeventsandratesinTable2(Keeling&Rohani,2008).

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Table2Transitionratesandchangeinstatespaceoftheprocess tothenexteventinsmalltime,

2.3 Method3:Withdemography(Pharmacologicaltreatment) Therearemanyways to improveandcontrol the infection fromspreading to thepopulation.These control measure help to minimize the risk of transmission between infectious andsusceptible humans. Hence, the intervention method for managing common cold ispharmacologicaltreatment.DeterministicSISmodelwithdemography(pharmacologicaltreatment)Birth anddeath rateparameters, was added into the equationsbelow.Thepopulation as awhole is assumed to be constant, S(t) + I(t) =N. This system is described by the followingequations:

Hence,thediscretetimedeterministicSISmodelhastheform:

where is a fixed timed interval (in days), and Where the total population S(t) + I(t) = N is constant. Let all theparameters are positive, and . Let S(t) be the number of susceptibleindividuals in the population at time t and I(t) be the number of infected individuals in thepopulation at time t. is the number of births or deaths per individuals during the timeinterval. istherecoveryratefrompharmacologicaltreatmenttakenbyinfectedindividualstorecoverinthetimeinterval, beforeenteragaininthesusceptibleclassbecausetherearenolong‐lastingimmunity.StochasticSISmodelwithdemography(pharmacologicaltreatment)

Thestochasticversionof theMarkovianSIS(withpharmacological treatment)model isdefinedbythefollowingeventsandratesasinTable3(Chowelletal.,2009).

Event Transition TransitionrateBirthofsusceptible Deathofsusceptible Infectionofsusceptible & Recoveryofinfection & Deathofinfection

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Table3Transitionratesandchangeinstatespaceoftheprocess tothenexteventinsmalltime, .

3. RESULTSDiscrete timedeterministicmodel andstochasticmodelsare formulatedandanalysed forSISmodels with different population sizes for two groups which are UniMAP’s students andUniMAP’s staffs. The discrete time model is directly applicable to particular disease such ascommon cold. The population of UniMAP’s students comprise of approximately 12500individualswhileUniMAP’sstaffsconsistofapproximately450individuals.Therewillbethreemethods (without demography, without demography pharmacological treatment and withdemographypharmacologicaltreatment)testedinSISmodels.3.1 Method1:Withoutdemography

(a) (b)

Figure4.ThedeterministicSISmodelwithoutdemographyfor(a)UniMAP’sstudentsand(b)UniMAP’sstaffs.

Event Transition Transitionrate

Birthofsusceptible Deathofsusceptible Infectionofsusceptible & Recoveryofinfectionfrompharmacologicaltreatment

&

Deathofinfection

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3.2 Method2:Withoutdemography(Withoutpharmacologicaltreatment)

(a) (b)

Figure5.(a)ThedeterministicSISmodelwithdemographyforUniMAP’sstudentsand(b)100simulationsofthestochasticSISmodelusingGillespie’sMethod.

(a) (b)

Figure6.(a)ThedeterministicSISmodelwithdemographyforUniMAP’sstaffsand(b)100simulationsofthestochasticSISmodelusingGillespie’sMethod.

3.3 Method3:Withdemography(Withpharmacologicaltreatment)

(a) (b)

Figure7.ComparisonbetweenwithoutdemographyandwithdemographydeterministicSISmodelof(a)

UniMAP’sstudentsand(b)UniMAP’sstaffs.

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(a) (b)

Figure8.ComparisondeterministicSISmodelwithandwithoutpharmacologicaltreatmentfor(a)UniMAP’sstudentsand(b)UniMAP’sstaffs.

3.4 ComparisonforthreemethodsbetweenUniMAP’sstudentsandUniMAP’sstaffs

Table4ComparisonforthreemethodsonUniMAP’sstudents

Table5ComparisonforthreemethodsonUniMAP’sstudents

4. CONCLUSIONSItcanbeconcludedthatthebehaviourofSISmodelswithandwithoutdemographyarenearlythesame.StochasticmodelssuchasMarkovchainmodelprovidesalternativetodeterministicmodels and ease some of the problems with the deterministic formulations (Allen & Allen,2003).Knowledgeofthesimilaritiesanddifferencesbetweenmodelsisusefulinselectingthecorrect formulation. In this case, thedifference is just thenumberof infected indemographywhich slightly decreases compared towithout demography before it reaches the equilibrium

Withoutdemography WithdemographyPharmacologicaltreatmentWith With

µ ‐ 0.0133β 0.3333γ 0.1429 ‐ρ ‐ 0.3 2.3324 2.1338 1.0638

Withoutdemography WithdemographyPharmacologicaltreatmentWith With

µ ‐ 0.0133β 0.3333γ 0.10 ‐ρ ‐ 0.3 3.333 2.9417 1.0638

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state.Thisisduetothepresenceofbirthanddeathrateinthepopulation.GillespiealgorithmwasusedtosimulatestochasticmodelsinRbydrawingaprocessrandomlyfromalleventsinprocessaccordingtotheirrespectiveprobabilities.Thebasicreproductivenumber playsanimportantrolewhetherthediseaseiseliminatedorpersistsinpopulation.ReproductionnumberofallSISmodelswerederivedfromDFEanditcanbe concluded that if the disease will remain and continuously spread out in thepopulation which agrees with the findings of Heatcote (1989), Allen and Burgin (2000) andChitnis (2011). In this case, numerical test shown that when all reproductive number in allmodelsaregreaterthan1,virtuallyeveryoneinpopulationwillsoonerorlaterremainsinfectedasthediseasewillcontinuouslyspreadoutintopopulation(Haran,2009).By implementing pharmacological treatment that increases the recovery rate, it helps toreducethebasicreproductiverateofadisease.Thisshowsthatiftherecoveryrateishigh,thenumberof infectedwillcontinuouslydecreaseand lesscompared towithoutpharmacologicaltreatment. So, if recovery rate is larger, then the infected individuals that are recovering inlargernumbersthanthesamesizedpoolwouldrecoverotherwisewhichisinagreementwithTassier(2013).Thisisbecauseofthecasethattheyspeeduptherateofrecovery.Reproductivenumber, can be reduced through an increase in the recovery rate through medication(Chitnis,2011).REFERENCES[1] Allan,G.M.,&Arroll,B.(2014).Preventionandtreatmentofthecommoncold:making

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APPENDIXIf any, theappendix shouldappeardirectlyafter the referenceswithoutnumbering, andon anewpage.

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