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JDynDiffEquat(2008)20:699–717DOI10.1007/s10884-008-9111-8ThresholdDynamicsforCompartmentalEpidemicModelsinPeriodicEnvironmentsWendiWang·Xiao-QiangZhaoReceived:5August2007/Publishedonline:23April2008©SpringerScience+BusinessMedia,LLC2008AbstractThebasicreproductionratioanditscomputationformulaeareestablishedforalargeclassofcompartmentalepidemicmodelsinperiodicenvironments.Itisprovedthatadiseasecannotinvadethedisease-freestateiftheratioislessthanunityandcaninvadeifitisgreaterthanunity.Itisalsoshownthatthebasicreproductionnumberofthetime-averagedautonomoussystemisapplicableinthecasewhereboththematrixofnewinfectionrateandthematrixoftransitionanddissipationwithininfectiouscompartmentsarediagonal,butitmayunderestimateandoverestimateinfectionrisksinothercases.Theglobaldynamicsofaperiodicepidemicmodelwithpatchstructureisanalyzedinordertostudytheimpactofperiodiccontactsorperiodicmigrationsonthediseasetransmission.KeywordsCompartmentalmodels·Reproductionratio·Periodicity·ThresholddynamicsMathematicsSubjectClassification(2000)34D20·37B55·92D301IntroductionThebasicreproductionnumberofaninfectiousdiseaseisafundamentalconceptinthestudyofdiseasetransmissions.Itistheexpectednumberofsecondarycasesproduced,inacompletelysusceptiblepopulation,byatypicalinfectiveindividual.Usually,thebasicrepro-ductionnumberdefinesthethresholdbehaviorforclassicalepidemicmodels.Itisacommoncasethatadiseasediesoutifthebasicreproductionnumberislessthanunityandthediseaseisestablishedinthepopulationifitisgreaterthanunity.Forautonomousepidemicmodels,W.WangDepartmentofMathematics,SouthwestUniversity,Chongqing400715,People’sRepublicofChinae-mail:wendi@swu.edu.cnX.-Q.Zhao(B)DepartmentofMathematicsandStatistics,MemorialUniversityofNewfoundland,St.John’s,NL,CanadaA1C5S7e-mail:xzhao@math.mun.ca123700JDynDiffEquat(2008)20:699–717Diekmannetal.[7],vandenDriesscheandWatmough[31]presentedgeneralapproachesforthecalculationsofbasicreproductionnumbers.Computationsofbasicreproductionnum-bersforspecificinfectiousdiseasesarecarriedoutin[20]forsexualdiseases,in[13]fortuberculosisinpossums,in[12]fordenguefever,in[15,24,33,40]forSARS.Furthermore,basicreproductionnumberswerestudiedin[1,2]fortheepidemicmodelswithpopulationtravelingamongcitieswheretheresidencesofindividualsaremaintained,andin[32,34–36]forthepatchymodelswithouttherecordofresidenceofindividuals.Itiswell-knownthatperiodicfluctuationsarecommonintheevolutionofdiseasetrans-missions.Contactratesvaryseasonallyforchildhooddiseasesbecauseofopeningandclosingofschools[5,9,26–28].Periodicchangesinbirthratesofpopulationsareevidencedinmanybiologicalworks,see,e.g.,[6,23,39].Vaccinationprogramisalsoasourceofperiodicity[10].Wereferto[3,4,8,14,17,22,25,30,37,38]andreferencesthereinforothertypesofperiodicepidemicmodels.Anaturalandimportantproblemassociatedwithperiodicepidemicmod-elsistodefineandcomputetheirbasicreproductionnumbers.Intuitively,onemayexpecttousethebasicreproductionnumberofthetime-averagedautonomoussystemofaperiodicepidemicmodeloveratimeperiod.Unfortunately,thisaveragebasicreproductionnumberisapplicableonlyincertaincircumstances,butoverestimatesorunderestimatesinfectionrisksinmanyothercases(seeexamplesinSect.3).Theeffectivereproductionnumberisalsousedintheliterature,whichisdefinedastheaveragenumberofsecondarycasesarisingfromasingletypicalinfectiveintroducedattimetintothepopulation[11].Itsmagnitudeisausefulindicatorofboththeriskofanepidemicandtheeffortrequiredtocontrolaninfection.However,thisnumberisnotathresholdparametertodeterminewhetherthedis-easecaninvadethesusceptiblepopulationsuccessfully.Recently,BacaërandGuernaoui[4]presentedageneraldefinitionofthebasicreproductionnumberinaperiodicenvironment.Thepurposeofourcurrentpaperistoestablishthebasicreproductionratioforalargeclassofperiodiccompartmentalepidemicmodelsandshowthatitisathresholdparameterforthelocalstabilityofthedisease-freeperiodicsolution,andevenfortheglobaldynamicsundercertaincircumstances.Theremainingpartsofthispaperareorganizedasfollows.Inthenextsection,wepresentthetheoryofthebasicreproductionratioforperiodiccompartmentalmodels.Section3pro-videsthreeexamplestoillustratetheapplicabilityofthebasicreproductionnumberofthetime-averagedsystems.InSect.4,weobtainathresholdconditionfortheglobalpersistenceandextinctionofdiseases.Basedonthisresult,weanalyzeanepidemicmodelwithperiodicpopulationdispersalandperiodiccontactrates.2TheBasicReproductionRatioWeconsideraheterogeneouspopulationwhoseindividualscanbegroupedintonhomoge-neouscompartments.Letx=(x1,...,xn)T,witheachxi≥0,bethestateofindividualsineachcompartment.Weassumethatthecompartmentscanbedividedintotwotypes:infectedcompartments,labeledbyi=1,...,m,anduninfectedcompartments,labeledbyi=m+1,...,n.DefineXstobethesetofalldisease-freestates:Xs:={x≥0:xi=0,∀i=1,...,m}.LetFi(t,x)betheinputrateofnewlyinfectedindividualsintheithcompartment,V+i(t,x)betheinputrateofindividualsbyothermeans(forexample,births,immigrations),andV−i(t,x)betherateoftransferofindividualsoutofcompartmenti(forexample,deaths,recoveryandemigrations).Thus