基本再生數(shù)課件

ThresholdDynamicsforCompartmentalEpidemicModelsinPeriodicEnvironments●Introduction●Thebasicreproductionratio●Threeexamples●ThresholddynamicsinapatchymodelThresholdDynamicsforCompart1●IntroductionThebasicreproductionratioistheexpectednumberofsecondarycasesproduced,inacompletelysusceptiblepopulation,byatypicalinfectiveindividual.Autonomousepidemicmodels[7,31]SpecificinfectiousdiseasesSexualdiseases[20]Tuberculosisinpossums[13]Denguefever[12]SARS[15,24,33,40]Peopletravelamongcities[1,2]Patchymodels[32,34-36]●IntroductionThebasicreprod2Periodicfluctuations(contactrates,birthrates,vaccinationprogram)Intuitively,onemayexpecttousethebasicreproductionnumberofthetime-averagedautonomoussystemofaperiodicepidemicmodeloveratimeperiod.Unfortunately,thisaveragebasicreproductionnumberisapplicableonlyincertaincircumstances,butoverestimatesorunderestimatesinfectionrisksinmanyothercases.Theeffectivereproductionnumberisalsousedintheliterature,whichisdefinedastheaveragenumberofsecondarycasesarisingfromasingletypicalinfectiveintroducedattimetintothepopulation[11].Itsmagnitudeisausefulindicatorofboththeriskofanepidemicandtheeffortrequiredtocontrolaninfection.However,thisnumberisnotathresholdparametertodeterminewhetherthediseasecaninvadethesusceptiblepopulationsuccessfully.Recently,Baca?randGuernaouipresentedageneraldefinitionofthebasicreproductionnumberinaperiodicenvironment[4].Thepurposeofourcurrentpaperistoestablishthebasicreproductionratioforalargeclassofperiodiccompartmentalepidemicmodelsandshowthatitisathresholdparameterforthelocalstabilityofthedisease-freeperiodicsolution,andevenfortheglobaldynamicsundercertaincircumstances.Periodicfluctuations(contact3●ThebasicreproductionratioWeconsideraheterogeneouspopulationwhoseindividualscanbegroupedintonhomogeneouscompartments.Letwitheachxi≥0,bethestateofindividualsineachcompartment.Weassumethatthecompartmentscanbedividedintotwotypes:infectedcompartments,labeledbyi=1,...,m,anduninfectedcompartments,labeledbyi=m+1,...,n.DefineXstobethesetofalldisease-freestates:Xs:={x≥0:xi=0,?i=1,...,m}.betheinputrateofnewlyinfectedindividualsintheithcompartment.

betheinputrateofindividualsbyothermeans(forexample,births,immigrations)

betherateoftransferofindividualsoutofcompartmenti(forexample,deaths,recoveryandemigrations)●Thebasicreproductionratio4Thediseasetransmissionmodelisgovernedbyanon-autonomousordinarydifferentialsystem:Thediseasetransmissionmodel5考慮周期線性系統(tǒng)。其中，連續(xù)，是以T為周期的周期函數(shù)。記其基本解矩陣為。關于其零解的穩(wěn)定性討論起至關重要的作用。引理：存在非奇異可微周期矩陣p(t)，以及一個常數(shù)矩陣Q，使得考慮周期線性系統(tǒng)。其中6基本再生數(shù)課件7基本再生數(shù)課件8有序Banach空間：設E為Banach空間，P為E中的閉凸錐，則可由P引出E中的序關系使E按構成有序Banach空間。此時錐稱為E的正元錐。有序Banach空間：使E按構成有序Banach空9基本再生數(shù)課件10Ascoli-Arzelatheorem:是列緊的當且僅當F為一致有界的且是等度連續(xù)的。Ascoli-Arzelatheorem:是列緊的當且僅當11基本再生數(shù)課件12基本再生數(shù)課件13基本再生數(shù)課件14基本再生數(shù)課件15基本再生數(shù)課件16基本再生數(shù)課件17基本再生數(shù)課件18基本再生數(shù)課件19基本再生數(shù)課件20基本再生數(shù)課件21基本再生數(shù)課件22基本再生數(shù)課件23基本再生數(shù)課件24基本再生數(shù)課件25●Threeexamples●Threeexamples26基本再生數(shù)課件27基本再生數(shù)課件28基本再生數(shù)課件29基本再生數(shù)課件30基本再生數(shù)課件31基本再生數(shù)課件32基本再生數(shù)課件33●Thresholddynamicsinapatchymodel●Thresholddynamicsinapatc34基本再生數(shù)課件35基本再生數(shù)課件36基本再生數(shù)課件37基本再生數(shù)課件38基本再生數(shù)課件39基本再生數(shù)課件40基本再生數(shù)課件41基本再生數(shù)課件42ThresholdDynamicsforCompartmentalEpidemicModelsinPeriodicEnvironments●Introduction●Thebasicreproductionratio●Threeexamples●ThresholddynamicsinapatchymodelThresholdDynamicsforCompart43●IntroductionThebasicreproductionratioistheexpectednumberofsecondarycasesproduced,inacompletelysusceptiblepopulation,byatypicalinfectiveindividual.Autonomousepidemicmodels[7,31]SpecificinfectiousdiseasesSexualdiseases[20]Tuberculosisinpossums[13]Denguefever[12]SARS[15,24,33,40]Peopletravelamongcities[1,2]Patchymodels[32,34-36]●IntroductionThebasicreprod44Periodicfluctuations(contactrates,birthrates,vaccinationprogram)Intuitively,onemayexpecttousethebasicreproductionnumberofthetime-averagedautonomoussystemofaperiodicepidemicmodeloveratimeperiod.Unfortunately,thisaveragebasicreproductionnumberisapplicableonlyincertaincircumstances,butoverestimatesorunderestimatesinfectionrisksinmanyothercases.Theeffectivereproductionnumberisalsousedintheliterature,whichisdefinedastheaveragenumberofsecondarycasesarisingfromasingletypicalinfectiveintroducedattimetintothepopulation[11].Itsmagnitudeisausefulindicatorofboththeriskofanepidemicandtheeffortrequiredtocontrolaninfection.However,thisnumberisnotathresholdparametertodeterminewhetherthediseasecaninvadethesusceptiblepopulationsuccessfully.Recently,Baca?randGuernaouipresentedageneraldefinitionofthebasicreproductionnumberinaperiodicenvironment[4].Thepurposeofourcurrentpaperistoestablishthebasicreproductionratioforalargeclassofperiodiccompartmentalepidemicmodelsandshowthatitisathresholdparameterforthelocalstabilityofthedisease-freeperiodicsolution,andevenfortheglobaldynamicsundercertaincircumstances.Periodicfluctuations(contact45●ThebasicreproductionratioWeconsideraheterogeneouspopulationwhoseindividualscanbegroupedintonhomogeneouscompartments.Letwitheachxi≥0,bethestateofindividualsineachcompartment.Weassumethatthecompartmentscanbedividedintotwotypes:infectedcompartments,labeledbyi=1,...,m,anduninfectedcompartments,labeledbyi=m+1,...,n.DefineXstobethesetofalldisease-freestates:Xs:={x≥0:xi=0,?i=1,...,m}.betheinputrateofnewlyinfectedindividualsintheithcompartment.

betheinputrateofindividualsbyothermeans(forexample,births,immigrations)

betherateoftransferofindividualsoutofcompartmenti(forexample,deaths,recoveryandemigrations)●Thebasicreproductionratio46Thediseasetransmissionmodelisgovernedbyanon-autonomousordinarydifferentialsystem:Thediseasetransmissionmodel47考慮周期線性系統(tǒng)。其中，連續(xù)，是以T為周期的周期函數(shù)。記其基本解矩陣為