基本再生数ppt课件

ThresholdDynamicsforCompartmentalEpidemicModelsinPeriodicEnvironments,IntroductionThebasicreproductionratioThreeexamplesThresholddynamicsinapatchymodel,1,Introduction,Thebasicreproductionratioistheexpectednumberofsecondarycasesproduced,inacompletelysusceptiblepopulation,byatypicalinfectiveindividual.,Autonomousepidemicmodels7,31,Specificinfectiousdiseases,Sexualdiseases20Tuberculosisinpossums13Denguefever12SARS15,24,33,40,Peopletravelamongcities1,2Patchymodels32,34-36,2,Periodicfluctuations(contactrates,birthrates,vaccinationprogram)Intuitively,onemayexpecttousethebasicreproductionnumberofthetime-averagedautonomoussystemofaperiodicepidemicmodeloveratimeperiod.Unfortunately,thisaveragebasicreproductionnumberisapplicableonlyincertaincircumstances,butoverestimatesorunderestimatesinfectionrisksinmanyothercases.Theeffectivereproductionnumberisalsousedintheliterature,whichisdefinedastheaveragenumberofsecondarycasesarisingfromasingletypicalinfectiveintroducedattimetintothepopulation11.Itsmagnitudeisausefulindicatorofboththeriskofanepidemicandtheeffortrequiredtocontrolaninfection.However,thisnumberisnotathresholdparametertodeterminewhetherthediseasecaninvadethesusceptiblepopulationsuccessfully.Recently,BacarandGuernaouipresentedageneraldefinitionofthebasicreproductionnumberinaperiodicenvironment4.Thepurposeofourcurrentpaperistoestablishthebasicreproductionratioforalargeclassofperiodiccompartmentalepidemicmodelsandshowthatitisathresholdparameterforthelocalstabilityofthedisease-freeperiodicsolution,andevenfortheglobaldynamicsundercertaincircumstances.,3,Thebasicreproductionratio,Weconsideraheterogeneouspopulationwhoseindividualscanbegroupedintonhomogeneouscompartments.Letwitheachxi0,bethestateofindividualsineachcompartment.Weassumethatthecompartmentscanbedividedintotwotypes:infectedcompartments,labeledbyi=1,.,m,anduninfectedcompartments,labeledbyi=m+1,.,n.DefineXstobethesetofalldisease-freestates:Xs:=x0:xi=0,i=1,.,m.betheinputrateofnewlyinfectedindividualsintheithcompartment.betheinputrateofindividualsbyothermeans(forexample,births,immigrations)betherateoftransferofindividualsoutofcompartmenti(forexample,deaths,recoveryandemigrations),4,Thediseasetransmissionmodelisgovernedbyanon-autonomousordinarydifferentialsystem:,5,考虑周期线性系统。其中，连续，是以T为周期的周期函数。记其基本解矩阵为。关于其零解的稳定性讨论起至关重要的作用。引理：存在非奇异可微周期矩阵p(t)，以及一个常数矩阵Q，使得,6,7,8,有序Banach空间：设E为Banach空间，P为E中的闭凸锥，则可由P引出E中的序关系,使E按构成有序Banach空间。此时锥,称为E的正元锥。,9,10,Ascoli-Arzelatheorem:,是列紧的当且仅当F为一致有界的且是等度连续的。,11,12,13,14,