基本再生数

ThresholdDynamicsforCompartmentalEpidemicModelsinPeriodicEnvironments●Introduction●Thebasicreproductionratio●Threeexamples●Thresholddynamicsinapatchymodel●IntroductionThebasicreproductionratioistheexpectednumberofsecondarycasesproduced,inacompletelysusceptiblepopulation,byatypicalinfectiveindividual.Autonomousepidemicmodels[7,31]SpecificinfectiousdiseasesSexualdiseases[20]Tuberculosisinpossums[13]Denguefever[12]SARS[15,24,33,40]Peopletravelamongcities[1,2]Patchymodels[32,34-36]Periodicfluctuations(contactrates,birthrates,vaccinationprogram)Intuitively,onemayexpecttousethebasicreproductionnumberofthetime-averagedautonomoussystemofaperiodicepidemicmodeloveratimeperiod.Unfortunately,thisaveragebasicreproductionnumberisapplicableonlyincertaincircumstances,butoverestimatesorunderestimatesinfectionrisksinmanyothercases.Theeffectivereproductionnumberisalsousedintheliterature,whichisdefinedastheaveragenumberofsecondarycasesarisingfromasingletypicalinfectiveintroducedattimetintothepopulation[11].Itsmagnitudeisausefulindicatorofboththeriskofanepidemicandtheeffortrequiredtocontrolaninfection.However,thisnumberisnotathresholdparametertodeterminewhetherthediseasecaninvadethesusceptiblepopulationsuccessfully.Recently,BacaërandGuernaouipresentedageneraldefinitionofthebasicreproductionnumberinaperiodicenvironment[4].Thepurposeofourcurrentpaperistoestablishthebasicreproductionratioforalargeclassofperiodiccompartmentalepidemicmodelsandshowthatitisathresholdparameterforthelocalstabilityofthedisease-freeperiodicsolution,andevenfortheglobaldynamicsundercertaincircumstances.●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)Tnxxxx),...,,(21),(xtFi),(xti),(xtiThediseasetransmissionmodelisgovernedbyanon-autonomousordinarydifferentialsystem: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xtAdtdx)(nnijtatA))(()()()(tATtA)(t.)()(Qtetpt的零解稳定性将xtAdtdx)(零解的稳定性。转化为Qydtdy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,Pxyyx使E按构成有序Banach空间。此时锥xExP称为E的正元锥。Ascoli-Arzelatheorem:]1,0[CF是列紧的当且仅当F为一致有界的且是等度连续的。●Threeexamples●Thresholddynamicsinapatchymodel