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1、ThresholdDynamicsforCompartmentalEpidemicModelsinPeriodicEnvironments Introduction Thebasicreproductionratio Threeexamples Thresholddynamicsinapatchymodel Introduction Thebasicreproductionratioistheexpectednumberofsecondarycasesproduced inacompletelysusceptiblepopulation byatypicalinfectiveindividua
2、l Autonomousepidemicmodels 7 31 Specificinfectiousdiseases Sexualdiseases 20 Tuberculosisinpossums 13 Denguefever 12 SARS 15 24 33 40 Peopletravelamongcities 1 2 Patchymodels 32 34 36 Periodicfluctuations contactrates birthrates vaccinationprogram Intuitively onemayexpecttousethebasicreproductionnum
3、berofthetime averagedautonomoussystemofaperiodicepidemicmodeloveratimeperiod Unfortunately thisaveragebasicreproductionnumberisapplicableonlyincertaincircumstances butoverestimatesorunderestimatesinfectionrisksinmanyothercases Theeffectivereproductionnumberisalsousedintheliterature whichisdefinedast
4、heaveragenumberofsecondarycasesarisingfromasingletypicalinfectiveintroducedattimetintothepopulation 11 Itsmagnitudeisausefulindicatorofboththeriskofanepidemicandtheeffortrequiredtocontrolaninfection However thisnumberisnotathresholdparametertodeterminewhetherthediseasecaninvadethesusceptiblepopulati
5、onsuccessfully Recently Baca randGuernaouipresentedageneraldefinitionofthebasicreproductionnumberinaperiodicenvironment 4 Thepurposeofourcurrentpaperistoestablishthebasicreproductionratioforalargeclassofperiodiccompartmentalepidemicmodelsandshowthatitisathresholdparameterforthelocalstabilityofthedis
6、ease freeperiodicsolution andevenfortheglobaldynamicsundercertaincircumstances Thebasicreproductionratio Weconsideraheterogeneouspopulationwhoseindividualscanbegroupedintonhomogeneouscompartments Letwitheachxi 0 bethestateofindividualsineachcompartment Weassumethatthecompartmentscanbedividedintotwot
7、ypes infectedcompartments labeledbyi 1 m anduninfectedcompartments labeledbyi m 1 n DefineXstobethesetofalldisease freestates Xs x 0 xi 0 i 1 m betheinputrateofnewlyinfectedindividualsintheithcompartment betheinputrateofindividualsbyothermeans forexample births immigrations betherateoftransferofindi
8、vidualsoutofcompartmenti forexample deaths recoveryandemigrations 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 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 Ascoli Arzelatheorem 是列紧的当且仅当F为一致有界的且是等度连续的 Threeexamples Thresholddynamicsinapatchymodel
