基本再生数PPT课件

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1、2021/7/241Threshold Dynamics for Compartmental EpidemicModels in Periodic Environments Introduction The basic reproduction ratio Three examples Threshold dynamics in a patchy model2021/7/242 IntroductionThe basic reproduction ratio is the expected number of secondary cases produced,in a completely s

2、usceptible population,by a typical infective individual.Autonomous epidemic models 7,31Specific infectious diseasesSexual diseases 20Tuberculosis in possums 13Dengue fever 12SARS 15,24,33,40People travel among cities 1,2Patchy models 32,34-362021/7/243Periodic fluctuations(contact rates,birth rates,

3、vaccination program)Intuitively,one may expect to use the basic reproduction number of the time-averaged autonomous system of a periodic epidemic model over a time period.Unfortunately,this average basic reproduction numberis applicable only in certain circumstances,but overestimates or underestimat

4、es infection risks in many other cases.The effective reproduction number is also used in the literature,which is defined as the average number of secondary cases arising from a single typical infective introduced at time t into the population 11.Its magnitude is a useful indicator of both the risk o

5、f an epidemic and the effort required to control an infection.However,this number is not a threshold parameter to determine whether the diseasecan invade the susceptible population successfully.Recently,Bacar and Guernaoui presented a general definition of the basic reproduction number in a periodic

6、 environment4.The purpose of our current paper is to establish the basic reproduction ratio for a large class of periodic compartmental epidemic models and show that it is a threshold parameter for the local stability of the disease-free periodic solution,and even for the global dynamics under certa

7、in circumstances.2021/7/244 The basic reproduction ratioWe consider a heterogeneous population whose individuals can be grouped into n homogeneous compartments.Let with each xi 0,be the state of individuals in each compartment.We assume that the compartments can be divided into two types:infected co

8、mpartments,labeled by i=1,.,m,and uninfected compartments,labeled by i=m+1,.,n.Define Xs to be the set of all disease-free states:Xs:=x 0:xi=0,i=1,.,m.be the input rate of newly infected individuals in the ith compartment.be the input rate of individuals by other means(for example,births,immigration

9、s)be the rate of transfer of individuals out of compartment i(for example,deaths,recovery and emigrations)Tnxxxx),.,(21),(xtFi),(xti),(xti2021/7/245The disease transmission model is governed by a non-autonomous ordinary differential system:2021/7/246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

10、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Qydtdy2021/7/2472021/7/2482021/7/249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。2021/7/24102021/7/2411Ascoli-Arzela theorem:1,0CF 是列紧的当且仅当F为一致有界的且是等度连续的

11、。2021/7/24122021/7/24132021/7/24142021/7/24152021/7/24162021/7/24172021/7/24182021/7/24192021/7/24202021/7/24212021/7/24222021/7/24232021/7/24242021/7/24252021/7/2426 Three examples2021/7/24272021/7/24282021/7/24292021/7/24302021/7/24312021/7/24322021/7/24332021/7/2434 Threshold dynamics in a patchy model2021/7/24352021/7/24362021/7/24372021/7/24382021/7/24392021/7/24402021/7/24412021/7/2442