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1、Threshold Dynamics for Compartmental EpidemicModels in Periodic Environments, Introduction The basic reproduction ratio Three examples Threshold dynamics in a patchy model, Introduction,The basic reproduction ratio is the expected number of secondary cases produced, in a completely susceptible popul
2、ation, by a typical infective individual.,Autonomous epidemic models 7, 31,Specific infectious diseases,Sexual diseases 20Tuberculosis in possums 13Dengue fever 12SARS 15, 24, 33, 40,People travel among cities 1, 2Patchy models 32, 34-36,Periodic fluctuations (contact rates, birth rates, vaccination
3、 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 underestimates inf
4、ection 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 of an
5、 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 cert
7、ain circumstances., The basic reproduction ratio,We 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 compa
8、rtments, 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 othe
9、r means (for example, births, immigrations) be the rate of transfer of individuals out of compartment i (for example, deaths, recovery and emigrations),The disease transmission model is governed by a non-autonomous ordinary differential system:,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-Arzela theorem:,是列紧的当且仅当F为一致有界的且是等度连续的。, Three examples, Threshold dynamics in a patchy model,
