基本再生數(shù)實用PPT課件PPT課件

1、 IntroductionThe basic reproduction ratio is the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual.Autonomous epidemic models 7, 31Specific infectious diseasesSexual diseases 20Tuberculosis in possums 13Dengue fever 12SARS 15, 24,

2、33, 40People travel among cities 1, 2Patchy models 32, 34-36第1頁/共42頁Periodic fluctuations (contact rates, birth rates, 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. Unfortu

3、nately, this average basic reproduction numberis applicable only in certain circumstances, but overestimates or underestimates 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

4、a single typical infective introduced at time t into the population 11. Its magnitude is a useful indicator of both the risk of 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 po

5、pulation successfully.Recently, Bacar and Guernaoui presented a general definition of the basic reproduction number in a periodic 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 i

6、s a threshold parameter for the local stability of the disease-free periodic solution, and even for the global dynamics under certain circumstances.第2頁/共42頁 The basic reproduction ratioWe consider a heterogeneous population whose individuals can be grouped into n homogeneous compartments. Let with e

7、ach xi 0, be the state of individuals in each compartment. We assume that the compartments can be divided into two types: infected compartments, 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 : x

8、i = 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, immigrations) be the rate of transfer of individuals out of compartment i (for example, deaths, recovery and emigrations)Tnxxxx),.,(21

9、),( xtFi), ( xti), ( xti第3頁/共42頁The disease transmission model is governed by a non-autonomous ordinary differential system:第4頁/共42頁考慮周期線性系統(tǒng) 。其中， 連續(xù)， 是以T為周期的周期函數(shù)。記其基本解矩陣為 。關于其零解的穩(wěn)定性討論起至關重要的作用。引理：存在非奇異可微周期矩陣p(t)，以及一個常數(shù)矩陣Q，使得xtAdtdx)(nnijtatA)()()()(tATtA)(t.)()(Qtetpt 的零解穩(wěn)定性將xtAdtdx)(零解的穩(wěn)定性。轉化為Qydtdy

10、第5頁/共42頁第6頁/共42頁第7頁/共42頁有序Banach空間：設E為Banach空間，P為E中的閉凸錐，則可由P引出E中的序關系,Pxyyx使E按 構成有序Banach空間。此時錐xExP稱為E的正元錐。第8頁/共42頁第9頁/共42頁Ascoli-Arzela theorem:1 ,0CF 是列緊的當且僅當F為一致有界的且是等度連續(xù)的。第10頁/共42頁第11頁/共42頁第12頁/共42頁第13頁/共42頁第14頁/共42頁第15頁/共42頁第16頁/共42頁第17頁/共42頁第18頁/共42頁第19頁/共42頁第20頁/共42頁第21頁/共42頁第22頁/共42頁第23頁/共42頁第24頁/共42頁 Three examples第25頁/共42頁第26頁/共42頁第27頁/共42頁第28頁/共42頁第29頁/共42頁