基本再生数.ppt

● Threshold dynamics in a patchy model * Threshold Dynamics for Compartmental Epidemic Models 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 population, by a typical infective individual. Autonomous epidemic models [7, 31] Specific infectious diseases Sexual diseases [20] Tuberculosis in possums [13] Dengue fever [12] SARS [15, 24, 33, 40] People travel among cities [1, 2] Patchy models [32, 34-36] 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. Unfortunately, this average basic reproduction number is 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 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 disease can invade the susceptible population successfully. Recently, Baca?r and Guernaoui presented a general definition of the basic reproduction number in a periodic environment[4]. 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 certain circumstances. ● The basic reproduction ratio We consider a heterogeneous population whose individuals can be gro