Dynamic Behaviors For Two Classes Of Discretized SEIR Epidemic Models

Epidemic dynamics is an important branch of the biological mathematics. Gener-ally, mathematical models on epidemic disease are usually established according to the mechanism of epidemic transmission. Further, based on the theoretical and mathemat-ical results, the dynamic behaviours of epidemic models have been studied. Finally, the causes and key factors of the epidemics are analysed, and some feasible advices for con-trolling the epidemics are raised. In this paper, we discretize two classes of continuous SEIR epidemic models by applying suitable Euler discretization method, then derive the corresponding discretized SEIR epidemic models. By comparison theorem and sta-bility theory for difference equations, we analyze the dynamic behaviours of systems, which include the stability of the disease-free equilibrium and the permanence of the systems. Further, we verify our results by numerical simulations.In chapter one, we introduce the background and significance of epidemic dynamics, and give some basic definitions and preliminaries of the paper.In chapter two, we study the dynamic behaviours of a discretized SEIRS epidemic model with time delays. By using backward Euler discretization method, we establish the discretized SEIRS model, and study the positivity and upper positive boundedness of the solution sequences. We prove that if the crucial value Ro is less than 1, the disease-free equilibrium is globally asymptotically stable by Lyapunov method. Further, we consider the permanence of the system, and obtain the sufficient conditions which guarantee the permanence of the system.Since the standard incidence rate is more realistic than the bilinear incidence rate when we consider epidemic models, in chapter three, we discuss a class of SEIR epidemic model with standard incidence rate. The discretized epidemic model is established by mixed Euler discretization method. It is proven that if R0< 1, the disease-free equilibrium is globally asymptotically stable. Further, the permanence of the discrete SEIR epidemic model is considered, and such a discretized epidemic model is permanent if Ro> 1.In chapter four, we summary our works, and point out the shortcomings for the paper. At last, we make a prospect for the future work.