The Dynamics Analysis Of Two Class Of Discrete SI Epidemic Models

This dissertation focuses on two classes of discrete-time epidemic dynamic models. Some dynamical behaviors such as stability, persistence, and bifurcation phenomena of the models are studied and the corresponding sufficient conditions are obtained.In the first chapter, the research background and general situation of epidemic,the research significance and progress of discrete epidemic models are introduced.Meanwhile, the main task of this paper is summarized.In the second chapter, we give some important definitions and lemma.In the third chapter, we study a class of discrete SI epidemic model with bilinear incidence rate. We get a class of discrete SI epidemic model by using the method of mixture Euler discrete. The basic reproduction number R₀ determining the dynamic properties is obtained. The sufficient conditions ensuring stability, uniform persistence and bifurcation of the model are obtained by regarding R₀ as a parameter.Numerical simulations are given to illustrate the theoretical results. With the resultsas follows: when h is a positive number which is sufficiently small, a 3bK,if R₀ ?1, the disease-free equilibrium is globally asymptotically stable. And if R₀ >1,the epidemic equilibrium is permanent. At the same time, by choosing appropriate parameters, and using the Flip-bifurcation theory and Neimark-Sacker branch existence theory, bifurcation of the model is discussed, and the corresponding numerical simulations are given to illustrate the theoretical results.In the fourth chapter, we obtain a class of discrete Lotka-Volterra model through discretizing the corresponding continuous-time susceptible population with the Logistic growth pattern. By studying the discrete model, we get the existence and stability of the equilibria. Some sufficient conditions for the globally asymptotical stability of the disease-free equilibrium are given, and the permanence of the epidemic equilibrium is also proved. At the same time, by choosing appropriate parameters, and using the Flip-bifurcation theory and Neimark-Sacker branch existence theory, bifurcation of the model is discussed, and the corresponding numerical simulations are given to illustrate the theoretical results.