Two Diffusive SIS Epidemic Models With Saturated Incidence Function And Spontaneous Infection In A Heterogeneous Environment

In this thesis,we are concerned with a diffusive SIS epidemic model with saturated incidence function and spontaneous infection in a heterogeneous environment.In the case that the disease-induced mortality rate is positive every-where,we show that the disease always goes to extinction and the susceptible individuals will be stabilized at a constant,zero or positive,depending on the others parameters.This paper includes four chapters as follow:In chapter 1,we introduce the backgrounds and the main content of this paper.In chapter 2,in the case that the disease-induced mortality rate is negligible,we consider the model in a spatially heterogeneous and time-periodic environment and establish a threshold dynamics for the disease extinction or persistence in terms of the basic reproduction number;in particular,in the homogeneous environment,by constructing suitable Lyapunov function,we examine the global attractivity of the diseasefree equilibrium and endemic equilibrium.In chapter 3,if the movement(migration)rate of the susceptible or infected population is small,this paper analyze their asymptotic profile.Compared to the case that saturated incidence function and spontaneous infection are ignored,theoretical study of this paper here shows that spontaneous infection can enhance persistence of infections disease.In chapter 4,we discuss the theoretical results and biological significance of this paper.