Study Of Two Kinds Of SIRS Epidemic Models With Delays

This paper addresses two time-delayed SIRS epidemic models. One is a model with a nonlinear incidence rate. The other is the one with a linear incidence rate. Immunity gained by experiencing the disease is temporary; whenever infected, the disease individuals will return to the susceptible class after a fixed period of time. The following questions are studied. First the local and global stabilities of the infection-free equilibrium are analyzed, respectively. Second, the endemic equilib-rium is formulated in terms of the incidence rate, and its locally asymptotic sta-bility are proved. Third by using a Lyapunov functional, a sufficient criterion for the globally stability of the endemic equilibrium is established. Last we studied the unchanged straight line and the super critical bifurcation of the first model. Our results obtained here extend some corresponding conclusions given in the related references.There are three chapters in this paper. The first one contains a brief introduc-tion to the process and significance of the study of the epidemic model, and the main problems discussed in this paper are also presented. The second one investi-gates the stability, the existence of invariant sets and the bifurcations of the SIRS epidemic model with the nonlinear incidence rate. The last one is devoted to the global analysis of the dynamics of the SIRS epidemic model with the linear incidence rate.