The Study Of Some Classes Of Epidemic Models In A Polluted Environment

In this paper, we mainly study the dynamics of some classes of epidemic model in a polluted environment. The article includes four chapters.The preface is in chapter 1. we introduce the research background of this article, the main task and some important preliminaries.In Chapter 2, the dynamics of SIR epidemic model with saturated incidence rate in a polluted environment is explored. In a polluted environment, considering the biological pop-ulation infected with some kinds of diseases and hunted by human beings, we formulate two SIR pollution-epidemic models with continuous and impulsive external effects, respectively, and investigate the dynamics of such systems. We assume that only the susceptible population is hunted by human beings. For the continuous system, we obtain the global asymptotical stabil-ity of equilibria. For the impulsive system, by using the comparison theorem and the analysis method, we show that under different conditions the disease-free periodic solution is globally asymptotically attractive, or the system is permanent. Numerical simulations are presented to support and complement the theoretical findings.In Chapter 3, a nonautonomous epidemic model with distributed delay in a polluted envi-ronment is studied. We establish some sufficient conditions on the permanence and extinction of the disease by using inequality analytical techniques. By a Lyapunov functional method, we also obtain some sufficient conditions for global asymptotic stability of this model. Finally, numerical simulations are presented to explain the mathematical conclusions.In Chapter 4, the stability and Hopf bifurcation of a delayed SEIR pollution-epidemic model with saturated incidence rate and saturated treatment function is studied. The eigen-value method and Routh-Hurwitz criterion are applied to analyze the stability of discase-free equilibrium and endemic equilibrium. Choosing the delay as a bifurcation parameter, we tablish a set of sufficient conditions for the existence of Hopf bifurcation. Finally, numerical simulations are presented to explain the mathematical conclusions.