The Study Of Some Classes Of Epidemic Models With Information Variable

In this paper, we mainly study the dynamics of some classes of epidemic model with information variable in population. The article includes four chapters.The preface is in chapter1, we introduce the research background of this article, the main task and some important preliminaries.In Chapter2, the stability of the SIR epidemic model with information variable and limited medical resources was studied. When the basic reproduction ratio B0<1, there exists the disease-free equilibrium and when the basic reproduction ratio B0>1, we obtain the sufficient conditions of the existence of the endemic equilibrium. The local asymptotical stability of equilibrium is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion. We also discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov function and the endemic equilibrium by autonomous convergence theorem. A numerical analysis is given to show the effectiveness of the main results.In Chapter3, Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease free equilibrium is discussed. If the basic reproduction ratio B0<1, we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. If B0>1, we obtain sufficient conditions under which the endemic equilibrium E*of system is locally asymptotically by using the Routh-Hurwitz criterion. And we also researched the existence of Hopf bifurcation and obtained the stability and the direction of the Hopf bifurcation by using the normal theory and the center theoretical results. In the end, numerical simulations are carried out to explain the mathematical conclusions.In Chapter4, the dynamics of an SIR epidemic model with information variable and ex-ternal effects in a polluted environment revisited. In a polluted environment, considering the biological population 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. We obtained the sufficient conditions of the existence of the endemic equilibrium. Then, the local asymptotical stability of equilibrium is verified by ana-lyzing the corresponding characteristic equations and using the Lyapunov functional. Numerical simulations were carried out to illustrate the main theoretical results.