The Study Of Some Classes Of Epidemic Models With Saturated Incidence Rate

In this paper, we mainly study the dynamics of some classes of epidemic model with saturated incidence rate in population. The article includes four chapters.The preface is in chapter 1, we introduce the research background of this article, overseas and domestic research status, the main task and some important preliminaries.In Chapter 2, the dynamics of SEIR, epidemic model with saturated incidence rate and saturated treatment function is explored. The basic reproduction number is given. The ex-isting threshold conditions of all kinds of the equilibrium points and backward bifurcation are obtained. The local asymptotical stability of equilibria are verified by analyzing the eigenvalues. We also discuss the global asymptotical stability of the endemic equilibrium by autonomous convergence theorem. Numerical simulations are presented to support and complement the theoretical findings.In Chapter 3, a delayed SIR epidemic model with information variable, saturated incidence rate and saturated treatment function is studied. The eigenvalue method is applied to analyze the stability of disease-free equilibrium and endemic equilibrium. Choosing the delay as a bifur-cation parameter, we establish a set of sufficient conditions for the existence of Hopf bifurcation. Moreover, the stability of bifurcating periodic solutions and the direction of Hopf bifurcation are obtained by using the normal form theory and center manifold argument. Finally, numerical simulations are presented to explain the mathematical conclusions.In Chapter 4, an SIR epidemic model with periodic variant saturated incidence and time-varying pulse vaccination rate is studied. The threshold value which determines permanence and extinction of the disease is established. Our results imply that the disease will die out eventually if the basic reproduction number is less than unity, whereas the diseases will persist if the basic reproduction number is larger than 1+a*A/d. Finally, numerical simulations are presented to support our main results.