| In this paper, we mainly study the dynamics of some classes of epidemic models with impulse effects and delays. 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 dynamical behavior of an SIRS epidemic model with birth pulse, pulse vaccination and saturated cure rate is studied by means of both theoretical and numerical ways. First, we investigate the existence and stability of the infection-free periodic solution and the condition for the existence of bifurcation. Second, we analyze the bifurcation of the epidemic period solution. Then we construct a poincare map, the poincare map and center manifold theorem are used to discuss Flip bifurcation. Finally, numerical simulations for phase portraits and periodic solutions which are illustrated with an example are in good agreement with the theoretical analysis.In Chapter 3, we formulate and analyze a new delayed SEIRS epidemic model with pulse vaccination, nonlinear incidence rate and saturation incidence rate. We obtain the sufficient conditions for the global a.ttractivity of the infection-free periodic solution and permanence of the system through comparison theorem. Finally, numerical simulations verify the theoretical results by using MATLAB.In Chapter 4, an SVEIR epidemic model with distributed delay is discussed. We obtain the basic reproduction number R0 and equilibria. The global asymptotic stability of equilibria is proved by using Lyapunov method and LaSalle’s invariance principle. It’s shown that the disease-free equilibria is globally asymptotically stable if R0≤1, while when R0>1, there exists a unique positive endemic equilibrium which is globally asymptotically stable. |
