The Study Of Some Classes Of Epidemic Models With Nonlinear Birth In Population

In this paper, we mainly study the dynamics of some classes of epidemic model with nonlinear birth 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:an epidemic model involving nonlinear birth in population and vertical transmission was studied. When R0<1, the disease-free equilibrium was stable, while if R0>1, the disease-free equilibrium was unstable. We researched the existence of Hopf bifurcation and obtained the stability and direction of the Hopf bifurcation by using the normal theory and the center manifold theorem. Numerical simulations were carried out to illustrate the main theoretical results.In Chapter3:an epidemic model involving nonlinear birth in population and stage struc-ture was studied. The stability of disease-free equilibrium was verified by Routh-Hurwitz cri-terion and LaSalle’s invariance principle. We researched the existence of Hopf bifurcation and obtained the stability and direction of the Hopf bifurcation by using the normal theory and the center manifold theorem. In the end, numerical simulations were carried out to illustrate the main theoretical results.In Chapter4:a new SIR epidemic model with double epidemic hypothesis and delays is proposed, which is a system of impulsive functional differential equations with time delays and with vertical transmission. Using the discrete dynamical system determined by the stroboscopic map. we obtain the exact disease-free periodic solution. We also show that the disease-free periodic solution of this system is globally attractive when R0<1and the system is permanent under R0>1. Numerical simulations were carried out to illustrate the main theoretical results.