Study Of Two Kinds Of Epidemic Models

In this paper, two kinds of epidemic models are studied.Firstly, we consider the SIRS epidemic model with standard incidence rate under the hypotheses of vertical transmission and the vaccination of susceptibles and new-borns. It has been shown that this model always has a disease-free equilibrium, and has a unique endemic equilibrium under some given conditions. Then, by considering the distribution of the equilibriums and using Hurwitz and center manifold theorems, the local asymptotical stability and supercritical bifurcation of the equilibriums are given. At last, the global asymptotical stability of the endemic equilibrium is achieved by means of the formula of Stokes.Secondly, we study an SIQS epidemic model with nonlinear incidence rate and constants inflows, both for susceptibles and infectives. The nonlinear incidence rate, together with the introduction of constants inflows, brings us much more difficulties. It shows that this model not only has a disease-free equilibrium, but also has an endemic equilibrium when no infective inflow exits, and it has only a unique endemic equilibrium when the infective inflow exits. The local asymptotical stability of the equilibriums is derived by the joint application of the Hurwitz and center manifold theorems. Fur-thermore, the global asymptotical stability of the endemic equilibrium is also given by using the Liapunov function.Our results obtained here extend some corresponding conclusions given in the related references.