Qualitative Analysis Of Two Epidemic Models

In this paper, we study two epidemic models in chapter two and chapter three, respectively.In chapter two, we study a SEIQR epidemic model with a class of nonlinear incidence rate. The model always exhibits the disease-free equilibrium, and the unique endemic equilibrium turns up if and only if the basic reproduction number R0>1. It is shown that if R0≤1, the disease-free equilibrium is globally asymptotically stable and if R0>1, the disease-free equilibrium is unstable. Moreover, we show that if R0>1,the disease is uniformly persistent and the unique endemic equilibrium is globally asymptotically stable under certain condition H which we will give in§2.2. Numerical simulations are carried out to illustrate the feasibility of the obtained results and the effect of quarantine to eliminate the disease.In chapter three, we consider a SIR epidemic model with non-monotone incidence rate and time delay. Through analysis we get the relative of the existence of the disease free equilibrium and endemic equilibrium between the basic reproduction number R0. We show that the disease-free equilibrium is globally asymptotically stable when R0≤1, and if R0>1, the disease free equilib-rium is unstable but the disease is persistent, and the endemic equilibrium is globally asymptoti-cally stable if the parameters satisfied some conditions.