Qualitative Analysis Of Two Epidemic Models

In this paper, two kinds of epidemic models are studied.Firstly, we study an SEIR epidemic models with general 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.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, a unique endemic equilibrium exists and is globally asymptotically stable in the interior of the feasible region under certain condition H which we will give in2.2. Numerical simulations are carried out to illustrate the feasibility of the obtained results and the effect of quarantine to control the disease.Secondly, we consider an SEIR epidemic model with saturated recovery rate. The recovery rate can describe the effect of delayed treatment when the number of infected individuals is getting larger and the medical condition is limited. It is shown that this model has two thresholds and a backward bifurcation will take place for some conditions. Furthermore, global dynamics are shown by geometric approaches. Numerical simulations are presented to illustrate the results and suggest that giving the patients timely treatment, improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of disease.