Stability and Hopf Bifurcation of an SIS Model with Species Logistic Growth and Saturating Infect Rate
具有种群Logistic增长及饱和传染率的SIS模型的稳定性和Hopf分支

In this paper, an SIS infective model with species Logistic growth and saturating infective rate is studied. The author discusses the existence and the globally asymptotical stability of the equilibrium, and obtains the threshold value at which disease is eliminated, which is just the basic rebirth number $R_{0}=1$. The author proves that when$R_{0}<1$, the non-disease equilibrium is globally asymptotically stable; when $R_{0}>1$ and $\alpha K\leq 1$, the positive equilibrium is globally asymptotically stable; when $R_{0}>1$ and $ \Delta =0 $, a Hopf bifurcation occurs near the positive equilibrium; when $R_{0}>1$ and $ \Delta <0 $, the system has a unique limit cycle which is stable near the outside of the positive equilibrium.