关键词: 平衡点, 全局渐近稳定, 极限环, Hopf分支

Abstract:

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.

Key words: Equilibrium, Global asymptotic stability, Limit cycle, Hopf bifurcation