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

该文研究一类具有种群Logistic增长及饱和传染率的SIS传染病模型,讨论了平衡点的存在性及全局渐近稳定性,得到疾病消除的阈值就是基本再生数$R_{0}=1$. 证明了,当$R_{0}<1$ 时,无病平衡点全局渐近稳定;当$R_{0}>1$ 且$\alpha K\leq 1$ 时,正平衡点全局渐近稳定;当$R_{0}>1$ 且$\Delta ={0}$ 时,系统在正平衡点附近发生Hopf分支;当$R_{0}>1$ 且$\Delta <{0}$ 时,系统在正平衡点外围附近存在唯一稳定的极限环.

Stability and Hopf Bifurcation of an SIS Model with Species Logistic Growth and Saturating Infect Rate

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.