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

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

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αK≤1时,正平衡点全局渐近稳定;当R_0>1且△=0时,系统在正平衡点附近发生Hopf分支;当R_0>1且△<0时,系统在正平衡点外围附近存在唯一稳定的极限环.

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 R0 = 1. The author proves that when R0 〈 1, the nondisease equilibrium is globally asymptotically stable; when R0 〉 1 and αK ≤ 1, the positive equilibrium is globally asymptotically stable; when Ro 〉 1 and Δ = 0, a Hopf bifurcation occurs near the positive equilibrium; when R0 〉 1 and Δ〈 0, the system has a unique limit cycle which is stable near the outside of the positive equilibrium.