具有预防接种的非线性传染率传染病模型的稳定性

The Stability for an Epidemic Model with Vaccinal Immunity and Nonlinear Infectious Rate

本文讨论了一个采取预防接种措施的非线性传染率传染病模型,得到了决定疾病流行与否的阈值θ,当θ>1时,仅存在无病平衡点E0,是渐近稳定的,当θ<1时,存在两个平衡点:无病平衡点E0和地方病平衡点E+,其中无病平衡点E0不稳定。在不考虑免疫的丧失或者不考虑因病死亡的因素的情况下,当θ>1时E0全局渐近稳定;当θ<1时E+全局渐近稳定。

An epidemic model with vaccinal immunity and nonlinear infectious rate is analyzed in this paper. Derived is a threshold θ which determines the existence of the infectious disease. When θ 〉 1, there only exists disease free equilibrium point Eo, which is asymptotically stable; when θ〈 1, there exist two equilibrium points, the disease free equilibrium point Eo and endemic equilibrium point E＋, in which the Eo is unstable. Provided that there is no case of losing immunity or death of diseases, when θ 〉 1, the E0 is globally asymptotically stable, and when θ 〈 1, the E＋ is globally asymptotically stable.