一类带有周期参数的SIS传染病模型

通过对经典的SIS传染病模引入周期性变化的疾病传播参数,建立了一类具有周期性变化参数的SIS传染病模型。借助微分方程比较定理和稳定性理论,对其进行定性分析,得到了决定疾病灭绝与否以及模型动力学形态的阈值。在该阈值之下,模型的无病周期解是全局渐近稳定的,这意味着疾病最终灭绝；在该阈值之上,模型的无病周期解是不稳定的,同时模型还存在全局渐近稳定的地方病周期解,这意味着疾病将持续存在于种群之中,并且染病者的数量呈周期性变化。

A Kind of SIS Epidemic Model with Periodic Parameters

The spread of some communicable diseases often has a certain periodicity. By incorporating the periodic parameters of disease transmission into the classical SIS epidemic model, an SIS epidemic model with periodic parameters is established. By means of the comparison theorem and the stability theory of ordinary differential equations, the threshold determining whether the disease dies out or not and determining the dynamical behaviors of the model is obtained via qualitative analysis. When the threshold is negative, the disease-free periodic solution of the model is globally asymptotic stable, which implies that the disease dies out eventually. When the threshold is positive, the disease-free periodic solution of the model is unstable, and the model still has a unique endemic periodic solution that is globally stable. This implies that the disease persists in the population, and that the number of the infected individuals will change with a certain periodicity.