具扩散SIR传染病模型的平衡态的稳定性分析

Stability analysis of a class of SIR endemic model with diffusion

考虑一类具Laplace扩散项的SIR传染病模型.通过对其线性化系统的特征值进行理论分析,研究模型常数平衡态的稳定性问题.当基本再生数小于1时,无疾平衡态是局部渐近稳定的;当基本再生数大于1时,无疾平衡态不稳定且地方病平衡态局部渐近稳定.理论分析和数值模拟结果表明,如果要控制传染病的爆发,可以采取降低易感者与感染者的接触,提高治愈效率以及隔离感染者等方式来降低基本再生数的数值.

This article considers a class of SIR endemic model with diffusion.The stability of the constant steady states of the model is considered by analyzing the characteristic values of its linearized system.The results show that the disease-free steady state is locally stable when the basic reproduction number is less than 1,while the disease-free steady state is unstable and the endemic steady state is locally stable when the basic reproduction number is greater than 1.Both of theoretical analysis and the simulations suggest that some strategies can be done to control the outbreak of the epidemics,such as decreasing the contact between susceptibles and infectives,increasing the cure efficiency or isolating infectives to decrease the basic reproduction number.