基于两斑块和人口流动的SIR传染病模型的稳定性

根据传染病动力学原理，考虑人口在两斑块上流动且具有非线性传染率，建立了一类基于两斑块和人口流动的SIR传染病模型.利用常微分方程定性与稳定性方法，分析了模型永久持续性和非负平衡点的存在性，通过构造适当的Lyapunov函数和极限系统理论，获得无病平衡点和地方病平衡点全局渐近稳定的充分条件.研究结果表明:基本再生数是决定疾病流行与否的阈值，当基本再生数小于等于1时，感染者逐渐消失，病毒趋于灭绝；当基本再生数大于1并满足永久持续条件时，感染者持续存在且病毒持续流行并将成为一种地方病.

Stability of an SIR Epidemic Model With 2 Patches and Population Movement

Based on the epidemic dynamics,in view of the population movement between 2 patches and the nonlinear infection rate,a class of SIR epidemic model with 2 patches and population movement was established.With the qualitative method and the stability method for ordinary differential equations,the permanence of the model and the existence of nonnegative equilibrium points were analyzed.Through construction of proper Lyapunov functions and according to the limit system theory,the sufficient conditions for the global asymptotic stability of the disease-free equilibrium points and the endemic equilibrium points were obtained.The results show that,the basic reproduction number makes a threshold to determine wether the disease spreads or not.When the basic reproduction number is less than or equal to 1,the infection will gradually disappear,the virus will tend to be extinct;when the dominant regeneration number of the virus is greater than 1 and satisfies the permanence conditions,the infection will persist,and the virus will continue to prevail and become an endemic disease.