具有标准发生率的随机SIR传染病模型的建立及平稳分布与疾病灭绝研究

研究了一类具有标准发生率以及考虑随机扰动与系统变量成正比的随机SIR传染病模型.首先,对于任意的正的初值,系统存在唯一的全局正解以及通过构造合适的随机李雅普诺夫函数,得到了模型遍历平稳分布存在的充分条件.其次,给出了疾病灭绝的充分条件,并与模型遍历平稳分布存在的充分条件作对比,得出了在特定条件下随机SIR模型的阈值.最后通过数值模拟验证了结果的正确性.

The Stationary Distribution and Extinction of a Stochastic SIR Epidemic Model with a Standard Incidence

In this paper,we study a class of a random SIR epidemic model with a standard incidence and the random disturbance being proportional to the state variables.Firstly,for any positive initial value,the system has a unique global positive solution and we construct a suitable stochastic Lyapunov function and obtain a sufficient condition for the existence of an ergodic stationary distribution.Secondly,some sufficient conditions of the extinction of the disease are given.Furthermore,we compare with the sufficient condition of the existence of the stationary distribution,and give a threshold of the stochastic SIR model under certain conditions.Finally,the correctness of the results is verified by numerical simulation.