考虑了一类具有饱和发生率的随机SIRS流行病模型. 通过定义停时及利用李雅普诺夫函数, 得到了随机SIRS流行病模型的解是全局存在唯一的, 接着分析了解沿无病平衡点及地方病平衡点的渐近行为. 在适当的参数条件下, 证明了随机SIRS流行病模型具有遍历的平稳分布及解渐近服从三维正态分布这一主要结论. 最后, 数值模拟验证了所得到的主要结果.

A stochastic SIRS epidemic model with saturated incidence is discussed in this paper. By defining the stopping time and using Lyapunov function, the unique global positive solution to the stochastic SIRS model is derived. Then the asymptotic behaviors of the stochastic SIRS model around the disease-free equilibrium and the endemic equilibrium are analyzed. Furthermore, we prove that, under moderate parametric conditions, the stochastic SIRS model has a unique stationary distribution with ergodicity and the solution asymptotically follows a 3-dimensional normal distribution. Some numerical simulations are carried out to clarify our results.