In this paper, by introducing environmental perturbation, we extend an epidemic model with graded cure, relapse, and nonlinear incidence rate from a deterministic framework to a stochastic differential one. The existence and uniqueness of positive solution for the stochastic system is verified. Using the Lyapunov function method, we estimate the distance between stochastic solutions and the corresponding deterministic system in the time mean sense. Under some acceptable conditions, the solution of the stochastic system oscillates in the vicinity of the disease-free equilibrium if the basic reproductive number , while the random solution oscillates near the endemic equilibrium, and the system has a unique stationary distribution if . Moreover, numerical simulation is conducted to support our theoretical results.

中文翻译：

---

具有复发和治愈的非线性随机流行病模型的渐近行为和平稳分布

在本文中，通过介绍环境扰动，我们将具有分级治愈，复发和非线性发生率的流行病模型从确定性框架扩展到随机差分模型。验证了随机系统正解的存在性和唯一性。使用李雅普诺夫函数法，我们在时间平均意义上估计了随机解与相应的确定性系统之间的距离。在某些可接受的条件下，如果基本生殖数为零，则随机系统的解在无病平衡附近振荡。，而随机解在地方均衡附近振荡，并且如果。此外，进行了数值模拟以支持我们的理论结果。