Due to the continuous interference of environmental white noise, the dynamical behaviors of a stochastic SIQR epidemic model with mean-reverting Ornstein–Uhlenbeck process and standard incidence are considered. Several conclusions can be verified after dimensionality reduction. We study the existence and uniqueness of positive solution by construct a nonnegative Lyapunov function at first. Then, a sufficient condition for extinction of the diseases is derived by constructing a suitable Lyapunov function. In addition, we also obtain the stationary distribution of the model by constructing a complex Lyapunov function. Particularly, we propose the exact local expression of the density function of the random model near the unique endemic equilibrium. Finally, the numerical simulations illustrate our above theoretical results.

由于环境白噪声的持续干扰，考虑了具有均值回归Ornstein-Uhlenbeck过程和标准发病率的随机SIQR流行病模型的动力学行为。降维后可以验证几个结论。我们首先通过构造一个非负Lyapunov函数来研究正解的存在性和唯一性。然后，通过构造合适的Lyapunov函数，导出了疾病消退的充分条件。此外，我们还通过构造一个复杂的 Lyapunov 函数来获得模型的平稳分布。特别是，我们提出了随机模型在独特地方性平衡附近的密度函数的精确局部表达式。最后，数值模拟说明了我们的上述理论结果。