Focusing on the unpredictability of person-to-person contacts and the complexity of random variations in nature, this paper will formulate a stochastic SIR epidemic model with nonlinear incidence rate and general stochastic noises. First, we derive a stochastic critical value ${\mathcal{R}}_0^S$ related to the basic reproduction number ${\mathcal{R}}_0$. Via our new method in constructing suitable Lyapunov function types, we obtain the existence and uniqueness of an ergodic stationary distribution of the stochastic system if ${\mathcal{R}}_0^S>1$. Next, via solving the corresponding Fokker-Planck equation, it is theoretically proved that the stochastic model has a log-normal probability density function when another critical value ${\mathcal{R}}_0^H>1$. Then the exact expression of the density function is obtained. Moreover, we establish the sufficient condition ${\mathcal{R}}_0^C<1$ for disease extinction. Finally, several numerical simulations are provided to verify our analytical results. By comparison with other existing results, our developed theories and methods will be highlighted to end this paper.

针对人与人接触的不可预测性和自然界随机变化的复杂性，本文将构建具有非线性发病率和一般随机噪声的随机 SIR 流行模型。首先，我们推导出一个随机临界值${\mathcal{电阻}}_0^{秒}$ 与基本再生数有关 ${\mathcal{电阻}}_0$. 通过我们构建合适的 Lyapunov 函数类型的新方法，我们获得了随机系统的遍历平稳分布的存在性和唯一性，如果${\mathcal{电阻}}_0^{秒}>1$. 接下来，通过求解相应的 Fokker-Planck 方程，理论上证明了随机模型在另一个临界值时具有对数正态概率密度函数${\mathcal{电阻}}_0^H>1$. 然后得到密度函数的精确表达式。此外，我们成立充分条件${\mathcal{电阻}}_0^C<1$为灭病。最后，提供了几个数值模拟来验证我们的分析结果。通过与其他现有结果的比较，将突出我们发展的理论和方法来结束本文。