The aim of this paper is to study the stochastic SIR equation with general incidence functional responses and in which both natural death rates and the incidence rate are perturbed by white noises. We derive a sufficient and almost necessary condition for the extinction and permanence for an epidemic system with multi noises \begin{equation*} \begin{cases} dS(t)=\big[a_1-b_1S(t)-I(t)f(S(t),I(t))\big]dt + \sigma_1 S(t) dB_1(t) -I(t)g(S(t),I(t))dB_3(t),\\ dI(t)=\big[-b_2I(t) + I(t)f(S(t),I(t))\big]dt + \sigma_2I(t) dB_2(t) + I(t)g(S(t),I(t))dB_3(t). \end{cases} \end{equation*} Moreover, the rate of all convergences of the solution are also established. A number of numerical examples are given to illustrate our results

中文翻译：

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随机 SIR 流行模型的持久性和消亡

本文的目的是研究具有一般发生率函数响应的随机 SIR 方程，其中自然死亡率和发生率都受到白噪声的干扰。我们推导出具有多噪声的流行系统的灭绝和持久性的充分和几乎必要条件 \begin{equation*} \begin{cases} dS(t)=\big[a_1-b_1S(t)-I(t) f(S(t),I(t))\big]dt + \sigma_1 S(t) dB_1(t) -I(t)g(S(t),I(t))dB_3(t),\ \ dI(t)=\big[-b_2I(t) + I(t)f(S(t),I(t))\big]dt + \sigma_2I(t) dB_2(t) + I(t) g(S(t),I(t))dB_3(t)。\end{cases} \end{equation*} 此外，还建立了解决方案的所有收敛率。给出了一些数值例子来说明我们的结果