Environmental noise is unavoidable in the spread of infectious diseases. In this paper, we present a mathematical system to investigate the impact of environmental noise on disease transmission dynamics. The model incorporates Brownian noise, Markovian switching noise and nonlinear incidence. The results show that the long-term dynamics of the stochastic system is determined by a threshold parameter which is closely related to the stochastic noise. If the threshold is greater than zero all solutions converge exponentially to a unique invariant probability distribution, while if the threshold is less than zero the infectious diseases are extinct at an exponential rate and the level of susceptible individuals converges weakly to a unique invariant probability distribution. The threshold parameter also provides essential guidelines for accessing control of the diseases and implies that the environmental noise may be beneficial to contain the infectious diseases. The results extend and generalize previous work in understanding the dynamics of stochastic epidemic models with Markov switching. The theoretical approach can also be applied to the stochastic systems driven by white noise.

传染病的传播不可避免地会产生环境噪声。在本文中，我们提出了一个数学系统来研究环境噪声对疾病传播动力学的影响。该模型包含布朗噪声，马尔可夫切换噪声和非线性入射。结果表明，随机系统的长期动力学是由与随机噪声密切相关的阈值参数确定的。如果阈值大于零，则所有解决方案均会以指数形式收敛至唯一不变的概率分布，而如果阈值小于零，则传染病将以指数速率灭绝，易感个体的水平会微弱地收敛至唯一不变的概率分布。阈值参数还提供了控制疾病的基本准则，并暗示环境噪声可能有利于控制传染病。结果扩展并归纳了以前的工作，以理解马尔可夫切换的随机流行病模型的动力学。理论方法也可以应用于由白噪声驱动的随机系统。