带Markov切换的随机SIQS模型的渐近行为

Asymptotic behavior of a stochastic SIQS model with Markov switching

研究了一类带有Markov切换和饱和发生率的随机SIQS传染病模型。首先通过构造适当的Lyapunov函数,得到模型全局正解的存在唯一性;其次利用Markov链的遍历性,得到疾病灭绝和均值持久的充分条件;最后运用数值模拟验证了理论结果。结果表明,若传染病在一个状态的子系统中是随机持久的,但在另一个状态的子系统中是随机灭绝的,则传染病在混合系统中既可能随机持久也可能随机灭绝,其结果依赖于Markov链在每个状态内停留的概率。电报噪声对疾病传播具有重要的影响。隔离对疾病传播具有抑制作用,从而隔离染病者更有助于控制疾病的传播。

A stochastic SIQS epidemic model with Markovian switching and saturated incidence is investigated. First, the existence and uniqueness of the global positive solution of the model are proved by constructing the suitable Lyapunov functions. Then by using the ergodic property of Markov chains, the sufficient conditions of extinction and persistence in the mean of the disease are obtained. At last, the theoretical results are verified by numerical simulations. The results show that if one of the subsystems is stochastically persistent, and another is stochastically extinct, then the hybrid system may be either stochastically extinct or persistent, and the result depends on the probability that the Markov chain remains in each status. Telegraph noise has a major impact on disease transmission. It can be seen that isolation has an inhibitory effect on disease transmission, so the isolation of infected individuals is more helpful to control the spread of the disease.