Abstract: This paper discusses a stochastic SIQS epidemic model with the quarantined-adjusted incidence. We obtain that, the stochastic model admits a unique and global solution. Our research reveals that, when the intensities of the white noises are large enough, the solution of the stochastic model around the disease-free equilibrium will be extinct, and the density of the infective individuals will exponentially approach zero. When the intensities of the white noises are small enough, the positive solution of the stochastic model obeys a unique stationary distribution around the endemic equilibrium. Further, Under some sufficient conditions, the solution will asymptotically follow a three-dimensional normal distribution if the endemic equilibrium is stable, and the mean and the variance can be expressed by formulation. Moreover, the numerical simulations demonstrate the properties of the solution and give good explanations to our model.

Key words: Stochastic SIQS epidemic model, Extinction, Stationary distribution, Normal distribution, Lyapunov functions