Dynamic behavior and sliding mode control on a stochastic epidemic model with alertness and distributed delay

In this paper, a stochastic SEIR epidemic model with alertness and distributed delay is studied. The dynamic behavior of the model is analyzed and an integral sliding mode controller is designed. By using Lyapunov function method, sufficient conditions for the existence of unique global positive solution and stationary distribution are obtained. To analyze stochastic stability and stochastic bifurcation through singular boundary theory, we apply stochastic center manifold and stochastic averaging method to transform the model into a one-dimensional Markov diffusion process. The integral sliding mode controller is designed to confine the state trajectories of system to a sufficiently small band around the sliding mode surface, and prevent large outbreak of disease. Numerical simulations verify the correctness of the theoretical analysis and the effectiveness of the controller. •A stochastic SEIR epidemic model with alertness and distributed delay is studied.•The sufficient condition for the existence of stationary distribution is obtained.•The stochastic stability and bifurcation are analyzed using singular boundary theory.•An integral sliding mode controller is designed to prevent the outbreaks of disease.