Stability analysis of an epidemic model with diffusion and stochastic perturbation

In this paper, we investigate the stability of an epidemic model with diffusion and stochastic perturbation. We first show both the local and global stability of the endemic equilibrium of the deterministic epidemic model by analyzing corresponding characteristic equation and Lyapunov function. Second, for the corresponding reaction–diffusion epidemic model, we present the conditions of the globally asymptotical stability of the endemic equilibrium. And we carry out the analytical study for the stochastic model in details and find out the conditions for asymptotic stability of the endemic equilibrium in the mean sense. Furthermore, we perform a series of numerical simulations to illustrate our mathematical findings.

► We investigate the stability analysis of an modified epidemic model.
► For the deterministic model, we obtain both local and global stability of a solution.
► For the diffusion model, we get a condition of the globally asymptotical stability.
► For the stochastic model, we study the existence, boundedness and permanence.
► We prove the solution is stochastic asymptotically stable and perform some figures.