具有非线性发生率的SIRS反应扩散传染病模型的动力学研究

We focus on a diffusive SIRS epidemic model with nonlinear incidence rate under constant coefficients in the paper. Firstly, we define the basic reproduction number and show the existence and uniqueness of the equilibrium for the system. Next, we expose the stability of equilibrium in terms of the relationship between $\mathcal{R}_0$ and 1: in a normal case, disease-free equilibrium $E_0$ is locally asymptotically stale; especially, when the incidence is the standard incidence, by constructing the Lyapunov function, we show disease-free equilibrium $E_0$ and the epidemic equilibrium $E^*$ are globally asymptotically stable for $\mathcal{R}_0<1$ and $\mathcal{R}_0>1$, res\\-pectively. Moreover, we support our conclusions through numerical simulation.