Stability analysis of a delayed sir epidemic model with diffusion and saturated incidence rate

Abstract

In this paper, we investigate the effect of spatial diffusion and delay on the dynamical behavior of the SIR epidemic model. The introduction of the delay in this model makes it more realistic and modelizes the latency period. In addition, the consideration of an SIR model with diffusion aims to better understand the impact of the spatial heterogeneity of the environment and the movement of individuals on the persistence and extinction of disease. First, we determined a threshold value $$R_0$$ of the delayed SIR model with diffusion. Next, By using the theory of partial functional differential equations, we have shown that if $$R_0<1$$ , the unique disease-free equilibrium is asymptotically stable and there is no endemic equilibrium. Moreover, if $$R_0>1$$ , the disease-free equilibrium is unstable and there is a unique, asymptotically stable endemic equilibrium. Next, by constructing an appropriate Lyapunov function and using upper–lower solution method, we determine the threshold parameters which ensure the the global asymptotic stability of equilibria. Finally, we presented some numerical simulations to illustrate the theoretical results.