Stability for Delay SEIR Epidemic Models with Saturated Incidence Rates and Diffusion

Abstract

AbstractIn this paper, we investigate the effect of spatial diffusion and delay on the dynamical behavior of the SEIR 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 SEIR 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 \(\mathscr {R}_{0}\) of the delayed SEIR model with diffusion. Next, By using the theory of partial functional differential equations, we have shown that the unique disease-free equilibrium and the endemic equilibrium are asymptotically stable. Moreover,we determine, using Lyapunov functionals, conditions by which the disease-free equilibrium and the endemic equilibrium are globally asymptotically stable. Also some numerical simulations are given to illustrate the theoretical results.