Spatiotemporal Dynamics in a Reaction–Diffusion Epidemic Model with a Time-Delay in Transmission

Abstract

In this paper, we investigate the effects of time-delay and diffusion on the disease dynamics in an epidemic model analytically and numerically. We give the conditions of Hopf and Turing bifurcations in a spatial domain. From the results of mathematical analysis and numerical simulations, we find that for unequal diffusive coefficients, time-delay and diffusion may induce that Turing instability results in stationary Turing patterns, Hopf instability results in spiral wave patterns, and Hopf–Turing instability results in chaotic wave patterns. Our results well extend the findings of spatiotemporal dynamics in the delayed reaction–diffusion epidemic model, and show that time-delay has a strong impact on the pattern formation of the reaction–diffusion epidemic model.