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

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.