Pattern formation of an epidemic model with time delay

• We presented an epidemic model with spatial diffusion and time delay.
• We find two different types of instability.
• Delay can induce regular patterns.
• The interaction of diffusion and time delay may give rise to rich dynamic.

One of the central issues in epidemiology is the study of the distribution of disease. And time delay widely exists in the process of disease spread. Thus, in this paper, we presented an epidemic model with spatial diffusion and time delay. By mathematical analysis, we find two different types of instability. One is the diffusion induced instability, and the other one is delay induced instability. Moreover, we derive the corresponding patterns by performing a series of numerical simulations. The obtained results show that the interaction of diffusion and time delay may give rise to rich dynamics in epidemic systems.