Traveling waves for a nonlocal delayed reaction-diffusion SIR epidemic model with demography effects

Considering the importance of time delay phenomenon in disease transmission and the interdependence of time delay and spatial location, a reaction-diffusion SIR epidemic model with nonlocal time delay (or time-space time delay) is proposed and the travelling wave solutions are discussed. Specifically, we define the basic reproduction number R0 and the critical wave speed c∗. For every wave speed c≥c∗, the existence of travelling wave solutions is studied by using the upper–lower solutions, the fixed-point theorem and some limit techniques when R0>1. The nonexistence of travelling waves when R0>1 for any 00 is also proved. Finally, the influence of time-delay on disease propagation, especially the critical wave speed, is discussed through analysis and simulations.