Periodic traveling waves for a diffusive SIR epidemic model with general nonlinear incidence and external supplies.

In this paper, we propose a non-autonomous reaction–diffusion SIR infectious disease model with nonlinear incidence, taking fully into account the effects of periodic environmental factors as well as population dynamics on disease transmission in space, and investigate the existence of periodic traveling wave solutions satisfying boundary conditions. Specifically, we first define the basic reproductive number R 0 and critical wave speed c ∗ , which will directly determine the existence of periodic traveling waves. Then, by considering a truncation problem and using fixed-point theorem, some estimation and limit techniques, the sufficient conditions on the existence of periodic traveling waves satisfying some boundary conditions are deduced for every wave speed c > c ∗ when R 0 > 1 , and the nonexistence of periodic traveling waves is also obtained for any c > 0 when R 0 < 1. Finally, some numerical examples are given to verify the theoretical results. • A reaction–diffusion SIR epidemic model in time-periodic environment is proposed. • The existence of periodic traveling waves is established by using fixed-point theorem. • The nonexistence of periodic traveling waves is also obtained. [ABSTRACT FROM AUTHOR]