Traveling waves in a delayed SIR epidemic model with nonlinear incidence

•We study the traveling waves of a diffusive SIR epidemic model with a general nonlinear incidence.•We obtain the threshold dynamics for spatial spread of the disease.•We show that c* is the minimum wave speed for the existence of traveling wave solutions.

We establish the existence and non-existence of traveling wave solutions for a diffusive SIR model with a general nonlinear incidence. The existence proof is shown by introducing an auxiliary system, applying Schauder’s fixed point theorem and then a limiting argument. The nonexistence proof is obtained by two-sided Laplace transform when the speed is less than the critical velocity. Numerical simulations support the theoretical results. We also point out the effects of the delay and the diffusion rate of the infective individuals on the spreading speed.