Wave propagation in a diffusive SIR epidemic model with spatiotemporal delay

In this paper, we propose a SIR epidemic model incorporating Laplacian diffusion and the spatiotemporal delay to model the transmission of communicable diseases. The existence and nonexistence of traveling wave solutions for the model are investigated. It is found that the threshold dynamics are determined by the basic reproduction number R0 and the minimum wave speed c. By introducing an auxiliary system and Schauder's fixed point theorem, we establish the existence of traveling wave solutions for the model if R0>1 and c>c. Employing Fubini theorem and the two-sided Laplace transform, we obtain the nonexistence of traveling wave solutions for the model if R01 or 0<c<c. Our results cover and improve some results in the literature.