Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission

• The diffusion ability of the infected individuals can accelerate the wave speed.
• The incubation period can slow down the propagation.
• The non-locality of interaction would speed up the spread of the disease.

This paper is concerned with the existence and nonexistence of traveling wave solutions of a nonlocal dispersal SIR model with nonlocal delayed transmission. The threshold dynamics are determined by the so-called basic reproduction number of the corresponding reaction system and the critical wave speed. Our results imply that (i) the diffusion ability of the infected individuals can accelerate the wave speed; and (ii) the incubation period can slow down the propagation while the non-locality of interaction would speed up the spread of the disease. In particular, we remove the usual condition that the nonlocal dispersal kernel function is compacted supported.