Traveling waves in a nonlocal dispersal SIR model with non-monotone incidence.

• The novelty of this work lies in the construction of super-sub solutions and the proof of the complete continuity of operator. • Most importantly and differently, the authors obtain the existence of wave front with critical speed, by verifying the boundedness of component R. • The result provides a method for some non-monotone systems. It is well-known that the nonlocal dispersal operator has the advantage of capturing short-range as well as long-range factors for the dispersal of the spices by choosing the kernel function properly, and is also capable to include spatial dispersal strategies of the species beyond random (local) diffusion. This paper is concerned with the existence and nonexistence of traveling wave solutions for a nonlocal dispersal Kermack-McKendrick epidemic model with non-monotone incidence, which is a non-monotone system. The method of sub and super solutions combined with Schauder's fixed-point theorem is applied to establish the existence of positive traveling waves as the wave speed is over critical speed. We further prove the existence of traveling waves with critical speed and the nonexistence of bounded positive traveling waves by the delicate analysis method. The main difficulty is to get the boundedness of traveling waves caused by the nonlocal dispersal operator. [ABSTRACT FROM AUTHOR]