Monostable waves for a delayed system with sign-changing dispersal kernels

A delayed infectious disease system with sign-changing dispersal kernels is considered. The existence of weak traveling wave solutions is firstly established by applying upper and lower solutions and Schauder’s fixed point theorem. Compared with nonnegative kernel functions, a technique condition is essential to guarantee the nonnegativity of traveling waves and a more detailed analysis is necessary to determine the smallest positive eigenvalue of the characteristic equation corresponding to the linearized system because of the specificity of sign-changing kernels. In addition, the strong traveling waves is also obtained for larger wave speed after the asymptotic boundary conditions of traveling waves at the positive infinity are testified, but the verification becomes relatively complicated for the coupling of different unknown functions and it is solved by additional requirement on the nonlinear functions. Notably, this might be the first attempt to investigate the monostable waves of delayed systems with sign-changing dispersal kernels.