LAUSR

In this paper, we propose a reaction‐diffusion system to describe the spread of infectious diseases within two population groups by self and criss‐cross infection mechanism. Firstly, based on the eigenvalues, we give two methods for the calculation of the critical wave speed c∗. Secondly, by constructing a pair of upper‐lower solutions and using the Schauder fixed‐point theorem, we prove that the system admits positive traveling wave solutions, which connect the initial disease‐free equilibrium (u10,0,u30,0) at t = −∞, but the traveling waves need not connect the final disease‐free equilibrium (u1∗,0,u3∗,0) at t = +∞. Hence, we study the asymptotic behaviors of the traveling wave solutions to show that the traveling wave solutions converge to (u1∗,0,u3∗,0) at t = +∞. Finally, by the two‐sided Laplace transform, we establish the nonexistence of traveling waves for the model. The approach in this paper provides an effective method to deal with the existence of traveling wave solutions for the nonmonotone reaction‐diffusion systems consisting of four equations.