Traveling waves for a diffusive SIR epidemic model with delay in the diffusion term

This paper investigates the existence of traveling waves in a diffusive SIR model with delay incorporated in the diffusion terms and a nonlinear incidence rate with delay. By employing a cross-iteration scheme and partial monotonicity conditions, we establish that the existence of quasi-upper and lower solutions, along with suitable super and sub-solutions, provides sufficient conditions for the existence of a traveling wavefront. This existence result is obtained via Schauder’s fixed-point theorem. Furthermore, given an appropriate basic reproduction number, the traveling wavefront transitions from the disease-free steady state to the endemic steady state. To illustrate our approach, we explicitly construct super- and sub-solutions for a specific model.