Traveling Waves in a Reaction-Diffusion Addiction Epidemic Model with Distributed Delays

This study examines whether a traveling wave solution for an addiction reaction-diffusion epidemic model with dispersed delays exists or not. The primary characteristic of the model is the potential weakness in the standard comparison principle, which prevents many known conclusions from being applied. If the basic reproduction number of the model, indicated by \(R_{0}\), is larger than one, there is a minimal wave speed, denoted by \(c^{*}>0\), such that the system admits a nontrivial traveling wave solution with wave speed \(c\) when \(c\geq c^{*}\). However, for either \(R_{0}>1\) and \(c< c^{*}\), or \(R_{0}<1\), there is no nontrivial traveling wave solution.