Traveling Waves in a Hybrid Reaction‐Diffusion‐Difference SEIR Epidemic Model With Nonlocal Transmission and Protection Phase With Delay

Overview & Synopsis

In this paper, we study the existence and non‐existence of traveling waves for a SEIR epidemic model with diffusion. This system is coupled to an age‐structured protection equation that takes into account a proportion of susceptible individuals who protect themselves from the disease through medical treatment or vaccination with temporary immunity, for example. The model includes a compartment of exposed individuals to represent diseases with an incubation period. In addition, the model contains a nonlinear saturated incidence rate to describe the interaction between susceptible and infected individuals. By solving the equation associated with the exposed compartment, we obtain a nonlocal integral representation of the exposed population as a function of the nonlinear incidence rate. This reformulation reduces the model to an hybrid system of reaction‐diffusion equations coupled to a continuous difference equation that includes a delay and a nonlocal term, reflecting the movement of individuals during the protection and exposure phases. For wave speeds above a certain threshold, we construct lower and upper solutions for the resulting hybrid system. Finally, we apply Schauder's fixed point theorem in an appropriate function space to establish the existence of progressive wave solutions.