Periodic Wave Train Solutions of the Age-Structured SIRS Epidemic Model with Diffusion

In this paper, we conduct a mathematical analysis of an age-structured SIRS infectious disease model, which incorporates a general nonlinear infection rate and accounts for the diffusion of the disease. The main objective of this study is to investigate the existence of periodic wave train solutions for the age-structured SIRS epidemic model with diffusion by utilizing the Hopf bifurcation theory for nondensely defined Cauchy problem and the persistence of nondegenerate Hopf bifurcation of singularly perturbed second-order equations. The outcomes indicate that the temporal oscillations derived from the age-structured part would interact with the spatial diffusion to result in the existence of periodic wave train solutions. At the same time, to illustrate our theoretical analysis more vividly, numerical simulations are performed.