Analysis on mixed types of waves for an SIR epidemic model with infection–age structure and spatial diffusion

Abstract

In Wu et al. (2021), Wu et al. studied an SIR epidemic model incorporating infection–age structure and spatial diffusion. It focused on the existence of traveling wave solutions when the diffusion coefficients met a technical condition (i.e., $d_3\le 2d_2$). Moreover, the question of the existence of traveling wave solutions with $c=c^{\ast }$ remains open. This paper employs an approach rooted in the Schauder’s fixed-point theory, along with sophisticated upper and lower solution techniques, to show the existence of the super-critical and critical traveling wave solutions. Notably, it is established that the existence of this traveling wave solution is independent of the relative diffusivity ratio.