Spatio-temporal dynamics of an age-structured reaction-diffusion system of epidemic type subjected by Neumann boundary condition

This paper is concerned with the spatio-temporal dynamics of an age-structured reaction-diffusion system of KPP-epidemic type (SIS), subject to Neumann boundary conditions and incorporating $L^1$ blow-up type death rate. We first establish the existence of time dependent solutions using age-structured semigroup theory. Afterward, the basic reproduction number $\mathcal{R}_0$ is derived by linearizing the system around the disease-free equilibrium state. In the case $\mathcal{R}_0<1$, the existence, uniqueness and stability of disease-free equilibrium are shown by using $ω$-limit set approach of Langlais \cite{langlais_large_1988}, combined with the technique developed in recent works of Zhao et al. \cite{zhao_spatiotemporal_2023} and Ducrot et al. \cite{ducrot_age-structured_2024}. We highlight that the absence of a general comparison principle for the age-structured SIS-model and non-separable variable mortality rate prevent the direct application of the semi-flow technique developed in \cite{ducrot_age-structured_2024} to study the long time dynamics.