An infection age-space structured SIR epidemic model with Neumann boundary condition

In this paper, we are concerned with an SIR epidemic model with infection age and spatial diffusion in the case of Neumann boundary condition. The original model is constructed as a nonlinear age structured system of reaction–diffusion equations. By using the method of characteristics, we reformulate the model into a system of a reaction–diffusion equation and a Volterra integral equation. For the reformulated system, we define the basic reproduction number R0 by the spectral radius of the next generation operator, and show that if R0<1, then the trivial disease-free steady state is globally attractive, whereas if R0>1, then the disease in the system is persistent. Moreover, under an additional assumption that there exists a finite maximum age of infectiousness, we show the global attractivity of a constant endemic steady state for R0>1.