Global threshold dynamics of an infection age-structured SIR epidemic model with diffusion under the Dirichlet boundary condition

Abstract

In this paper, we are concerned with the global asymptotic behavior of an infection age-structured SIR epidemic model with diffusion in a general n-dimensional bounded spatial domain under the homogeneous Dirichlet boundary condition. By using the method of characteristics, we reformulate the model into a system of a reaction-diffusion equation and a Volterra integral equation. We define the basic reproduction number R0 by the spectral radius of a compact positive linear operator and show that if R0<1, then the disease-free steady state is globally attractive, whereas if R0>1, then a positive endemic steady state exists and the system is uniformly persistent. By numerical simulation for the 2-dimensional case, we show that R0 depends on the shape of the spatial domain. This result is in contrast with the case of the homogeneous Neumann boundary condition, in which R0 is independent of the spatial domain.