Travelling wave solutions for an infection-age structured epidemic model with external supplies

Abstract

The aim of this paper is to investigate the spatial invasion of some infectious disease. The contamination process is described by the age since infection. Compared with the classical Kermack and McKendrick's model, the vital dynamic is not omitted, and we allow some constant input flux into the population. This problem is rather natural in the context of epidemic problems and it has not been studied. Here we prove an existence and non-existence result for travelling wave solutions. We also describe the minimal wave speed. We are able to construct a suitable Lyapunov like functional decreasing along the travelling wave allowing to derive some qualitative properties, namely their convergence towards equilibrium points at x = ±∞.