The global structure of a spatial model of infectious disease

Abstract

In this paper we study an SI model of infectious diseases that takes into account spatial inhomogeneities, resulting in a system of reaction–convection–diffusion equations on a bounded domain. The convection process is included to account for social interaction, as modelled by the location of a focal point or den where the population will tend to aggregate. We show that a vertical bifurcation of steady–state solutions occurs in this model when birth rate is taken as the bifurcation parameter, from which emanates a global secondary branch, which then bifurcates at infinity. Subsequently, we use singular perturbation techniques to give a description of the limiting spatial structure along this branch in large and small parameter limits. Finally, the results are illustrated numerically on some biologically relevant cases.