Metapopulation dynamics as a contact process on a graph

Abstract

Metapopulation dynamics and epidemiology of some diseases on sessile hosts (like pathogen fungi/tree interactions) can be modelled by contact processes on a graph. There exist several recent work on the structure of a graph, with classical notion like degree, diameter, small world property, etc., but quite fewer work on dynamics on graphs. Here, we model time discrete contact process on graphs with known degree distribution, with local mortality and spread along the edges with constant diffusion coefficient. We show that the knowledge of the degree distribution only leads to a mean field approximation only of the contact process. The fraction of infected hosts (or occupied patches) is modelled as a function of the degree of the vertex. As a consequence of mean field approximation, the stationary distribution at equilibrium belongs to a universal family of curves, which is independent of the degree distribution. An analytic description of this family is given. Each curve in the family is identified by one parameter only, which can be the mean infection rate (or occupancy rate) for the whole graph, obtained as a solution of an implicit equation containing the degree distribution. Finally, an exact equation relating the density of occupancy and the probability for types of edges is given, as a first step towards pair approximation on graphs.