Cluster variation approximations for a contact process living on a graph

A model classically used for modelling the spread of an infectious diseases in a network is the time-continuous contact process, which is one simple example of an interacting particles system. It displays a non-equilibrium phase transition, related to the burst of an epidemic within a population in case of an accidental introduction. Several studies have recently emphasized the role of some geometrical properties of the graph on which the contact process lives, like the degree distribution, for quantities of interest like the singlet density at equilibrium or the critical value of the infectivity parameter for the emergence of the epidemics, but this role is not yet fully understood. As the contact process on a graph still cannot be solved analytically (even on a 1D lattice), some approximations are needed. The more naive, but well-studied approximation is the mean field approximation. We explore in this paper the potentiality of a finer approximation: the pair approximation used in ecology. We give an analytical formulation on a graph of the site occupancy probability at equilibrium, depending on the site degree, under pair approximation and another dependence structure approximation. We point out improvements brought about in the case of realistic graph structures, far from the well-mixed assumption. We also identify the limits of the pair approximation to answer the question of the effects of the graph characteristics. We show how to improve the method using a more appropriate order 2 cluster variation method, the Bethe approximation.