Cluster approximations for epidemic processes: a systematic description of correlations beyond the pair level

The spread of a virus is an example of a dynamic process occurring on a discrete spatial arrangement. While the mean-field approximation reasonably reproduces the spreading behaviour for topologies where the number of connections per node is either high or strongly fluctuating and for those that show small-world features, it is inaccurate for lattice structured populations. Various improvements upon the ordinary pair approximation based on a number of assumptions concerning the higher-order correlations have been proposed. To find approaches that allow for a derivation of their dynamics remains a great challenge. By representing the population with its connectivity patterns as a homogeneous network, we propose a systematic methodology for the description of the epidemic dynamics that takes into account spatial correlations up to a desired range. The equations that the dynamical correlations are subject to are derived in a straightforward way, and they are solved very efficiently due to their binary character. The method embeds very naturally spatial patterns such as the presence of loops characterizing the square lattice or the tree-like structure ubiquitous in random networks, providing an improved description of the steady state as well as the invasion dynamics.