Cluster approximations for probabilistic systems: a new perspective of epidemiological modelling

Especially in lattice structured populations, homogeneous mixing represents an inadequate assumption. 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 which 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 treelike structure ubiquitous in random networks, providing an improved description of the steady state as well as the invasion dynamics.