Cluster approximations for infection dynamics on random networks

TL;DR

This paper develops and compares cluster approximation methods for modeling infection dynamics on large random networks, deriving analytical power spectra of fluctuations and identifying when higher-order approximations are necessary.

Contribution

It introduces an uncorrelated triplet approximation that improves modeling accuracy beyond the pair approximation for certain parameter regimes.

Findings

Power spectrum can be analytically derived from the master equation.

Pair approximation accurately predicts fluctuations in the thermodynamic limit.

Triplet approximation captures behavior where pair approximation fails.

Abstract

In this paper, we consider a simple stochastic epidemic model on large regular random graphs and the stochastic process that corresponds to this dynamics in the standard pair approximation. Using the fact that the nodes of a pair are unlikely to share neighbors, we derive the master equation for this process and obtain from the system size expansion the power spectrum of the fluctuations in the quasi-stationary state. We show that whenever the pair approximation deterministic equations give an accurate description of the behavior of the system in the thermodynamic limit, the power spectrum of the fluctuations measured in long simulations is well approximated by the analytical power spectrum. If this assumption breaks down, then the cluster approximation must be carried out beyond the level of pairs. We construct an uncorrelated triplet approximation that captures the behavior of the system in a region of parameter space where the pair approximation fails to give a good quantitative or even qualitative agreement. For these parameter values, the power spectrum of the fluctuations in finite systems can be computed analytically from the master equation of the corresponding stochastic process.