Cluster approximations for infection dynamics on random networks

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.

the notation has been changed; Ref. [26] and a new paragraph in Section IV have been added

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• Collective oscillations of excitable elements: order parameters, bistability and the role of stochasticity
• Collective behavior of coupled nonuniform stochastic oscillators
• Population dynamics on random networks: simulations and analytical models
• The effects of heterogeneity on stochastic cycles in epidemics