  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.
