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We carry out an analytic investigation of stochastic oscillations in asusceptible-infected-recovered model of disease spread on a network of $n$cities. In the model a fraction $f_{jk}$ of individuals from city $k$ commuteto city $j$, where they may infect, or be infected by, others. Starting from acontinuous time Markov description of the model the deterministic equations,which are valid in the limit when the population of each city is infinite, arerecovered. The stochastic fluctuations about the fixed point of these equationsare derived by use of the van Kampen system-size expansion. The fixed pointstructure of the deterministic equations is remarkably simple: a uniquenon-trivial fixed point always exists and has the feature that the fraction ofsusceptible, infected and recovered individuals is the same for each cityirrespective of its size. We find that the stochastic fluctuations have ananalogously simple dynamics: all oscillations have a single frequency, equal tothat found in the one city case. We interpret this phenomenon in terms of theproperties of the spectrum of the matrix of the linear approximation of thedeterministic equations at the fixed point.

Stochastic oscillations in models of epidemics on a network of cities