Stochastic fluctuations in epidemics on networks.

The effects of demographic stochasticity on the long-term behaviour of endemic infectious diseases have been considered for long as a necessary addition to an underlying deterministic theory. The latter would explain the regular behaviour of recurrent epidemics and the former the superimposed noise of observed incidence patterns. Recently, a stochastic theory based on a mechanism of resonance with internal noise has shifted the role of stochasticity closer to the centre stage, by showing that the major dynamic patterns found in the incidence data can be explained as resonant fluctuations, whose behaviour is largely independent of the amplitude of seasonal forcing, and by contrast very sensitive to the basic epidemiological parameters. Here we elaborate on that approach, by adding an ingredient which is missing in standard epidemic models, the 'mixing network' through which infection may propagate. We find that spatial correlations have a major effect on the enhancement of the amplitude and the coherence of the resonant stochastic fluctuations, providing the ordered patterns of recurrent epidemics, whose period may differ significantly from that of the small oscillations around the deterministic equilibrium. We also show that the inclusion of a more realistic, time-correlated recovery profile instead of exponentially distributed infectious periods may, even in the random-mixing limit, contribute to the same effect.