Fluctuations and oscillations in a simple epidemic model

TL;DR

This paper demonstrates that simple stochastic epidemic models with spatial correlations can exhibit oscillations due to stochastic fluctuations and long-lasting immunity, explaining observed epidemic oscillations.

Contribution

It identifies two mechanisms for oscillations in epidemic models, linking stochastic resonance and immunity effects, supported by simulations and phase diagram analysis.

Findings

Oscillations arise from stochastic fluctuations in large parameter ranges.

Persistent oscillations occur in models with long-lasting immunity.

The mechanisms explain the ubiquity of epidemic oscillations in real data.

Abstract

We show that the simplest stochastic epidemiological models with spatial correlations exhibit two types of oscillatory behaviour in the endemic phase. In a large parameter range, the oscillations are due to resonant amplification of stochastic fluctuations, a general mechanism first reported for predator-prey dynamics. In a narrow range of parameters that includes many infectious diseases which confer long lasting immunity the oscillations persist for infinite populations. This effect is apparent in simulations of the stochastic process in systems of variable size, and can be understood from the phase diagram of the deterministic pair approximation equations. The two mechanisms combined play a central role in explaining the ubiquity of oscillatory behaviour in real data and in simulation results of epidemic and other related models.