Stochastic oscillations in models of epidemics on a network of cities

TL;DR

This paper analytically investigates stochastic oscillations in a networked epidemic model, revealing that fluctuations exhibit a single frequency and a uniform fixed point structure across cities, regardless of their size.

Contribution

It provides a novel analytical framework for understanding stochastic oscillations in epidemic models on networks of cities, extending previous single-city analyses.

Findings

Stochastic fluctuations oscillate at a single frequency.

The fixed point has identical susceptible, infected, recovered fractions across cities.

Oscillation dynamics are governed by the spectrum of the linearized system.

Abstract

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