A stochastic SIR model on a graph with epidemiological and population dynamics occurring over the same time scale

TL;DR

This paper introduces a stochastic SIR model on a graph that captures epidemic spread and population dynamics simultaneously, providing analytical and numerical tools validated with real cattle trade data.

Contribution

It develops a novel stochastic SIR model with density-dependent population transitions and offers new analytical bounds and numerical methods for epidemic outbreak probabilities.

Findings

Branching approximation results for epidemic process

Numerical computation method for outbreak probability

Exponential lower bound for extinction time and epidemic size

Abstract

We define and study an open stochastic SIR (Susceptible -- Infected -- Removed) model on a graph in order to describe the spread of an epidemic on a cattle trade network with epidemiological and demographic dynamics occurring over the same time scale. Population transition intensities are assumed to be density-dependent with a constant component, the amplitude of which determines the overall scale of the population process. Standard branching approximation results for the epidemic process are first given, along with a numerical computation method for the probability of a major epidemic outbreak. This procedure is illustrated using real data on trade-related cattle movements from a densely populated livestock farming region in western France (Finist\`ere) and epidemiological parameters corresponding to an infectious epizootic disease. Then we exhibit an exponential lower bound for the extinction time and the total size of the epidemic in the stable endemic case as a scaling parameter goes to infinity using results inspired by the Freidlin-Wentzell theory of large deviations from a dynamical system.