The relationships between message passing, pairwise, Kermack-McKendrick and stochastic SIR epidemic models

TL;DR

This paper develops a general stochastic SIR epidemic model on networks with correlated infectious periods and contact times, proving the existence of a unique message passing solution that bounds the expected epidemic size and connecting it to classical models.

Contribution

It introduces a novel message passing framework for correlated stochastic SIR models, proving solution uniqueness and establishing bounds and connections to classical epidemic models.

Findings

Message passing system has a unique feasible solution.

Cycles in the network inhibit epidemic spread.

Message passing converges to Kermack-McKendrick equations for large networks.

Abstract

We consider a very general stochastic model for an SIR epidemic on a network which allows an individual's infectious period, and the time it takes to contact each of its neighbours after becoming infected, to be correlated. We write down the message passing system of equations for this model and prove, for the first time, that it has a unique feasible solution. We also generalise an earlier result by proving that this solution provides a rigorous upper bound for the expected epidemic size (cumulative number of infection events) at any fixed time $t>0$. We specialise these results to a homogeneous special case where the graph (network) is symmetric. The message passing system here reduces to just four equations. We prove that cycles in the network inhibit the spread of infection, and derive important epidemiological results concerning the final epidemic size and threshold behaviour for a major outbreak. For Poisson contact processes, this message passing system is equivalent to a non-Markovian pair approximation model, which we show has well-known pairwise models as special cases. We show further that a sequence of message passing systems, starting with the homogeneous one just described, converges to the deterministic Kermack-McKendrick equations for this stochastic model. For Poisson contact and recovery, we show that this convergence is monotone, from which it follows that the message passing system (and hence also the pairwise model) here provides a better approximation to the expected epidemic size at time $t>0$ than the Kermack-McKendrick model.