Elementary proof of convergence to the mean-field model for the SIR process

TL;DR

This paper provides an elementary and rigorous proof that the stochastic SIR process on complete contact networks converges to its mean-field ODE model as the network size grows, using simple mathematical tools.

Contribution

It extends an elementary convergence proof from SIS to SIR models, simplifying the theoretical understanding of mean-field limits for epidemic processes.

Findings

Convergence of stochastic SIR to mean-field ODE proven with simple methods

Approach applicable to other linear compartmental models

Proof avoids complex semigroup and martingale techniques

Abstract

The susceptible-infected-recovered (SIR) model has been used extensively to model disease spread and other processes. Despite the widespread usage of this ordinary differential equation (ODE) based model which represents the mean-field approximation of the underlying stochastic SIR process on contact networks, only few rigorous approaches exist and these use complex semigroup and martingale techniques to prove that the expected fraction of the susceptible and infected nodes of the stochastic SIR process on a complete graph converges as the number of nodes increases to the solution of the mean-field ODE model. Extending the elementary proof of convergence for the SIS process introduced by Armbruster and Beck (2016) to the SIR process, we show convergence using only a system of three ODEs, simple probabilistic inequalities, and basic ODE theory. Our approach can also be generalized to many other types of compartmental models (e.g., susceptible-infected-recovered-susceptible (SIRS)) which are linear ODEs with the addition of quadratic terms for the number of new infections similar to the SI term in the SIR model.