An elementary proof of convergence to the mean-field equations for an epidemic model

TL;DR

This paper presents a simple proof demonstrating that the stochastic SIS epidemic model on a complete network converges to its mean-field ODEs as the network size grows, using basic ODEs and moment inequalities.

Contribution

It offers the first elementary proof of convergence for the SIS model to mean-field equations and provides a lower bound on the expected infected fraction.

Findings

Convergence in mean-square on finite intervals is established.

A novel lower bound on the expected infected fraction is derived.

The proof simplifies previous complex approaches.

Abstract

It is common to use a compartmental, fluid model described by a system of ordinary differential equations (ODEs) to model disease spread. In addition to their simplicity, these models are also the mean-field approximations of more accurate stochastic models of disease spread on contact networks. For the simplest case of a stochastic susceptible-infected-susceptible (SIS) process (infection with recovery) on a complete network, it has been shown that the fraction of infected nodes converges to the mean-field ODE as the number of nodes increases. However the proofs are not simple, requiring sophisticated probability, partial differential equations (PDE), or infinite systems of ODEs. We provide a short proof in this case for convergence in mean-square on finite time-intervals using a system of two ODEs and a moment inequality and also obtain the first lower bound on the expected fraction of infected nodes.