A refined mean field approximation of synchronous discrete-time population models

TL;DR

This paper analyzes the accuracy of discrete-time mean field approximations for synchronous population models, extending previous results and introducing a refined method that improves approximation quality for small populations.

Contribution

It extends mean field accuracy results to synchronous models and introduces a refined approximation method with quantifiable error bounds.

Findings

Expected performance indicators are $O(1/N)$-accurate.

The refined mean field approximation improves accuracy for small populations.

Simple expressions for asymptotic error enable better approximation quality.

Abstract

Mean field approximation is a popular method to study the behaviour of stochastic models composed of a large number of interacting objects. When the objects are asynchronous, the mean field approximation of a population model can be expressed as an ordinary differential equation. When the objects are (clock-) synchronous the mean field approximation is a discrete time dynamical system. We focus on the latter.We study the accuracy of mean field approximation when this approximation is a discrete-time dynamical system. We extend a result that was shown for the continuous time case and we prove that expected performance indicators estimated by mean field approximation are $O(1/N)$-accurate. We provide simple expressions to effectively compute the asymptotic error of mean field approximation, for finite time-horizon and steady-state, and we use this computed error to propose what we call a \emph{refined} mean field approximation. We show, by using a few numerical examples, that this technique improves the quality of approximation compared to the classical mean field approximation, especially for relatively small population sizes.