Mean-Field approximation and Quasi-Equilibrium reduction of Markov Population Models

TL;DR

This paper analyzes the relationship between mean field and quasi-equilibrium reductions in Markov Population Models, revealing that their order of application affects the resulting models and providing conditions for their interchangeability.

Contribution

It introduces a framework that explicitly distinguishes and allows arbitrary sequencing of mean field and quasi-equilibrium reductions in Markov Population Models.

Findings

Double limits do not generally commute.

Inverting the sequence of reductions changes the model outcome.

Conditions are provided for when the order of reductions can be exchanged.

Abstract

Markov Population Model is a commonly used framework to describe stochastic systems. Their exact analysis is unfeasible in most cases because of the state space explosion. Approximations are usually sought, often with the goal of reducing the number of variables. Among them, the mean field limit and the quasi-equilibrium approximations stand out. We view them as techniques that are rooted in independent basic principles. At the basis of the mean field limit is the law of large numbers. The principle of the quasi-equilibrium reduction is the separation of temporal scales. It is common practice to apply both limits to an MPM yielding a fully reduced model. Although the two limits should be viewed as completely independent options, they are applied almost invariably in a fixed sequence: MF limit first, QE-reduction second. We present a framework that makes explicit the distinction of the two reductions, and allows an arbitrary order of their application. By inverting the sequence, we show that the double limit does not commute in general: the mean field limit of a time-scale reduced model is not the same as the time-scale reduced limit of a mean field model. An example is provided to demonstrate this phenomenon. Sufficient conditions for the two operations to be freely exchangeable are also provided.