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

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