 Fluid limit of the SIRS Markov chain

For convergent initial proportions, the scaled SIRS Markov chain converges uniformly in probability on each fixed finite interval to the solution of these equations. The Poisson time-change representation of a Markov chain writes it as its integrated drift plus centered Poisson errors. Bounded rates, Poisson concentration and the Gronwall inequality control the error. This finite-time limit does not by itself justify an approximation over time intervals growing with the population size.