Fluid limit of the SIRS Markov chain ($\dot{s}=-\lambda si+\nu r,\dot{i}=\lambda si-\gamma i,\dot{r}=\gamma i-\nu r$)

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.