Equivalent probability density moments determine equivalent epidemics in an SIRS model with temporary immunity

In an SIRS compartment model for a disease we consider the effect of different probability distributions for remaining immune. We show that to first approximation the first three moments of the corresponding probability densities are sufficient to well describe oscillatory solutions corresponding to recurrent epidemics. Specifically, increasing the fraction who lose immunity, increasing the mean immunity time, and decreasing the heterogeneity of the population all favor the onset of epidemics and increase their severity. We consider six different distributions, some symmetric about their mean and some asymmetric, and show that by tuning their parameters such that they have equivalent moments that they all exhibit equivalent dynamical behavior.