Improving estimates of waning immunity rates in stochastic SIRS models with a hierarchical framework

Most disease pathogens require onward transmission for their continued persistence. It is necessary to have continuous replenishment of the population of susceptibles, either through births, immigration, or waning immunity in recovered individuals. Consider the introduction of an unknown infectious disease into a fully susceptible population where it is not known how long immunity is conferred once an individual recovers. If the disease takes off, the number of infectives will typically decrease to a low level after the first major outbreak. During this period, the disease dynamics will be highly influenced by stochastic effects and there is a non-zero probability that the epidemic will die out. This is known as an epidemic fade-out. If the disease does not die out, the susceptible population may be replenished by the waning of immunity, and a second wave may start. In this study, we describe an experiment where we generated synthetic outbreak data from independent stochastic SIRS models in multiple communities. Some of the outbreaks faded-out and some did not. By conducting Bayesian parameter estimation independently on each outbreak, as well as under a hierarchical framework, we investigated if the waning immunity rate could be correctly identified. When the outbreaks were considered independently, the waning immunity rate was incorrectly estimated when an epidemic fade-out was observed. However, the hierarchical approach improved the parameter estimates. This was particularly the case for those communities where the epidemic faded out.