Estimation of the probability of epidemic fade-out from multiple outbreak data

Abstract

Deterministic epidemic models, such as the SIRS model or an SIR model with demography, that allow for replenishment of susceptibles typically display damped oscillatory behaviour. If the population is initially fully susceptible, once an epidemic takes off a distinct trough will exist between the first and second waves of infection where the number of infectious individuals falls to a low level. Epidemic dynamics are, however, influenced by stochastic effects, particularly when the number of infectives is low. At the beginning of an epidemic, stochastic die-out is possible and well characterised through use of a branching process approximation to the full non-linear stochastic dynamics. Conditional on an epidemic taking off, stochastic extinction is highly unlikely during the first epidemic wave, but the probability of extinction increases again as the wave declines. Extinction during this period, prior to a potential second wave of infection, is defined as epidemic fade-out. We consider a set of observed epidemics, each distinct and having evolved independently, in which some display fade-out and some do not. While fade-out is necessarily a stochastic phenomenon, in general the probability of fade-out will depend on the model parameters associated with each epidemic. Accordingly, we ask whether time-series data for the epidemics contain sufficient information to identify the key driver(s) of different outcomes--fade-out or otherwise--across the sub-populations supporting each epidemic. We apply a Bayesian hierarchical modelling framework to synthetic data from an SIRS model of epidemic dynamics and demonstrate that we can 1) identify when the sub-population specific model parameters supporting each epidemic have significant variability and 2) estimate the probability of epidemic fade-out for each sub-population. We demonstrate that a hierarchical analysis can provide more accurate and precise estimates of the probability of fade-out than is possible if considering each epidemic in isolation. Our methods may be applied more generally, to both epidemiological and other biological data to identify where differences in outcome--fade-out or recurrent infection/waves are purely due to chance or driven by underlying changes in the parameters driving the dynamics.