External Seminar: Simon Spencer (Warwick)

[Statistics & Probability Seminar]

Simulating quasi-stationary distributions for epidemics with Sequential Monte Carlo samplers

(Joint work with Adam Griffin and Gareth Roberts.)

A quasi-stationary distribution (QSD) for a stochastic process with an absorbing state is the equilibrium distribution conditional on the absorbing state not yet being reached. Quasi-stationary distributions present numerous analytical challenges due to fact that they do not always exist and when they do exist they are not necessarily unique. In epidemic models, the quasi-stationary distribution might represent the equilibrium distribution of the number of infective individuals prior to disease eradication. Although analytical results for epidemic QSDs exist for simple models such as the Susceptible-Infective-Susceptible (SIS) model, they have proved extremely challenging to generalise to more realistic models, for example when there is more than one possible disease state. Even simulating from the QSD becomes extremely challenging because each simulation must be run for long enough to “forget” its starting state whilst also avoiding the absorbing state.

This talk will describe how to overcome most of these difficulties through the use of Sequential Monte Carlo samplers. We introduce several novel resampling techniques that promote particle diversity and allow much more efficient exploration of the tails of the QSD. Using some toy examples we demonstrate substantial improvements in efficiency compared with a rejection sampling strategy. Finally we illustrate the power of our approach by exploring the QSD for a multidimensional epidemic model that would not have been possible using existing techniques.