Fast and efficient Bayesian inference for stochastic epidemic models

Abstract

Epidemics are inherently stochastic in nature, and stochastic kinetic models (SKMs) provide an appropriate way to describe and analyse such phenomena. Given temporal data consisting of, for example, the number of new infections or removals in a given time window, a continuous-time discrete-valued Markov process provides a natural description of the dynamics of each model component, typically taken to be the number of susceptible, exposed, infected or removed individuals. Fitting the resulting SEIR model to time-course data is a challenging task due to the problem of partial observations and, consequently, the intractability of the observed data likelihood. Whilst sampling based inference schemes such as Markov chain Monte Carlo are routinely applied, their computational cost typically restricts analysis to data sets of no more than a few thousand infected cases. Moreover, upon receipt of new data, these schemes typically need to be restarted from scratch. This thesis addresses these issues via two complementary approaches. First, we develop a sequential inference scheme that makes use of a computationally cheap approximation of the most natural Markov process model. Crucially, the resulting model allows a tractable conditional parameter posterior which can be summarised in terms of a set of low dimensional statistics. This is used to rejuvenate parameter samples in conjunction with a novel bridge construct for propagating state trajectories conditional on the next observation of cumulative incidence. The resulting inference framework also allows for stochastic infection and reporting rates. Second, we tackle the intractability of the observed data likelihood in a batch inference setting. We adopt a stochastic differential equation (SDE) representation of the underlying epidemic dynamics by matching the infinitesimal mean and variance to the drift and diffusion coefficients of an Itˆo SDE. We then approximate the SDE to give a tractable Gaussian process, that is, the linear noise approximation (LNA). Unless the observation model linking the LNA to the data is both linear and Gaussian, the observed data likelihood remains intractable. To circumvent this, we marginalise over the latent process by enforcing a Gaussian approximation of the observation model and use a forward filter to efficiently calculate the resulting approximation of the observed data likelihood. The proposed inference methodology is illustrated using both real and synthetic data sets. Where possible, we compare against competing approaches.