Emulating a gravity model to infer the spatiotemporal dynamics of an infectious disease

Extremely contagious, acute, immunizing childhood infections like measles can exhibit spatiotemporal dynamics that depend on the nature of spatial contagion and spatiotemporal variations in population structure and demography. We study a metapopulation model for regional measles dynamics that uses a gravity coupling model and a time series susceptible- infected-recovered (TSIR) model for local dynamics. Standard maximum likelihood or Bayesian inference for this model is infeasible as there are potentially tens of thousands of latent variables in the model and each evaluation of the likelihood is expensive. We develop an efficient discretized MCMC algorithm for Bayesian inference with these expensive likelihood evaluations. However, we find through a simulation study that parameter estimates are biased and simulations at the obtained parameter settings do not explain some important biological characteristics of the data. We propose fitting a Gaussian process (GP) model to forward simulations of the gravity model at a number of parameter settings. Based on the GP-based emulator we perform a full Bayesian analysis of a given data set. This approach allows us to conveniently study posterior distributions of the key parameters of the gravity model and has number of advantages over the classic likelihood based inference.