  The dynamics that govern disease spread are hard to model because infections
are functions of both the underlying pathogen as well as human or animal
behavior. This challenge is increased when modeling how diseases spread between
different spatial locations. Many proposed spatial epidemiological models
require trade-offs to fit, either by abstracting away theoretical spread
dynamics, fitting a deterministic model, or by requiring large computational
resources for many simulations. We propose an approach that approximates the
complex spatial spread dynamics with a Gaussian process. We first propose a
flexible spatial extension to the well-known SIR stochastic process, and then
we derive a moment-closure approximation to this stochastic process. This
moment-closure approximation yields ordinary differential equations for the
evolution of the means and covariances of the susceptibles and infectious
through time. Because these ODEs are a bottleneck to fitting our model by MCMC,
we approximate them using a low-rank emulator. This approximation serves as the
basis for our hierarchical model for noisy, underreported counts of new
infections by spatial location and time. We demonstrate using our model to
conduct inference on simulated infections from the underlying, true spatial SIR
jump process. We then apply our method to model counts of new Zika infections
in Brazil from late 2015 through early 2016.
