  Probabilistic models for infectious disease dynamics are useful for
understanding the mechanism underlying the spread of infection. When the
likelihood function for these models is expensive to evaluate, traditional
likelihood-based inference may be computationally intractable. Furthermore,
traditional inference may lead to poor parameter estimates and the fitted model
may not capture important biological characteristics of the observed data. We
propose a novel approach for resolving these issues that is inspired by recent
work in emulation and calibration for complex computer models. Our motivating
example is the gravity time series susceptible-infected-recovered (TSIR) model.
Our approach focuses on the characteristics of the process that are of
scientific interest. We find a Gaussian process approximation to the gravity
model using key summary statistics obtained from model simulations. We
demonstrate via simulated examples that the new approach is computationally
expedient, provides accurate parameter inference, and results in a good model
fit. We apply our method to analyze measles outbreaks in England and Wales in
two periods, the pre-vaccination period from 1944-1965 and the vaccination
period from 1966-1994. Based on our results, we are able to obtain important
scientific insights about the transmission of measles. In general, our method
is applicable to problems where traditional likelihood-based inference is
computationally intractable or produces a poor model fit. It is also an
alternative to approximate Bayesian computation (ABC) when simulations from the
model are expensive.
