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.