  In this paper we introduce a novel method to conduct inference with models
defined through a continuous-time Markov process, and we apply these results to
a classical stochastic SIR model as a case study. Using the inverse-size
expansion of van Kampen we obtain approximations for first and second moments
for the state variables. These approximate moments are in turn matched to the
moments of an inputed generic discrete distribution aimed at generating an
approximate likelihood that is valid both for low count or high count data. We
conduct a full Bayesian inference to estimate epidemic parameters using
informative priors. Excellent estimations and predictions are obtained both in
a synthetic data scenario and in two Dengue fever case studies.
