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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. 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.