  We present a new Bayesian inference method for compartmental models that
takes into account the intrinsic stochasticity of the process. We show how to
formulate a SIR-type Markov jump process as the solution of a stochastic
differential equation with respect to a Poisson Random Measure (PRM), and how
to simulate the process trajectory deterministically from a parameter value and
a PRM realisation.
  This forms the basis of our Data Augmented MCMC, which consists in augmenting
parameter space with the unobserved PRM value. The resulting simple
Metropolis-Hastings sampler acts as an efficient simulation-based inference
method, that can easily be transferred from model to model. Compared with a
recent Data Augmentation method based on Gibbs sampling of individual infection
histories, PRM-augmented MCMC scales much better with epidemic size and is far
more flexible.
  PRM-augmented MCMC also yields a posteriori estimates of the PRM, that
represent process stochasticity, and which can be used to validate the model.
If the model is good, the posterior distribution should exhibit no pattern and
be close to the PRM prior distribution. We illustrate this by fitting a
non-seasonal model to some simulated seasonal case count data. Applied to the
Zika epidemic of 2013 in French Polynesia, our approach shows that a simple
SEIR model cannot correctly reproduce both the initial sharp increase in the
number of cases as well as the final proportion of seropositive.
  PRM-augmentation thus provides a coherent story for Stochastic Epidemic Model
inference, where explicitly inferring process stochasticity helps with model
validation.
