  We consider a continuous-time Markov chain model of SIR disease dynamics with
two levels of mixing. For this so-called stochastic households model, we
provide two methods for inferring the model parameters---governing
within-household transmission, recovery, and between-household
transmission---from data of the day upon which each individual became
infectious and the household in which each infection occurred, as would be
available from first few hundred studies. Each method is a form of Bayesian
Markov Chain Monte Carlo that allows us to calculate a joint posterior
distribution for all parameters and hence the household reproduction number and
the early growth rate of the epidemic. The first method performs exact Bayesian
inference using a standard data-augmentation approach; the second performs
approximate Bayesian inference based on a likelihood approximation derived from
branching processes. These methods are compared for computational efficiency
and posteriors from each are compared. The branching process is shown to be an
excellent approximation and remains computationally efficient as the amount of
data is increased.
