Prior distribution over initial infective?

Suppose we have a set of epidemic event times I,R, given covariates (i.e. location, age, sex, etc) X and parameters \theta. Let the epidemic generating process be F such that I,R \sim F with probability density function f(I,R | \theta).

For an epidemic, we are interested in making inference on \pi(\theta | I,R) \propto f(I,R | \theta)f(\theta) where f(\theta) is the prior distribution on the parameters. Having made inference, we may wish to check our model by assessing whether the posterior predictive distribution of the epidemic 'matches' the observed data. i.e. we are interested in: f(I^\prime,R^\prime | I, R) = \int_{\Theta} f(I^\prime,R^\prime | \theta)\pi(\theta | I,R) \mathrm{d}\theta

The notation f(I,R | \theta) is, however, deceiving since we have to explicitly condition on an initial event (for an SIR epidemic that's an infection event) in order to start our epidemic process. In other words, we need f(I,R | \theta,\kappa) where \kappa is the identity of the initial event.