  We study the stochastic susceptible-infected-recovered (SIR) model with
time-dependent forcing using analytic techniques which allow us to disentangle
the interaction of stochasticity and external forcing. The model is formulated
as a continuous time Markov process, which is decomposed into a deterministic
dynamics together with stochastic corrections, by using an expansion in inverse
system size. The forcing induces a limit cycle in the deterministic dynamics,
and a complete analysis of the fluctuations about this time-dependent solution
is given. This analysis is applied when the limit cycle is annual, and after a
period-doubling when it is biennial. The comprehensive nature of our approach
allows us to give a coherent picture of the dynamics which unifies past work,
but which also provides a systematic method for predicting the periods of
oscillations seen in whooping cough and measles epidemics.
