It is known from simulations that stochastic epidemic models such as SIR, SIRS produce sustained oscillations, whereas the corresponding deterministic models produce damped oscillations. We use multiscale analysis to identify a parameter range in which stochastic SIR is approximated by a stochastic combination of sinusoids, where the coefficient processes run on a slow time scale. Ornstein-Uhlenbeck excursions form an envelope of the epidemic oscillations.