  We investigate stochastic extinction in an epidemic model and the impact of
random vaccinations in large populations. We show that, in the absence of
vaccinations, the effective entropic barrier for extinction displays scaling
with the distance to the bifurcation point, with an unusual critical exponent.
Even a comparatively weak Poisson-distributed vaccination leads to an
exponential increase in the extinction rate, with the exponent that strongly
depends on the vaccination parameters.
