  We show that the simplest stochastic epidemiological models with spatial
correlations exhibit two types of oscillatory behaviour in the endemic phase.
In a large parameter range, the oscillations are due to resonant amplification
of stochastic fluctuations, a general mechanism first reported for
predator-prey dynamics. In a narrow range of parameters that includes many
infectious diseases which confer long lasting immunity the oscillations persist
for infinite populations. This effect is apparent in simulations of the
stochastic process in systems of variable size, and can be understood from the
phase diagram of the deterministic pair approximation equations. The two
mechanisms combined play a central role in explaining the ubiquity of
oscillatory behaviour in real data and in simulation results of epidemic and
other related models.
