  We consider a stochastic Susceptible-Exposed-Infected-Recovered (SEIR)
epidemiological model with a contact rate that fluctuates seasonally. Through
the use of a nonlinear, stochastic projection, we are able to analytically
determine the lower dimensional manifold on which the deterministic and
stochastic dynamics correctly interact. Our method produces a low dimensional
stochastic model that captures the same timing of disease outbreak and the same
amplitude and phase of recurrent behavior seen in the high dimensional model.
Given seasonal epidemic data consisting of the number of infectious
individuals, our method enables a data-based model prediction of the number of
unobserved exposed individuals over very long times.
