  We extend a technique of approximation of the long-term behavior of a
supercritical stochastic epidemic model, using the WKB approximation and a
Hamiltonian phase space, to the subcritical case. The limiting behavior of the
model and approximation are qualitatively different in the subcritical case,
requiring a novel analysis of the limiting behavior of the Hamiltonian system
away from its deterministic subsystem. This yields a novel, general technique
of approximation of the quasistationary distribution of stochastic epidemic and
birth-death models, and may lead to techniques for analysis of these models
beyond the quasistationary distribution. For a classic SIS model, the
approximation found for the quasistationary distribution is very similar to
published approximations but not identical. For a birth-death process without
depletion of susceptibles, the approximation is exact. Dynamics on the phase
plane similar to those predicted by the Hamiltonian analysis are demonstrated
in cross-sectional data from trachoma treatment trials in Ethiopia, in which
declining prevalences are consistent with subcritical epidemic dynamics.
