  We develop a theory of first passage processes in stochastic non-equilibrium
systems of birth-death type using two closely related epidemiological models as
examples. Our method employs the probability generating function technique in
conjunction with the eikonal approximation. In this way the problem is reduced
to finding the optimal path to extinction: a heteroclinic trajectory of an
effective multi-dimensional classical Hamiltonian system. We compute this
trajectory and mean extinction time of the disease numerically and uncover a
non-monotone, spiral path to extinction of a disease. We also obtain analytical
results close to a bifurcation point, where the problem is described by a
Hamiltonian previously identified in one-species population models.
