  The quasi-deterministic limit of the generic extinction transition is
considered within the framework of standard epidemiological models. The
susceptible-infected-susceptible (SIS) model is known to exhibit a transition
from extinction to spreading, as the infectivity is increased, described by the
directed percolation equivalence class. We find that the distance from the
transition point, and the prefactor controlling the divergence of the
(perpendicular) correlation length, both scale with the local population size,
$N$, with two novel universal exponents. Different exponents characterize the
large $N$ behavior of the susceptible-infected-recovered (SIR) model, which
belongs to the dynamic percolation class. Extensive numerical studies in a
range of systems lead to the conjecture that these characteristics are generic
and may be used in order to classify the high density limit of any stochastic
process on the edge of extinction.
