  Many real epidemics of an infectious disease are not straightforwardly super-
or sub-critical, and the understanding of epidemic models that exhibit such
complexity has been identified as a priority for theoretical work. We provide
insights into the near-critical regime by considering the stochastic SIS
logistic epidemic, a well-known birth-and-death chain used to model the spread
of an epidemic within a population of a given size $N$. We study the behaviour
of the process as the population size $N$ tends to infinity. Our results cover
the entire subcritical regime, including the "barely subcritical" regime, where
the recovery rate exceeds the infection rate by an amount that tends to 0 as $N
\to \infty$ but more slowly than $N^{-1/2}$. We derive precise asymptotics for
the distribution of the extinction time and the total number of cases
throughout the subcritical regime, give a detailed description of the course of
the epidemic, and compare to numerical results for a range of parameter values.
We hypothesise that features of the course of the epidemic will be seen in a
wide class of other epidemic models, and we use real data to provide some
tentative and preliminary support for this theory.
