  We study the Susceptible-Infected-Susceptible model of the spread of an
endemic infection. We calculate an exact expression for the mean number of
transmissions for all values of the population and the infectivity. We derive
the large-N asymptotic behavior for the infectivitiy below, above, and in the
critical region. We obtain an analytical expression for the probability
distribution of the number of transmissions, n, in the critical region. We show
that this distribution has a $n^3/2$ singularity for small n and decays
exponentially for large n. The exponent decreases with the distance from
threshold, diverging to infinity far below and approaching zero far above.
