Epidemic size in the SIS model of endemic infections

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 size (N) 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 an n^-3/2 singularity for small n and decays exponentially for large n. The exponent decreases with the distance from the threshold, diverging to ∞ far below and approaching 0 far above.