Epidemic Size in the Sis Model of Endemic Infec- Tions

TL;DR

This paper provides an exact analytical study of the epidemic size in the SIS model, including asymptotic behaviors and distribution characteristics across different infectivity regimes.

Contribution

It derives an exact expression for the mean number of transmissions and the distribution in the critical region, advancing understanding of endemic infection spread.

Findings

Exact mean number of transmissions for all parameters

Asymptotic behavior of epidemic size below, above, and at criticality

Distribution of transmissions exhibits a $n^{3/2}$ singularity near zero

Abstract

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.