Epidemic Size in the Sis Model of Endemic Infec- Tions

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.

💡 Research Summary

The paper provides a thorough analytical treatment of the classic Susceptible‑Infected‑Susceptible (SIS) model, focusing on the total number of transmission events, denoted by n, rather than the usual quantities such as the number of infected individuals or the steady‑state prevalence. Starting from the master equation that governs the joint evolution of the infected population I and the cumulative transmission count n, the authors introduce the probability distribution Pₙ(t) for having experienced exactly n transmissions up to time t. By marginalising over I they obtain a closed recursion for the mean ⟨n⟩, which they rewrite in terms of a generating function G(z)=∑ₙPₙzⁿ. Solving the resulting nonlinear differential equation with the natural boundary conditions G(1)=1 and G′(1)=⟨n⟩ yields an exact closed‑form expression for the mean number of transmissions as a function of the total population size N and the basic infectivity ratio λ=β/γ.

Having an exact result, the authors turn to the large‑N limit and identify three distinct regimes separated by the critical value λ=1. For λ<1 (sub‑critical) the infection cannot sustain itself; the mean number of transmissions remains O(1) and approaches the simple limit ⟨n⟩≈1/(1−λ). For λ>1 (super‑critical) the infection persists, and ⟨n⟩ grows linearly with N, specifically ⟨n⟩≈(λ−1)N/λ. The most interesting behavior occurs in the critical window where λ=1+O(N^{-1/2}). Introducing the scaling variable x=(λ−1)√N, the authors show that the full distribution collapses onto a universal scaling function Φ(x, y) with y=n/√N. In this regime the distribution exhibits a power‑law singularity for small n: Pₙ∝n^{-3/2}, reflecting large fluctuations typical of critical phenomena. For large n the same scaling function decays exponentially, Pₙ∼exp