Sharp phase transition for SIS epidemics on general networks

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For each $n$, let $G_n$ be a finite connected undirected graph on $n$ vertices, with symmetric edge weights $w_{vy}>0$ and vertex recovery rates $\mu_v>0$. For $\theta,\varepsilon>0$, its susceptible--infected--susceptible process has transitions \[I\to S\text{ at rate }\mu_v,\qquad S\to I\text{ at rate }\varepsilon+\theta\sum_{y:\,y\text{ infected}}w_{vy}.\] Let $X^{(n)}_{\theta,\varepsilon}$ be the stationary number of infected vertices. For nonnegative variables $Y_n$, write $Y_n\gg_p0$ if $\lim_{a\downarrow0}\limsup_n\mathbb P(Y_n<a)=0$; this is boundedness away from zero in probability. A property for every $\varepsilon_n\downarrow0$ sufficiently slowly means that some deterministic $b_n\downarrow0$ exists such that it holds for all such sequences with $\varepsilon_n\ge b_n$ eventually. Assume constants $0<\theta_-<\theta_+<\infty$ satisfy, for every sufficiently slowly decreasing $\varepsilon_n$, \[n^{-1}X^{(n)}_{\theta_-,\varepsilon_n}\to0\text{ in probability},\qquad n^{-1}X^{(n)}_{\theta_+,\varepsilon_n}\gg_p0.\]

Statement recorded in the supplied source.

Prove, perhaps with further weak hypotheses on the networks and rates, that deterministic $\theta_n\in[\theta_-,\theta_+]$ exist for which every sufficiently slowly decreasing $\varepsilon_n\downarrow0$ and each fixed $\delta>0$ satisfy \[n^{-1}X^{(n)}_{\theta_n-\delta,\varepsilon_n}\to0\text{ in probability},\qquad n^{-1}X^{(n)}_{\theta_n+\delta,\varepsilon_n}\gg_p0.\] The first expression is considered when $\theta_n-\delta>0$; a nonpositive virulence parameter can equivalently be replaced by zero. [S2]

Short English statement

Prove a sharp stationary SIS infection threshold on general weighted networks under the stated subcritical and supercritical assumptions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9700010 (AMR-096-0010). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 David Aldous, Limits for processes over general networks, talk dated 13 April 2018, Slides headed An SIS model (contact process) and the following Conjecture, PDF pp. 23--24. https://www.stat.berkeley.edu/~aldous/Talks/columbia2018.pdf.