The effects of individual versus community-influenced isolation on SIS epidemic persistence on finite random graphs

TL;DR

This paper compares how individual versus community-driven isolation strategies impact the duration of SIS epidemic persistence on finite random graphs, revealing significant differences in epidemic longevity based on the type of isolation.

Contribution

It introduces two new models of isolation in SIS epidemics, analyzing their effects on epidemic persistence and demonstrating how community influence can cause phase transitions in infection duration.

Findings

Isolation model leads to at least stretched exponential persistence time.

Vigilance model shows phase transition in persistence time based on infection rate.

Community-influenced isolation can significantly reduce epidemic duration.

Abstract

The contact process, or SIS epidemic, is a continuous-time Markov process used to model the spread of infection on a graph. Each vertex is either healthy or infected, and each infected vertex independently infects each of its healthy neighbors at rate $\lambda$ and recovers at rate $1$. We study the contact process in the presence of additional intervention measures by introducing a third possible state for vertices, which we call isolated. Vertices may enter the isolated state either because of individual decisions or due to community-influenced decisions, which leads to two distinct models that we call the isolation model and the vigilance model, respectively. In the isolation model, infected vertices self-isolate at rate $\alpha$. In the vigilance model, each healthy vertex causes each of its infected neighbors to isolate at rate $\alpha$. Unlike the usual contact process, these models lack the key features of attractiveness and existence of a dual, which makes analyzing them more challenging. We study the persistence times of the infection on large, finite, degree-heterogeneous random graphs. We show that the infection in the isolation model persists for at least stretched exponential time in the size of the graph for all values of $\alpha$ and $\lambda$. By contrast, in the vigilance model, for every fixed $\alpha$ the persistence time of the infection exhibits a phase transition in $\lambda$: for small $\lambda$ the infection persists for at most a linear time in the size of the graph, while for large $\lambda$ the infection persists exponentially long. This contrast demonstrates that individual versus community-influenced isolation can substantially affect the persistence of an epidemic.