Open Access
2022 The effect of avoiding known infected neighbors on the persistence of a recurring infection process
Shirshendu Chatterjee, David Sivakoff, Matthew Wascher
Author Affiliations +
Electron. J. Probab. 27: 1-40 (2022). DOI: 10.1214/22-EJP836


We study a generalization of the classical contact process (SIS epidemic model) on a directed graph G. Our model is a continuous-time interacting particle system in which at every time, each vertex is either healthy or infected, and each oriented edge is either active or inactive. Infected vertices become healthy at rate 1 and pass the infection along each active outgoing edge at rate λ. At rate α, healthy individuals deactivate each incoming edge from their infected neighbors, and an inactive edge becomes active again as soon as its tail vertex becomes healthy. When α=0, this model is the same as the classical contact process on a static graph. We study the persistence time of this epidemic model on the lattice Z, the n-cycle Zn, and the n-star graph. We show that on Z, for every α>0, there is a phase transition in λ between almost sure extinction and positive probability of indefinite survival; on Zn we show that there is a phase transition between poly-logarithmic and exponential survival time as the size of the graph increases. On the star graph, we show that the survival time is nΔ+o(1) for an explicit function Δ(α,λ) whenever α>0 and λ>0. In the cases of Z and Zn, our results qualitatively match what has been shown for the classical contact process, while in the case of the star graph, the classical contact process exhibits exponential survival for all λ>0, which is qualitatively different from our result. This model presents a challenge because, unlike the classical contact process, it has not been shown to be monotonic in the infection parameter λ or the initial infected set.

Funding Statement

DS and MW were partially supported by the NSF grant CCF–1740761. SC acknowledges partial support from NSF grant DMS 2154564.


SC and DS are grateful to the Mathematical Biosciences Institute, where some of this work was conducted. The authors are thankful to the anonymous referee for many thoughtful suggestions, which have led to substantial improvements in the presentation and articulation of this paper.


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Shirshendu Chatterjee. David Sivakoff. Matthew Wascher. "The effect of avoiding known infected neighbors on the persistence of a recurring infection process." Electron. J. Probab. 27 1 - 40, 2022.


Received: 24 November 2020; Accepted: 27 July 2022; Published: 2022
First available in Project Euclid: 22 August 2022

MathSciNet: MR4474536
zbMATH: 1507.60126
Digital Object Identifier: 10.1214/22-EJP836

Primary: 60K35

Keywords: contact process , epidemics on networks , evolving networks , SIS epidemic

Vol.27 • 2022
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