Epidemics with avoidance and isolation on ℤd
Abstract
The contact process with avoidance is a generalization of the classical contact process (SIS epidemic) that introduces a mechanism for healthy individuals to avoid their infected neighbors. Let G be a directed graph. At each time t, each vertex is either healthy or infected and each edge is either active or inactive. Each infected vertex infects each healthy neighbor across each active edge at rate λ and recovers at rate $1$, while each active edge pointing from an infected vertex to a healthy vertex becomes inactive at rate α. An inactive edge becomes active when its tail vertex recovers. This model has been previously studied on ℤ, the n-cycle ℤn, and the n-star graph; here we extend the study of this model to lattices ℤd, d ≥ 2. We show that for every d ≥ 2 and fixed α> 0, there exist constants λ(α,d)- and λ(α,d)+ such that for all λ< λ(α,d)- the infection dies out almost surely while for all λ> λ(α,d)+ the infection persists indefinitely with positive probability. Furthermore, we show that both λ(α,d)- and λ(α,d)+ scale like 1/d as d → ∞ and that there exists a constant C(α) such that when λ> C(α)/d the process has a nontrivial invariant measure for d sufficiently large. Our methods and most of our results also apply to the SIRS model and a related model in which infected vertices enter an isolated state at rate α and transition from both isolated and infected to healthy at rate $1$.
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BibTeX
@misc{heeszel2026epidemics,
title = {{Epidemics with avoidance and isolation on $\mathbbZ^d$}},
author = {Andrew Heeszel and Matthew Wascher},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03359},
eprint = {2609.03359},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03359},
}
RIS
TY - UNPB
TI - Epidemics with avoidance and isolation on ℤ^d
AU - Andrew Heeszel
AU - Matthew Wascher
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.03359
UR - https://arxiv.org/abs/2609.03359
AB - The contact process with avoidance is a generalization of the classical contact process (SIS epidemic) that introduces a mechanism for healthy individuals to avoid their infected neighbors. Let G be a directed graph. At each time t, each vertex is either healthy or infected and each edge is either active or inactive. Each infected vertex infects each healthy neighbor across each active edge at rate λ and recovers at rate $1$, while each active edge pointing from an infected vertex to a healthy vertex becomes inactive at rate α. An inactive edge becomes active when its tail vertex recovers. This model has been previously studied on ℤ, the n-cycle ℤ_n, and the n-star graph; here we extend the study of this model to lattices ℤ^d, d ≥ 2. We show that for every d ≥ 2 and fixed α> 0, there exist constants λ(α,d)^- and λ(α,d)^+ such that for all λ< λ(α,d)^- the infection dies out almost surely while for all λ> λ(α,d)^+ the infection persists indefinitely with positive probability. Furthermore, we show that both λ(α,d)^- and λ(α,d)^+ scale like 1/d as d → ∞ and that there exists a constant C(α) such that when λ> C(α)/d the process has a nontrivial invariant measure for d sufficiently large. Our methods and most of our results also apply to the SIRS model and a related model in which infected vertices enter an isolated state at rate α and transition from both isolated and infected to healthy at rate $1$.
ER -