{
  "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 $\\mathbb{Z}$, the $n$-cycle $\\mathbb{Z}_n$, and the $n$-star graph; here we extend the study of this model to lattices $\\mathbb{Z}^d$, $d \\geq 2$. We show that for every $d \\geq 2$ 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 $d \\rightarrow \\infty$ 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 $1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03359",
  "author_total": 2,
  "authors": [
    {
      "name": "Andrew Heeszel"
    },
    {
      "name": "Matthew Wascher"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ed0pnwvbzc6bv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03359",
  "fulltext_url": null,
  "id": "ed0pnwvbzc6bv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03359",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2739|Mathematical and Theoretical Epidemiology and Ecology Models"
    ]
  ],
  "openalex": "W7208748008",
  "pdf_url": "https://arxiv.org/pdf/2609.03359",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ed0pnwvbzc6bv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.03359",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03359"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Epidemics with avoidance and isolation on $\\mathbb{Z}^d$",
  "title_text": "Epidemics with avoidance and isolation on ℤ^d",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/ed0pnwvbzc6bv",
  "venue": "arXiv",
  "versions": []
}