发表机构
University of California, Davis; Case Western Reserve University（加州大学戴维斯分校; 凯斯西储大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究$\boldsymbol{\text{Z}^d}$上带规避和隔离的流行病接触过程，确定感染消失与持续的阈值λ，证明阈值随$\boldsymbol{1/d}$缩放，结果可推广至SIRS等模型。
AI 中文摘要
带规避机制的接触过程是经典接触过程（SIS流行病）的推广，它引入了健康个体规避受感染邻居的机制。设$\text{G}$为有向图，在每个时刻$\text{t}$，每个顶点要么健康要么受感染，每条边要么活跃要么不活跃。每个受感染顶点以速率$\boldsymbol{\text{λ}}$通过每条活跃边感染其健康邻居，以速率1恢复；同时，每条从受感染顶点指向健康顶点的活跃边以速率$\boldsymbol{\text{α}}$变为不活跃，不活跃边则在其尾顶点恢复时变为活跃。该模型此前已在$\boldsymbol{\text{Z}}$、$\boldsymbol{\text{n}}$环$\boldsymbol{\text{Z}_n}$和$\boldsymbol{\text{n}}$星图上研究，本文将其研究扩展至$\boldsymbol{\text{d} \text{≥} 2}$的晶格$\boldsymbol{\text{Z}^d}$。研究表明，对每个$\boldsymbol{\text{d} \text{≥} 2}$及固定$\boldsymbol{\text{α} \text{>} 0}$，存在常数$\boldsymbol{\text{λ(α,d)}^-}$和$\boldsymbol{\text{λ(α,d)}^+}$：当$\boldsymbol{\text{λ} \text{<} \text{λ(α,d)}^-}$时，感染几乎必然消失；当$\boldsymbol{\text{λ} \text{>} \text{λ(α,d)}^+}$时，感染以正概率持续存在。此外，还证明$\boldsymbol{\text{λ(α,d)}^-}$和$\boldsymbol{\text{λ(α,d)}^+}$在$\boldsymbol{\text{d} \text{→} \text{∞}}$时均按$\boldsymbol{1/d}$缩放，且存在常数$\boldsymbol{\text{C(α)}}$，当$\boldsymbol{\text{λ} \text{>} \text{C(α)/d}}$且$\boldsymbol{\text{d}}$足够大时，该过程具有非平凡不变测度。本文的方法及多数结果也适用于SIRS模型和相关模型（受感染顶点以速率$\boldsymbol{\text{α}}$进入隔离状态，从隔离态和感染态以速率1恢复为健康态）。
英文摘要
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 $1$.
Comments16 pages, 1 figure, fixed typos in abstract and added some additional discussion to the introduction
