The effect of avoiding known infected neighbors on the persistence of a recurring infection process
The effect of avoiding known infected neighbors on the persistence of a recurring infection process
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DOI：
10.1214/22-ejp836
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发表时间：
2020-11
影响因子：
1.4
通讯作者：
S. Chatterjee;David J Sivakoff;M. Wascher
中科院分区：
文献类型：
--
作者：
S. Chatterjee;David J Sivakoff;M. Wascher
研究了有向图$G$中经典接触过程（SIS流行病模型）的推广。我们的模型是一个连续时间相互作用的粒子系统，在每个时刻，每个顶点要么是健康的，要么是感染的，每个定向边缘要么是活动的，要么是不活动的。受感染的顶点以$1$的速率恢复健康，并以$\lambda$的速率沿每个活跃的外向边缘传播感染。至少$\alpha$，健康的个体会使来自被感染的邻居的每条边缘失效。研究了该流行病模型在格子$\mathbb{Z}$、$n$ -循环$\mathbb{Z}_n$和$n$ -星图上的持续时间。我们表明，在$\mathbb{Z}$上，对于每一个$\alpha>0$，在$\lambda$上存在一个介于几乎肯定灭绝和正的不确定生存概率之间的相变；在$\mathbb{Z}_n$上，我们表明，随着图的大小增加，多对数生存时间和指数生存时间之间存在相变。在星形图上，我们表明，对于显式函数$\Delta(\alpha,\lambda)$，每当$\alpha>0$和$\lambda>0$时，生存时间为$n^{\Delta+o(1)}$。在$\mathbb{Z}$和$\mathbb{Z}_n$的情况下，我们的结果在质量上与经典接触过程的结果相匹配，而在星图的情况下，经典接触过程对所有$\lambda > 0$都表现出指数生存，这与我们的结果在质量上不同。该模型提出了一个挑战，因为与经典接触过程不同，它在感染参数$\lambda$或初始感染集中并未显示为单调的。
We study a generalization of the classical contact process (SIS epidemic model) in 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 $\lambda$. At rate $\alpha$, healthy individuals deactivate each incoming edge from their infected neighbors. We study the persistence time of this epidemic model on the lattice $\mathbb{Z}$, the $n$-cycle $\mathbb{Z}_n$, and the $n$-star graph. We show that on $\mathbb{Z}$, for every $\alpha>0$, there is a phase transition in $\lambda$ between almost sure extinction and positive probability of indefinite survival; on $\mathbb{Z}_n$ 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 $n^{\Delta+o(1)}$ for an explicit function $\Delta(\alpha,\lambda)$ whenever $\alpha>0$ and $\lambda>0$. In the cases of $\mathbb{Z}$ and $\mathbb{Z}_n$, 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 $\lambda > 0$, 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 $\lambda$ or the initial infected set.