We study epidemic spreading according to a \emph{Susceptible-Infectious-Recovered} (for short, \emph{SIR}) network model known as the {\em Reed-Frost} model, and we establish sharp thresholds for two generative models of {\em one-dimensional small-world graphs}, in which graphs are obtained by adding random edges to a cycle. In $3$-regular graphs obtained as the union of a cycle and a random perfect matching, we show that there is a sharp threshold at $.5$ for the contagion probability along edges. In graphs obtained as the union of a cycle and of a $\mathcal{G}_{n,c/n}$ Erd\H{o}s-R\'enyi random graph with edge probability $c/n$, we show that there is a sharp threshold $p_c$ for the contagion probability: the value of $p_c$ turns out to be $\sqrt 2 -1\approx .41$ for the sparse case $c=1$ yielding an expected node degree similar to the random $3$-regular graphs above. In both models, below the threshold we prove that the infection only affects $\mathcal{O}(\log n)$ nodes, and that above the threshold it affects $\Omega(n)$ nodes. These are the first fully rigorous results establishing a phase transition for SIR models (and equivalent percolation problems) in small-world graphs. Although one-dimensional small-world graphs are an idealized and unrealistic network model, a number of realistic qualitative phenomena emerge from our analysis, including the spread of the disease through a sequence of local outbreaks, the danger posed by random connections, and the effect of super-spreader events.

中文翻译：

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一维小世界网络上SIR模型的敏锐阈值

我们根据称为{\ em Reed-Frost}模型的\ emph {Sceptceptible-Infectious-Recovered}（简称\ emph {SIR}）网络模型研究流行病的传播，并为两个{\ em一维小世界图}，其中的图是通过将随机边添加到循环中获得的。在$ 3 $-作为周期的结合和随机完美匹配而获得的正则图中，我们表明在$ .5 $处有一个沿边缘的传染概率的尖锐阈值。在以周期和$ \ mathcal {G} _ {n，c / n} $ Erd \ H {o} sR \'enyi随机图为边概率$ c / n $的并集获得的图中，我们显示传染概率有一个尖锐的阈值$ p_c $：$ p_c $的值原来是$ \ sqrt 2 -1 \ approx。稀疏情况$ c = 1 $的41 $产生了一个预期的节点度，类似于上面的随机$ 3 $-规则图。在这两个模型中，低于阈值我们证明感染仅影响$ \ mathcal {O}（\ log n）$节点，超过阈值则影响$ \ Omega（n）$节点。这些是在小世界图中为SIR模型（和等效的渗流问题）建立相变的第一个完全严格的结果。尽管一维小世界图是一个理想化且不现实的网络模型，但我们的分析还是出现了许多现实的定性现象，包括疾病通过一系列局部爆发的传播，随机连接造成的危险以及影响超级传播者事件。低于阈值，我们证明感染仅影响$ \ mathcal {O}（\ log n）$节点，并且超过阈值时，它会影响$ \ Omega（n）$节点。这些是在小世界图中为SIR模型（和等效的渗流问题）建立相变的第一个完全严格的结果。尽管一维小世界图是一个理想化且不现实的网络模型，但我们的分析还是出现了许多现实的定性现象，包括疾病通过一系列局部爆发的传播，随机连接造成的危险以及影响超级传播者事件。低于阈值，我们证明感染仅影响$ \ mathcal {O}（\ log n）$节点，并且高于阈值时，它会影响$ \ Omega（n）$节点。这些是在小世界图中为SIR模型（和等效的渗流问题）建立相变的第一个完全严格的结果。尽管一维小世界图是一个理想化且不现实的网络模型，但我们的分析还是出现了许多现实的定性现象，包括疾病通过一系列局部爆发的传播，随机连接造成的危险以及影响超级传播者事件。