AI 中文总结
本文研究图上的僵尸感染模型，通过耦合技术严格分析其单调性与感染无限传播概率，得出树图上该概率随λ单调、部分有界度图不单调等结论。
AI 中文摘要
我们研究一种图上的随机SIR模型变体，该模型此前已在物理学文献中被提出用于模拟僵尸爆发，在此称为僵尸感染模型（ZIM）。在该模型中，图的每个节点初始时为易感、感染或移除状态。与SIR模型类似，易感节点以λ倍于其感染邻居数量的速率被感染；此外，在ZIM中，感染节点以1倍于其易感邻居数量的速率被移除。该过程呈现出丰富且有时违背直觉的行为。通过结合各种耦合技术，我们对该模型进行了严格的数学分析，重点关注单调性性质和感染无限传播的概率。我们的主要结果之一是，对于树图上的过程，该概率随λ的增加而单调变化；但对于某些有界度图，该概率是连续的却不单调。我们还确定了一般图上该概率的边界，并对完全图、正则树和d维整数格推导了更精确的结果。
英文摘要
We study a variant of the stochastic SIR model on graphs that has previously been introduced in the physics literature for modelling zombie outbreaks and here referred to as the Zombie Infection Model (ZIM). In this model, initially each node of a graph is either susceptible, infected or removed. As in the SIR model, a susceptible node becomes infected at rate $λ$ times the number of its infected neighbours. Moreover, in the ZIM, an infected node is removed at rate $1$ times the number of its susceptible neighbours. This process exhibits rich and sometimes counterintuitive behaviour. By combining various coupling techniques, we provide a rigorous mathematical analysis of the model, focusing on monotonicity properties and the probability of the infection spreading indefinitely. One of our main results is that this probability is monotone with respect to an increase of $λ$ for the process on trees, but that there are graphs of bounded degree for which it is continuous and yet not monotone. We also establish bounds on this probability for the process on general graphs, and derive more precise results for complete graphs, regular trees, and the $d$-dimensional integer lattice.
Comments56 pages, 12 figures. Submitted to the Electronic Journal of Probability
