The End Time of SIS Epidemics Driven by Random Walks on Edge-Transitive Graphs
The End Time of SIS Epidemics Driven by Random Walks on Edge-Transitive Graphs
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DOI：
10.1007/s10955-020-02547-7
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发表时间：
2019-03
影响因子：
1.6
通讯作者：
Daniel R. Figueiredo;G. Iacobelli;S. Shneer
中科院分区：
文献类型：
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作者：
Daniel R. Figueiredo;G. Iacobelli;S. Shneer
网络流行病是一种普遍存在的模型，可以代表不同的现象并在各个领域都有应用。在其众多特征中，一个基本问题是流行病停止传播的时间。我们研究了由在有限图上根据独立连续时间随机游走移动的代理引起的 SIS 流行病的这一特征：代理可以被感染（I）或易感（S），并且当两个具有不同流行状态的代理在一个节点中相遇时就会发生感染。经过随机恢复时间后，受感染的代理返回到状态 S 并且可以再次被感染。流行病结束（EoE）表示所有智能体首次处于状态 S，因为此后不会再发生进一步的感染，流行病也会停止。对于边传递图上两个智能体的情况，我们通过将 EoE 的拉普拉斯变换与两个随机游走的相遇时间的拉普拉斯变换相关联，将 EoE 描述为网络结构的函数。有趣的是，这一分析显示了网络结构的影响与流行病动态之间的分离。然后，我们研究不同参数缩放下 EoE 的渐近行为（在图的大小上渐进），识别 EoE 分布收敛到适当随机变量或无穷大的状态。我们还强调了不同图结构对 EoE 的影响，在完全图、完全二分图和环下对其进行表征。
Network epidemics is a ubiquitous model that can represent different phenomena and finds applications in various domains. Among its various characteristics, a fundamental question concerns the time when an epidemic stops propagating. We investigate this characteristic on a SIS epidemic induced by agents that move according to independent continuous time random walks on a finite graph: agents can either be infected (I) or susceptible (S), and infection occurs when two agents with different epidemic states meet in a node. After a random recovery time, an infected agent returns to state S and can be infected again. The end of epidemic (EoE) denotes the first time where all agents are in state S, since after this moment no further infections can occur and the epidemic stops. For the case of two agents on edge-transitive graphs, we characterize EoE as a function of the network structure by relating the Laplace transform of EoE to the Laplace transform of the meeting time of two random walks. Interestingly, this analysis shows a separation between the effect of network structure and epidemic dynamics. We then study the asymptotic behavior of EoE (asymptotically in the size of the graph) under different parameter scalings, identifying regimes where EoE converges in distribution to a proper random variable or to infinity. We also highlight the impact of different graph structures on EoE, characterizing it under complete graphs, complete bipartite graphs, and rings.