[论文解读] Multifractal scaling analyses of the spatial diffusion pattern of COVID-19 pandemic in Chinese mainland

Yuqing Long, Yanguang Chen+1 more|arXiv (Cornell University)|Oct 5, 2020

COVID-19 epidemiological studies参考文献 23被引用 4

一句话总结

本研究利用实时地级市层面数据，对中国大陆地区COVID-19的空间扩散应用多重分形标度分析。通过ArcGIS与miu加权最小二乘法，揭示了具有反S形广义关联维数和右偏斜局部奇异性谱的多重分形结构，表明在四个不同疫情阶段中，存在自相似、全局聚集且分层扩散的模式。

ABSTRACT

Revealing spatiotemporal evolution regularity in the spatial diffusion of epidemics is helpful for preventing and controlling the spread of epidemics. Based on the real-time COVID-19 datasets by prefecture-level cities, this paper is devoted to exploring the multifractal scaling in spatial diffusion pattern of COVID-19 pandemic and its evolution characteristics in Chinese mainland. The ArcGIS technology and box-counting method are employed to extract spatial data and the least square regression based on rescaling probability (miu-weight method) is used to calculate fractal parameters. The results show multifractal distribution of COVID-19 pandemic in China. The generalized correlation dimension spectrums are inverse S-shaped curves, but the fractal dimension values significantly exceed the Euclidean dimension of embedding space when moment order q<<0. The local singularity spectrums are asymmetric unimodal curves, which slant to right. The fractal dimension growth curves are shown as quasi S-shaped curves. From these spectrums and growth curves, the main conclusions can be drawn as follows: First, self-similar patterns developed in the process of COVID-19 pandemic, which seem be dominated by multifractal scaling law. Second, the spatial pattern of COVID-19 across China can be characterized by global clustering with local disordered diffusion. Third, the spatial diffusion process of COVID-19 in China experienced four stages, i.e., initial stage, the rapid diffusion stage, the hierarchical diffusion stage, and finally the contraction stage. This study suggests that multifractal theory can be utilized to characterize spatio-temporal diffusion of COVID-19 pandemic, and the case analyses may be instructive for further exploring natural laws of spatial diffusion.

研究动机与目标

• 揭示中国内地COVID-19大流行在空间扩散中的时空规律。
• 探究多重分形标度定律是否支配传染病病例的空间分布。
• 识别不同的扩散阶段，并利用分形几何刻画其空间模式。
• 评估多重分形理论在建模复杂传染病传播动力学中的适用性。
• 为理解疾病扩散中全局聚集与局部无序的复合特征提供定量框架。

提出的方法

• 利用ArcGIS从实时地级市分辨率的疫情数据集中提取确诊病例的空间数据。
• 应用盒计数法估算不同矩阶下空间分布的分形维数。
• 基于重标度概率（miu加权法）的最小二乘回归用于计算广义关联维数与局部奇异性谱。
• 分析不同矩阶（q）下的广义关联维数谱，尤其关注q << 0以评估罕见的高强度事件。
• 从多重分形形式化中推导出局部奇异性谱，以评估空间模式中奇异性分布。
• 生成分形维数增长曲线，追踪空间复杂性随时间的演变，识别出不同的扩散阶段。

实验结果

研究问题

• RQ1 中国内地COVID-19的空间扩散模式是否表现出多重分形标度特性？
• RQ2 广义关联维数谱与局部奇异性谱如何刻画疫情的空间结构？
• RQ3 在时空演化中，疫情的关键阶段有哪些？它们如何反映扩散动力学的变化？
• RQ4 空间模式在多大程度上由全局聚集与局部无序主导？
• RQ5 分形维数如何随时间演变？其增长曲线揭示了疫情发展的哪些特征？

主要发现

• 广义关联维数谱呈现反S形曲线，表明病例空间分布中存在多重分形标度特性。
• 对于矩阶q << 0，分形维数值显著超过嵌入空间的欧几里得维数，反映出极端的空间异质性以及罕见的高强度聚集。
• 局部奇异性谱为右偏斜的单峰曲线，表明空间模式主要由少数强奇异性主导，而非均匀分布。
• 分形维数增长曲线呈准S形，表明其从初始低复杂性动态演变为高复杂性并最终趋于稳定。
• 疫情进程可划分为四个明确阶段：初始阶段、快速扩散阶段、分层扩散阶段和收缩阶段，各阶段具有独特的分形特征。
• 结果证实，中国COVID-19的空间扩散受多重分形标度定律支配，其主导特征为全局聚集与局部无序扩散。