Evolution of Scaling Emergence in Large-Scale Spatial Epidemic Spreading

Lin Wang,Kan Zhang,Xiang Li,Yan Zhang,Yi-Qing Zhang

doi:10.1371/journal.pone.0021197

Abstract

BackgroundZipf's law and Heaps' law are two representatives of the scaling concepts, which play a significant role in the study of complexity science. The coexistence of the Zipf's law and the Heaps' law motivates different understandings on the dependence between these two scalings, which has still hardly been clarified.Methodology/Principal FindingsIn this article, we observe an evolution process of the scalings: the Zipf's law and the Heaps' law are naturally shaped to coexist at the initial time, while the crossover comes with the emergence of their inconsistency at the larger time before reaching a stable state, where the Heaps' law still exists with the disappearance of strict Zipf's law. Such findings are illustrated with a scenario of large-scale spatial epidemic spreading, and the empirical results of pandemic disease support a universal analysis of the relation between the two laws regardless of the biological details of disease. Employing the United States domestic air transportation and demographic data to construct a metapopulation model for simulating the pandemic spread at the U.S. country level, we uncover that the broad heterogeneity of the infrastructure plays a key role in the evolution of scaling emergence.Conclusions/SignificanceThe analyses of large-scale spatial epidemic spreading help understand the temporal evolution of scalings, indicating the coexistence of the Zipf's law and the Heaps' law depends on the collective dynamics of epidemic processes, and the heterogeneity of epidemic spread indicates the significance of performing targeted containment strategies at the early time of a pandemic disease.

Highlights

Scaling concepts play a significant role in the field of complexity science, where a considerable amount of efforts is devoted to understand these universal properties underlying multifarious systems[1,2,3,4]
We have identified the relation between the Zipf’s law and the Heaps’ law from the perspective of coevolution between the scalings and large-scale spatial epidemic spreading
We illustrate the temporal evolution of the scalings: the Zipf’s law and the Heaps’ law are naturally shaped to coexist at the early stage of the epidemic at both the global and the U.S levels, while the crossover comes with the emergence of their inconsistency at a larger time before reaching a stable state, where the Heaps’ law still exists with the disappearance of strict Zipf’s law

Summary

Introduction

Scaling concepts play a significant role in the field of complexity science, where a considerable amount of efforts is devoted to understand these universal properties underlying multifarious systems[1,2,3,4]. Two representatives of scaling emergence are the Zipf’s law and the Heaps’ law. G.K. Zipf, sixty years ago, found a power law distribution for the occurrence frequencies of words within different written texts, when they were plotted in a descending order against their rank[5]. Sixty years ago, found a power law distribution for the occurrence frequencies of words within different written texts, when they were plotted in a descending order against their rank[5] This frequency-rank relation corresponds to a power law probability distribution of the word frequencies[32]. Zipf’s law and Heaps’ law are two representatives of the scaling concepts, which play a significant role in the study of complexity science. The coexistence of the Zipf’s law and the Heaps’ law motivates different understandings on the dependence between these two scalings, which has still hardly been clarified

Methods

Results

Conclusion