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. 2023 Nov 21;18(11):e0294445.
doi: 10.1371/journal.pone.0294445. eCollection 2023.

Heavy-tailed distributions of confirmed COVID-19 cases and deaths in spatiotemporal space

Peng Liu  1 , Yanyan Zheng  2
Affiliations

Heavy-tailed distributions of confirmed COVID-19 cases and deaths in spatiotemporal space

Peng Liu et al. PLoS One. .

Abstract

This paper conducts a systematic statistical analysis of the characteristics of the geographical empirical distributions for the numbers of both cumulative and daily confirmed COVID-19 cases and deaths at county, city, and state levels over a time span from January 2020 to June 2022. The mathematical heavy-tailed distributions can be used for fitting the empirical distributions observed in different temporal stages and geographical scales. The estimations of the shape parameter of the tail distributions using the Generalized Pareto Distribution also support the observations of the heavy-tailed distributions. According to the characteristics of the heavy-tailed distributions, the evolution course of the geographical empirical distributions can be divided into three distinct phases, namely the power-law phase, the lognormal phase I, and the lognormal phase II. These three phases could serve as an indicator of the severity degree of the COVID-19 pandemic within an area. The empirical results suggest important intrinsic dynamics of a human infectious virus spread in the human interconnected physical complex network. The findings extend previous empirical studies and could provide more strict constraints for current mathematical and physical modeling studies, such as the SIR model and its variants based on the theory of complex networks.

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Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1
Fig 1. The US county-level distributions of the numbers of cumulative confirmed COVID-19 cases.
The star markers are the empirically estimated probability density P(N) of a county to have N cumulative confirmed cases by one day. The red dashed curves and legends are the fit results using theoretical distributions. The values of R (p-value) of the log-likelihood ratio tests are shown as the black text in the bottom left corners of each panel.
Fig 2
Fig 2. The US county-level distributions of the numbers of cumulative confirmed COVID-19 deaths.
The star markers are the empirically estimated probability P(N) of a county to have N cumulative confirmed deaths by one day. The red dashed curves and legends are the fit results using theoretical distributions. The values of R (p-value) of the log-likelihood ratio tests are shown as the black text in the bottom left corners of each panel.
Fig 3
Fig 3. The US county-level distributions of the numbers of daily confirmed COVID-19 cases.
The star markers are the empirically estimated probability density P(N) of a county to have N daily confirmed cases on one day. The red dashed curves and legends are the results of the lognormal fits. For ease of comparison, the results of the stretched exponential fit are presented as the blue dashed curve and legend in the top middle panel. The values of R (p-value) of the log-likelihood ratio tests are shown as black text in the bottom left corners of each panel.
Fig 4
Fig 4. The US county-level distributions of the numbers of daily confirmed COVID-19 deaths.
The star markers are the empirically estimated probability density P(N) of a county to have N daily confirmed deaths on one day. The red dashed curves and legends are the fit results using theoretical distributions. The values of R (p-value) of the log-likelihood ratio tests are shown as the black text in the bottom left corners of each panel.
Fig 5
Fig 5. The Chinese city-level distributions of the numbers of cumulative confirmed COVID-19 cases.
The star markers are the empirically estimated probability density P(N) of a city to have N cumulative confirmed cases by one day. The red dashed curves and legends are the results of the lognormal fits. The values of R (p-value) of the log-likelihood ratio tests are shown as the black text in the bottom left corners of each panel.
Fig 6
Fig 6. State/province-level distributions of the numbers of cumulative confirmed COVID-19 cases.
The star markers are the empirically estimated probability density P(N) of a state/province to have N cumulative confirmed cases by one day. The red dashed curves and legends are the results of the stretched exponential fits. For ease of comparison, the results of the lognormal fits are presented by blue dashed curves and legends. The values of R (p-value) of the log-likelihood ratio tests are shown as the black text in the bottom left corners of each panel.
Fig 7
Fig 7. State/province-level distributions of the numbers of cumulative confirmed COVID-19 deaths.
The star markers are the empirically estimated probability density P(N) of a state/province to have N cumulative confirmed deaths. The red dashed curves and legends are the fit results using theoretical distributions. The values of R (p-value) of the log-likelihood ratio tests are shown as the black text in the bottom left corners of each panel.
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References

    1. World Health Organization. Coronavirus disease (COVID-19) pandemic; 2023. https://www.who.int/emergencies/diseases/novel-coronavirus-2019. Retrieved 6 October 2023.
    1. Sun J, He WT, Wang L, Lai A, Ji X, Zhai X, et al.. COVID-19: Epidemiology, evolution, and cross-disciplinary perspectives. Trends in Molecular Medicine. 2020;26(5):483–495. doi: 10.1016/j.molmed.2020.02.008 - DOI - PMC - PubMed
    1. Hu B, Guo H, Zhou P, Shi ZL. Characteristics of SARS-CoV-2 and COVID-19. Nature Reviews Microbiology. 2021;19:141–154. doi: 10.1038/s41579-020-00459-7 - DOI - PMC - PubMed
    1. Coronaviridae Study Group of the International Committee on Taxonomy of Viruses. The species severe acute respiratory syndrome-related coronavirus: Classifying 2019-nCoV and naming it SARS-CoV-2. Nature Microbiology. 2020;5:536–544. doi: 10.1038/s41564-020-0695-z - DOI - PMC - PubMed
    1. COVID-19 Excess Mortality Collaborators. Estimating excess mortality due to the COVID-19 pandemic: A systematic analysis of COVID-19-related mortality, 2020–21. The Lancet. 2022;399:1513–1536. doi: 10.1016/S0140-6736(21)02796-3 - DOI - PMC - PubMed