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. 2020 Nov:412:132649.
doi: 10.1016/j.physd.2020.132649. Epub 2020 Jul 16.

On the emergence of a power law in the distribution of COVID-19 cases

Brendan K Beare  1 , Alexis Akira Toda  2
Affiliations

On the emergence of a power law in the distribution of COVID-19 cases

Brendan K Beare et al. Physica D. 2020 Nov.

Abstract

The first confirmed case of Coronavirus Disease 2019 (COVID-19) in the US was reported on January 21, 2020. By the end of March, 2020, there were more than 180,000 confirmed cases in the US, distributed across more than 2000 counties. We find that the right tail of this distribution exhibits a power law, with Pareto exponent close to one. We investigate whether a simple model of the growth of COVID-19 cases involving Gibrat's law can explain the emergence of this power law. The model is calibrated to match (i) the growth rates of confirmed cases, and (ii) the varying lengths of time during which COVID-19 had been present within each county. Thus calibrated, the model generates a power law with Pareto exponent nearly exactly equal to the exponent estimated directly from the distribution of confirmed cases across counties at the end of March.

Keywords: COVID-19; Coronavirus; Gibrat’s law; Mathematical modeling of epidemics; Power law; Tauberian theorem.

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Conflict of interest statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Figures

Fig. 1
Fig. 1
Log–log plot of confirmed COVID-19 cases against tail probabilities across US counties on March 31, 2020. The tail probability of a county is the proportion of all counties matching or exceeding its number of COVID-19 cases. The Pareto fit was obtained by applying the Hill estimator to the top 6.2% of counties by number of cases. The estimated Pareto exponent is ζˆ=0.930, with a standard error of 0.081.
Fig. 2
Fig. 2
Ordinary least squares estimates of β0t,β1t,β2t,β3t in Eq. (5) for the 28 days between March 3 and March 30 inclusive, with 95% confidence bands. In panel (2(a)) we also display the pooled mean growth rate of 0.180.
Fig. 3
Fig. 3
Histogram of growth rates of confirmed COVID-19 cases, using data from all days up to the end of March and all counties with at least 10 confirmed cases and a positive growth rate of cases. The gamma fit was obtained by the method of maximum likelihood. The nonparametric fit was obtained by Gaussian kernel smoothing.
Fig. 4
Fig. 4
Histogram of days-since-outbreak on March 31, using data from all counties reporting at least one confirmed COVID-19 case by March 31. The truncated logistic fit was obtained by the method of maximum likelihood.
Fig. 5
Fig. 5
The Pareto exponent ζ is the unique positive real z at which the Laplace transform M(z) is equal to 1∕q.
See this image and copyright information in PMC

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