福井大学 学術研究院工学系部門 研究報告 第 70 巻 2021 年 10 月
Mem. Fac. Eng., Univ. Fukui, Vol. 70（ October 2021 ）23

Analysis on the Distribution of COVID-19 Cases by Country－ 2. Robust Power-Law Distribution Observed during the First Half of 2021－
Hidetaka TOBITA*(Received August 20, 2021)
Part 1 of this series[1] analyzed the global cases of COVID-19 during April through November,
2020. In this second report, the data from November 14, 2020 to August 7, 2021 are used for the
analysis. Four important findings on the rank-size relationship reported in Part 1 are reconfirmed.
(1) The top countries approximately follow the power-law distribution, with rank∝size−α. (2) The
overall distribution can be represented by the lognormal distribution. (3) These two characteristics
are reproduced reasonably well by the discrete-time stochastic model that employs the weekly
reproduction rate µ. (4) The distribution of µ can be approximated by the gamma distribution. Theµ-distribution determined for the time period of November 14, 2020 through May 22, 2021
predicts that the stationary distribution after a long time leads to form the power-law tail with α =1.24, while the actual data show α = 0.9 during the whole period. This persistent characteristic is
supported by the Monte Carlo simulation, showing the robustness of the power-law distribution
once established. In order to keep the number average size finite, the magnitude of α must be larger
than 1, which corresponds the condition with the average of µ smaller than 1. In addition to thecases by country, the distribution of the number of deaths by country is investigated to find that thecharacteristics described in item (1) and (2) apply.
Key Words : COVID-19, Ranking Plot, Power Law, Lognormal Distribution, Pandemic,
Reproduction Rate, Gamma Distribution
It was found[1] that the power-law relationship, rank∝x−αapplies for the top countries, and α is approximately in

1. Introduction

the range of 0.7 – 0.9 during the time period of April[1]

the global cases ofthrough November, 2020. The power law for the

COVID-19 reported on the website of Center for

COVID-19 distribution was also reported by the other

Systems Science and Engineering at Johns Hopkins

researcher.[3] When the power-law distribution rank∝x−α

In Part 1 of this series,

University[2]

were used to analyze the distribution of thecases by country, i.e., the rank–size relationship. Because
the rank is proportional to the number fraction of theapplies, Equation (1) leads to give N(x) as follows.

N (x) ∝ x −α −1 for large x’s.

(2)

countries whose number of cases is larger than a given

The power-law distribution is sometimes referred to assize, the rank corresponds to the upper probability

scale free, because the average can go infinity. The

distribution CN(x) as follows.

scale-free power-law distribution is a pathway capable of∞rank ∝ CN (x) = ∫ N (x) dx .
x(1)connecting finite and infinite sizes smoothly, and appears
frequently in the theory of phase transition.[4,5] Because

In Eq. (1), N(x) shows the number-based probability

the pandemic is a phenomena that the infection prevails

density function (pdf), and x is the number of cases in aendlessly, it may be related with the scale-free power law

country. The rank-size relationship directly shows the

distribution.

cumulative distribution function graphically.

Although the upper tail probability distribution
conforms to the power law, it was found[1] that the whole* Department of Materials Science and Engineering

distribution, including the lower-ranked countries, can be24represented by the lognormal distribution rather well.
The following simple discrete-time model[6] was used
to examine the data. The increase of cases during arate, µ obtained from the data between t = 0 and 27. The
distribution can be approximated by the gamma
distribution, represented by the following equation.

certain week is represented by Δxt, and the increase inf (µ) =

the next week is by Δxt+1. The newly infected would be
infectious, and let µt be the weekly reproduction rate,

⎛ µ⎞
µ m−1
exp ⎜ − ⎟ .
m
⎝ η⎠
Γ(m)η

(6)

Two parameters, m and η are determined to be m =defined by:µt = Δxt+1 Δxt .

(3)

16.3 and η = 0.0609.

The weekly reproduction rate µ corresponds to an
approximate representation of the effective reproduction
number in epidemiology,[7] sometimes referred to byusing a symbol, Rt.
With this simple model, the number of cases xt isrepresented by the following equations.

xt+1 = xt + Δxt .

(4)

Δxt = µt−1Δxt−1 .

(5)During the time period of April 20 through November
11, 2020, the distribution of µ for the top 120 countries

Fig. 1 Relationship between the rank of the country andthe number of cases for the country. During this time
period, the USA is ranked the first, and rank = 1 for theUSA.

showed that the average of µ is 1.14, which is larger than1, and the distribution cannot reach the stationary state.[6]On the other hand, however, the Monte Carlo (MC)simulation showed[1] that the distribution that conforms
to the above-mentioned two different types distribution
simultaneously can be formed during the transient period,i.e., the power-law for the upper tail distribution and thelognormal for the whole region.
In this subsequent report, the data were collected
every week for the period of Nov. 14, 2020 (0th week, t =Fig. 2 Fitted curves by the lognormal distribution.

th0) through Aug. 7, 2021 (38 week, t = 38). From May
15, 2021 (t = 26), the numbers of deaths by country arealso collected.
2. Results and Discussion
2.1 Cases by Country
Fig. 1 and 2 show the rank-size relationship on thedesignated dates, from t = 0 (Nov. 14, 2020) to 27 (May22, 2021). Fig. 1 shows that the upper tail distribution
conforms to the power law, while the whole distribution
is represented by the lognormal distribution, as shown in
Fig. 2. The power exponent, α was determined by theleast square fit of the double logarithmic plot for the top
30 countries. Two characteristics found in Part 1[1] arereconfirmed.
Fig. 3 shows the distribution of weekly reproduction

Fig. 3 Distribution of the weekly reproduction rate µ forthe time period from t = 0 to 27. The solid curve is thefitted gamma distribution.
The average of the gamma distribution is given by(7)µ = mη ,25

which leads toµ= 0.993 and the distribution

converges to the stationary state.
[6]

stable and is about 0.9 throughout the time period. Does
this discrepancy from the stationary state distribution

reported that the present discrete-timemeans that the present discrete-time stochastic model

model leads to a stationary power-law tail with rank∝

cannot be applied? To examine this, another MC

Yamamoto−αx , if the following equation has a unique positive
solution, α.
∞simulation was conducted.
The value of x1 was set to be equal to the actual data at∫ µ f (µ) d µ = 1 .
α0t = 1, and Δx0 was determined from the difference of x1(8)and x0 for each country. The simulation was done for N =Note that the left-hand side represents the expectedvalue of µα.

179 countries where Δx0 is larger than 0. The distribution
at t = 27 was simulated.

For the gamma distribution, Equation (8) can becalculated to give∞α
∫ µ f (µ) d µ =0η α Γ(m + α )
= 1.
Γ(m)

(9)

For the case with m = 16.3 and η = 0.0609, one
obtains α = 1.24.
To confirm if the stationary state α is equal to 1.24,
the MC simulation was conducted. To determine thestationary state α accurately, the number of clusters (i.e.,
the total number of countries for the present problem)
5

Fig. 5 Obtained α-values from t = 0 to 27.

was set to be N = 10 . The initial values used were x0 = 0
and Δx-1 = 1 for all clusters. The random number that

The MC simulation was repeated for 100 times, andfollows the gamma distribution with m = 16.3 and η =Fig. 6 shows all the simulated results, together with the

0.0609 were used.

actual data shown by the red line. The data (red) is

Fig. 4 shows the MC simulation results at t = 103(blue) and t = 104 (red, broken). Note that the y-axis

within the range of 100 simulated results (colored dots).

shows the upper probability distribution CN(x), definedby Equation (1). Both curves overlap each other, and the
distribution has already reached the stationary state at t =103. It is confirmed that the upper tail follows thepower-law distribution with α = 1.24.

Fig. 6 Data at t = 27 (red) and the simulated results(colored dots).

Fig. 4 MC simulation results to confirm the validity of α
= 1.24 at the stationary state.
In spite of the fact that the stationary distribution has atail with α = 1.24, the α-values shown in Fig. 1 are about
0.9 for all cases. Fig. 5 shows the determined values foreach week up to the 27th week. The α-value is quite

Fig. 7 Comparison between the data at t = 27 (red) andthe average of simulated results (blue).

26

The blue line in Fig. 7 shows the average of thesimulated results repeated 100 times, which agrees
reasonably well with the actual data (red). The simulated
results (blue) show two important characteristics: (1) the
upper tail follows the power-law distribution with α =0.9, as shown in Fig. 8a, and (2) the whole distributionagrees well with the lognormal distribution, as shown in
Fig. 8b.
Fig. 9 Simulation results at t = 27 and 100. Both
distributions possess the power law tail with α = 0.9.

Fig. 10 Lognormal fit for the simulation results at t =100.

Fig. 8 MC simulation results (blue) at t = 27. Red line
shows (a) the power law with α = 0.9 and (b) the
lognormal fit.
Although the convergent stationary distribution is thepower-law distribution with α = 1.24, as shown in Fig. 4,
the MC simulation shows that a stable power-law
distribution with α = 0.9 holds throughout 27 weeks,
which agrees with the actual data.
In order to further examine the robustness of thepower law tail with α = 0.9, the simulation for t = 100
was conducted. As shown in the green line in Fig. 9, thepower law tail with α = 0.9 is preserved even after 100
weeks. The power law tail is remarkably robust. As
shown in Fig. 10, the overall lognormal distribution is
also kept after 100 weeks.
Fig. 11 shows the actual data of α’s for the period
from t = 28 to 38, and it is shown that α = 0.9 persists.
The power law distribution once formed seems to berather robust and the power exponent α cannot bechanged easily.

Fig. 11 Obtained α-value for the cases by country during
the time period from t = 28 to 38.
2.2 Deaths by Country
Starting from May 15, 2021 (t = 26), the numbers ofdeaths by country were also collected. Fig. 12 shows the
distribution of the number of deaths at t = 38. The
distribution shows (1) the power-law tail and (2) good fit
tothelognormal

distribution.

The

fundamental

characteristics are the same as for the cases by country.
The power exponent determined from the top 30
countries is α = 0.85 at t = 38.
Assuming that the fatality rate is almost the same forall countries, the number of deaths is expected to havethe same power-law tail with α = 0.9. Fig. 13 shows thepower exponent for the deaths by country. The values arecompared with those for the cases by country. The

27α-values for the deaths ( ≅ 0.85) and the cases ( ≅ 0.9)
are almost the same, but the deaths show slightly smaller
α’s consistently. In general, smaller α indicates a wider
disparity. This may imply the disparity of medical
systems among countries.

science,[8] the gel (critical) point is reached when theweight-average molecular weight goes to infinity.
Various power-law relationships hold at the vicinity ofgel point,[4,5] which is known as the universality of thecritical phenomena.
In order to cause polymeric gelation, the formation ofcross-linkage that connects two different clusters is
required.[9] The molecular weight distribution at the gelpoint follows the power law, with α ≤ 2 at which theweight average goes to infinity.
On the other hand, in the branched structure of
infection tree, no cross-linkage is formed. The elements
are connected in one direction, and there exists only one
start point in each tree. The expected size of the tree

Fig. 12 Relationship between the rank of the country andthe number of deaths for the country.

when one picks up such a start point randomly is thenumber-average size.[10] For the infection tree, it isreasonable to consider that the critical point is reachedwhen the number average goes to infinity. The numberof cases in a country is the sum of a finite number of
infection trees, and therefore, the critical point will be
the same as for the tree. In the distribution having a
power-law tail, the number average goes to infinity whenthe magnitude of power exponent is α ≤ 1 .
Assuming that the µ-distribution follows the gamma
distribution, Equation (8) reduces to Equation (9). The

Fig. 13 Obtained α-value for the deaths by country, incomparison with cases by country, during the time period
from t = 26 to 38.
Fig. 14 shows the fatality rate by country. Fatality
rates of several countries are much higher than the
average.

condition that yields α = 1 is given by the following
equation.
ηm = µ = 1 .

(10)

The number average goes to infinity when the averageof µ reaches 1. In fact, Yamamoto[6] showed that α = 1,
which is known as the Zipf’s law, is obtained when the
average of µ is 1, irrespective of the type ofµ-distribution. In epidemiology, usual measure to
suppress epidemic is to make the effective reproduction
number, which corresponds to µ, smaller than 1.[7] The
condition to make α > 1 conforms to this usual criterion.
In Part 1 of this series,[1] the conditions to make thepower exponent α > 2 was considered, in order tosuppress the pandemic. It is now shown that α > 1 would
be a better criterion for the suppression.
The µ data for 0 < t < 27 give m = 16.3 and η = 0.0609,

Fig. 14 Fatality rate by country as of August 7, 2021 (t =38). The country order in the x-axis is the rank of theas shown in Fig. 3. In this case, µ = 0.993 and α =1.24. Because µ < 1 , the pandemic is expected to becases by country.

terminated if the present µ-distribution continues. The

2.3 Power Exponent at the Critical Point

1.24) of the stationary distribution is significantly larger

In the standard theory of gelation in polymerµ -value (= 0.993) is very close to 1, but the α-value (=than 1.
As could be recognized from Fig. 3 the values of m =28

16.3 and η = 0.0609 are approximate. Assuming m =while the actual data show α = 0.9 persistently

16.3, it is straightforward to show that η = 0.05955 gives

throughout the investigated time period. This persistent

α = 2 from Equation (9). The values of m = 16.3 and η =0.05955 leads to η m = µ = 0.971 .
Fig. 15 shows these two types of gamma distribution
graphically. The difference is surprisingly small. It
would be difficult to distinguish these two curves fromthe actual data. Considering small differences in the
gamma distribution, the prevention criterion proposed inPart 1, based on α = 2, might be still acceptable: (1) the
average µ must be made smaller than 1 and (2) the time
period for µ > 2 must be made negligibly small. Safercondition of item (2) might be “the time period for µ >1.5 should be controlled to make as small as possible”.

α-value agrees with the MC simulation results. The
power-law distribution once formed is quite stable, andthe power exponent cannot be changed easily.
The number of deaths by country was investigated forthe time period of t = 26 through 38. The relationshipbetween the rank and the number of deaths shows thesimilar characteristics with the number of cases. (1) The
distribution possesses the power-law tail and (2) the
distribution is well fit to the lognormal distribution. The
power exponent of the tail distribution was α = 0.85,
which is slightly smaller than that for the cases bycountry.
In order to keep the number average size finite, themagnitude of α must be larger than 1, which corresponds
the condition with the average of µ smaller than 1. Inorder to suppress the pandemic, (1) the average µ must
be made smaller than 1 and (2) the time period for µ >1.5 should be controlled to make as small as possible.
References

Fig. 15 Two types of gamma distribution. The red curveshows the case with m = 16.3 and η = 0.0609, and theblue curve is the case with m = 16.3 and η = 0.05955.
3. Conclusions
Following the analysis of the global cases of
COVID-19 reported in Part 1 of this series,[1] the data
during November 14, 2020 (elapsed weeks, t = 0)
through August 7, 2021 (t = 38) were used for the
analysis in this report. Four important findings on therelationship between the rank and the number of cases bycountry reported in Part 1 are reconfirmed for the present
time period. (1) The top countries approximately follow
the power-law distribution, with rank∝size−α. (2) The
overall distribution is represented well by the lognormal
distribution. (3) These two characteristics can be
reproduced

reasonably

well

by

the

discrete-time

stochastic model that employs the weekly reproduction
rate µ. (4) The distribution of µ can be approximated by
the gamma distribution.
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