福井大学 学術研究院工学系部門 研究報告 第 69 巻 2021 年 3 月
Mem. Fac. Eng. Univ. Fukui, Vol. 69（ March 2021 ）37

Analysis on the Distribution of COVID-19 Cases by Country
Hidetaka TOBITA* and Mitsuki FUJII**(Received December 21, 2020)
The global cases of COVID-19 reported on the website of Center for Systems Science andEngineering at Johns Hopkins University are used to analyze the distribution by country. The
relationship between rank and size (ranking plot) is approximately represented by the power-law
distribution for the top countries, while the overall distribution follows the lognormal distribution.
Assuming a continuous power-law distribution, the power exponent determined from the data leads
to make the weight average infinity, which would be considered as gelation or pandemic. The
distribution is rationalized by the discrete-time stochastic model that employs the weekly
reproduction rate µ. The distribution of µ obtained for the time period, between April 20 andNovember 11, 2020 could be approximated by the gamma distribution. By assuming a gamma
distribution for µ with various pairs of average and variance, the required conditions to stop the
pandemic are (1) the average µ must be smaller than unity and (2) the time period for µ > 2 must
be negligibly small.
Key Words : COVID-19, Ranking Plot, Power Law, Lognormal Distribution, Pandemic,
Weekly Reproduction Rate

1. Introduction
On April 7, 2020, the Japanese government declared astate of emergency for the rapid spread of COVID-19 inJapan. On that weekend, one of the authors (HT) was
staring abstractedly at the website of Center for Systems
Science and Engineering at Johns Hopkins University,[1]listing a large number of the COVID-19 cases by country.
Simply because there are so many numbers, HT

Fig. 1 Conventional ranking plot.

examined the numbers, and tried the ranking plot,
graphically showing the relationship between rank andFig. 1 shows a conventional type of ranking plot insize. Figure 1 shows the double logarithmic plot made atwhich the y-axis shows the size and the x-axis shows thethat time. Interestingly, the top countries appear torank. Because the rank is proportional to the number

follow the power-law relationship. Recalling his own

fraction of countries whose size is larger than theexperience that a power law was found also in thecorresponding size, when the axes are exchanged to plot[2]

infection tree of SARS in 2003,

and intrigued by the

Zipf’s law like behavior, HT called for a volunteer tocollect the data, and an undergraduate student, MFapplied for this project.

the rank on the y-axis, the ranking plot shows the upperprobability distribution given as follows.
∞

Rank ∝ ∫ N (x) dx .
x

(1)

In Eq. (1), N(x) shows the number-based probability* Department of Materials Science and Engineering**Department of Materials Science and Biotechnology,

School of Engineering

density function (pdf). In the rest of this article, the rank
is shown on the y-axis, representing the upperprobability distribution.

38

The collected data from the website[1] during thethe onset of gelation in the polymer science.[4] In orderperiod from April 11 to November 11, 2020 are analyzed.

for the expected cluster size to stay finite, α must beAbylarger than 2. On the other hand, however, the obtainedis used to rationalize the formation processnecessary conditions to stop the pandemic are proposed.

α-values are always smaller than 2 during the wholeinvestigated period. This could be interpreted as
pandemic.

2. Analysis of Distribution Data

2.2 Lognormal Distribution

2.1 Power-Law Distribution

be represented by the power law reasonably well, thediscrete-time

Yamamoto[3]

stochasticof the obtained distribution.

model

proposed

Based on the analysis, theAlthough the upper tail probability distribution could
Fig. 2 shows the ranking plot on the designated date.
The power law seems to apply for the top countries,
irrespective of the dates.
∞∫ N (x) dx ∼ x−αxwhole distribution, including the lower-ranked countries,
is not.
Fig. 3 shows the least-square fit by using thefor large x’s.

(2)

Fig. 2 Ranking plots on the designated dates.

lognormal distribution. The whole distribution is well
represented by the lognormal distribution.

Fig. 3 Fitted curves by using the lognormal distribution.

The power exponent, α is approximately in the range

Recently, it has been argued that many power-law

of 0.7 – 0.9. Based on Eq. (2), the upper tail of the

distributions reported earlier both for the social andnumber-based pdf, N(x) is represented by:natural sciences are just apparent, and do not fully

N (x) ∼ x −α −1 for large x’s.
Aninteresting

characteristic

(3)
of

the

conform to a strict power law,[5] and many of them are
better represented by the lognormal distribution.[6] It is

power-law

not straightforward to distinguish these two types of

distribution is that when the pdf is converted to that on a

distribution. In this article, we do not go into the details

weight basis W(x), the power exponent changes by one.

of statistical discrimination. We just report the following

W (x) ∼ x −α for large x’s.

(4)Equation (4) shows that the power exponent of the
ranking plot is directly equal to the power exponent of

two notable characteristics of the ranking plot of
COVID-19: (1) the top countries appear to follow thepower-law distribution, and (2) the overall distribution
could be represented by the lognormal distribution.

the weight-based pdf.
Assuming a continuous distribution, the weightxw = ∫ xW (x) dx ∼ ∫ x −α +1 dx .
0

to Elucidate the

Ranking Plot

average is obtained from the following equation.
∞

3. Model-Based Investigation(5)

In COVID-19, the patients are isolated once they are

Equation (5) shows that when the exponent (–α +1) isfound to be positive. Therefore, the infectious period isgreater than or equal to –1, the integration goes tolimited. Although there may be many hidden patients

infinity. Therefore, the critical value of α is 2.

that are not counted as the cases reported on the Johns

The weight average is the expected size of the cluster
when a unit is selected randomly, which corresponds to

Hopkins’ website, we consider only the numbers
reported therein.

39

We employ a discrete-time stochastic model. Suppose

because the frequency to have µ > 4 is so small, we

the increase of cases during a certain period of time isΔxt, and the increase in the next time period is Δxt+1. The

neglected such data to determine the µ distribution. The
solid line shows the interpolation curve, connecting the

newly infected would be infectious, and let µt be thecenter points of the histogram. This curve is used for theapparent reproduction rate, defined by:Monte Carlo (MC) simulation.

µt = Δxt+1 Δxt .

(6)With the present simple model, the number of cases xt
is represented by the following equations.

xt+1 = xt + Δxt .

(7)

Δxt = µt−1Δxt−1 .

(8)

Yamamoto[3]

investigated the present model, and−α

In the MC simulation, the total number of countries is
set to be N = 190, which is equal to that of the obtained
data for the most of the investigated period. The initialvalues are set to be x0 = 0 and Δx-1 = 1 for all countries.
Fig. 5 shows the MC simulation results after 50 weeks.
Even though the initial values are the same for all
countries, a wide variation among countries is generated.
The upper figure, (a) shows that a power-law distribution
applies for the top countries, with α = 0.8, which isfound that x has a stationary power-law tail with x , if
the following equation possesses a unique positive

approximately the same magnitude of α-value as shownin Figure 2. The lower figure, (b) shows that the wholesolution.

distribution is represented reasonably well by theE( µtα ) = 1 .

(9)lognormal distribution. The important characteristics
observed for the COVID-19 data are satisfied. Note that

where E represents the expectation of the distribution.

the distribution obtained for each trial of the MC

Therefore, Eq. (9) is equivalent to:simulation∞∫ µ f (µ) d µ = 1 .
0

α

(10)

is

different,

however,

these

two

characteristics always apply, at least, approximately. It isshown that the present discrete-time stochastic model

where f is the pdf of µ.
The reported COVID-19 data showed a weekly

can generate the distribution that satisfies two importantfluctuation: the number tends to be small every Monday.

simultaneously.

characteristics

of

the

COVID-19

distribution

In addition, one week would be a reasonable infectious
time period, and we employ the weekly reproduction
rate.

Fig. 4 Distribution of the weekly reproduction rate µ.
The histogram shown in Fig. 4 is the pdf of the weekly

Fig. 5 MC simulation results at t = 50 weeks.

reproduction rate µ observed during the period, from
Fig. 6 shows the distribution after 100 weeks. Two

April 20 to November 11, 2020. The data for the top 120
countries on October 17 are used. Note that the data ofcharacteristics are satisfied also at t = 100 weeks. The

the lower ranked countries tend to involve Δxt = 0, as

magnitude of α is about 0.6, which is smaller than thatwell as µ with extraordinary large magnitude. In addition,

shown in Figure 5. In fact, the magnitude of α decreases

40with time. Assuming a continuous distribution, thewe consider the requirements to stop the pandemic. As

weight average of the distribution goes to infinity when

discussed earlier, in order to keep the weight-averagethe α is less than or equal to 2, and a shift to a smaller

cases finite, the condition, α > 2 is needed.
Fig. 7 shows the least-square fit of the µ data, byusing the gamma distribution, which is represented by

α-value implies that the situation is getting worse if thepresent µ distribution is preserved.

the following equation.

f (µ) =⎛ µ⎞µ m−1
exp ⎜ − ⎟ .
⎝ η⎠Γ(m)η m(11)

The gamma distribution fits the data reasonably well.
The determined values of two parameters are m = 10.79
and η = 0.1003.

Fig. 6 MC simulation results at t = 100 weeks.
Fig. 7 Least-square fit of the µ data by using the gamma
Yamamoto[3] reported that Eq. (10) has a positive

distribution.

βsolution α, if E(lnµ) < 0 and lim E( µ ) = ∞ . In our
β →∞obtained data, E(lnµ) = 0.0523 > 0, and therefore, thedistribution cannot reach the stationary state. It would bereasonable to think that if the average of µ, E(µ) is larger
than unity, the number of cases continues to increase.
The present data shows that E(µ) = 1.14 > 1.
The numerical integration of Eq. (10), by using thecontinuous function f(µ) shown in the solid curve in Fig.

With the gamma distribution, Eq. (10) is calculated toobtain the following equation.
∞α∫ µ f (µ) d µ =0η α Γ(m + α )
= 1.
Γ(m)

(12)

Now, let us seek the values of m and η at the criticalpoint, i.e., α = 2. Note that in order to keep the weightaverage of µ finite, α must be larger than 2. When α = 2,
Eq. (12) reduces to:4, leads to show that the trivial solution, α = 0 is theunique solution. There is no unique positive solution.

η 2 (m + 1)m = 1 .

This would be the reason for the continuous decrease of

The parameter, m is represented by using the number

α with time, and the distribution cannot reach thestationary state.
The present MC simulation shows that the present
stochastic model satisfies two notable characteristics inthe ranking plot simultaneously during the transientperiod.
4. Suggestions to Stop the Pandemic(13)

average of µ, E( µ ) = µ , as follows.

m= µ η.

(14)

By substituting Eq. (14) into Eq. (13), one obtains:η=1− µ 2.
µ(15)

Figure 8 shows the calculated results of Eq. (15). The
right axis shows the variance σ2 of µ, which is given by:

Assuming the present statistical model is valid and theupper tail of the ranking plot conforms to the power law,σ 2 = mη 2 = ηµ

(16)

41

0.8 seems to apply, which may show the statisticalself-similarity with the world data, shown in Fig. 2. It
would be reasonable to think that the above criteria forthe prevention measure are valid also for the local area.

Fig. 8 Magnitudes of η and σ2 as a function of µ at thecritical condition with α = 2.
As shown in Fig. 8, when the average is close to unity,
the µ distribution must be quite narrow with a very small

Fig. 10 Ranking plot by the prefectures in Japan.

2

variance, σ . On the other hand, if the average is smallenough, relatively broad distribution is allowed.

5. Conclusions

Fig. 9 shows the critical distribution that leads to α = 2
for µ = 0.9, 0.8 and 0.7. A common characteristic for

The global cases of COVID-19 reported on thethese curves is that the probability to have the time

website of Center for Systems Science and Engineering

period to make µ > 2 is rather small.

at Johns Hopkins University[1] are used to analyze the
distribution by country. The relationship between rank
and size (ranking plot) is approximately represented by
the power-law distribution for the top countries.
Assuming a continuous power-law distribution, thepower exponent determined from the data shows that theweight average diverges to infinity. This fact may imply
the pandemic, which corresponds to the present situationin 2020.
On the other hand, the overall distribution, including
the lower ranked countries, agrees reasonably with thelognormal distribution.

2

Fig. 9 Magnitudes of η and σ as a function of µ at theHere, we report two notable characteristics in thecritical condition with α = 2.

global ranking plot of COVID-19 during the year 2020:(1) the top countries appear to follow the power-law

On the basis of the results shown in Fig. 8 and Fig. 9,
to stop the pandemic, (1) the average µ must be
controlled to make µ < 1, and (2) the time period for µ

distribution, and (2) the overall distribution could be
represented by the lognormal distribution.
These two characteristics are reproduced well by thediscrete-time stochastic model that employs the weekly

to be greater than 2 must be small enough. When the

reproduction rate µ, during the transient period. On theµ-value increases to approach 2, some strong measure toprevent the infection spread must be undertaken to makeother hand, the present µ distribution predicts that theµ smaller.
Fig. 10 shows the ranking plot in Japan. The x-axis
shows the number of cases by the prefecture. Because
the total number of prefectures is only 47 and the data
points are too small, the statistical analysis is difficult to
conduct. However, the power law with about α = 0.7 –infection status will get worse, without changing the µdistribution.
The present µ distribution can be approximated by the
gamma distribution. By assuming a gamma distribution
is valid even when the infection situation changes, therequired conditions to stop the pandemic are (1) the
average µ must be made smaller than unity and (2) the42time period for µ > 2 must be kept small enough.
References[1] COVID-19 Dashboard by the Center for Systems
ScienceandEngineering

at

Johns

Hopkins

University, <https://coronavirus.jhu.edu/map.html>,(2020/04/11-11/11).
[2] H. Tobita: Mem. Fac. Eng. Univ. Fukui, 52, 37-41(2004).
[3] K. Yamamoto: Phys. Rev. E, 89, 042115 (2014).
[4] P.J. Flory: Principles of Polymer Chemistry, Cornell
University Press, Ithaca, USA (1953), Chap. 9.
[5] L. Benguigui and M. Marinov: arXiv:1507.0348(2015).
[6] A. D. Broido and A. Clauset: Nat. Comm. 10, 1017
(2019).