Content uploaded by Toshikazu Kuniya

Author content

All content in this area was uploaded by Toshikazu Kuniya on Mar 17, 2020

Content may be subject to copyright.

Journal of

Clinical Medicine

Article

Prediction of the Epidemic Peak of Coronavirus

Disease in Japan, 2020

Toshikazu Kuniya

Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan;

tkuniya@port.kobe-u.ac.jp

Received: 24 February 2020; Accepted: 10 March 2020; Published: 13 March 2020

Abstract:

The ﬁrst case of coronavirus disease 2019 (COVID-19) in Japan was reported on 15 January

2020 and the number of reported cases has increased day by day. The purpose of this study is to give a

prediction of the epidemic peak for COVID-19 in Japan by using the real-time data from 15 January to

29 February 2020. Taking into account the uncertainty due to the incomplete identiﬁcation of infective

population, we apply the well-known SEIR compartmental model for the prediction. By using a

least-square-based method with Poisson noise, we estimate that the basic reproduction number for

the epidemic in Japan is

R0=

2.6 (95%CI, 2.4–2.8) and the epidemic peak could possibly reach the

early-middle summer. In addition, we obtain the following epidemiological insights: (1) the essential

epidemic size is less likely to be affected by the rate of identiﬁcation of the actual infective population;

(2) the intervention has a positive effect on the delay of the epidemic peak; (3) intervention over a

relatively long period is needed to effectively reduce the ﬁnal epidemic size.

Keywords: COVID-19; SEIR compartmental model; basic reproduction number

1. Introduction

In December 2019, the ﬁrst case of respiratory disease caused by a novel coronavirus was

identiﬁed in Wuhan City, Hubei Province, China. The outbreak of the disease is ongoing worldwide

and the World Health Organization named it coronavirus disease 2019 (COVID-19) on 11 February

2020 [

1

]. In Japan, the ﬁrst case was reported on 15 January 2020 and the number of reported

laboratory-conﬁrmed COVID-19 cases per week has increased day by day (see Table 1).

Table 1. Number of newly reported COVID-19 cases in Japan until 1 March 2020 [2].

Week Number of Newly Reported Cases Number of Accumulated Cases

12 January–18 January 1 1

19 January–25 January 2 3

26 January–1 February 14 17

2 February–8 February 8 25

9 February–16 February 28 53

17 February–23 February 79 132

24 February–1 March 107 239

As seen in Table 1, the number of newly reported cases per week has increased and a serious

outbreak in Japan is a realistic outcome. One of the greatest public concerns is whether the epidemic

continues until summer so that it affects the Summer Olympics, which is planned to be held in

Tokyo. The purpose of this study is to give a prediction of the epidemic peak of COVID-19 in Japan,

which might help us to act appropriately to reduce the epidemic risk.

J. Clin. Med. 2020,9, 789; doi:10.3390/jcm9030789 www.mdpi.com/journal/jcm

J. Clin. Med. 2020,9, 789 2 of 7

The epidemic data as shown in Table 1would have mainly twofold uncertainty. The ﬁrst one is

due to the fact that asymptomatic infected people could spread the infection [

3

]. The second one is

due to the lack of opportunity for the diagnostic test as sufﬁciently simple diagnostic test kits have

not been developed yet and the diagnosis in the early stage in Japan was mainly restricted to people

who visited Wuhan [

4

]. In this study, taking into account such uncertainty, we apply a simple and

well-known mathematical model for the prediction. More precisely, we assume that only

p

(0

<p≤

1)

fraction of infective individuals can be identiﬁed by diagnosis.

2. Methods

2.1. Model

We apply the following well-known SEIR compartmental model (see, e.g., [

5

]) for the prediction.

(S0(t) = −βS(t)I(t),E0(t) = βS(t)I(t)−εE(t),

I0(t) = εE(t)−γI(t),R0(t) = γI(t),t>0, (1)

where

S(t)

,

E(t)

,

I(t)

and

R(t)

denote the susceptible, exposed, infective and removed populations at

time

t

, respectively.

β

,

ε

and

γ

denote the infection rate, the onset rate and the removal rate, respectively.

Note that 1

/ε

and 1

/γ

imply the average incubation period and the average infectious period, respectively.

Let the unit time be 1 day. Based on the previous studies [

6

,

7

], we fix 1

/ε=

5, and thus,

ε=

0.2 and

γ=

0.1, respectively. We fix

S+E+I+R

to be 1 so that each population implies the proportion

to the total population. We assume that one infective person is identified at time

t=

0 among total

N=1.26 ×108number of people in Japan [8]. That is, Y(0) = pI(0)×1.26 ×108=1, where

Y(t) = p I(t)×1.26 ×108

denotes the number of infective individuals who are identiﬁed at time

t

. Thus, we obtain

I(

0

) =

1

/(p×

1.26

×

10

8)

. We assume that there is no exposed and removed populations at

t=

0, that is,

E(0) = R(0) = 0, and hence,

S(0) = 1−E(0)−I(0)−R(0) = 1−1

p×1.26 ×108.

It was estimated in [9] that 77 cases were conﬁrmed among the possible 940 infected population

in February in Hokkaido, Japan. Based on this report, we assume that

p

ranges from 0.01 to 0.1.

The basic reproduction number

R0

, which means the expected value of secondary cases produced by

one infective individual [

10

], is calculated as the maximum eigenvalue of the next generation matrix

FV−1[11], where

F="0βS(0)

0 0 #,V="ε0

−ε γ #.

Thus, we obtain

R0=βS(0)

γ=β

γ1−1

p×1.26 ×108. (2)

2.2. Sensitivity of the Basic Reproduction Number

It is obvious that the basic reproduction number

R0

is independent from the onset rate

ε

.

The sensitivity of R0to other parameters β,γand pare calculated as follows:

Aβ=β

R0

∂R0

∂β =1, Aγ=γ

R0

∂R0

∂γ =−1, Ap=p

R0

∂R0

∂p=1

p×1.26 ×108−1, (3)

J. Clin. Med. 2020,9, 789 3 of 7

where

Aβ

,

Aγ

and

Ap

denote the normalized sensitivity indexes with respect to

β

,

γ

and

p

, respectively.

We see from Equation

(3)

that the

k

time’s increase in

β

(resp.

γ

) results in the

k

(resp.

k−1

) time’s

increase in

R0

. In particular, we see from the third equation in Equation

(3)

that

Ap≈

0 if

p≥

1.0

×

10

−6

.

This implies that the identiﬁcation rate pin a realistic range almost does not affect the size of R0.

2.3. Estimation of the Infection Rate

Let

y(t)

,

t=

0, 1,

. . .

, 45 be the number of daily reported cases of COVID-19 in Japan from 15

January (t=0) to 29 February (t=45) 2020. We perform the following least-square-based procedure

with Poisson noise to estimate the infection rate β.

Description 1.

(P1) Fix β>0and calculate the numerical value of Y(t), t =0, 1, . . . , 45 by using model Equation (1).

(P2) Calculate

˜

Y(t) = Y(t) + qY(t)e(t) = Y(t) + (Poisson noise),t=0, 1, . . . , 45,

where

e(t)

,

t=

0, 1,

. . .

, 45 denote random variables from a normal distribution with mean zero and variance 1[

12

].

(P3) Calculate J (β) = ∑45

t=0[y(t)−˜

Y(t)]2.

(P4) Run (P1)–(P3) for 0.2 ≤β≤0.4 and ﬁnd β∗such that J(β∗) = min0.2≤β≤0.4 J(β).

(P5) Repeat (P1)–(P4) 10, 000 times and obtain the distribution of β∗.

(P6) Approximate the distribution of β∗by a normal distribution and obtain a 95% conﬁdence interval.

Note that for the reason stated above, the value of 0.01

≤p≤

0.1 does not affect this estimation

procedure. By (P1)–(P6), we obtain a normal distribution with mean 0.26 and standard derivation 0.01.

Thus, we obtain an estimation of

β

as 0.26 (95%CI, 0.24–0.28) (see Figure 1). Moreover, by Equation

(2)

,

we obtain an estimation of R0as 2.6 (95%CI, 2.4–2.8) (see Table 2).

Figure 1.

Comparison of

Y(t)

with the estimated infection rate

β

and the number of daily reported

cases of COVID-19 in Japan from 15 January (t=0) to 29 February (t=45).

Table 2. Parameter values for model Equation (1).

Parameter Description Value Reference

βInfection rate 0.26 (95%CI, 0.24–0.28) Estimated

R0Basic reproduction number 2.6 (95%CI, 2.4–2.8) Estimated

εOnset rate 0.2 [6]

γRemoval rate 0.1 [7]

NTotal population in Japan 1.26 ×108[8]

pIdentiﬁcation rate 0.01–0.1 [9]

J. Clin. Med. 2020,9, 789 4 of 7

3. Results

3.1. Peak Prediction

We deﬁne the epidemic peak

t∗

by the time such that

Y

attains its maximum in 1 year, that is,

Y(t∗) = max0≤t≤365 Y(t)

. We ﬁrst set

p=

0.1. In this case, we obtain the following ﬁgure on the long

time behavior of Y(t)for β=0.28, 0.26 and 0.24.

We see from Figure 2that the estimated epidemic peak is

t∗=

208 (95%CI, 191–229). That is,

starting from 15 January (

t=

0), the estimated epidemic peak is 10 August (

t=

208) and the uncertainty

range is from 24 July (t=191) to 31 August (t=229).

(a)β=0.28 (t∗=191) (b)β=0.26 (t∗=208) (c)β=0.24 (t∗=229)

Figure 2.

Time variation of the number

Y(t)

of infective individuals who are identiﬁed at time

t

(0 ≤t≤365) for p=0.1. The dot lines represent the epidemic peak t∗.

We next set p=0.01. In this case, we obtain the following ﬁgure.

We see from Figure 3that the estimated epidemic peak is

t∗=

179 (95%CI, 165–197). That is,

starting from January 15 (t=0), the estimated epidemic peak is July 12 (t=179) and the uncertainty

range is from June 28 (

t=

165) to July 30 (

t=

197). In contrast to

R0

, the epidemic peak and the

(apparent) epidemic size are sensitive to the identiﬁcation rate

p

. Note that the essential epidemic size,

which is characterized by R0, is almost the same in both of p=0.1 and p=0.01.

(a)β=0.28 (t∗=165) (b)β=0.26 (t∗=179) (c)β=0.24 (t∗=197)

Figure 3.

Time variation of the number

Y(t)

of infective individuals who are identiﬁed at time

t

(0 ≤t≤365) for p=0.01. The dot lines represent the epidemic peak t∗.

3.2. Possible Effect of Intervention

We next discuss the effect of intervention. In Japan, school closure has started in almost all

prefectures from the beginning of March [

13

] and many social events have been cancelled off to reduce

the contact risk. However, the exact effect of such social efforts is unclear and might be limited as the

proportion of young people to the whole infected people of COVID-19 seems not so high (2% of 72, 314

reported cases in China [

14

]). In this simulation, we assume that such social efforts successfully reduce

the infection rate

β=

0.26 to 75% during a period from 1 March (

t=

46) to a planned day (

t=T≥

47).

In what follows, we ﬁx p=0.01.

J. Clin. Med. 2020,9, 789 5 of 7

First, we set

T=

77, that is, the intervention is carried out for 1 month (from 1 March to 1 April).

In this case, the epidemic peak

t∗

is delayed from 179 (12 July) to 190 (23 July). However, the epidemic

size is almost the same. On the other hand, if

T=

220, that is, the intervention is carried out for 6

months (from 1 March to 1 September), then the epidemic peak

t∗

is delayed from 179 (12 July) to 243

(14 September) and the epidemic size is effectively reduced (see Figure 4).

Figure 4.

Time variation of the number

Y(t)

of infective individuals who are identiﬁed at time

t

(0

≤t≤

365) for

p=

0.01 and no intervention, 1 month intervention (

T=

77) and 6 months

intervention (T=220). The dot lines represent the epidemic peak.

More precisely, we see from Figure 5a that the epidemic peak

t∗

is delayed almost linearly for

47 ≤T≤239 and ﬁxed to t∗=237 for T≥240.

(a) (b)

Figure 5.

The relation between the planned ﬁnal day for intervention

T

and (

a

) the epidemic peak

t∗

;

(b) the number of accumulated cases at time t=365: pR(365)×1.26 ×108.

This implies that the intervention has a positive effect on the delay of the epidemic peak, which

would contribute to improve the medical environment utilizing the extra time period. On the other

hand, we see from Figure 5b that the number of accumulated cases at

t=

365, which is calculated as

pR(

365

)×

1.26

×

10

8

, is monotonically decreasing and converges to 0.99

×

10

6

as

T

increases. However,

it almost does not change for small

T≤

180. This implies that the intervention over a relatively long

duration is required to effectively reduce the ﬁnal epidemic size.

4. Discussion

In this study, by applying the SEIR compartmental model to the daily reported cases of COVID-19

in Japan from 15 January to 29 February, we have estimated that the basic reproduction number

R0

is

2.6 (95%CI, 2.4–2.8) and the epidemic peak could possibly reach the early-middle summer. Of course,

this kind of long range peak prediction would contain the essential uncertainty due to the possibility

of some big changes in the social and natural (climate) situations. Nevertheless, our result suggests

that the epidemic of COVID-19 in Japan would not end so quickly. This might be consistent with the

WHO’s statement on 6 March 2020 that it is a false hope that COVID-19 will disappear in the summer

like the ﬂu [15].

J. Clin. Med. 2020,9, 789 6 of 7

The estimated value of the basic reproduction number

R0

in this study is not so different from

early estimations: 2.6 (95%CI, 1.5–3.5) [

16

], 2.90 (95%CI, 2.32–3.63) [

17

], 3.11 (95%CI, 2.39–4.13) [

18

],

3.58 (95%CI, 2.89–4.39) [

19

] and 3.28 (average of estimations in 12 studies) [

20

]. In addition, in this

study, we have obtained the following epidemiological insights:

•

The essential epidemic size, which is characterized by

R0

, would not be affected by the

identiﬁcation rate pin a realistic parameter range 0.01–0.1, in particular, p≥1.0 ×10−6.

•

The intervention exactly has a positive effect on the delay of the epidemic peak, which would

contribute to improve the medical environment utilizing the extra time period.

•Intervention over a relatively long period is needed to effectively reduce the ﬁnal epidemic size.

The ﬁrst statement implies that underestimation of the actual infective population would not

contribute to the reduction of the essential epidemic risk. Correct information based on an adequate

diagnosis system would be desired for people to act appropriately.

Funding:

This research was funded by the Japan Society for the Promotion of Science (JSPS) KAKENHI grant

number 19K14594.

Acknowledgments:

The author would like to thank the associate editor and the anonymous reviewers for their

helpful comments that allowed me to improve the manuscript.

Conﬂicts of Interest: The author declares no conﬂict of interest.

References

1.

Centers for Disease Control and Prevention. Coronavirus Disease 2019 (COVID-19). Available online:

https://www.cdc.gov/coronavirus/2019-ncov/index.html (accessed on 23 February 2020).

2.

World Health Organization. Coronavirus Disease 2019 (COVID-19) Situation Reports. Available online: https:

//www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports/ (accessed on 5 March 2020).

3.

Rothe, C.; Schunk, M.; Sothmann, P.; Bretzel, G.; Froeschl, G.; Wallrauch, C.; Zimmer, T.; Thiel, V.; Jankem, C.;

Guggemos, W.; et al. Transmission of 2019-nCoV Infection from an Asymptomatic Contact in Germany.

New Eng. J. Med. 2020, doi:10.1056/NEJMc2001468.

4.

NHK World Japan. Japan Sets up Emergency Measures for Coronavirus. Available online: https://www3.

nhk.or.jp/nhkworld/en/news/20200213_07/ (accessed on 17 February 2020).

5.

Inaba, H. Age-Structured Populatin Dynamics in Demography and Epidemiology; Springer: Berlin/Heidelberg,

Germany, 2017.

6.

Linton, N.M.; Kobayashi, T.; Yang, Y.; Hayashi, K.; Akhmetzhanov, A.R.; Jung, S.; Yuan, B.; Kinoshita, R.;

Nishiura, H. Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections

with right truncation: A statistical analysis of publicly available case data. J. Clin. Med.

2020

,9, 538,

doi:10.3390/jcm9020538.

7.

Sun, H.; Qiu, Y.; Yan, H.; Huang, Y.; Zhu, Y.; Chen, S.X. Tracking and predicting COVID-19 epidemic in

China mainland. medRxive 2020, doi:10.1101/2020.02.17.20024257.

8.

Statistics Bureau Japan. Population Estimates Monthly Report January. Available online: https://web.

archive.org/web/20190623053923/http://www.stat.go.jp/english/data/jinsui/tsuki/index.html (accessed

on 5 March 2020).

9.

Bloomberg. Japan’s Hokkaido may Have 940 Infected, Researcher Says. Available online: https://www.

bloomberg.com/news/articles/2020-03-03/japan-s-hokkaido-could-have-up-to-940-infected-researcher-says

(accessed on 5 March 2020).

10.

Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A.J. On the deﬁnition and the computation of the basic

reproduction ratio

R0

in models for infectious diseases in heterogeneous populations. J. Math. Biol.

1990

,28,

365–382.

11.

van den Driessche, P.; Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for

compartmental models of disease transmission. Math. Biosci. 2002,180, 29–48.

12.

Capaldi, A.; Behrend, S.; Berman, B.; Smith, J.; Wright, J.; Lloyd, A.L. Parameter estimation and uncertainty

quantiﬁcation for an epidemic model. Math. Biosci. Eng. 2012,9, 553–576.

J. Clin. Med. 2020,9, 789 7 of 7

13.

The Japan Times. Nearly All Prefectures in Japan Shut Schools Amid Coronavirus Outbreak. Available

online: https://www.japantimes.co.jp/news/2020/03/02/national/japan-prefectures-shut-schools-covid-

19/#.XmIKE0BuKUk (accessed on 6 March 2020).

14.

Wu, Z.; McGoogan, J.M. Characteristics of and imoprtant lessons from the coronavirus disease 2019

(COVID-19) outbreak in China. J. Am. Med. Assoc. 2020, doi:10.1001/jama.2020.2648.

15.

CNBC. It’s a ‘false hope’ Coronavirus will Disappear in the Summer like the Flu, WHO Says. Available

online: https://www.cnbc.com/2020/03/06/its-a-false-hope-coronavirus-will-disappear-in-the-summer-

like-the-ﬂu-who-says.html (accessed on 8 March 2020).

16.

Imai, N.; Cori, A.; Dorigatti, I.; Baguelin, M.; Connelly, C.A.; Riley, S.; Ferguson, N.M. Report 3: Transmissibility

of 2019-nCoV; Imperial College London: London, UK, 2020.

17.

Liu, T.; Hu, J.; Kang, M.; Lin, L.; Zhong, H.; Xiao, J.; He, G.; Song, T.; Huang, Q.; Rong, Z.; et al. Transmission

dynamics of 2019 novel coronavirus (2019-nCoV). bioRxive 2020, doi:10.1101/2020.01.25.919787.

18.

Read, J.M.; Bridgen, J.R.E.; Cummings, D.A.T.; Ho, A.; Jewell C.P. Novel coronavirus 2019-nCoV: Ealry estimation

of epidemiological parameters and epidemic predictions. medRxive 2020, doi:10.1101/2020.01.23.20018549.

19.

Zhao, S.; Lin, Q.; Ran, J.; Musa, S.S.; Yang, G.; Wang, W.; Lou, Y.; Gao, D.; Yang, L.; He, D.; et al. Preliminary

estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020:

A data-driven analysis in the early phase of the outbreak. Int. J. Infect. Dis. 2020,92, 214–217.

20.

Liu, Y.; Gayle, A.A.; Wilder-Smith, A.; Rocklöv, J. The reproductive number of COVID-19 is higher compared

to SARS coronavirus. J. Travel. Med. 2020, doi:10.1093/jtm/taaa021.

c

2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access

article distributed under the terms and conditions of the Creative Commons Attribution

(CC BY) license (http://creativecommons.org/licenses/by/4.0/).