Prediction of the Epidemic Peak of Coronavirus
Disease in Japan, 2020
Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan;
Received: 24 February 2020; Accepted: 10 March 2020; Published: 13 March 2020
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
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
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
]. 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 .
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 [
]. 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 [
]. 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
fraction of infective individuals can be identiﬁed by diagnosis.
We apply the following well-known SEIR compartmental model (see, e.g., [
]) 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)
denote the susceptible, exposed, infective and removed populations at
denote the infection rate, the onset rate and the removal rate, respectively.
Note that 1
imply the average incubation period and the average infectious period, respectively.
Let the unit time be 1 day. Based on the previous studies [
], we fix 1
5, and thus,
0.1, respectively. We fix
to be 1 so that each population implies the proportion
to the total population. We assume that one infective person is identified at time
0 among total
N=1.26 ×108number of people in Japan . 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
. Thus, we obtain
. We assume that there is no exposed and removed populations at
0, that is,
E(0) = R(0) = 0, and hence,
S(0) = 1−E(0)−I(0)−R(0) = 1−1
It was estimated in  that 77 cases were conﬁrmed among the possible 940 infected population
in February in Hokkaido, Japan. Based on this report, we assume that
ranges from 0.01 to 0.1.
The basic reproduction number
, which means the expected value of secondary cases produced by
one infective individual [
], is calculated as the maximum eigenvalue of the next generation matrix
0 0 #,V="ε0
−ε γ #.
Thus, we obtain
p×1.26 ×108. (2)
2.2. Sensitivity of the Basic Reproduction Number
It is obvious that the basic reproduction number
is independent from the onset rate
The sensitivity of R0to other parameters β,γand pare calculated as follows:
∂β =1, Aγ=γ
∂γ =−1, Ap=p
p×1.26 ×108−1, (3)
J. Clin. Med. 2020,9, 789 3 of 7
denote the normalized sensitivity indexes with respect to
We see from Equation
time’s increase in
) results in the
. In particular, we see from the third equation in Equation
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
. . .
, 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.
(P1) Fix β>0and calculate the numerical value of Y(t), t =0, 1, . . . , 45 by using model Equation (1).
Y(t) = Y(t) + qY(t)e(t) = Y(t) + (Poisson noise),t=0, 1, . . . , 45,
. . .
, 45 denote random variables from a normal distribution with mean zero and variance 1[
(P3) Calculate J (β) = ∑45
(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
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
as 0.26 (95%CI, 0.24–0.28) (see Figure 1). Moreover, by Equation
we obtain an estimation of R0as 2.6 (95%CI, 2.4–2.8) (see Table 2).
with the estimated infection rate
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 
γRemoval rate 0.1 
NTotal population in Japan 1.26 ×108
pIdentiﬁcation rate 0.01–0.1 
J. Clin. Med. 2020,9, 789 4 of 7
3.1. Peak Prediction
We deﬁne the epidemic peak
by the time such that
attains its maximum in 1 year, that is,
Y(t∗) = max0≤t≤365 Y(t)
. We ﬁrst set
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
208 (95%CI, 191–229). That is,
starting from 15 January (
0), the estimated epidemic peak is 10 August (
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)
Time variation of the number
of infective individuals who are identiﬁed at time
(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
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 (
165) to July 30 (
197). In contrast to
, the epidemic peak and the
(apparent) epidemic size are sensitive to the identiﬁcation rate
. 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)
Time variation of the number
of infective individuals who are identiﬁed at time
(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 [
] 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 [
]). In this simulation, we assume that such social efforts successfully reduce
the infection rate
0.26 to 75% during a period from 1 March (
46) to a planned day (
In what follows, we ﬁx p=0.01.
J. Clin. Med. 2020,9, 789 5 of 7
First, we set
77, that is, the intervention is carried out for 1 month (from 1 March to 1 April).
In this case, the epidemic peak
is delayed from 179 (12 July) to 190 (23 July). However, the epidemic
size is almost the same. On the other hand, if
220, that is, the intervention is carried out for 6
months (from 1 March to 1 September), then the epidemic peak
is delayed from 179 (12 July) to 243
(14 September) and the epidemic size is effectively reduced (see Figure 4).
Time variation of the number
of infective individuals who are identiﬁed at time
0.01 and no intervention, 1 month intervention (
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
is delayed almost linearly for
47 ≤T≤239 and ﬁxed to t∗=237 for T≥240.
The relation between the planned ﬁnal day for intervention
) the epidemic peak
(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
365, which is calculated as
, is monotonically decreasing and converges to 0.99
it almost does not change for small
180. This implies that the intervention over a relatively long
duration is required to effectively reduce the ﬁnal epidemic size.
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
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 .
J. Clin. Med. 2020,9, 789 6 of 7
The estimated value of the basic reproduction number
in this study is not so different from
early estimations: 2.6 (95%CI, 1.5–3.5) [
], 2.90 (95%CI, 2.32–3.63) [
], 3.11 (95%CI, 2.39–4.13) [
3.58 (95%CI, 2.89–4.39) [
] and 3.28 (average of estimations in 12 studies) [
]. In addition, in this
study, we have obtained the following epidemiological insights:
The essential epidemic size, which is characterized by
, 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.
This research was funded by the Japan Society for the Promotion of Science (JSPS) KAKENHI grant
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.
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