MBE, 18(5): 6506–6526.
DOI: 10.3934/mbe.2021323
Received: 19 May 2021
Accepted: 29 June 2021
http://www.aimspress.com/journal/MBE

Published: 28 July 2021

Research article

SIR model-based verification of effect of COVID-19 Contact-Confirming
Application (COCOA) on reducing infectors in Japan
Yuto Omae1, * , Yohei Kakimoto1 , Jun Toyotani1 , Kazuyuki Hara1 , Yasuhiro Gon2 and Hirotaka
Takahashi3College of Industrial Technology, Nihon University, Izumi, Narashino, Chiba, Japan
Nihon University School of Medicine, Ooyaguchi, Itabashi, Tokyo, Japan
3
Research Center for Space Science, Advanced Research Laboratories, Tokyo City University,
Todoroki, Setagaya, Tokyo, Japan

1

2* Correspondence: Email: oomae.yuuto@nihon-u.ac.jp.
Abstract: As of April 2021, the coronavirus disease (COVID-19) continues to spread in Japan. To
overcome COVID-19, the Ministry of Health, Labor, and Welfare of the Japanese government developed
and released the COVID-19 Contact-Confirming Application (COCOA) on June 19, 2020. COCOA
users can know whether they have come into contact with infectors. If persons who receive a contact
notification through COCOA undertake self-quarantine, the number of infectors in Japan will decrease.
However, the effectiveness of COCOA in reducing the number of infectors depends on the usage rateof COCOA, the rate of fulfillment of contact condition, the rate of undergoing the reverse transcription
polymerase chain reaction (RT-PCR) test, the false negative rate of the RT-PCR test, the rate of infection
registration, and the self-quarantine rate. Therefore, we developed a Susceptible-Infected-Removed(SIR) model to estimate the effectiveness of COCOA. In this paper, we introduce the SIR model andreport the simulation results for different scenarios that were assumed for Japan.
Keywords: contact tracing app; COCOA; COVID-19; SARS-CoV-2; SIR model; simulation in Japan

1. Introduction
As of April 2021, the coronavirus disease (COVID-19) continues to spread in Japan. On June 19,
2020, the Japanese government released the COVID-19 Contact-Confirming Application (COCOA), asmartphone app [1] that was developed to let COCOA users know whether they had come into contact
with any COVID-19 infectors. The contact notification process comprises the following steps:
1. An individual with a pre-existing severe acute respiratory distress syndrome coronavirus 2

6507

Figure 1. COVID-19 Contact-Confirming Application (the screenshot of the app is shownin [1]).
(COVID-19) infection and an uninfected individual both install COCOA on their smartphones(the infector is unaware of the infection because he/she has not undergone the reverse transcription polymerase chain reaction (RT-PCR test) for COVID-19).
2. The uninfected individual is in the proximity (<1 m) of the individual with a pre-existing COVID19 infection for 15 minutes or longer. Through this process, the uninfected person becomes a new
infector.
3. At a later date, the individual with the pre-existing COVID-19 infection who is the source of
infection undergoes an RT-PCR test.
4. In the case of a positive result for COVID-19, the individual with a pre-existing COVID-19 infection who is the source of infection registers this result in the COCOA.
5. Then, individuals who came into contact with the individual with a pre-existing COVID-19 infection who is the source of infection within less than 2 weeks from the date of the positive result get
a contact notification via COCOA.
The display of the COCOA is shown in Figure 1. If COCOA users click on the “check close contacts”
button to the top-left, they receive information on whether they had any contact with the infectors.
Infectors who become aware of a positive RT-PCR test result can register this result by using the lower“Register positive test result” button. Through these features, COCOA users can become aware of thepossibility of infection through close contact with infectors with a pre-existing COVID-19 infection. If
these exposed individuals decrease venturing outside and initiate self-quarantine, the number of total
infectors in the community will decrease.
Worldwide, apps that are similar to COCOA have been developed (e.g., England [2], China, India, and Israel [3]) as various countries consider that the contact notification function is important forovercoming COVID-19. There are some important parameters that can decrease the number of new infectors as a function of COCOA. It is essential that the pre-existing infector and the close contact bothuse COCOA; therefore, the usage rate of COCOA is an important parameter (the term “close contact”
refers to a person who has been in close contact with a pre-existing COVID-19 infector). Moreover,Mathematical Biosciences and Engineering

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because the condition by which COCOA ascertains a positive contact with a pre-existing infector iscontact for 15 minutes or longer at a distance of less than 1 m, the rate of new infectors who meet theabovementioned condition is important. Individuals who have undergone an RT-PCR test and have received positive results only could register their result in the COCOA. Therefore, the rate of PCR testing
and the false-negative rate of PCR tests are important parameters. Furthermore, the duration of delay
from the time of infection to undergoing an RT-PCR test is important. If the delay is too long, the close
contact would not receive the “infector contact” notification from COCOA for a long duration. If theclose contact becomes an infector through contact with pre-existing infectors, he/she may infect anotherindividual. Moreover, COCOA cannot send the contact notification to a close contact if the infector who
is the source of infection does not register a positive result in COCOA, even if he/she has received apositive result on the RT-PCR test and is aware of the infection. Therefore, the rate of registration of
positive results is an important parameter. If the pre-existing infector who was the source of infection
had maintained appropriate correspondence (using COCOA, getting the RT-PCR test done, registering
positive results, and so on), the close contact can receive contact notification from COCOA. However, if
a person in close contact does not undertake self-quarantine, the effectiveness of COCOA in decreasing
the number of COVID-19 infectors does not occur. Therefore, the rate of self-quarantine of individuals
who receive contact notifications from COCOA is another important parameter. Thus, there are some
important parameters to estimate the effectiveness of COCOA.
Simulation is a desirable approach to clarify the effectiveness of COCOA. Many studies on infection
have used simulation (e.g., [4–7]). Recently, COVID-19 infection simulations have been reported fromtarget countries, such as China [8–10], India [11], The Netherlands [12], Korea, Italy, France [13],Nigeria [14], Arizona [15], Australia [16], South Carolina [17], Scandinavia [18], and Brazil [19, 20].
Moreover, Niwa et al. reported the effectiveness of airport quarantine [21]. Kurahashi et al. reported
the effectiveness of remote working and staggered commuting [22]. Vasconcelos et al. [23] and Macêdo
et al. [24] analyzed the fatality curve of COVID-19 infectors by the simulation approach.
Recently, simulation studies have been conducted to estimate the effectiveness of contact-tracing
apps. Various approaches [25–29] have been evaluated; however, the target of these researches were
not COCOA. As an exception, Omae et al. [30] verified the effectiveness of COCOA using multi-agent
simulation (MAS). However, because the calculation cost of MAS is high, this simulation scale is small(approximately 1000 agents), and undertaking trial and error in the various scenarios is difficult. Moreover, Omae et al. [30] did not consider the rate of RT-PCR test results and false negative rates. Kurita
et al. [31] reported the effectiveness of COCOA in reducing the number of infectors; however, they did
not consider the rate of registration of positive results and the false negative rate of RT-PCR tests.
Against this background, we developed a Susceptible-Infected-Removed (COCOA-SIR) model toestimate the effectiveness of COCOA in decreasing infectors to simulate infection spread in Japan. The
COCOA-SIR model has the following important parameters for estimating the effect of COCOA.
1. the usage rate of COCOA
2. the rate of new infectors who are infected by contact with other infectors of less than 1 m and for
15 minutes or longer.
3. the rate of undergoing RT-PCR test
4. the delay from infection timing to undergoing an RT-PCR test
5. the false negative rate of RT-PCR test
6. the rate of registration of a COVID-19 positive result
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7. the rate of self-quarantine of individuals who receive the contact notification from COCOA
The SIR model is used for infection epidemic simulation of COVID-19 (e.g., [32–35]) and, therefore,
we adopted the SIR model as our research approach.
2. Basic SIR model
The basic SIR model [36] is represented as,
dS (t)= −β(t)S (t)I(t),dtdI(t)= β(t)S (t)I(t) − γI(t),dtdR(t)= γI(t),
dt

(2.1)
(2.2)
(2.3)

where S (t), I(t), R(t) are the number of susceptible (not infected), infected, and removed individuals
at time t, respectively. “Removed persons” means individuals who are recovering from a COVID-19
infection or those who died of the infection. β(t) is the infectivity of time t, and γ is the recovery rate.
If people in the community are careful about preventing infection, then the infectivity is low. However,
if individuals in the community are not careful about infection, then the infectivity is high. The degreeof care differs based on various situations. Therefore, β(t) is the function of time t.
Based on the above-described differential equations of Equations (2.1)-(2.3), we can understand theinfection spread dynamics continuously. However, the number of new COVID-19 infectors is reported
once a day in Japan. Therefore, it is desirable to express this parameter through a discrete-time equation
rather than via a differential equation. Thus, we use,
S t+1 = S t − β0t S t It ,It+1 = It + β0t S t It − γ0 It ,

(2.4)
(2.5)

Rt+1 = Rt + γ0 It ,(2.6)

as discrete-time equations for simulating infection spread. Where S t , It , and Rt are the numbers ofsusceptible (not infected), infected, and removed persons on day t, respectively. β0t is the infectivity
on day t, and γ0 is the recovery rate per day. Assuming that the infection period follows exponential
distribution, the recovery rate γ0 is the reciprocal of the average infection period [37, 38]. For example,
if the average infection period is 7 days, γ0 is 1/7. Moreover, it is necessary to set S 0 , I0 , and R0 .
In Equation (2.4), the first term is the number of susceptible individuals on day t, and the second
term represents the number of individuals who are susceptible to virus infection from day t to day t + 1.
In Equation (2.5), the first term is the number of infected persons on day t, the second term is thenumber of newly infected persons from day t to day t + 1, and the third term represents the number of
infected persons who are excluded due to recovery following virus infection (or death) from day t to day
t + 1. In Equation (2.6), the first term is the number of recovered individuals on day t, and the second
term represents the number of newly recovered persons based on recovery from COVID-19 infection(or death) from day t to day t + 1.
An effective reproduction number is an important index for the virus-spread simulation. The effective
reproduction number refers to the number of additional persons infected by an infected person during
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the entire infection period [39]. In Equation (2.5), an infector infects β0t S t persons per day, and theaverage infection period is (γ0 )−1 . Therefore, we can define the effective reproduction number rt on dayt as
β0t S trt = 0 .
γ(2.7)

If the effective reproduction number rt > 1, then β0t S t It − γ0 It > 0. Therefore, It+1 is larger than It .
In contrast, if the effective reproduction number is rt < 1, It+1 is less than It . For decreasing rt , it isrequired to decrease β0t . To decrease β0t , it is important to wear a mask, disinfect fingers, decrease goingout, and so on.
3. COCOA-SIR modelWith regard to the originality of our research, we propose the COCOA-SIR model, which can represent the effectiveness of the COCOA developed by Japan. After this section, we use two variables
i, t to represent “day” i represents the timing of the occurrence of a new infector. t is a variable that
represents the current day.
3.1. The effectiveness of COCOA on decreasing infectors
By the second term of Equation (2.5) of the basic SIR model, the number of new infectors occurringfrom day t to day t + 1 is defined by the value obtained by multiplying the number of infectors It on day tand β0t S t . Thus, the infector It on day t generates new infectors. If COCOA is present in the community,
we consider that a percentage of individuals who receive notification of contact with infectors fromCOCOA would undertake self-quarantine. Here, the number of infectors who receive notifications ofcontact with infectors through COCOA and undertake self-quarantine is defined as Itnotice . Then, thenumber of infectors that infect other persons on day t is It − Itnotice instead of It . Therefore, in the caseof existing COCOA in society, the number of infectors on day t + 1 is defined by
It+1 = It + β0t S t (It − Itnotice ) − γ0 It .

(3.1)

This means that the infectors It − Itnotice who did not receive a COCOA notification of contact with
infectors (or did not take self-quarantine in spite of receiving it) would generate new infectors.
Herein, we consider the details of Itnotice . We explain the process flow until the new infectors receive
a COCOA notification of contact, based on Figure 2. This is an example of how a newly infected person
reproduces the infection by contact with an individual with a preexisting infection from days 3 to 4.
In this case, to send the contact notification via COCOA, both individuals need to use COCOA andsatisfy the condition of contact of COCOA (<1 m and ≥15 minutes). Moreover, an individual witha preexisting infection needs to undergo an RT-PCR test and to obtain a positive result, as well as
input the fact into COCOA. In the case shown in Figure 2, on day 7, an individual with the preexisting
infection fulfills these conditions. At this time, an individual who is newly infected from days 3 to 4
can receive a contact notification via COCOA. If a newly infected individual who receives the contact
notification appropriately undertakes self-quarantine, he/she will not infect other individuals after day
7. With regard to the function of COCOA [1], the app retains the notification-related information forMathematical Biosciences and Engineering

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Figure 2. Conditions under which an infector who receives the contact notification fromCOCOA does not infect other individuals: (i, t) = (3, 7).
Table 1. The elements of Pi,t .
Variables Meaning
puse
Usage rate of COCOA
15min
p
The rate of new infectors infected by contact with other infectors less than 1
m and for 15 minutes or longer.
ppcr
The rate of undergoing an RT-PCR test
i,t
false
p
false-negative rate of the RT-PCR test
registpRate of registration of the COVID-19-positive resulthomepRate of self-quarantine of an individual who receives a notification of contact
through COCOA

14 days after the contact with an individual with a preexisting infection. Therefore, an individual who
is newly infected from days 3 to 4 receives the notification via COCOA until day 17.
Based on the abovementioned discussions, it is necessary to fulfill various conditions to ensure theMathematical Biosciences and Engineering

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effectiveness of COCOA in decreasing new infectors. Furthermore, the timing when the infected individual receives the contact notification through COCOA is the time at which the individual with thepreexisting infection undergoes the RT-PCR test, receives a positive test result, and inputs this result
into COCOA. Therefore, the probability that a newly infected individual who appears on day i will
receive a contact notification on day t, and will not infect others from day t to day t + 1 is defined by

pcr


 (puse )2 × p15min × pi,t (1 − pfalse ) × pregist × phome (t ≤ i + 14)
Pi,t = .
(3.2)
 0 (otherwise)
Where, puse is the usage rate of COCOA, p15min is the rate of satisfying the conditions of infective contact
of more than 15 minutes at a distance less than 1 m, pfalse is the false-negative rate of the RT-PCR test.
pregist is the rate of registration of a COVID-19-positive result, and phome is the rate of self-quarantineof a person who receives a contact notification via COCOA (shown in Table 1). There are two puse in
Equation (3.2) because both an individual with a preexisting infection and a close contact use COCOA.
The period of sending the contact notification was 14 days from the contact and time of infection [1].
Therefore, if the current day t is more than 14 days from day i, Pi,t is zero.
Next, we explain ppcr
i,t in Equation (3.2). As a function of COCOA, the timing of sending the contact
notification to the close contact is not the timing of contact with the infected person who is the sourceof infection (it is not on day i). The timing when the close contact can receive the contact notification
is when the infected person who is the source of infection registers a positive result on the RT-PCR test
in COCOA. To represent the described delay time, we define the probability of the individual with thepreexisting infection obtaining an RT-PCR test on day t (who is the source of a new infected person
occurring on day i) as
ppcr
i,t =

ppcr
max
,
1 + exp {−a(t − (i + 1) − b)}(3.3)

where, ppcr
max is the maximum probability of the individual undergoing an RT-PCR test. As it is a sigpcr
moid function, ppcr
i,t ∈ (0, pmax ). where the parameter a is the gradient of the sigmoid function and theparameter b is the number of days that half of the infected individuals of ppcr
max undergo the RT-PCR
pcr
test. For example, we show the results for pmax = 0.2 and (a, b) = (1.25, 3), (4.25, 3), (1.25, 6), (4.25, 6)
in Figure 3. The horizontal axis t − (i + 1) represents the number of days elapsed from the infection
timing (day i) of individuals in close contact. Because t = i + 1 is the next day of the infection timingby contact with infectors, ppcr
i,t=i+1 is a small value. The reason is that we assume the rate of undergoing
RT-PCR test gradually increases depending on the elapsed time. We can understand that the larger thea is, the larger is the gradient of ppcr
i,t . Moreover, b is the number of days that half of the infected personsof ppcrundergothe
RT-PCR
test.
As the parameters a, b represent the delay time from infection timingmax
to the RT-PCR test, we call a, b the delaying parameters. For your information, ppcr
i,t is not more thanpcr
pcr
pmax . Therefore, in the case of Figure 3 (pmax = 0.2), 80% of individuals with preexisting infection who
were the source of infection did not undergo an RT-PCR test regardless of how much time had elapsed.
Moreover, there is a problem in that the RT-PCR test is negative even if the individual is infected (false
negative). In this case, because infected persons cannot be aware of infection, they cannot register positive results into COCOA even if they are infected. As the false negative rate is pfalse , the 1 − pfalse in
Equation (3.2) is the correct positive rate.
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Figure 3. The relationship between the rate of undergoing the RT-PCR test and the days
elapsed.
We consider Itnotice , which is the number of infectors who receive contact notifications on day t basedon the above-described discussions. We define the number of new infectors occurring from day i to day
i + 1 as Ii+ . As this is the second term of Equation (3.1), Ii+ is
Ii+ = β0i S i (Ii − Iinotice ).

(3.4)

Here, Ii+ , which is the infectors generated from day i to day i + 1, and is recovered by the rate γ0 per day.
Therefore, the number of unrecovered infected persons on day t is,Ii+ (1 − γ0 )t−(i+1) .

(3.5)

In the case of COCOA existing in the community, some of Ii+ (1 − γ0 )t−(i+1) will receive a contact notification on COOCA. As it is the rate of satisfying the event, as shown in Figure 2, the number of infectors
who have received contact notifications on COCOA is
Pi,t Ii+ (1 − γ0 )t−(i+1) .

(3.6)

In other words, the number of persons who were infected on day i and have not recovered from infection
by day t and have received the contact notification is Pi,t Ii+ (1 − γ0 )t−(i+1) .
These abovementioned discussions indicate only new infectors who are generated from day i to day
i + 1. As of day t, these new infectors exist for t − 1 days. Therefore, as of day t, the number of new
infectors who have not recovered from infection and have received the contact notification in COCOA
and will not infect others by undertaking self-quarantine is
Itnotice =t−1
XPi,t Ii+ (1 − γ0 )t−(i+1)(3.7)

Pi,t β0i S i (Ii − Iinotice )(1 − γ0 )t−(i+1) .

(3.8)

i=0

=

t−1
X
i=0

Note that, by using Equation (3.2), the summation range of Equations (3.7) and (3.8) can also be fromt − 14 to t − 1 instead of from 0 to t − 1 when satisfying t ≥ 14. In Equation (3.8), Itnotice includes
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noticenotice. As I0notice denotes the number of infectors who haveI{0,··· , t−1} . Therefore, we cannot calculate I0
received contact notification on COCOA on day 0, we set

I0notice = const.

(3.9)

In other words, we can set I0notice to any number on the simulation starting time. If there is no particularreason, set I0notice to 0. This means that there are no individuals who have received contact notifications
at the simulation starting time. By substituting Itnotice calculated by Equation (3.8) for Equation (3.1),
we can calculate the infectors It+1 on day t + 1 by considering the effect of COCOA on decreasing thenumber of infectors.
Thus, we arrange Equation (3.1) as,
It+1 = It + β0t S t It − γ0 It − β0t S t Itnotice .

(3.10)

Comparing Equation (2.5) of the basic SIR model and Equation (3.10) of the proposed SIR model,
the first, second, and third terms are the same, and the fourth term is then added. In other words, weconsider the fourth term β0t S t Itnotice of Equation (3.10) as the effect of COCOA on decreasing the numberof new infectors. Therefore, we define the effect of COCOA in decreasing the number of new infectors
on day t to day t + 1 as
Etapp = β0t S t Itnoticet−1
X= β0t S t
Pi,t β0i S i (Ii − Iinotice )(1 − γ0 )t−(i+1) .

(3.11)(3.12)

i=0This indicates that the effect of COCOA is affected not only by Pi,t but also by various variables.
3.2. Proposed SIR model
Based on the above discussion, to estimate the effect of COCOA in decreasing the number of infectors, we propose the COCOA-SIR model, defined as
S t+1 = S t − β0t S t It + Etapp ,It+1 =
Rt+1 =

It + β0t S t It − γ0 It − Etapp ,Rt + γ 0 I t .

(3.13)(3.14)(3.15)

Equation (3.14) to estimate the number of infectors It+1 is obtained by substituting Etapp defined by
Equation (3.11) for Equation (3.10). Moreover, because the number of new infectors decreases by Etapp ,Equation (3.13) to estimate the number of susceptible persons S t+1 has the term Etapp . The number ofrecovered persons Rt+1 is the same as in the basic SIR model. Here, when we set the coefficient Pi,t , ∀i, taffects the effect of COCOA as 0, Etapp = 0 by Equation (3.12). In this case, the COCOA-SIR model
defined by Equations (3.13), (3.14), and (3.15) are in the same form as the basic SIR model defined by
Equations (2.4), (2.5), and (2.6). Therefore, we consider the COCOA-SIR model an extension of thebasic SIR model.
The simulation results of the basic SIR and COCOA-SIR models are shown in Figure 4. The simulation parameters are rt = 2 as the effective reproduction number, S 0 = 9999, I0 = 1, R0 = 0 as the
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Figure 4. The comparison of the basic SIR and the COCOA-SIR model.
number of initial susceptible, infected, and recovered individuals, γ0 = 1/7 as the recovery rate (theaverage recovery period is 7 days), and the simulation period is 200 days. We set puse = 1, p15min = 0.5,
false
ppcr
= 0.1, pregist = 1 and phome = 1 for parameters of COCOA. Moreover, the delaying
max = 0.5, pparameters were adopted as (a, b) = (4, 3). From Figure 4, it was confirmed that the number of infectors
decreased.
4. Simulations in Japan
4.1. Objective and outline
In this section, we compare the simulations in the case of the presence and absence of COCOA
using the COCOA-SIR model and actual infector-open data in Japan [40]. COCOA was released onJune 19, 2020 [41] in Japan. Open data [40] obtained after June 19, 2020 cannot be used because it
is impossible to assume a Japan where the COCOA does not exist. Moreover, there are some days inwhich new infectors were reported to be zero, before February 12, 2020. This means that fluctuations in
the number of newly infected people are unstable. Therefore, we use the open data [40] of reported new
infectors in Japan from February 12, 2020 to June 18, 2020 for simulations (the maximum simulation
period: tmax = 128 days).
4.2. Simulations parameters4.2.1. New infectors considering infectors who are unaware of infection
The open data [40] of the number of new infectors reported by Japan is only the number of infectors
who were aware of infection based on the results of the RT-PCR test. There will be many infectors who
are unaware of the infection. Therefore, we define the number of new infectors on day t, considering
the amount of both individuals who are unaware and aware of infection as
It+,all = CIt+,find ,(4.1)

where It+,find is the number of new infectors on day t reported in Japan, and C is a non-finding coefficient.
Tsuchiya estimated that the number of new infectors, considering the amount of being unaware andaware of infection, was 23 times larger than that reported by Japan [39]. Therefore, we adopted C = 23.
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4.2.2. Recover rate γ0
Kobayashi et al. [32] reported that γ0 ranges from 0.13 to 0.17 as depicting the spreading of COVID19 infection in Japan; therefore, we adopted γ0 = 0.15, which is the median. In this study, because we
assume that the infection period follows an exponential distribution according to reports from previous
research [37] of the SIR model, we set γ0−1 = 1/0.15 ' 6.67 days as the average infection period. The
function of the exponential distribution in the case of setting t as the infective period is
f (t) = γ0 exp(−γ0 t).

(4.2)

By integrating the exponential distribution, the probability of recovering from 0 to x days is:
Z x
Pr(x) =
f (t)dt = 1 − exp(−γ0 x).

(4.3)

0

For example, when adopting γ0 = 0.15, we obtain Pr(x = 20) ' 0.950. This means that approximately
95% of COVID-19 infectors recover within 20 days.
4.2.3. The numbers of initial susceptible, infected and recovered persons S 0 , I0 , R0
To perform simulations using the SIR model, it is necessary to set the numbers of initial susceptible,
infected, and recovered individuals (S 0 , I0 , and R0 ). As mentioned above, the starting time of thesimulation was February 12, 2020, and the average infection period was 6.67 days. Therefore, we
calculated the total number of new infectors reported in Japan for 7 days (from February 5 to February11) prior to February 12, 2020. Subsequently, we multiplied the non-finding coefficient C to the total
new infectors, and we adopted it as I0 . As a result, the initial infectors I0 were 46 individuals in Japan.
Moreover, because the total population of Japan is approximately 1.2 × 108 persons, we adopted thevalue of subtracting I0 from 1.2 × 108 as the initial number of susceptible (non-infected) individuals S 0 .
Because of the starting time of the simulation, we assumed the initial number of recovered individuals
to be zero (i.e., R0 = 0). In conclusion, we set S 0 + I0 + R0 = 1.2 × 108 , which is the total population ofJapan.
4.2.4. infectivity β0t
To perform the simulation using the COCOA-SIR model, it is necessary to set the infectivity β0t .
As mentioned above, we used the open data of periods without COCOA. Therefore, we searched for
infectivity β0t that minimizes the absolute error between the new infectors β0t S t It using the basic SIR
model shown in Equation (2.5) and the new infectors It+,all reported by Japan. Thus, we adopted the
optimal infectivity β0 opt
t as,
+,all
0
β0 opt|.
t = Argmin |βt S t It − It
β0t

(4.4)

Moreover, by using Equation (2.7), we calculated the optimal effective reproduction number rtopt as
rtopt =
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β0 opt
t St
.
γ0

(4.5)
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Figure 5. The new infectors included both reported and unreported infected individuals inJapan (upper side: black line) and the output of the SIR model (upper side: yellow line). The
value at the bottom of the figure means the estimated optimal reproduction number rtopt . The
black dashed line means that the reproduction number is 1.
The line charts of new infectors in Japan It+,all and new infectors reproduced by the SIR model using the
optimal reproduction number rtopt are shown in Figure 5. The black line on the upper side is It+,all , andthe yellow line on the upper side represents new infectors reproduced by the SIR model. As they arevery similar, we adopted β0 opt
t to simulate the spread of infection and to verify the effect of COCOA in
reducing infectors in Japan. The blue line at the bottom of Figure 5 indicates the optimal reproduction
number rtopt solved by Equation (4.5).
4.2.5. The usage rate of COCOA
According to the Japanese government, the number of COCOA users was 23.1 million on January
8, 2021, in Japan [41]. This means that the usage rate of COCOA in Japan is 0.1925 because the total
population in Japan is 1.2 × 108 . Therefore, we adopted puse = 0.1925. Moreover, we adopted puse =0.6, 0.9 to perform simulations in the case of the successful promotion of COCOA for the Japanese
population.
4.2.6. The rate of undergoing RT-PCR test
Tsuchiya [39] reported that the estimated number of total infectors in Japan may be 23 times thenumber of infected individuals. In other words, the rate of RT-PCR testing in Japan may be approximately 1/23 ' 0.0435 for all infectors. Therefore, we adopted ppcr
max = 0.0435 as the maximum rate of
the PCR test. Next, we consider the delaying parameters of undergoing the RT-PCR test. It is difficult todetermine the evidence-based values for the delaying parameters a, b in Equation (3.3). Therefore, weMathematical Biosciences and Engineering

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adopted (a, b) = (1.25, 3) as the optimistic scenario and (a, b) = (1.25, 6) as the pessimistic scenario,
respectively (these forms are shown in Figure 3). This means that half of ppcr
max underwent the RT-PCR
pcr
test by day 3 or 6. Moreover, we adopted pmax = 0.3, 0.6, 0.9 to perform simulations in the case of
successful PCR tests in the Japanese population.
4.2.7. The registration rate of positive result to COCOA
The total number of registrations of positive results on the RT-PCR test to COCOA is 5297 individuals on December 25, 2020, and 6929 individuals on January 8, 2021, [41]. Therefore, the number ofregistrations of positive results for these 2 weeks is 6929 − 5297 = 1632 individuals. Moreover, thenumber of new infectors for 2 weeks from December 25, 2020, to January 8, 2021, was 63383 individuals [40]. As the usage rate of COCOA puse is 0.1925, the estimated total number of infectors who use
COCOA may be approximately 63383 × 0.1925 ' 12201 individuals for these 2 weeks. In this case,
we can estimate pregist = 1632/12201 ' 0.1337 as the registration rate of the positive result inputted to
COCOA. Moreover, we adopted pregist = 0.3, 0.6, 0.9 to perform simulations in the case of successfulpromotion of the registration of RT-PCR positive results to the Japanese population.
4.2.8. False negative rate of PCR test
Kucirka et al. [42] reported a false-negative rate of the COVID-19 RT-PCR test. These authors
indicated that the false negative rate depended on the days elapsed from the time of infection, and the
median value was 0.38. Therefore, we adopted pfalse = 0.38 as the false negative rate.
4.2.9. The rate of satisfying the contact condition and self-quarantine
The conclusive condition of the COCOA of contact with infectors is that individuals are in proximity
to the infectors by less than 1 m and for more than 15 minutes. However, it is difficult to know whatpercentage of infected people have been infected by satisfying these conditions. In other words, it isdifficult to set the evidence-based value to p15min . Therefore, we adopted p15min = 0.6 as the optimistic
scenario and p15min = 0.3 as the pessimistic scenario. Moreover, it is difficult to know what percentage of
close contacts who receive a contact notification from COCOA undertake self-quarantine appropriately.
Therefore, we adopted phome = 0.3, 0.6, 0.9 to obtain the results for various scenarios.
4.3. Results and discussions
We calculated the COCOA-SIR model, which consists of Equations (3.13), (3.14) and (3.15), using
Python 3.7.3 [43], Numpy 1.18.5 [44]. The simulation results of the optimistic and pessimistic scenarios are presented in Tables 2 and 3, respectively. The optimistic and pessimistic scenarios indicatethat the simulations adopted (a, b, p15min ) = (1.25, 3, 0.6) and (a, b, p15min ) = (1.25, 6, 0.3), respectively.
Column 1 means scenario ID. snoapp represents Japan, which is the absence of COCOA, which meansthat the actual number of infectors are considered to be infectors who were unaware of the infection.
s1actual , · · · , s3actual are the scenarios wherein parameters close to reality are adopted in the case of COCOA
use
in Japan. s160,90 , · · · , s12= 0.6, 0.9).
60,90 are the scenarios for the successful promotion of COCOA (p13
Thus, s60,90 , · · · , s60,90 are the scenarios wherein parameters close to reality are adopted, with the exception of the usage rate of COCOA, and s460,90 , · · · , s12
60,90 are the scenarios in which all parameters
improved. Moreover, sbest is the best scenario for decreasing infectors. “total infectors” in Table 2 andMathematical Biosciences and Engineering

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Table 2. The total infectors Itmax + Rtmax of ending the timing of simulation in the cases of the
optimistic scenarios (a, b, p15min ) = (1.25, 3, 0.6) in Japan.
ID
snoapps1actuals2actuals3actual
s160
s260
s360
s460
s560
s660
s760
s860
s960
s10
60
s11
60
s12
60
s190
s290
s390
s490
s590
s690
s790
s890
s990
s10
90
s11
90
s12
90
sbest

puse
0.000
0.193
0.193
0.193
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.900
0.900
0.900
0.900
0.900
0.900
0.900
0.900
0.900
0.900
0.900
0.900
1.000

ppcr
max
0.000
0.044
0.044
0.044
0.044
0.044
0.044
0.300
0.300
0.300
0.600
0.600
0.600
0.900
0.900
0.900
0.044
0.044
0.044
0.300
0.300
0.300
0.600
0.600
0.600
0.900
0.900
0.900
1.000

pregist
0.000
0.134
0.134
0.134
0.134
0.134
0.134
0.300
0.300
0.300
0.600
0.600
0.600
0.900
0.900
0.900
0.134
0.134
0.134
0.300
0.300
0.300
0.600
0.600
0.600
0.900
0.900
0.900
1.000

phome total infectors reductions reduction rate[%]0.000
40604600.00
0.300
405986
60
0.01
0.600
405925
121
0.03
0.900
405865
181
0.04
0.300
405458
588
0.14
0.600
404870
1176
0.29
0.900
404284
1762
0.43
0.300
397030
9016
2.22
0.600
388199
17847
4.40
0.900
379549
26497
6.53
0.300
371079
34968
8.61
0.600
338913
67133
16.53
0.900
309339
96707
23.82
0.300
331284
74762
18.41
0.600
26941513663233.65
0.900
21834918769746.23
0.300
404724
1323
0.33
0.600
403405
2641
0.65
0.900
402091
3956
0.97
0.300
386019
20027
4.93
0.600
366909
39137
9.64
0.900
348677
57370
14.13
0.300
331284
74762
18.41
0.600
26941513663233.65
0.900
21834918769746.23
0.300
25571015033737.02
0.600
158220247827
61.03
0.900
95976
310070
76.36
1.000
31714
37433392.19

3 is the total number of infectors at the ending time of simulation (t = tmax ). “reductions” in Table 2
and 3 means that the number of decreasing infectors in comparison with the scenario snoapp and the“reduction rate” represents that rate.
First, we compare snoapp and s1actual , · · · , s3actual . In both the optimistic and pessimistic scenarios, therate of the number of decreasing infectors was only 0.00 % to 0.04 %. Therefore, we consider that theeffect of COCOA on decreasing infectors is small in the actual conditions on January 2021 in Japan.
Next, we check the cases of successful increase in the usage rate of COCOA to 60% (s160 , · · · , s12
60 ).
pcr
registhome
In these cases, where pmax , p,p
are high rates and the optimistic scenarios, the total infectors
decrease by 46.23% (e.g., the scenario of s12
60 ). However, even if the usage rate of COCOA is 60%,
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Table 3. The total infectors Itmax + Rtmax of ending the timing of simulation in the cases of thepessimistic scenarios (a, b, p15min ) = (1.25, 6, 0.3) in Japan.
ID
snoapps1actuals2actuals3actual
s160
s260
s360
s460
s560
s660
s760
s860
s960
s10
60
s11
60
s12
60
s190
s290
s390
s490
s590
s690
s790
s890
s990
s10
90
s11
90
s12
90
sbest

puse
0.000
0.193
0.193
0.193
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.600
0.900
0.900
0.900
0.900
0.900
0.900
0.900
0.900
0.900
0.900
0.900
0.900
1.000

ppcr
max
0.000
0.044
0.044
0.044
0.044
0.044
0.044
0.300
0.300
0.300
0.600
0.600
0.600
0.900
0.900
0.900
0.044
0.044
0.044
0.300
0.300
0.300
0.600
0.600
0.600
0.900
0.900
0.900
1.000

pregist
0.000
0.134
0.134
0.134
0.134
0.134
0.134
0.300
0.300
0.300
0.600
0.600
0.600
0.900
0.900
0.900
0.134
0.134
0.134
0.300
0.300
0.300
0.600
0.600
0.600
0.900
0.900
0.900
1.000

phome total infectors reductions reduction rate[%]0.000
40604600.00
0.300
406032
14
0.00
0.600
406017
29
0.01
0.900
406002
44
0.01
0.300
405904
142
0.04
0.600
405762
284
0.07
0.900
405620
426
0.11
0.300
403850
2196
0.54
0.600
401664
4382
1.08
0.900
399488
6558
1.62
0.300
397322
8724
2.15
0.600
388757
17290
4.26
0.900
380347
25699
6.33
0.300
386640
19406
4.78
0.600
368020
38026
9.37
0.900
350158
55888
13.76
0.300
405726
320
0.08
0.600
405407
640
0.16
0.900
405087
959
0.24
0.300
401119
4927
1.21
0.600
396243
9803
2.41
0.900
391417
14630
3.60
0.300
386640
19406
4.78
0.600
368020
38026
9.37
0.900
350158
55888
13.76
0.300
363485
42562
10.48
0.600
324726
81321
20.03
0.900
28946411658228.71
1.000
22610817993844.31

registin these cases, ppcr
, phome are low (e.g., scenarios s1,2,3
max , p60 ), the effect of COCOA on decreasing,12), there are similar trends. In previous
infectors is insufficient. In the case of a usage rate of 90% (s1,···90
studies [30], the objective usage rate of the contact tracing app was 60% or higher. However, we foundregistthat ppcr, phome are also important parameters. The reasons for the different results from themax , pprevious studies are discussed. Omae et al. [30] verified the effect of COCOA on decreasing infectors
using a multi-agent simulation. However, there were unrealistic assumptions, such as one assumptionthat all infectors underwent PCR tests and the false negative was zero [30].
We consider the results described in this report to reflect the specifications of COCOA as muchas possible, and the conditions were set to be realistic. Therefore, the results shown in Tables 2 and 3

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Figure 6. It+ , S t , It , Rt of the optimistic scenario (a, b, p15min ) = (1.25, 3, 0.6) in Japan.

appear to be reliable. As for which is right, the optimistic or pessimistic scenario, it is difficult to decide
this conclusively. However, it is important to promote not only the usage rate of COCOA, but also therate of RT-PCR testing, the registration of positive results, and of self-quarantine to improve the effectof COCOA on decreasing the total number of infectors.
The line charts of some scenarios are shown in Figure 6 and 7. Figure 6 and 7 mean the optimistic
and pessimistic scenarios, respectively. These indicate that snoapp is very similar to s1actual . Therefore,
we consider that there will be little effect of COCOA on decreasing the number of infectors in thecase of a small usage rate of COCOA. However, in the case of success in improving the rate of usage of
COCOA, usage of RT-PCR tests, registration of positive results, and undertaking of self-quarantine (the12scenarios: s12
60 , s90 , sbest ), Figure 6, and 7 indicate that COCOA has a significant effect on decreasing thenumber of infectors. We consider the COCOA developed by Japan to be very important in overcoming
the spread of COVID-19.
Caution must be exercised when explaining the explained simulations. The SIR model implies complete mixing of the population. However, this premise is not a realistic assumption because Japan is asufficiently large country. For this point, we consider that this is not a problem because the COVID-19
outbreak in different locations in Japan started at approximately the same time (Postnikov [45] discussed the relationship between estimating infectors of the country as a whole and local disjoint subpopulations by the SIR model).
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Figure 7. It+ , S t , It , Rt of the pessimistic scenario (a, b, p15min ) = (1.25, 6, 0.3) in Japan.
5. Conclusion
In this research, we proposed a COCOA-SIR model to verify the effect of the contact tracing app
false
COCOA that was developed by Japan. The important parameters puse , p15min , ppcr
, pregist , phome
i,t , pshown in Table 1 were included in the COCOA-SIR model. Moreover, we performed simulations ofvarious scenarios assumed for Japan. The results improved an understanding of the effect of COCOA
on decreasing the number of infectors.
We show the future directions for research below. We have used the assumed numbers (e.g.,a, b, p15min ) to perform the simulation and, therefore, we will analyze and find more desirable numbers.
In this research, we used an SIR model. However, we consider the SEIR model, which can represent the
incubation period, may be a more desirable model for such a simulation. Therefore, we plan to proposeand develop an SEIR model-based simulation to clarify the effect of COCOA on decreasing the numberof infectors.
Acknowledgments
This work was supported in part by the Telecommunications Advancement Foundation (Grant No.
20203002 to Y. Omae) and a JSPS Grant-in-Aid for Scientific Research (C) (Grant No. 21K04535 to J.
Toyotani].
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Conflict of interest
The authors have no conflict of interest.
References
1. Ministry of Health, Labour, and Welfare, Request to install the COVID-19 contact-confirming
application (COCOA) 2021, Available from: https://www.mhlw.go.jp/content/10900000/
000773753.pdf.
2. Department of Health and Social Care, NHS. NHS COVID-19 App 2020, Available from: https://www.nhsx.nhs.uk/covid-19-response/nhs-covid-19-app/.
3. Anti-Covid-19 Tech Team, Trends in each country regarding the introduction of contact confirmation applications, 2020, Available from: https://cio.go.jp/sites/default/files/
uploads/documents/techteam_20200508_02.pdf
4. F. Yang, Q. Yang, X. Liu, P. Wang, SIS evolutionary game model and multi-agent simulation of an
infectious disease emergency, Technol. Health Care, 23 (2015), 603–613.
5. J. B. Dunham, An agent-based spatially explicit epidemiological model in MASON, J. Artif. Soc.
Social Simul., 9 (2005).
6. H. Hirose, Pandemic simulations by MADE: A combination of multi-agent and differential equations, with a novel influenza a (H1N1) case, Information, 16 (2013), 5365–5390.
7. D. Chumachenko, V. Dobriak, M. Mazorchuk, I. Meniailov, K. Bazilevych, On agent-based approach to influenza and acute respiratory virus infection simulation, 14th International Conference
on Advanced Trends in Radioelectronics, Telecommunications, and Computer Engineering, (2018),
192–195, doi: 10.1109/TCSET.2018.8336184
8. C. Hou, J. Chen, Y. Zhou, L. Hua, J. Yuan, S. He, et al., The effectiveness of quarantine in Wuhan
city against coronavirus disease 2019 (COVID-19): a well-mixed SEIR model analysis, J. Med.
Virol, 92 (2020), 841–848, doi: 10.1002/jmv.25827
9. B. Prasse, M. A. Achterberg, L. Ma, P. V. Mieghem, Network-inference-based prediction of the
COVID-19 epidemic outbreak in the Chinese province Hubei, Appl. Network Sci., 5 (2020), doi:
10.1007/s41109-020-00274-2
10. Q. Yang, C. Yi, A. Vajdi, L. W. Cohnstaedt, H. Wu, X. Guo, et al., Short-term and long-term
mitigation evaluations for the COVID-19 epidemic in Hubei province, China, Infect. Dis. Modell.,
5 (2020), 563–574.
11. K. Chatterjee, K. Chatterjee, A. Kumar, S. Shankar, Healthcare impact of COVID-19 epidemic in
India: a stochastic mathematical model, Med. J. Armed Forces India, 76 (2020), 147–155.
12. M. A. Achterberg, B. Prasse, L. Ma, S. Trajanovski, M. Kitsak, P. V. Mieghem, Comparing theaccuracy of several network-based COVID-19 prediction algorithms, Int. J. Forecast., (2020), doi:
10.1016/j.ijforecast.2020.10.001
13. Z. Liu, P. Magal, G. Webb, Predicting the number of reported and unreported cases of COVID-19
epidemics in China, South Korea, Italy, France, Germany, and the United Kingdom, J. Theor. Biol,
509 (2020), doi: 10.1016/j.jtbi.2020.110501
Mathematical Biosciences and Engineering

Volume 18, Issue 5, 6506–6526.

6524

14. E. A. Iboi, O. Sharomi, C. N. Ngonghala, A. B. Gumel, Mathematical modeling and analysis of COVID-19 pandemic in Nigeria, Math. Biosci. Eng., 17 (2020), 7192–7220, doi:
10.3934/mbe.2020369
15. H. Wang, N. Yamamoto, Using a partial differential equation with Google Mobility data to predict
COVID-19 in Arizona, Math. Biosci. Eng., 17 (2020), 4891–4904, doi: 10.3934/mbe.2020266
16. S. Wang, Y. Liu, T. Hu, Examining the change of human mobility adherent to social restriction
policies and its effect on COVID-19 cases in Australia, Int. J. Environ. Res. Public Health, 17(2020), doi: 10.3390/ijerph17217930
17. C. Zeng, J. Zhang, Z. Li, X. Sun, B. Olatosi, S. Weissman, X. Li, Spatial-temporal relationshipbetween population mobility and COVID-19 outbreaks in South Carolina: time series forecasting
analysis, J. Med. Internet Res., 23 (2021), doi:10.2196/27045
18. M. Sulyokab, M. D. Walker, Mobility and COVID-19 mortality across Scandinavia: A modelingstudy, Travel Med. Infect. Dis., 41 (2021), doi: 10.1016/j.tmaid.2021.102039
19. I. F. F. dos Santos, G. M. A. Almeida, F. A. B. F. de Moura, Adaptive SIR model for propagationof SARS-CoV-2 in Brazil, Phys. A, 569 (2021), doi: 10.1016/j.physa.2021.125773
20. I. F. F. dos Santos, G. M. A. Almeida, F. A. B. F. de Moura, Evolution of SARS-CoV-2 inthe state of Alagoas-Brazil via an adaptive SIR model, Int. J. Mod. Phys. C, 32 (2021), doi:
10.1142/S0129183121500406
21. M. Niwa, Y. Hara, S. Sengoku, K. Kodama, Effectiveness of social measures against COVID19 outbreaks in several Japanese regions analyzed by system dynamic modeling, SSRN 3653579(2020), doi: 10.2139/ssrn.3653579
22. S. Kurahashi, Estimating effectiveness of preventing measures for 2019 novel coronavirus diseases(COVID-19), Trans. Jpn. Soc. Artif. Intell., 35 (2020), D-K28-1, doi: 10.1527/tjsai.D-K28
23. G. L. Vasconcelos, A. M. S. Macêdo, G. C. Duarte-Filho, A. A. Brum, R. Ospina, F. A. G. Almeida,
Power law behaviour in the saturation regime of fatality curves of the COVID-19 pandemic, Sci.
Rep., 11 (2021), article number: 4619, doi: 10.1038/s41598-021-84165-1
24. A. M. S. Macêdo, A. A. Brum, G. C. Duarte-Filho, F. A. G. Almeida, R. Ospina, G. L. Vasconcelos,
A comparative analysis between a SIRD compartmental model and the Richards growth model,
medRxiv, doi: 10.1101/2020.08.04.20168120
25. R. Hinch, W. Probert, A. Nurtay, M. Kendall, C. Wymant, M. Hall, et al., Effective configurationsof a digital contact tracing app: a report to NHSX, 2020. Available from: https://045.medsci.
ox.ac.uk/files/files/report-effective-app-configurations.pdf.
26. E. Hernandez-Orallo, P. Manzoni, C. T. Calafate, J. C. Cano, Evaluating how smartphone contact
tracing technology can reduce the spread of infectious diseases: The case of COVID-19, IEEE
Access, 8 (2020), 99083–99097.
27. M. E. Kretzschmar, G. Rozhnova, M. C. J. Bootsma, M. Boven, J. H. H. M. Wijgert, M. J. M.
Bonten, Impact of delays on effectiveness of contact tracing strategies for COVID-19: A modelingstudy, Lancet Public Health, 5 (2020), e452–e459.
Mathematical Biosciences and Engineering

Volume 18, Issue 5, 6506–6526.

6525

28. J. A. Kucharski, P. Klepac, A. J. K. Conlan, S. M. Kissler, M. L. Tang, H. Fry, et al., Effectiveness of
isolation, testing, contact tracing, and physical distancing on reducing transmission of SARS-CoV2 in different settings: A mathematical modeling study, Lancet Infect. Dis., 20 (2020), 1151–1160.
29. L. Ferretti, C. Wymant, M. Kendall, L. Zhao, A. Nurtay, L. Abeler-Dorner, et al., Quantifying
SARS-CoV-2 transmission suggests epidemic control with digital contact tracing, Science, 368(2020), doi: 10.1126/science.abb6936
30. Y. Omae, J. Toyotani, K. Hara, H. Takahashi, Effectiveness of COVID-19 contact-confirming application (COCOA) based on a multi-agent simulation, preprint, arXiv: 2008.13166.
31. J. Kurita, T. Sugawara, Y. Ohkusa, Effectiveness of COCOA, a COVID-19 contact notification
application, in Japan, preprint, medRxiv (2020) doi: 10.1101/2020.07.11.20151597
32. G. Kobayashi, S. Sugasawa, H. Tamae, T. Ozu, Predicting intervention effect for COVID19 in Japan: State space modeling approach, Biosci. Trends, 14 (2020), 174–181, doi:
10.5582/bst.2020.03133
33. M. S. Boudrioua, A. Boudrioua, Predicting the COVID-19 epidemic in algeria using the SIR model,
preprint, medrxiv (2020) doi: 10.1101/2020.04.25.20079467
34. B. Malavika, S. Marimuthu, M. Joy, A. Nadaraj, E. S. Asirvatham, L. Jeyaseelan, Forecasting
COVID-19 epidemic in India and high incidence states using SIR and logistic growth models,Clin. Epidemiol. Glob. Health, 9 (2020), 26–33.
35. I. Cooper, A. Mondal, C. G. Antonopoulos, A SIR Model Assumption for the spread of COVID-19
in different communities, Chaos Solitons Fractals, 139 (2020), doi: 10.1016/j.chaos.2020.110057
36. G. Nakamura, B. Grammaticos, M. Badoual, Confinement strategies in a simple SIR Model, Regul.
Chaot. Dyn., 25 (2020), 509–521.
37. M. Itakura, M. Asahi, T. Yamaguchi, Proposal of an infection model for foot-and-mouth disease
epidemic, Oper. Res. Manag. Sci. Res., 56 (2011), 728–734, Available from: http://www.orsj.
or.jp/archive2/or56-12/or56-12_728.pdf
38. H. Nishiura, H. Inaba, Prediction of infectious disease outbreak with particular emphasis on statistical issues using the transmission model, Proc. Inst. Stat. Math., 54 (2006), 461–480, Available
from: https://www.ism.ac.jp/editsec/toukei/pdf/54-2-461.pdf
39. T. Tsuchiya, A study on the spread of new coronavirus infections, Commun. Oper. Res. Soc. Jpn.,
66 (2021), 90–103, doi: 10.24545/00001758
40. Ministry of Health, Labour, and Welfare, Open data of positive result of COVID-19, 2021, Available from: https://www.mhlw.go.jp/content/pcr_positive_daily.csv.
41. Ministry of Health, Labour, and Welfare, COVID-19 Contact-Confirming Application (COCOA),
2020, Available from: https://www.mhlw.go.jp/stf/seisakunitsuite/bunya/cocoa_
00138.html.
42. L. M. Kucirka, S. A. Lauer, O. Laeyendecker, D. Boon, J. Lessler, Variation in false-negative rate ofreverse transcriptase polymerase Chain reaction-based SARS-CoV-2 tests by time since exposure,Ann. Intern. Med., 173 (2020), 262–267.
43. Python 3.7.3, 2019, Available from:python-373/
Mathematical Biosciences and Engineeringhttps://www.python.org/downloads/release/

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44. Numpy, 2021, Available from: https://numpy.org
45. E. B. Postnikov, Estimation of COVID-19 dynamics "on a back-of-envelope": Does the simplest
SIR model provide quantitative parameters and predictions?, Chaos Solitons Fractals, 135 (2020),
doi: 10.1016/j.chaos.2020.109841© 2021 the Author(s), licensee AIMS Press.
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is an open access article distributed under theterms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by/4.0)

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