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European Journal of Applied Sciences – Vol. 10, No. 5
Publication Date: October 25, 2022
DOI:10.14738/aivp.105.13087. Saito, T., & Shigemoto, K. (2022). A Logistic Curve in the SIR Model and Its Application to Deaths by COVID-19 in Japan. European
Journal of Applied Sciences, 10(5). 157-160.
Services for Science and Education – United Kingdom
A Logistic Curve in the SIR Model and Its Application to Deaths by
COVID-19 in Japan
Takesi Saito
Department of Physics, Kwansei Gakuin University
Sanda 669-1337, Japan
Kazuyasu Shigemoto
Tezukayama University, Nara 631-8501, Japan
ABSTRACT
An approximate solution of SIR equations is given, which leads to the logistic growth
curve in Biology. This solution is applied to fix the basic reproduction number α and
the removed ratio c, especially from data of accumulated number of deaths in
COVID-19. We then discuss the end of the epidemic. These results of logistic curve
results are compared with the exact results of the SIR model. Introduction
INTRODUCTION
The SIR model [1] in the theory of infection is powerful to analyze an epidemic about how it
spreads and how it ends [2–8]. The SIR model is composed of three equations for S, I and R,
where they are numbers for susceptible, infectives and removed, respectively. Three equations
can be solved completely by means of MATHEMATICA, if two parameters α and c are given,
where α is the basic reproduction number and c the removed ratio.
In Sec. 2 we would like to summarize some exact solutions of the SIR equations. These exact
solutions are applied to COVID-19 in Japan. Here, our policy is a little use of data of cases. In Sec.
3 we propose approximate solutions of SIR equations, based on the logistic growth curve in
Biology [9]. These approximate solutions have simple forms, so that they are very useful to
discuss an epidemic. The final section is devoted to concluding remarks. The logistic approach
is compared with exact solutions of the SIR model to our epidemic.
THE SIR MODEL IN THE THEORY OF INFECTION
Equations of SIR model are given by:
(1) ��(�)/�� = −��(�)�
(2) ��(�)/�� = ��(�)�(�) − ��(�)
(3) ��(�)/�� = ��(�)
where S, I and R are numbers for susceptible, infective and removed, respectively, b the
infection ratio and c the removed ratio. Our aim is to propose an approximate solution of these
equations, which leads to a logistic growth curve in Biology.
We first summarize some of exact solutions of the SIR model. From Eq. (1) and Eq. (3) we get
��/��=−��, (� = �/�), which is integrated to be
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 5, October-2022
Services for Science and Education – United Kingdom
(4) � = exp(−��)
where α stands for the basic reproduction number. In the same way, from Eq.(2) and Eq.(4)
, we have ��/��=�� − 1 = � exp(−��) − 1, which is integrated to be
(5) �(�) = 1 − � − exp (−��)
The solutions Eqs. (4) and (5) obey boundary conditions � = 1 ��� � = 0 �� � = 0. We
normalize the total number to be unity, i.e., � + � + � = 1. Since �(�) is 0 when � → ∞, we have
1 − �(∞) − exp[−��(∞)] = 0 from Eq.(5). Hence, it follows a useful formula
(6)
Some exact formulas at the peak � = � are summarized as follows:
(7) �(�) = ���/�
(8) �(�) = 1 − (1/�)−ln�/�
The first one is derived as follows: Since the peak point is given by Eq. (2), we get
�(�) = exp[−��(�)] = 1/�. This yields Eq.(7) directly. The equation (8) is derived from the
normalization condition � + � + � = 1.
Now let us consider an application of the above exact results in the SIR model to the first wave
of COVID- 19 in Japan (2020). In order to fix �, � and �, we use the data of deaths [10]. At May
2nd, 2020, the accumulated number of deaths D takes 492 and the new increased number of
deaths ∆�(�)/∆� takes the maximum value 34.
Then we find � to be May 2nd. Here, our policy is a little use of data of cases. Then we connect
�(�) with �(�) at � = � by the formula �(�) = ��(�), where r is the death rate, so that
(9)
Since ∆�(�)/∆� is fluctuating, we take 5-day average from April 30th to May 4th. The 5-day
values for ∆�(�)/∆� are (April 30th: 17), (May 1st: 26), (May 2nd: 34), (May 3rd :18), (May 4th:
11), and the 5-day average around May 2nd gives ∆�(�)/∆�(5 − ��� �������)=21.2. By using
�(�) = 492, we have (∆D/∆�)/D|T = 21.2/492 =0.043.
Next, let us fix � from the formula � = (∆�(�)/∆�)/�(�).We take the total average from Feb.16
to June 9, in order to compensate the fluctuation. Thus we get � = 0.041, which means that
almost all cases are rejected approximately after 25 days. Substituting � = 0.041 into Eq.(9), we
have an equation for �
(10) [1 − 1/α −ln (α)/α][α/ln (α)]=0.045/0.041=1.05
which gives a solution �(�����) =3.66.
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Saito, T., & Shigemoto, K. (2022). A Logistic Curve in the SIR Model and Its Application to Deaths by COVID-19 in Japan. European Journal of Applied
Sciences, 10(5). 157-160.
URL: http://dx.doi.org/10.14738/aivp.105.13087
To sum up, we have exact values of parameters: � =May 2nd, �(�����) = 3.66 and c = 0.041.
We then draw curves of �, � and � by means of MATHEMATICA in Fig. 1.
Figure 1: Graph of S, I and R for � = �. �� and � = �. ���.
A LOGISTIC CURVE FROM THE SIR MODEL
Let us consider an approximate SIR functions. The third equation Eq.(3) can be written as
(11) ��/��′=�(�!
) = 1 − � − exp(−��) , �! = ��
with t the true time. Let us expand the exponential factor in the second order of � = αR,
(12) exp(−�) ≅ 1 − � + �"/2
Then we have
(13) ��/��′ = 1 − � − (1 − �� + �"�"/2)=��(� − �)
with � = �"/2, � = 2(� − 1)/�". This equation is a type of the logistic growth curve in Biology
[9], easily solved as
(14) �(�) = #(%)
'()*+ (-.)
with � = ��(� − �)
where �� = � − 1 = � and � = �(∞) = 2�/�"=2�(�). Inserting �(�) into �(�) = (1/�)(��/
��), we have
(15) �(�) = /#(%)/"
'(1234 (.)
Where the peak value of �(�) is given by �(�) = ��(∞)/4 at � = �. Here we have a useful
formula
(16) �(∞) = 2�(�)
In the following, we consider an application of the logistic curve to COVID-19 in Japan. By using
Eq. (9), Eqs. (14) and (15), we have
(17) 56/57
6 |8 = 5#/57
# |8 = 9:(8)
#(8) = /9#(%)/;
#(%)/" = /9
"
According to the data of accumulated number of Deaths, we have (∆�/∆�)/D|8 =0.043, which
gives �(����) = � + 1 = 3.10. In this way, we have fixed parameters in the logistic curve: � =