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European Journal of Applied Sciences – Vol.10, No.5
Publication Date: October 25, 2022
DOI:10.14738/aivp.105.13261. Sato, T. & Shigemoto, K. (2022). A Logistic Curve in the SEIR Model and the Basic Reproduction Number of COVID-19 in Japan.
European Journal of Applied Sciences, 10(5).481-486.
Services for Science and Education – United Kingdom
A Logistic Curve in the SEIR Model and the Basic Reproduction
Number of COVID-19 in Japan
Takesi Saito
Department of Physics and Astronomy, School of Science
Kwansei Gakuin University, Sanda 669-1337, Japan
Kazuyasu Shigemoto
Tezukayama University, Nara 631-8501, Japan
ABSTRACT
The SEIR model is one of modified models of SIR, especially taken into account of
exposed people. SEIR equations can be solved numerically, but it is hard to obtain
analytically. Here, we propose some approximate solutions of SEIR equations, one
of which is related with the logistic formula in Biology. As the second aim, the SEIR
model is applied to the 7th-wave of COVID-19 in Japan. The basic reproduction
number (α) in the SEIR model is estimated for the Omicron wave. We make use of
data of the removed number �(�) rather than that of the infective number, because
the latter seems to be ambiguous. This analysis gives α=10 with γ=1 and σ=0.5.
Keywords: Variants of SARS-COV-2, SIR model, Epidemiology, Molecular biology
INTRODUCTION
The SIR model [1] in the theory of infection is powerful to analyze an epidemic about how it
spreads and how it ends [3-16]. In this article we consider the SEIR model [2] which is one of
modified models of SIR, especially taken into account of exposed people. SEIR equations can
be solved numerically, but it is hard to obtain analytically. In order to analyze infections,
analytic solutions are important. Here, therefore, we propose some approximate solutions of
SEIR equations, one of which is related with the logistic formula in Biology.
As our second aim, the SEIR model is applied to the 7th-wave of Omicron-COVID-19 in Japan.
The basic reproduction number (α) in the SEIR model is estimated for the 7th - wave. We
make use of data of the removed number �(�) rather than that of the infective number,
because the latter seems to be ambiguous. According to this analysis we have α=10 with γ
=1 and σ=0.5, where γ and σ are the removed and the exposed ratios, respectively.
A LOGISTIC FORMULA FROM THE SEIR MODEL
The sequential SEIR equations are given by
(1) ��(�)/�� = −��(�)�(�)
(2) ��(�)/�� = ��(�)�(�) − ��(�)
(3) ��(�)/�� = ��(�) − ��(�)
(4) ��(�)/�� = ��(�)
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European Journal of Applied Sciences (EJAS) Vol.10, Issue 5, October-2022
Services for Science and Education – United Kingdom
where �(�), �(�),�(�) and �(�) are numbers for susceptible, exposed, infectious and
recovered, respectively, α = �/� the basic reproduction number, � the exposed ratio and
� the removed ratio. Four numbers are normalized as
(5) S(t) + �(�) + �(�) + �(�) = 1
The SEIR equations can be solved numerically, when parameters �, � ��� � are given. In Fig.1
we give four curves for �(�), �(�),�(�) and �(�) with � = 10, � = 1 ��� � = 0.5.
Fig.1. Four curves for �(�), �(�),�(�) and �(�) with � = 10, � = 1 ��� � = 0.5.
Here we propose some approximate solutions of SEIR equations, one of which is related with
a logistic formula in Biology. From Eqs. (1) and (4) we get ��/�� = −��, which is integrated
to be
(6) �(�) = exp [−��(�)]
Hence we have from Eq. (5)
(7) �(�) + �(�) = 1 − �(�) − exp [−��(�)]
Now, from Eq. (3) �(�) can be expressed as
(8) �(�) = !
" ��(�)/�� + #
" �(�)
Substituting �(�) = �$!��(�)/�� of Eq. (4) into Eq. (8) above, we get
(9) �(�) = !
#" �%�(�)/��%+
!
" ��(�)/��
so that Eq. (7) reduces to the second order differential equation of �(�).
(10) �%�(�)/��%+(�+�)��(�)/��+��C�(�) + expD−��(�)E − 1F = 0 (Exact)
In order to find approximate solution of the exact equation (10), we consider three cases:
Case 1: |�%�(�)/��% |≪ |��(�)/�� |, �(�)
In this case the first term in Eq. (10) can be neglected so that we have
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Sato, T. & Shigemoto, K. (2022). A Logistic Curve in the SEIR Model and the Basic Reproduction Number of COVID-19 in Japan. European Journal
of Applied Sciences, 10(5).481-486.
URL: http://dx.doi.org/10.14738/aivp.105.13261
(11) ��(�)/��+ #"
!&" C�(�) + expD−��(�)E − 1F = 0 (Case 1)
When � ≫ �, we find that this equation reduces to the SIR equation:
(12) ��(�)/��+�C�(�) + expD−��(�)E − 1F = 0 (SIR)
Case 2: When ��(�) ≪ 1, we can use the approximate formula exp(−��) ≅ 1 − �� + �%�%/2,
then Eq. (10) reduces to
(13) �%�(�)/��%+(�+�) '((*)
'* − ��(� − 1)�(�) K1 − ,!
%(,$!)
�(�)L = 0 (Case 2)
Case 3: A combination of Case 1 and Case 2 gives
(14) ��(�)/�� − #"
!&" (� − 1)�(�) K1 − ,!
%(,$!)
�(�)L = 0 (Case 3 : Logistic equation)
Fig. 2: Curves of approximate solutions for Cases 1, 2 and 3, against the Exact equation. solution.
The Case 3 is for the logistic equation.
In Fig. 2, we draw curves of approximate solutions for Cases 1, 2 and 3, against the Exact
equation (10). Here we find that the contribution of the second order differential part,
�%�(�)/��%, into any case is very small. The last equation (14), Case 3, is nothing but the
logistic equation.