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European Journal of Applied Sciences – Vol. 10, No. 1
Publication Date: February 25, 2022
DOI:10.14738/aivp.101.11816. Saito, T. (2022). Area Theorem of Infection Curves for Large Basic Reproduction Number. European Journal of Applied Sciences,
10(1). 482-486.
Services for Science and Education – United Kingdom
Area Theorem of Infection Curves for Large Basic Reproduction
Number
Takesi Saito
Department of Physics, Kwansei Gakuin University
Sanda 669-1337, Japan
ABSTRACT
Aremarkable theorem is established in the SIR model in Epidemiology, that is, when
the basic reproduction number α is larger than 5, the area of any infection curve is
constant given by 1/c, where c is the removed ratio. From this theorem we see that
as α gets bigger, the infection curve raises sharply, but its half width decreases
inversely. Accordingly, the wave of Omicron will end with a shorter life-time than
that of the wave of Delta, because α for Omicron is larger than Delta’s which is
regarded as 5<α<9. A rough estimation of the 5-wave area of infection in Japan tells
us that the 6-wave will reach a peak on early in Feb., then it will be controlled
around early in March.
Keywords: Variants of SARS-COV-2, SIR model, Epidemiology, Molecular Biology
INTRODUCTION
The new COVID-19 variant 'Omicron' is spreading across the world. Its speed is very high since
it is extremely infectious. Fortunately, however, the number of deaths has remained low. In a
previous article [1], we have considered, based on the SIR equation [2], [3-14] in Epidemiology,
the reason why the death toll due to such a highly infectious virus is so small. In the present
article we would like also to consider how long-time such infections are living on.
We derive an area theorem from the SIR equation, that is. “the area S of any infection curve is
constant given by 1/c, c being the removed ratio, when the basic reproduction number α is
larger than 5.” From this theorem we see that as α gets bigger, the infection curve raises
sharply, but its half width decreases inversely. Accordingly, the wave of Omicron (α>9) will
end with a shorter life-time than that of the wave of delta (α=5~9). As another example,
Omicron in South Africa appeared on 11/24/2021, raised sharply at the middle of Dec., then
turned out to decrease suddenly.
FORMULATION AND RESULTS
Let us start with the 3rd equation of SIR model [2],
dR(t)/dt=cI(t), (1)
where R(t) is the removed number and I(t) the infective number, and c is the removed ratio.
We integrate Eq.(1) over t from 0 to infinity under the boundary condition, R(0)=0, then it
follows:
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Saito, T. (2022). Area Theorem of Infection Curves for Large Basic Reproduction Number. European Journal of Applied Sciences, 10(1). 482-486.
URL: http://dx.doi.org/10.14738/aivp.101.11816
R(∞)=c∫ �(�)�� !
" = ��. (2)
Here, S is a whole area of function I(t) of t from 0 to infinity. Thanks to the well-known formula
in the SIR model,
α = − #$ ['()(!)]
)(!) , (3)
where α is the basic reproduction number, it is possible to obtain R(∞) against α. In order to
see this more easily, let us put R(∞) = 1 − ε, α ε ≪ 1. Substituting this into Eq. (3), we get
ε = �(-�-. = �(- + 0(��). (4)
In Fig. 1, numerical values of εagainst α are given. We see thatεcan be regarded as zero for α>
5, hence R(∞)=1 for α>5. The error is 0(�(/ = 0.0076). This shows from Eq.(2) that the area
S of any infection curve takes a definite value of 1/c for α>5.
Fig. 1. Numerical values of εagainst α
In Table 1, we give numerical values of I(T), infection number at the peak t=T, and for half width
against α. Here we have used the formula in the SIR model
I(T) = 1 − '
- − 01-
- . (5)
One can see from Table 1 that as α gets bigger, I(T) increases, while the half-width decreases
inversely.
Accordingly, the 6-wave(Omicron) will go to the end faster than the 5-wave(Delta), because
the Omicron is regarded to have α>9 while the Delta has 9>α>5, hence the half-width of the
former becomes narrower than that of the latter. A rough estimation of the 5-wave area in Japan
[15] tells us that the 6-wave will reach a peak on early in Feb., then it will be controlled around
early in March.
Table 1. Numerical values of I(T), infection number at the peak, and half width against α
α 2.5 5 7 9 11 13
I(T) 0.23 0.5 0.6 0.64 0.69 0.73
half
width
3.29 1.7 1.3 1.15 1.12 1.07
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 2 4 6 8 10 12 14
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
Services for Science and Education – United Kingdom
CONCLUDING REMARKS
A remarkable theorem has been established in the SIR model in Epidemiology, that is, when the
basic reproduction number α is larger than 5, the area of any infection curve is constant given
by 1/c with error 0(0.0076/�), where c is the removed ratio.
The SIR equations for S, I and R can be solved by means of EXCEL, if the basic reproduction
number α and the removed ratio c are given. In Fig.2 we draw such curves for α=5 and c=1,
where S is black, susceptible), I is red, infective and R is blue, removed. We also draw them for
α=10 and c=1 in Fig.3. These cases are typical examples for the 5 wave and the 6 wave in Japan.
According to the area theorem, both areas of red curves in Fig.2 and Fig.3 are equal with each
other, though the latter curve raises more sharply than the former. A rough estimation of the 5-
wave area in Japan tells us that the 6-wave will reach a peak on early in Feb., then it will be
controlled around early in March.
The area theorem is also useful to check whether both data of infections, say, Delta and
Omicron, are right or not from the view point of the SIR equation. When both areas of infections
are equal with each other, both data are right, but when they are not, the data are not so.
Fig. 2. Curves for S(black, susceptible), I(red, infective) and R(blue, removed) for α=5 and c=1
0
0.2
0.4
0.6
0.8
1
1.2
0
0.45
0.9
1.35
1.8
2.25
2.7
3.15
3.6
4.05
4.5
4.95
5.4
5.85
6.3
6.75
7.2
7.65
8.1
8.55
9
9.45
9.9
10....
10.8
11....
11.7
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Saito, T. (2022). Area Theorem of Infection Curves for Large Basic Reproduction Number. European Journal of Applied Sciences, 10(1). 482-486.
URL: http://dx.doi.org/10.14738/aivp.101.11816
Fig. 3. Curves for S(black, susceptible), I(red, infective) and R(blue, removed) for α=10 and c=1
ACKNOWLEDGMENTS
The author would like to express his deep gratitude to K. Shigemoto for valuable discussions.
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0
0.2
0.4
0.6
0.8
1
1.2
0
0.42
0.84
1.26
1.68
2.1
2.52
2.94
3.36
3.78
4.2
4.62
5.04
5.46
5.88
6.3
6.72
7.14
7.56
7.98
8.4
8.82
9.24
9.66
10.08
10.5
10.92
11.34
11.76
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[15] Data published from the Ministry of Health, Labor and Welfare of Japan. See for example, ”Summary of the
New Coronavirus Infection” (in Japanese), https://hazard.yahoo.co.jp/article/20200207; ”Status of the Domestic
New Coronavirus Infection”(in Japanese), https://toyokeizai.net/sp/visual/tko/covid19/.