key: cord-1053399-u85rfwki
authors: Saito, T.; Kokado, A.
title: Logistic Formula in Biology and Its Application to COVID-19 in Japan
date: 2021-03-29
journal: nan
DOI: 10.1101/2021.03.24.21254279
sha: 32bcdf73a0deaa0afd11e93270c253bd06338e11
doc_id: 1053399
cord_uid: u85rfwki

A logistic formula in biology is applied, as the first principle, to analyze the second and third waves of COVID=19 in Japan.

In biology the logistic equation for a population N (t) is given by

, (2.1) where A, B and c are some parameters. The solution is easily obtained as

, z ≡ ABc(t − T ) . (2.2) Here T gives a peak of I(t), which is given by

, (2.3) that is, I(t) = AB 2 /4. The equation (2.1) can be regarded as the third equation of the SIR model ,

if we identify N (t) with R(t), where R(t) and I(t) are the removed number and the infectious number, respectively, and c the removed ratio. In previous works [10] [11] [12] [13] , our logistic formulas (2.2) and (2.3) have been driven approximately from the SIR model. In the present paper, however, we regard our logistic formulation as more fundamental rather than the SIR theory. Let us rewrite Eq. (2.2) in notations A = a/d, B = d and N (t) = R(t) as follows:

where d is the final total removed number, e. g., d = R(∞). Eq. (2.5) can be expressed as

Accordingly, for different times t n , t n+1 and t n+2 , (n = 1, 2, . . . ), we have 

(2.10)

Our logistic formula is applied to the second wave of COVID-19 in Japan. This provides a revise of previous work [12] The R(t) is the accumulated number of removed in the second wave in Japan, which is an average for 7 days in a middle at each t with standard deviations, where t is the date starting from June 20, 2020. The virus is now called the Tokyo type. We have subtracted the accumulated number 20507 on July 19 of removed in the first wave from that in the first and second waves. Table 1 : Date t and the removed number R(t) in the second wave in Japan [14] t D(t) t 1 =Aug. 7 n 1 = 11377 ± 2205 t 2 =Aug. 14 n 2 = 18949 ± 2652 t 3 =Aug. 21 n 3 = 27455 ± 2281 Substituting data in the Table 1 into Eq. (2.10) with n = 1, we have the equation for d

From Eq. (2.7) with n = 1 we get

Substituting the result ac = 0.1092 into Eq. (2.6), we have

Error estimations for d and T can be seen from Appendix. By using relative errors.,

The value of d = 45071 on Aug.17 is plotted in Fig. 1 . Values of d for the other n are also plotted in Fig. 1 .

In Fig. 2 we draw a red curve of Eq. (2.5) for R(t) calculated with the average value d = 44465 and ac = 0.1106 from Aug.13 to Aug.20, while the blue curve shows data for R(t). Both lines coincide well in a region before Aug.27. The value of d should be constant as seen between Aug. 13 and Aug.20 within errors 1200∼5000, but not before Aug.12. The deviation comes from the deviation between the red line and blue line in Fig. 2 , where the red line is the calculated one of R(t) and the blue line is its data. From this reason we abandon data outside of Aug.13∼20.

To sum up, the second wave started from July.20, 2020, and peaked at Aug.17 with its total removed number 22536 ± 568. These calculated values should be compared with actual data that the peak date is around Aug.17 with its total removed number 22460.

3 remix, or adapt this material for any purpose without crediting the original authors. preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally share, reuse, 

Our logistic formula is applied to the third wave of COVID-19 in Japan. This provides a revise of previous work [13] Table 2: Date and the removed number in the third wave in Japan [14] t D(t) t 1 =Jan. 29 n 1 = 245292 ± 11265 t 2 =Feb. 12 n 2 = 302185 ± 5537 t 3 =Feb. 26 n 3 = 330669 ± 3186

The R(t) is the accumulated number of removed in the third wave in Japan, which is an average for 7 days in a middle at each t with standard deviations, where t is the date starting from Oct. 11. We have subtracted the accumulated number 82810 of removed in the first and second waves from that in the first, second and third waves.

Substituting data in the Table 2 into Eq. (2.10) with n = 1, we have the equation for d Error estimations for d and T can be seen from Appendix. By using relative errors., The value of d = 350329 on Feb. 12 is plotted in Fig. 4 . Values of d for the other n are also plotted in Fig. 4 . In Fig. 5 we draw a red curve of Eq. (2.6) for R(t) calculated with the average value d = 348008 and ac = 0.0723 from Jan. 28 to March 15, while the blue curve shows data for R(t). Both lines coincide well in a region after Jan. 19. The value of d should be constant as seen after Jan. 24, but not before Jan. 24. The deviation comes from the deviation between the red line and blue line before Jan. 24 in Fig. 5 , where the red line is the calculated one of R(t) and the blue line is its data. From this reason we abandon data before Jan.24.

To sum up, the third wave started from Oct.11, 2020, and peaked at Jan. 17 with its total removed number 175165 ± 1657. These calculated values should be compared with actual data that the peak date is around Jan. 17 with its total removed number 177501. 

The logistic formula in biology is applied, as the first principle, to analyze the removed number by the second and third waves of COVID-19 in Japan.

The second wave started from July.20, 2020, and peaked at Aug.17 with its total removed number 22536 ± 568. These calculated values should be compared with actual data that the peak date is around Aug.17 with its total removed number 22460.

The third wave started from Oct.11, 2020, and peaked at Jan. 17 with its total removed 6 remix, or adapt this material for any purpose without crediting the original authors.

preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally share, reuse,

The copyright holder has placed this this version posted March 29, 2021. ; https://doi.org/10.1101/2021.03.24.21254279 doi: medRxiv preprint number 175165 ± 1657. These calculated values should be compared with actual data that the peak date is around Jan. 17 with its total removed number 177501. Results of the third wave have been obtained by using new data after the peak, Jan. 17. So, these are not a kind of prediction. However, we have succeeded to reproduce the peak data fairly well.

Secondly, the equation for the peak day T , ac(T − t 3 ) = ln F 3 , yields a formula

from which one can estimate δT .

A Contribution to the Mathematical Theory of Epidemics

Epidemic models and the dynamics of infectious diseases

The Mathematics of Infectious Diseases

Mathematical modelling of infectious diseases

Rich dynamics of an SIR epidemic model

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes

Mathematical Modeling of Epidemic Diseases; A Case Study of the COVID-19 Coronavirus

Notice sur la loi que la population poursuit dans son accroissement

A Logistic Curve in the SIR Model and Its Application to Deaths by COVID-19 in Japan

Modified SIR Model Yielding a Logistic Solution

An Application of Logistic Formula to Deaths by COVID-19 in Japan

A Logistic Formula and Its Application to Deaths by the Third Wave of COVID-19 in Japan

Labor and Welfare of Japan. See for example

We would like to thank K. Shigemoto for many valuable discussions and big supports.

From the formula F 1 F 3 = F 2 2 the relative error of removed number d is driven as