key: cord-0692971-hdxrq8ll
authors: Shigemoto, K.; Saito, T.
title: A Logistic Curve in the SIR Model and Its Application to Deaths by COVID-19 in Japan
date: 2020-06-26
journal: nan
DOI: 10.1101/2020.06.25.20139865
sha: ba34a2a38df705cf934839c710c1241a5f20796a
doc_id: 692971
cord_uid: hdxrq8ll

Approximate solutions of SIR equations are given, ased on a logistic growth curve in the Biology. These solutions are applied to fix the basic reproduction number $alpha$ and the removed ratio $c$, especially from data of accumulated number of deaths in Japan COVID-19. We then discuss the end of the epidemic. These logistic curve results are compared with the exact results of the SIR model. }

The SIR model [1] in the theory of infection is powerful to analyze an epidemic about how it spreads and how it ends [2] [3] [4] [5] [6] [7] [8] . The SIR model is composed of three equations for S, I and R, where they are numbers for susceptibles, 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 the 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.

Equations of the SIR model are given by

where S, I and R are numbers for susceptibles, infectives and removed, respectively, b the infection ratio and c the removed ratio. Here we propose an approximate solution of these equations, based on a logistic growth curve in the Biology. From Eq(2.1) and Eq.(2.3) we get dS/dR = −αS, (α = b/c), which is integrated to be S = exp(−αR), (2.4) where α stands for the basic reproduction number. In the same way, from Eq.(2.2) and Eq.(2.4) , we have dI/dR = αS − 1 = α exp(−αR) − 1, which is integrated to be

The solutions Eq.(2.4) and Eq.(2.5) satisfy boundary conditions S = 1 and I = 0 at R = 0. We normalize the total number to be unity, i.e., S

Hence, it follows a useful formula

Some exact formulas at the peak t = T are summarized as follows:

The first one is derived as follows: Since the peak point is given by

Now let us consider an application of the above exact results in the SIR model to COVID-19 in Japan. In order to fix T , α and c, we use the data of deaths [10] . At May 2nd, the accumulated number of deaths D takes 492 and the new increased number of deaths D(t)/ t takes the maximum value 34. Then we find T to be May 2nd. Here, our policy is a little use of data of cases. Then we connect D(t) with R(t) at t = T by the formula D(t) = rR(t), where r is the death rate, so that

Since D(t)/ t is fluctuating, we take 5-day average from April 30th to May 4th. 

with t the true time. Let us expand the exponential factor in the second order of x = αR,

Then we have

with A = α 2 /2, B = 2(α − 1)/α 2 . This equation is a type of the logistic growth curve in the Biology [9] , easily solved as

, z = ac(t − T ), (3.4) where AB = α − 1 = a and B = R(∞) = 2(α − 1)/α 2 = 2R(T ). Inserting R(t) into I(t) = (1/c)(dR(t)/dt), we have

where the peak value of I(t) is given by I(T ) = aR(∞)/4 at t = T .

Here we have useful formulas

In the following, we consider an application of the logistic curve to COVID-19 in Japan. By using Eq.(2.9), Eq.(3.4) and Eq.(3.5), we have

According to the data of accumulated number of deaths, we have ( D/ t)/D| T = 21.2/492 = 0.043, which gives α appr = a + 1 = 3.10. In this way, we have fixed parameters in the logistic curve: T =May 2nd, α appr = 3.10 and c = 0.041. The value α appr = 3.10 should be compared with α exact = 3.66. The error of our approximation for α is about 15%.

We have proposed logistic formulas in the theory of infection, which are approximate formulas driven from the SIR model. These formulas appear to be simple forms, therefore, very useful to analyze an epidemic. In logistic formulas we have determined the basic reproduction number α appr = 3.10 with the removed ratio c = 0.041, from data of the accumulated number of deaths in Japan COVID-19. One can see D(∞) = 984 from the approximate formula D(∞) = 2D(T ) = 2 × 492, that is, the final accumulated number of deaths is 984. We have also fixed exactly α and c, in the SIR model for Japan COVID-19. Once having the basic reproduction number α exact = 3.66 and the removed ratio c = 0.041, we can draw curves of S, I and R by means of MATHEMATICA in Fig. 1 .

The peak day of D(t)/ t is T =May 2nd from the data, which is the 77th day from Feb.15. Since D(t)/ t is almost left-right symmetric with respect to the peak day, the epidemic ends on 77 days after May 2nd, a middle of July. Finally if we compare α appr = 3.10 with α exact = 3.66, the error of our approximation for α is about 15%.

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

Labor and Welfare of Japan. See for example