key: cord-0299689-i6y8nfay
authors: Hashiguchi, K.
title: Modeling and analysis of COVID-19 infected persons during repeated waves in Japan
date: 2021-10-14
journal: nan
DOI: 10.1101/2021.10.11.21264869
sha: bb80afd1db7ebf5aef6d08c93948f002933148f4
doc_id: 299689
cord_uid: i6y8nfay

A model for estimating the number of COVID-19 infected persons (infecteds) is proposed based on the exponential function law of the SIR model. This model is composed of several equations expressing the number of infecteds, considering the onset after an incubation period, infectivity, wavy infection persistence with repeated infection spread and convergence with insufficient quarantine. This model is applied to the infection in Japan, which is currently suffering from the 5th wave, and the initial number of infecteds and various related parameters are calculated by curve fitting of the cumulative number of infecteds to the total cases in the database. As a minimum boundary of the number of infecteds for the infection continuation up to the 5th wave, the initial number of infecteds at the outbreak of infection is more than an order of magnitude higher than the actual initial cases. A convergence ratio (cumulative number of infecteds / total cases) at the end of the first wave is obtained as 1.5. The quarantine rate and social distancing ratio based on the SIQR model are evaluated, and the social distancing ratio increases sharply just after the government's declaration of emergency. The quarantine rate closely equals the positive rate by PCR tests, meaning that the number of infecteds, which had been unknown, is on the order of almost the same as the number of tests.

A new type of coronavirus, COVID-19, which took off in Wuhan, China, at the end of 2019 and quickly spread all over the world, has not been settled even after one year and a half, with repeated waves of spread and convergence of the infection.

The SIR model 1) proposed about 100 years ago is known as a mathematical model for describing the epidemic process of infection. It is a simple model that expresses the increase/decrease in the number of persons between groups S (Susceptible), I (Infected), and R (Recovered) with differential equations under the population conservation law. This model reproduced the plague epidemic of the early 20th century and is considered to help understand the current COVID-19 epidemic 2).

Furthermore, an improved SIQR model 3) was also proposed, in which the recovered group, R in the . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 14, 2021. ; https://doi.org/10.1101/2021.10.11.21264869 doi: medRxiv preprint SIR model was divided into Q (quarantined), and R (recovered). Odagaki 4) proposed SIQR model with two important parameters, the quarantine rate and the social distancing ratio. These explicitly introduced parameters are related with the exponent λ of an exponential function for the daily variation in the number of new infection-positive persons and contribute to the evaluation of infection control measures.

In the previous papers 5, 6) intended to analyze these parameters, mathematical analyses between these relationships were conducted on Taiwan and South Korea with measures of mass PCR tests in the early stages of infection, and Western countries with measures of lockdown, including Japan, and clarified the difference in the quarantine rate-social distancing ratio relationship caused by these countermeasures.

In this paper, a model estimating the number of infected persons, which has been unknown in the previous papers, is proposed with consideration of an incubation period after infection, subsequent infection rate, and the convergence ratio at the stage of wavy infection convergence. This model also presupposes the exponential spread of infection common to the SIR and SIQR models under the law of conservation of population. This model is applied to a database of infection status, and the daily variation of the infected persons is evaluated by fitting the calculated cumulative number of infected persons to the actual value of the cumulative number of newly detected infection-positive persons.

Using the results, the variation of infected persons, quarantine rate, and social distancing ratio are analyzed during five waves in Japan.

Basic equations in this estimation model are described separately for the derivation of the cumulative number of infected persons, and the derivation of the quarantine rate / social distancing ratio. Since the latter parameters have been described in detail in the previous papers, only the resulting equations are described.

Assuming the cumulative number of infected persons (hereinafter referred to as cumulative infecteds) on day increases exponentially, is expressed by the exponential function (1) using an exponent .

The number of newly infected persons ∆ is expressed by Eq. (2) as the increase in cumulative infecteds.

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted October 14, 2021. ; https://doi.org/10.1101/2021.10.11.21264869 doi: medRxiv preprint After infection, a latent disease declares itself after an incubation period. Assuming an incubation period follows the gamma distribution ｆ( ) with maximum onset on the 5th day after infection and ending on the 16th day 7) , which gives the shape parameter a = 5 and the scale parameter b = 1.25 in the gamma distribution formula. This gamma distribution curve is discretized, normalized so that the sum of ｆ( ) for 16 days is 1, and the ratio of onset on day is expressed by a function in Eq. (3). 

The cumulative infecteds is given by Eq. (6) .

In these basic equations, setting the initial infecteds at the outbreak of infection as , which is equal to the cumulative infecteds on the infection outbreak, and this initial value is substituted into Eq. (2). By repeating the sequential calculation from equations (4) to (6) . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 14, 2021. ; https://doi.org/10.1101/2021.10.11.21264869 doi: medRxiv preprint

The latter is defined as a ratio of the calculated cumulative infecteds to the total cases at the end of each wave. When = 1, excess infecteds becomes 0, which means all the cumulative infecteds are detected and quarantined as total cases during the period, resulting in termination of infection. Otherwise, excess infecteds with larger than 1 bring about the next spread of infection. By definition, both parameters can be varied within a range of = 0 ∽ 1、 ≥ 1.

The new cases is expressed by Eq. (7) using an exponent λ.

As already mentioned in the previous papers, the quarantine rate and the social distancing ratio (q and x, respectively in the previous papers) are shown by equations (8) and (9), respectively. The relationship of these parameters and in Eq. (7) is expressed by the following equations. 

This model was applied for the analysis of the infection status in Japan. Daily variation of new cases, and total cases, from the "Our World in Data" (January 1, 2020-August 4, 2021) database 8) were converted to per million population, as well as the number of PCR tests described later. As a preliminary preparation, the daily and were calculated by the logarithmic regression of Eq.

(1) and (7) using those converted data. In the regression calculation, the exponent was calculated as the moving-average logarithmic slope for 13 days, and of Eq. (3) was also calculated. The variation of new cases, total cases, and the corresponding exponents λ and α are shown in Fig. 1a , . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 14, 2021. ; https://doi.org/10.1101/2021.10.11.21264869 doi: medRxiv preprint calculation, value was varied in 0.1 step within the range of 1 to 2.3 and the calculation was executed to obtain for each value. In the 1st wave, all infected persons are assumed to be infectivity, then was set to be 1.

・ After the 1st wave

The same calculating procedure is adopted for subsequent waves setting the last value of previous wave as the initial value for the next wave. 3 Relationship between 1st wave convergence ratio, and best fit index, dv over the entire period (a), and initial value of infecteds, (b). Initial value varies linearly with convergence ratio.

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 14, 2021. ; https://doi.org/10.1101/2021.10.11.21264869 doi: medRxiv preprint

The relationship between (%) in Eq. (11) and value in the first wave is shown in Fig. 3a .

values obtained are all little over 0.1%, showing very good fittability. As shown in Fig. 3b , the initial value of infecteds increases linearly with the increase in the convergence ratio of the first wave. In the range of <1.5, becomes lower and decreases sharply as shown in Fig.2 before reaching the 5th wave. fittability with the small deviation from the total cases in the latter wave. Table 1 shows the and values for each wave including conditions other than those in Fig.4 . An increase in results in a decrease in , and subsequent convergence ratios remains close to 1.

For the case of value 1.5, the quarantine rate and the social distancing ratio are calculated using the equations (8) and (9) and shown in Fig. 5a , b. By definition, peaks of quarantine rate correspond to the peak of new cases in each wave and reach up to about 0.25 in the third wave, staying low in other waves. Higher makes increase in the later wave due to decrease in in later wave.

(b) for each value of 1.5, 1.9, and 2.3. Cumulative infecteds best fit to total cases. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted October 14, 2021. ; https://doi.org/10.1101/2021.10.11.21264869 doi: medRxiv preprint

The social distancing ratio exhibits the opposite tendency to the quarantine rate and tends to be maximum near the minimum quarantine rate point. This is because the sum of the quarantine rate and the social distancing ratio equals ∆ in Eq.

(9), and the variation in ∆ is small in a short period.

In The calculation for the best fit of the cumulative infecteds were executed for a lot of combinations of parameters, , for the 1st wave, and for each wave. Apart from this method, there is also a deductive way, using the linearity between and or as shown in Fig. 3b . Relational expressions between the initial value and are formulated for each for each wave, and the optimum solution is obtained while matching the waves using these relational expressions.

However, it is necessary to pay attention to the case where the incubation period expressed by Eq.

(3) spans two waves. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted October 14, 2021. ; https://doi.org/10.1101/2021.10.11.21264869 doi: medRxiv preprint introduced to guarantee the continuity and compatibility between neighboring waves. This model was applied to the infection in Japan, which is currently up to the 5th wave, and the initial infecteds and various parameters were calculated by curve fitting of the cumulative infecteds to the total cases in the database. ・ As a best-fit solution of the cumulative infecteds up to the 5th wave to the total cases, the initial infecteds of 0.6 / million population at the start of infection is more than an order of magnitude higher than the actual initial cases of 0.03 / million population. A convergence ratio 1.5 (cumulative infecteds / total cases) at the end of the first wave was obtained.

・ This result gives a minimum value of infecteds for the infection continuation up to the 5th wave.

Otherwise, the initial infecteds less than this lower limit may result in the termination of infection in the second or the third wave.

・ Although a convergence ratio of more than 1.5 in the first wave is allowed, the number of excess infecteds, in that case, is compensated by the decrease in the infection ratio (infectivity ratio of infecteds after the incubation period) in the second wave.

・ The isolation rate and social distancing ratio based on the SIQR model proposed by Odagaki were evaluated, and the social distancing ratio increases sharply just after the government's declaration of emergency. The quarantine rate closely equals the positive rate by PCR tests, meaning that the infecteds, which had been unknown, is the same level as the number of tests.

Suuri-Toukei54

Data COVID-19 dataset

The author would like to thank Professor Emeritus Ikuo Yoshihara of the University of Miyazaki for his detailed suggestions and discussions.