key: cord-0926405-53y45ui4
authors: Hashiguchi, K.
title: Dynamic Analysis of Social Distancing Ratio, Isolation Rate and Transmission Coefficient in COVID-19 Epidemic for Many Countries by SIQR Model
date: 2020-08-07
journal: nan
DOI: 10.1101/2020.08.04.20167882
sha: 76acd94d48e3d25060bdbc3a3f900643fe0903e4
doc_id: 926405
cord_uid: 53y45ui4

Recently SIQR model was proposed by Odagaki as the modification of conventional SIR model by adding the term for isolation of infected persons, Q(Quarantined). The exponent {lambda} of the exponential function expressing the number of newly tested positive persons was defined as an linear equation explicitly with three important parameters, transmission coefficient, social distancing ratio x and isolation rate q. In this study, applying this model to the number of positive persons in publicly available database, daily {lambda} values are regression analyzed, and social distancing ratio and isolation rate are derived. Analyses for 7 countries including Japan, Taiwan, South Korea, and western countries are performed and determine the dynamic locus of q-x relation on the q-x plane during epidemic propagation. Finally, the remaining parameter, the transmission coefficient is shown to closely relate to the maximum {lambda}, {lambda}max, and {lambda}max (transmission coefficient) is characterized as a specific value for each country. Then, the magnitude of {lambda}max is combined with the value of {lambda}min to influence the total number of new cases until the convergence stage.

quarantined, Q and the recovered, R. With additional group of uninfected, S (Susceptible), and community infected, I (Infected), the in and out of the number of people between each group were expressed in the differential equations with the overall condition of preservation of population.

Three important parameters, transmission coefficient, isolation rate and social distancing ratio, which are essential for infection control, were explicitly introduced to bring more concreteness to the expression of actual infection status and it is expected to contribute to evaluating the infection control.

In order to evaluate the validity of the model and to clarify the status of infection in several countries, the number of daily isolated persons(new cases) in a publicly available database are analyzed based on the SIQR model, and the social distancing ratio and isolation rate are derived.

Then the infection process in each country from the initial stage to the convergence stage are characterized by using these two parameters. Finally, the relationship between the number of new cases until the convergence of infection and the transmission coefficient, which is the third important parameter as specific value for each country in this model, are discussed.

Basic Equations in SIQR model and Matching Method in data-fitting In the SIQR model by Odagaki, the daily variation in the number of persons in each group of S, I, Q, and R is represented by four differential equations, and λ, corresponding to the exponent after integration, was defined in Equation (1), with combining social distancing ratio x, isolation rate q and cure rate γ.

βN is the transmission coefficient β multiplied by population N, which is an index of the degree of transmission through contact between uninfected and infected individuals. Since x and q are both time-dependent parameters, exact integration of the above mentioned differential equations are difficult. Therefore, Odagaki divided the infection period into the four stages, first stage of the initial rise of the infection, the second stage of the rapid increase period, the third stage of the plateau with the maximum value, and the fourth stage of the convergence. Assuming that the number of daily . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

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The copyright holder for this this version posted August 7, 2020. . https://doi.org/10.1101/2020.08.04.20167882 doi: medRxiv preprint quarantined persons (tested as positive and maybe isolated, hereafter new cases) ΔQ for each stage is represented by an exponential function with a different exponent, ΔQ and the number of infected persons in the community I are expressed by equations (2) and (3), respectively. Then Odagaki analyzed the social distancing ratio and the isolation rate separately for 4 stages.

In this analysis, the same approximation method was adopted except Odagaki's classification of the 4 stages which was replaced with a day-to-day basis.. The number of new cases ( ) on a daily basis was expressed by an logarithmic approximate expression (4) with time unit of a day and constant term of , and the exponent thereof was expressed by eq. (1) to proceed with the analysis.

Daily were regression analyzed using eq. (4) with day range [ − Δ , + Δ ], where is the current day, and Δ corresponds to day range. This procedure was carried out not in the 4 stages of Odagaki but in each day during the period, which leads to the dynamic capturing of variation of infection process.

In each of four groups of S, I, Q, and R, the degree of influence on infection spread may differ due to depopulation, regional differences such as age composition, and differences in countermeasures, even though, only the average behavior in each country was considered similarly with Odagaki's method.

Both the social distancing ratio x and the isolation rate q in eq. (1) are parameters that take values of 0 to 1 based on their definitions. When both are 0, that is, without any measures to suppress infection, the right side of eq. (1) becomes βN−γ, which is the maximum value λmax (= βN−γ), and the number of new cases increases at the fastest speed. This relationship is substituted into eq. (1) and transformed to obtain equation (5).

. CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

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The copyright holder for this this version posted August 7, 2020. . https://doi.org/10.1101/2020.08.04.20167882 doi: medRxiv preprint λmax is considered constant throughout the period of infection. γ is the cure rate of communityinfected individuals and is set at 0.04, following Makino 3) , and is not significantly influential during the following analysis. The units of λ, q and βN are same as (1/day), and x is dimensionless.

From equations (1) and (4), the daily new cases ΔQ is expressed as an exponential function that takes varying daily λ as an exponent. With obtaining daily λ, the right side ∆λ of eq. (5) is determined, using λmax, the maximum value of λ during the entire period. By sharing the value between x and q on the left side, it is possible to determine the social distancing ratio x and the isolation rate q. Figure   1 shows The terms frequently used throughout this paper are unified as follows.

New infected and isolated persons as new cases, 1st and 2nd period by Odagaki's paper by increasing stage, maximum number of new cases as peak, 4th period as converging stage, and the day of minimum number of new cases in the 4th period as the convergence day.

Derivation of the social distancing ratio x and isolation rate q from the daily λ value (Solution of the indefinite equation)

The indefinite equation with unknown parameters q and x with substituting daily λ value to eq. (5) was solved in the following manner, under the rules that the initial conditions are given and that neither q nor x leads to an abrupt and large discontinuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted August 7, 2020. . https://doi.org/10.1101/2020.08.04.20167882 doi: medRxiv preprint At the start of infection, x was set to zero, and sequential calculations were performed using the following procedure.

On a first day ; x1 = 0, and q1=λmax−λ1−x1 βN is calculated.

On the second day ; set q2 = q1 and calculate x2 = (λmax−λ2−q2)/βN. On the third day ; x3=x2 and q3=λmax−λ3−x3 βN is calculated.

In this way, the sequential calculations were carried out by fixing one of q and x and moving the other, imitating a bipedal walk. Thus, on the q-x plane, the calculation starts on a q-axis (x=0) and follow the (q, x) coordinates in a stepwise manner afterwards. In the rare cases where x becomes negative due to the same value of q adopted from the previous day, x is set to 0 and q is calculated again.

For the analysis, the number of new cases in more than 200 countries in the "Our World in Data" database 4) (data coverage from 1 st of Jan. 2020 to 11 th of July) were used. There is a big difference in the daily data due to differences in the data aggregation method depending on the country, so a moving average was taken on a weekly basis. In addition, the number of new cases was converted to per million population for comparison by country.

In the calculation of daily λ value, the range of days for regression, 2Δ + 1 was varied from 7 days up to 31 days with varying Δ as of 3, 9, 15. The period ofλ analysis was from the day when the cumulative number of new cases exceeds 10 persons, to the day when the number of new cases reaches the minimum after peak.

The following seven countries with different countermeasures were chosen for the analysis.

Japan with the social distancing policy, Taiwan and South Korea that quickly settled due to swift response, Sweden for mass infection, Italy, Germany and the United States suffering from explosive mass outbreak. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted August 7, 2020. Also shown in Table 1 is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted August 7, 2020. . Table 1 shows that the λmax value varies greatly depending on the regression range ± Δ , and as the range is expanded, the value becomes smaller due to the leveling-off effect. Hereafter, results for the range of λ calculation at 19 days with moderate leveling effect is discussed and the effect of the range will be discussed later.

Characteristic variation of social distancing ratio x and isolation rate q for each country The variation in both of q and x for Japan are described in Figure 4 as an example. For the sake of explanation, daily λ value curve in Fig. 1 and the corresponding equi-λ lines calculated by eq. (5) are shown in Fig. 4. 1) Starting from q=0.1 at which the horizontal axis (x=0) intersects with the equi-λ line (λ=0.1, Fig. 4b ) corresponding to λ=0.1 at the start of infection (Fig. 4a) . is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted August 7, 2020. . https://doi.org/10.1101/2020.08.04.20167882 doi: medRxiv preprint 2) Then, the daily λ suddenly decreases and becomes 0 (Fig. 4a) , both q and x increase slightly to the position intersecting the equi-λ line (λ=0).

3) Thereafter, when the daily λ increases and reaches λmax (Fig. 4a) , the q-x locus reaches the origin (∆λ = 0). In this way, the isolation rate q and the social distancing ratio x follow a complicated locus on the q-x plane corresponding to the variation in the daily λ value after the start of infection. Figure 5 shows the q-x loci of 7 countries. In the figure, the day points for the peak are indicated by dots.

Keeping in mind the common locus starting from on the q-axis as revealed in Fig. 4, Fig. 5 clearly shows the differences in q-x relationships between countries. In contrast to Japan and Taiwan, where the variation in isolation rates is small, the magnitude of the variation in isolation rates in Germany, Korea and the United States is noteworthy. In the cases of Sweden and Italy, λmax was reached on the first day of infection (Fig. 2b) , is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted August 7, 2020. . https://doi.org/10.1101/2020.08.04.20167882 doi: medRxiv preprint so their locus started from origin on the q-x plane. There is also a large difference in the straight line up to the right during the spread of infection to the convergence period. In Japan and Taiwan, the isolation rate remains low and the social distancing ratio increases, meanwhile in Germany and Korea the isolation rate increases significantly and the social distancing ratio also increases. The Since the calculated isolation rate and social distancing ratio are strongly influenced by the degree of variation in daily λ values, care should be taken in setting the regression range of daily λ values.

Taking Japan as an example, Figure 6 shows the relationship between daily λ value and the isolation rate and social distancing ratio when the regression range of daily λ values is changed from 7 to 19 and 31 days. As the regression range expands, the fluctuation in daily λ value becomes smaller, and the position of λmax shifts almost to the first day of infection (Fig. 6a) . The proximity of λmax day to the first day of infection means that the q-x locus curve starts near the origin, and this behavior is similar to those of Sweden and Italy, as described above. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted August 7, 2020. . https://doi.org/10.1101/2020.08.04.20167882 doi: medRxiv preprint Importance of the specific value βN for each country

The above analysis reveals that the magnitude of λmax affects the varying degree in q and x through the magnitude of the variation in the left-hand side of eq. (5). This suggests that the effect of the transmission coefficient βN (=λmax + γ), which is specific value for each country, on the variable width of q and x, and thus on the degree of influence of the countermeasure differs. In Japan and Taiwan, where λmax (βN) is small, the spread of infection is slow, while in other countries where λmax (βN) is large, the spread of infection is rapid.

In order to examine the effect of βN on the spread of infection, the relationship between λmax and total cases for the other many countries was examined. The latter were determined at the convergence day, which is more than 1 week before the last day of the database. The relationship of both is shown in Figure 7a . The calculation of both parameters were performed for about 100 countries with more than 400 new cases or more than 1 million populations among all 211 countries/regions in the database. Although there is a large variation, the lower λmax, the lower the upper limit of the total cases. This means that even at the same λmax the total cases fluctuate widely, is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted August 7, 2020. . https://doi.org/10.1101/2020.08.04.20167882 doi: medRxiv preprint and the range of fluctuation increases with λmax. In fact, data for the seven countries were denoted in red in Fig. 7 , the total cases in five countries with λmax > 0. 3 

Korea to the smaller in Germany, then Italy, Sweden, and the U.S., in the order in which the total cases increase according to the so-called national measures. The countries with lower total cases at fixed λmax, are considered to have taken effective measures within the range of λmax, that is, q and x, which are allowed by the specific value βN (λmax), and as a result, the total cases is considered to have been controlled at a low level.

In Fig. 7b , which is organized by λmin instead of λmax, the lower λmin (the larger the negative value), the lower the upper limit of the total cases. In other words, lower λmax and lower λmin are effective in reducing infection, which corresponds to the self-evident fact that the increasing rate of infection must be lower (low λmax) and the decreasing rate must be higher (low λmin).

Through the above examination on the relationship between the transmission coefficient βN and the total cases, the importance of the former was confirmed. As described in previous section, the transmission coefficient βN was closely related with λmax as a specific value for each country, which was assumed to be constant throughout the infection period. Although detailed examination of this coefficients is beyond the scope of this study, it can actually be interpreted as specific values that represent the unique characteristics of each country and society, including such as the lifestyle of interpersonal activities, and further research is needed.

For the purpose of verifying the validity of the newly proposed SIQR model, in which the exponent λ of the exponential function expressing the number of newly tested positive persons was defined as an linear equation explicitly with three important parameters, transmission coefficient, social distancing ratio x and isolation rate q, the trend of the number of isolated persons in publicly available database for COVID-19 epidemic were analyzed. These analyses were performed for 7 countries including Japan, Taiwan, South Korea and western countries.

Results are summarized as follows.

・Daily λ values were regression analyzed, and were shown to vary from positive value during the increasing stage of infection, then decrease through 0 (peak stage) to negative value (convergence . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity.

is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted August 7, 2020. . https://doi.org/10.1101/2020.08.04.20167882 doi: medRxiv preprint stage). During this epidemic period, λ reaches maximum value λmax which is closely related to the transmission coefficient βN.

・The isolation rate, q and the social distancing ratio, x were calculated by sequential calculations using the equation relating daily λ with the isolation rate and the social distancing ratio, and the locus of q-x relation with the progress of infection were determined.

・Japan and Taiwan exhibit higher social distancing ratio, South Korea and Germany show higher isolation rates, and Italy, Sweden, and the United States show intermediate characteristics.

・Transmission coefficient βN is lower in Japan and Taiwan, higher in 5 other countries. The cumulative number of isolated persons after convergence tends to decrease as the transmission coefficient becomes lower, which were confirmed in examination for more than 100 countries.

Data COVID-19 dataset

The author would like to thank professor emeritus Takashi Odagaki at Kyushu University for several comments and especially his SIQR model that inspired the author to start this analysis, and also professor emeritus Ikuo Yoshihara at University of Miyazaki for detailed and fruitful comments and discussions in preparing this paper.