key: cord-1043871-sxgo1sad authors: Hashiguchi, K. title: Examination of Isolation Rate in SIQR model for COVID-19 Epidemic date: 2020-09-03 journal: nan DOI: 10.1101/2020.09.01.20185611 sha: 03d1e0d967f4f0aa8fe52e32be031b27fb8f4a23 doc_id: 1043871 cord_uid: sxgo1sad Newly proposed SIQR model defines exponent {lambda} of exponential function expressing daily number of isolated persons as linear equation of isolation rate q and social distancing ratio x. In order to dynamically analyze the process of COVID-19 epidemic in seven countries by means of regression analyses of {lambda}, increasing rate of cumulative isolated persons(cases), IRCC, is proposed as practical index for the isolation rate q. IRCC is correlated with q in the form of q=C*IRCC, where C is a normalizing coefficient. At first, C is formulated in two modes, one is simple and the other complex, under the constraint conditions by definition 0[≤]x, q[≤]1, which give allowable narrow path of C between upper and lower boundaries. Then, the dynamic locus of q-x relation is analyzed for each of seven countries including Japan and the United States using formulated isolation rate q, and characteristic q-x behavior for each country is derived. At the same time, it is shown that specific path selection of C gives almost same linear loci of q-x relation derived by mathematical sequential method imitating a bipedal walk. In addition, increasing rates of cumulative PCR tests, IRCT, for six countries are discussed in relation with IRCC, and are shown that IRCT contributes to the promotion of the isolation rate via IRCC. The applicability of newly proposed SIQR model by Odagaki 1) was verified by dynamic analyses 2) using publicly available database for typical seven countries. , the exponent of exponential function for daily new cases, is defined as a linear expression of isolation rate q and social distancing ratio x, which is the basic equation in SIQR model and indefinite. This indefinite equation was solved mathematically by sequential method imitating so-called bipedal walk without any practical and physical consideration for the isolation rate 2) . As a result, the locus of q-x relation during the progress of infection for each country was able to be determined. The isolation rate must be a practical and physical meaningful parameter from a viewpoint of practical use. Therefore, the increasing rate of cumulative cases (IRCC/day) is proposed as an index with a practical meaning as the isolation rate. First in this work, the analyzing method and database used in the previous report are briefly reviewed. Then normalizing coefficient C is introduced to correlate isolation rate q and IRCC as q=C•IRCC. Further, C is formulated in two modes, one is simple and the other complex, under the constraint conditions by definition of 0≤x, q≤1, which give allowable narrow path of C during infection period between upper and lower boundaries of C. Then, the dynamic locus of q-x relation is analyzed for each of seven countries including Japan and the United States using formulated isolation rate q, and characteristic q-x behavior of each country is derived. At the same time, it is shown that specific path selection of C gives almost same loci of linear q-x relation derived mathematically in the previous report. In addition, increasing rates of cumulative PCR tests, IRCT, for six countries are discussed in relation with IRCC, and are shown that IRCT contributes to the promotion of the isolation rate via IRCC. The same analyzing method as previous analyses was taken, the important items of which are reviewed briefly in the following. ・Basic equation; The number of daily isolated persons (tested as positive and maybe isolated, hereafter new cases) ΔQ is expressed by logarithmic approximation formula (1) with constant term , and λ value was obtained as a coefficient of linear regression. The regression analyzed daily λ leads to the dynamic capturing of transition of infection process. ・Definition of exponent λ; In the SIQR model, the exponent λ of eq. (1) is defined by eq. (2). Here, βN is a value obtained by multiplying the transmission coefficient β by population N, an index of the degree of transmission through contact between uninfected and infected persons, x is a social distancing ratio, and q is an isolation rate. is the cure rate which was set equal to 0.04. The unit of λ is (1/day), q and βN have the same unit, and x is dimensionless. ・ Approximation in the model; In each of the four groups of S(Susceptible), I(Infected), All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 September 3, 2020. . https://doi.org/10.1101/2020.09.01.20185611 doi: medRxiv preprint Q(Quarantined), and R(Recovered), the degree of impact on infection may differ due to depopulation, regional differences such as age composition, and differences in measures, even though, only the average behavior in each country was considered similarly with Odagaki's method. Additionally, the epidemic process is regarded as same even though the possibility of genetically different viruses. ・Analytical procedure; Both the social distancing ratio x and the isolation rate q in eq. (2) 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. (2) becomes βN −γ, which gives the maximum value λmax (= βN−γ), and the number of new cases increases at the fastest speed. This relationship is substituted into eq. (2) and transformed to obtain eq. (3). βN associated with the maximum λ is considered as constant throughout the infection period. Using the right-hand side ∆λ of eq. (3) determined by and daily λ, it is possible to determine the social distancing ratio x and the isolation rate q. ・Terminology; The terms frequently used throughout this paper are unified as follows. Newly infected and isolated persons as new cases, maximum number of new cases as peak, and the day of minimum number of new cases after peak day as the convergence day. ・Database; The data for following 7 countries are analyzed using the daily isolated cases (new cases) and PCR tests in more than 200 countries/regions in the "Our World in Data" database 3) . The numbers of new cases and PCR tests were converted to per million people for comparison by country, and also were moving averaged with one week range. Japan with social distancing policy, Taiwan and South Korea that quickly settled due to swift response, Sweden for mass immunity, Italy, Germany, and the United States suffering from explosive mass outbreak. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 September 3, 2020. . https://doi.org/10.1101/2020.09.01.20185611 doi: medRxiv preprint In the previous analysis, the isolation rate q and the social distancing ratio x were derived as the mathematical solution of the indefinite equation 2) , and linear relation between q and x were obtained. There was no consideration of isolation rate from a practical and physical point of view in those analyses, therefore it is necessary to explore the physical basis of isolation rate based on the practical index for practical application of SIQR model. The isolation rate q defined in eq. (2) is the index representing the rate of isolation from the group of infected but unquarantined patients, which directly corresponds to the daily new cases. Considering two requirements for q, as of 0≤ q≤1 and the unit of (1/day), the increasing rate of daily new cases is not suitable because it can be positive or negative depending on the daily increase or decrease. Remaining candidate for isolation rate is the number of new cases per day, which is always 0 or more. However, this number fluctuates largely depending on the method of data tabulation even after a moving average. Therefore, the increasing rate of the cumulative cases (IRCC) calculated as a daily slope of the cumulative cases curve, is used as a candidate for isolation rate. IRCC is more suitable than the number of new cases per day for the purpose of capturing the overall trend with less fluctuation through a leveling effect by regression. IRCC is regression analyzed as the daily slope of the cumulative cases curve, and is shown in Figs. 1a and b. The start, peak and convergence days are shown as dots in the figure, and the days at maximum increasing rate are located close to the peak days. IRCC is assumed to be proportional to the isolation rate q. Under this assumption, the isolation rate increases rapidly in all countries from the start day of maximum value λmax, when q is zero, and the maximum isolation rate is recorded near the peak day. Thereafter, the isolation rate Fig. 1 Variation of increasing rate of cumulative cases (IRCC) for seven countries, days at start, peak and convergence are marked with dots. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 September 3, 2020. . https://doi.org/10.1101/2020.09.01.20185611 doi: medRxiv preprint decreases toward the convergence day. Let C be the coefficient that normalizes IRCC as the isolation rate q (0≤q≤1). Substituting eq. (4) into eq. (3), and rearranging gives eq. (5) for the social distancing ratio xi. There is a constraint of 0≤ xi ≤1, and eq. (5) is applied to this constraint to obtain eq. (6) for normalization of xi. for Japan (a) and Italy (b). is upper boundary curve for . Red and blue curves are for simple and complex mode, respectively. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 September 3, 2020. , dots marked at peak and convergence days Fig. 4a , b Daily variation of isolation rate q (a) and social distancing ratio x (b) for seven countries All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 September 3, 2020. . https://doi.org/10.1101/2020.09.01.20185611 doi: medRxiv preprint Germany at the end of Feb. may be caused by measures of each country's policy such as intensive PCR test implementation in the former and lockdown in the latter. Assessment of these effects by various measures on q and x requires more detailed examination relating to time series of each countermeasures. In Taiwan, the social distancing ratio exceeds 1 in the middle of May (Fig. 4b) , which corresponds to the decrease in Ci value crossing the lower boundary corresponding to this Substitution of eq. (7) into (4) gives ｑ ＝ • ＝∆ ∕ ｎ, which is substituted into eq. (3) or (5) to obtain eq. (8). Therefore, the q-x relationship is a straight line of slope (n-1)/βN, and the differences of the slope in Fig. 5 are explained by the transmission coefficient βN for each country. In this way, it became clear that the relationship between q and x varies greatly depending on the Fig. 5 Moving locus of (q, x) point on q-x plane for seven countries, for complex mode with normalizing coef. , dots marked at peak and convergence days All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 September 3, 2020. . https://doi.org/10.1101/2020.09.01.20185611 doi: medRxiv preprint method of determining Ci value that links the isolation rate and IRCC. There is no guideline yet for determination of Ci path, which remains as a future analysis. The only means of controlling the isolation rate is to rely on tests such as PCR. The isolation rate is examined based on the viewpoint of PCR tests in this section. Although the definition of the number of PCR tests differs from country to country, such as the number of people tested or the number of tests, they were treated equally in this analysis. to the peak day, and IRCC, which corresponds to the isolation rate, decreases after the peak day. Germany was excluded from this analysis because of an unusual trend shown in the first half of the infection stage due to a problem in data tabulation. The slope of the linear relationship between IRCC and IRCT up to the peak is 0.04 for Taiwan and South Korea, and 0.1 for Japan, while the slope is a little more than 0.2 for Sweden, the United States and Italy. This slope is an indicator that Fig. 7 Relationship between cumulative PCR tests and cumulative cases for seven countries Fig. 8 Relationship between increasing rate of cumulative cases (IRCC) and increasing rate of cumulative tests (IRCT) for seven countries, dots marked at peak days. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 September 3, 2020. high. In Taiwan, Japan and South Korea with lower positive ratio, epidemic infection was controlled at low level even though the test efficiency is low, due to the low transmission coefficient in the former two countries as described in the previous report, and also due to the intensive test implementation at the early stage of infection in South Korea (Fig. 4b) . Throughout the previous report and this analysis, the process of COVID-19 epidemic in the seven countries was dynamically analyzed by applying newly proposed SIQR model to publicly available database through the analyses of exponent λ of exponential function expressing daily new cases. As a result, the process from the start to the convergence of infection were clarified as the trends in isolation rate q and social distancing ratio x for each country. Through these analyses, the importance of dynamic parameters such as isolation rate q, increasing rate of cumulative cases(IRCC) and also increasing rate of cumulative PCR tests (IRCT), were emphasized in order to assess the dynamic transition of epidemic infection based on SIQR model. ・The increasing rate of cumulative cases (tested and isolated persons) IRCC was chosen and analyzed as a candidate for the isolation rate q with a practical and physical background, Fig. 9 Relationship between ratio of increasing rate of cumulative cases (IRCC) to increasing rate of cumulative tests (IRCT) and average positive ratio for six countries, Taiwan, Korea, Japan, Sweden, USA and Italy. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 September 3, 2020. . https://doi.org/10.1101/2020.09.01.20185611 doi: medRxiv preprint ・A normalization coefficient C was introduced in expression q=C•IRCC to normalize the isolation rate q, and the social distancing ratio x within the constraint of 0≤ q, x≤1. These constraints gave another constraint to normalization coefficient C with allowable narrow path between upper and lower boundaries. q and x were calculated for two types selected for C, a simple mode that traces the neutral line within this narrow path and a complex mode that traces the upper boundary curve in a similar fashion. ・Loci on the q-x plane in the simple mode shows a clear difference between countries. Taiwan and South Korea exhibit rapid increase in isolation rate in the early stages of infection, whereas countries in Europe exhibit the early increase in the social distancing ratio. ・In the complex mode in which the normalization coefficient C traces the upper boundary curve of the narrow path, all countries exhibit the linearly varying loci which is almost same with the result of the mathematical solution with the sequential calculation method simulating the bipedal walking in the previous report. ・The number of PCR tests was examined in relation with isolation rate, and the increasing rate of cumulative tests is found to be important as a means to promote the isolation rate. 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, and also Professor Emeritus Takashi Odagaki of Kyushu University for his SIQR model that inspired the author to complete these analyses.