key: cord-0824540-q3npj5kb authors: Takeshita, J.-i.; Murakami, M.; Kamo, M.; Naito, W.; Yasutaka, T.; Imoto, S. title: Quantifying the effect of isolation and negative certification on COVID-19 transmission date: 2022-02-22 journal: nan DOI: 10.1101/2022.02.20.22270449 sha: ac1c94ff106e4b985326cafcd6434348b0a56ae3 doc_id: 824540 cord_uid: q3npj5kb Background: Isolation of close contact people and negative test certification are used to manage the spread of new coronavirus infections worldwide. These effectively prevent the spread of infection in advance, but they can lead to a decline in socio-economic activity. Thus, the present study quantified the extent to which isolation and negative test certification respectively reduce the risk of infection. Methods: A discrete-time SEIR model was used as the infectious disease model, and equations for calculating the conditional probability of non-infection status given negative test results on two different days were derived. Results: The respective non-infection probabilities with two negative PCR test results, and with one negative PCR test result and one antigen test result, were quantified. By substituting initial parameters of the SEIR model into these probabilities, the present study revealed the following: (1) isolating close contact individuals can reduce by 80% the risk of infection during the first five days, but five more days are needed to reduce the risk 10% more, and seven more days to reduce the risk 20% more; and (2) if an individual with a negative PCR test result has a negative antigen test result the next day, then his or her infection probability is between 0.6% and 0.7%. Conclusions: Five-day isolation has a proportionally greater effect on risk reduction, compared to longer isolation; and thus, if an isolation period of longer than five days is contemplated, both the risk reduction and the negative effects from such increased isolation should be considered. Regarding negative test certification, our results provide those in managerial positions, who must decide whether to accept the risk and hold mass-gathering events, with quantitative information that may be useful in their decision-making. and recovered (R), with the infected status subdivided into symptomatic (I s ) and asymptomatic (I a ), in one of which the infected individual is registered. The average duration of each state, in days, was adopted from He et al. [8] , rounded to the nearest integer: after an individual is exposed to the coronavirus, they are in Status E for three days, Status P for two days, Status I s or I a for seven days, and then remain in Status R (Figure 1 ). Note that the first and second day of Status P are denoted by P 1 and P 2 , respectively; and there is a possibility of being infected by COVID-19 two or more times, but this possibility is omitted in the present study. Let S 0 , E 0 , P 1,0 , P 2,0 , and I 0 be the initial populations of Statuses S, E, P 1 , P 2 , and I, respectively; and let day 0 be the initial day. Then the populations of day k (k = 1, 2, . . . , 14) can be easily calculated as in Table 1 . The present study deals with two types of tests (PCR and antigen tests) for determining whether an individual is infected with COVID-19. Kucirka et al. [11] and Brummer et al. [2] reported that the sensitivity depended on an individual's status. Let a 1 , a 2 , and a I , then, be sensitivities of COVID-19 PCR tests for individual status P 1 , P 2 , and I, respectively. Then a 1 = 0.33, a 2 = 0.67, and a I = 0.80. Further, let b be the specificity of a COVID-19 PCR test for individual Statuses S and E, then b = 0.999. Table 2 summarizes the sensitivity and specificity of COVID-19 PCR tests. COVID-19 antigen tests have 0.7-times the sensitivity and the same specificity as PCR tests [3] . In other words, the sensitivities of antigen tests for individual Statuses P 1 , P 2 , and I are 0.7a 1 = 0.23, 0.7a 2 = 0.47, and 0.7a I = 0.56, respectively; while the specificity for individual Statuses S and E is b = 0.999. Table 3 summarizes the sensitivity and specificity of COVID-19 antigen tests. It should be remarked that the sensitivity of antigen tests varies with the type of test, and can be less than 0.7 [14] . Further, in the case of the omicron strain, one study has reported that the sensitivity of nasal antigen tests is low within three days from onset [1] . Hereafter, let ⊕ k and ⊖ k be situations in which an individual has a positive and a negative PCR test result, respectively, on day k (i = 0, . . . , 14); and P(X|Y ) represents the probability that the individual is in Status X in Situation Y , where X is one of S, E, P 1 , P 2 , I, R, or their union, and Y is one of ⊕ k , ⊖ k , or their intersection. This clause derives the non-infected probability when an individual has a negative test result on day 0. In other words, P(S 0 ∪ R 0 |⊖ 0 ) is derived. Note that cursive script P stands for the probability, while capital letter P stands for an individual's status. Let S ′ 0 := S 0 ∪ R 0 ; that is, S ′ 0 means a non-infection status on day 0. Since the overall status on day 0, say Ω 0 , equals S 0 ∪ E 0 ∪ P 1,0 ∪ P 2,0 ∪ I 0 ∪ R 0 , the following holds from Bayes' theorem. This clause derives the non-infection probability when an individual has a negative PCR test result on both day 0 and day k k (1 ≤ k ≤ 14). In other words, 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 February 22, 2022. ; https://doi.org/10.1101/2022.02.20.22270449 doi: medRxiv preprint non-infection status at day k (0 ≤ k ≤ 14). First, from Bayes' theorem, the following holds. Next, in order to calculate (1), P(S ′ k )P(⊖ 0 ∩ ⊖ k |S ′ k ) is derived. By the definitions of the conditional probability and Ω 0 , Then, from De Morgan's laws, the following holds. 4 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. In the same way, the following hold. Finally, by substituting (3)-(4) into (1), the desired probability can be derived. From Table 1 , the following relationships between the day 0 and day 1 statuses hold. P(S 1 ∪ R 1 ) = P(S 0 ∪ R 0 ) + 1 7 P(I 0 ), P(E 1 ) = 2 3 P(E 0 ), P(P 1,1 ) = 1 3 P(E 0 ), P(P 2,1 ) = P(P 1,0 ), and P(I 1 ) = P(P 2,0 ) + 6 7 P(I 0 ). Therefore, the members on the right-hand side of (1) can be calculated as follows. By substituting (5)-(6) and k = 1 into (1), the desired probability can be derived as follows. In the same way, the non-infection probability for an individual with a negative PCR test result on both day 0 and day k for any k(= 2, . . . , 14) can be derived. Only the results are shown in the following clauses. 3.1.2 The case where k = 2 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. 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 case where k = 9 3.1.10 The case where k = 10 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 case where k = 12, 13, and 14 Since every individual is in Status R after day 12, Since the sensitivity of COVID-19 antigen tests are 0.7 times that of COVID-19 PCR tests, we can obtain the probability P(S 1 ∪ R 1 ⊖ 0 ∩⊖ A 1 ) in a manner similar to the derivation of (7), but with ⊖ A 1 indicating that an individual has a negative antigen test result on day 1. This subsection demonstrates the respective infection probabilities for the following two-types of close contact people after a certain isolation period: (i) isolated when an individual with a positive PCR test result appears in the community; (ii) isolated when a symptomatic individual appears. In order to demonstrate numerical results, the present study applied the initial-population parameters in Kamo et al. [? ] . For case (i), P(S 0 ) = 0.9808, P(E 0 ) = 0.005111, P(P 1,0 ) = 0.0025502, P(P 2,0 ) = 0.003292, and P(Is 0 ) = 0.008218; 9 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 February 22, 2022. ; https://doi.org/10.1101/2022.02.20.22270449 doi: medRxiv preprint while, for case (ii), P(S 0 ) = 0.98075, P(E 0 ) = 0.0048505, P(P 1,0 ) = 0.001258, P(P 2,0 ) = 0.0010239, and P(Is 0 ) = 0.001211. By substituting a 1 , a 2 , a I , and (10) or (11) into (7)- (8) , and subtracting them from 1, we obtain estimates of the infection probabilities on day k for an individual with a negative PCR test result on both day 0 and day k. Table 4 shows these probabilities, and Figure 2 presents the same information using a scatter plot. In the graph, for x ̸ = 0, the x-axis represents the day when an individual takes the second PCR test, and the y-axis the infection probability for an individual with a negative PCR test result on both day 0 and day x. For x = 0, the corresponding y value represents the infection probability for an individual with a negative PCR test result on day 0. The dots show the infection probabilities for Case (i), and the squares for Case (ii). The solid and dashed lines connect the dots and squares, respectively. This clause assumes that 0.25 of the total population are close contact people, and p of the total number of infected people are close contact people. Then the initial-population parameters are, for Case (i), P(S 0 ) = 1 − 0.07668p, P(E 0 ) = 0.02044p, P(P 1,0 ) = 0.01020p, P(P 2,0 ) = 0.01316p, and P(Is 0 ) = 0.003287p; (12) while, for Case (ii), P(S 0 ) = 1 − 0.07669p, P(E 0 ) = 0.0194p, P(P 1,0 ) = 0.005033p, P(P 2,0 ) = 0.004096p, andP(Is 0 ) = 0.004846p. (13) When the proportion p is varied, Table 5 shows the initial populations of close contact people who are isolated at the appearance of an individual with a positive PCR test result (Case (i-Cls)) or who is symptomatic (Case (ii-Cls)). By substituting a 1 , a 2 , a I , and (12) or (13) Using the same conditions as in the previous clause, this clause deals with the situation where an individual takes a COVID-19 antigen test on day 1. Also, for each p, the infection probabilities of people with a negative PCR test result on day 1 and a negative antigen test result on day 0 are compared to those of people with negative PCR test results on both day 0 and day 1. By substituting a 1 , a 2 , a I , and (10) or (11) into (9), and subtracting it from 1, we obtain estimates of the infection probabilities on day 1 for an individual with a negative PCR test result on day 0 and a negative antigen test result on day 1. Table 8 shows these infection probabilities and the comparison with the case where an individual has negative PCR test results on both day 0 and day 1. In the table, ∆ stands for the increment in the infection probability when the test type on day 1 is changed from a PCR to an antigen test. By substituting a 1 , a 2 , a I , and (12) or (13) into (9), and subtracting it from 1, we obtain estimates of the infection probabilities on day 1 for an isolated individual with a negative PCR test result on day 0 and a negative antigen test result on day 1, for the cases where p = 0.25, 0.30, 0.50, 0.60, and 0.90. Table 9 shows these infection probabilities and the comparison with the case where an individual has a negative PCR test result on both day 0 and day 1, for both Case (i-Cls) and Case (ii-Cls). In the table, ∆ stands for a similar increment as in Table 8 . As noted in the Introduction, in Japan, close contact people previously had to be isolated for 14 days from the last day of contact, but this period has been reduced from 14 to 7 days with the appearance of the omicron strain. 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. Figure 2 shows that the infection probability decreases sharply from day 0 to day 5, remains roughly the same from day 6 to day 8, and then decreases from day 9 to day 12, for both Case (i) and Case (ii). This tendency implies that a five-day isolation period could be chosen to reduce the isolation period with less increased risk. Reducing the isolation period from 14 to 5 days corresponds to allowing for the risk of a 0.17% and a 0.14% increase in the infection probability for Case (i) and Case (ii), respectively. In other words, the expected number of infected people in a group of 1, 000 people would be less than two for both cases. Reducing the period from 7 to 5 days corresponds to the risk of a roughly 0.01% increase in the infection probability for both cases, meaning that the expected number of infected people in a group of 10, 000 people would be roughly one. These results suggest that if we consider a small group, the increased risk would be negligible in these cases. Next, when close contact people constitute 25% of a given group, Figures 3 and 4 show the same pattern as Figure 2 . Therefore, here too, 5 days are a candidate for a reduced isolation period. From Table 6 , the reduction in Case (i-Cls) corresponds to allowing for the risk of a 0.17%, 0.21%, 0.35%, 0.43%, and 0.65% increase in the infection probability for p = 0.25, 0.30, 0.50, 0.60, and 0.90, respectively. From Table 7 , the reduction in Case (ii-Cls) corresponds to allowing for the risk of a 0.14%, 0.17%, 0.28%, 0.34%, and 0.52% increase in the infection probability for p = 0.25, 0.30, 0.50, 0.60, and 0.90, respectively. As the infection probability on day 5 is less than 20% of that on day 0, the five-day isolation period and double test would result in an 80% risk reduction. In sum, the maximum expected number of infected people in a group of 1, 000 people is less than 7 in every case. If one wants to achieve 90% and 100% risk reduction, the isolation period should be 10 and 12 days, respectively. This implies that we can eliminate 80% of the risk during the first 5 days but need 5 more days to eliminate 10% more and 7 more days to eliminate 20% more. In summary, a five-day isolation period has a significantly greater per diem effect on risk reduction than longer isolation. Thus, if an isolation period of longer than five days is contemplated, both the risk reduction and the negative effects of such prolonged isolation should be considered; and in the case of the omicron strain, an even shorter period may be sufficient to reduce the risk, because the incubation period is shorter. Consider a team, such as a Japanese professional soccer or baseball team, and the situation where • all the members of the team take PCR tests on the day before a game; • an individual with a positive test result appears, but the other individuals have negative test results; and • the other individuals take antigen tests on game day, and individuals with negative antigen test results are allowed to participate in the game. By applying the study's results for day 1, we can discuss the effect of negative test certification in the above situation. Table 8 shows that the infection probability of an individual is between 0.6% and 0.7%; that is, the infection probability for an individual with negative antigen test certification is very low. For example, when 30 people with negative antigen test certification take part in a game, the probability that at least one of them is infected is about 20% because 1 − (1 − 0.006) 3 0 = 0.17 and 1 − (1 − 0.007) 3 0 = 0.23. It is up to the manager to decide whether to accept this risk and continue with the game, but our results will help such officials, by providing them with such quantitative information, which may be useful for their decision-making. If PCR tests are used instead of antigen tests, the expected infection probability decreases by only about 0.1%; that is, there is little difference between the two tests in this situation. It should be noted that our model assumes that the people in a given group are isolated until day 1, but they are not isolated strictly from day 0 to day 1 in the above scenario. However, people's behavior can typically be well managed in the case of small groups, which means that the probability of a new infection appearing in that one day is almost negligible. Our result shows that administering an antigen test on game day can be useful for determining whether a game should be held, and which team members should be allowed to participate in the game, when an individual with a positive test result appears before the game. Further, this serves as an example of how inexpensive and rapid antigen tests can be applied effectively to implement commercial activities and avoid economic shutdowns. The present study investigated, in the context of COVID-19 transmission: • the relationship between the length of the isolation period and the expected infection probability; and • the expected infection probability with negative antigen test certification. 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. With regard to the first item, the study results suggest that a five-day isolation period is effective from the view of risk management; and that, if a longer period is contemplated, then both the effect on the risk reduction and the possible negative effects due to the longer isolation should be considered. Regarding the second item, the results provide quantitative information that may be useful for the decision-making of officials in managerial positions. Finally, it should be noted that the study results were generated by a simple discrete-time SEIR model with averaged parameters based on the coronavirus information as of 2020; thus, the estimated infection probabilities and risk magnitudes are also averages as of 2020. If one wished to be especially careful, focusing on the 95th percentile or a sensitive population for example, the parameters should be reconsidered. Further, since viruses typically mutate, the parameters must be adjusted according to the latest situation. In particular, in the case of the omicron strain, it is important to utilize the latest data, as new information of the sort required to perform the same analysis as in the present study is being reported daily. Abbreviations COVID-19: coronavirus disease of 2019; PCR: polymerase chain reaction; SEIR model: Susceptible-Exposed-Infectious-Recovered model 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. 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. 16 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. Figure 3 : For x ̸ = 0, the x-axis represents the day when an individual takes the second PCR test, and the y-axis the infection probability for an individual with a negative PCR test result on both day 0 and day x, for Case (i-Cls). For x = 0, the corresponding y value represents the infection probability for an individual with a negative PCR test result on day 0, for Case (i-Cls). The dots, squares, diamonds, upward triangles, and downward triangles show the infection probabilities for p = 0.25, 0.30, 0.50, 0.60, 0.90, respectively, for Case (i-Cls); and the solid, short-dashed, dotted, dash-dotted, and long-dashed lines connect the dots, squares, diamonds, upward triangles, and downward triangles, 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. 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. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table 5 : Initial populations of close contact people who are isolated at the appearance of an individual with a positive PCR test result (Case (i-Cls)) or who is symptomatic (Case (ii-Cls), with various percentages of infected people in the close contact group (p). (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 23 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. 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The Japanese professional soccer league Not applicable Not applicable Not applicable Among the outside support for the study, WN and TY report a relationship with Kao Corporation (funding grants); and a relationship with the Nippon Professional Baseball Organization, Yomiuri Giants, Tokyo Yakult Swallows, Japan Professional Football League, and Japan Professional Basketball League (funding grants).Other activities: the study was conducted as part of a comprehensive research project, comprising members from two private companies, Kao Corporation and NVIDIA Corporation, Japan; however, no authors in the study belong to these companies. MM, MK, WN, TY, and SI have attended the new Coronavirus Countermeasures Liaison Council, jointly established by the Nippon Professional Baseball Organization and Japan Professional Football League, as experts without remuneration. WN and TY are advisors to the Japan National Stadium and Japan Professional Football League. The other authors declare no competing interests. The findings and conclusions of this article are solely the responsibility of the authors, and do not represent the official views of any institution. JT made substantial contributions to the design of the study and the analysis, and drafted the article. MM made substantial contributions to the conception and design of the study, and substantively revised the draft. MK made substantial contributions to the design of the study and the analysis. WN, TY, and SI made substantial contributions to the conception and design of the study. All the authors have approved the submitted version and agreed both to be personally accountable for the author's own contributions, and to ensure that questions related to the accuracy or integrity of any part of the study, even ones in which the author was not personally involved, are appropriately investigated and resolved, with the resolution documented in the literature.