key: cord-0771115-y0dezmi6
authors: Kaiser, A. K.; Kretschmer, D.; Leszczensky, L.
title: Social network-based strategies for classroom size reduction can help limit outbreaks of SARS-CoV-2 in high schools. A simulation study in classrooms of four European countries.
date: 2020-12-02
journal: nan
DOI: 10.1101/2020.11.30.20241166
sha: a79d5a10853214f84df21497ae018c9ec5837b46
doc_id: 771115
cord_uid: y0dezmi6

Background Until pharmaceutical measures are widely available to slow the spread of SARS-CoV-2, social distancing strategies are key to avert overwhelmed health systems. Since schools host large numbers of students in enclosed spaces, they are feared to produce infection clusters. With school closures coming at high social and economic costs, social distancing measures within schools are needed to make them as safe as possible. One widely discussed distancing measure in the school context is to use cohorting strategies, i.e., to split larger clusters such as classrooms into smaller groups that are instructed separately. In addition to facilitating social distancing within these cohorts, cohorting strategies also aim to prevent transmission across cohorts. However, little is known about which cohorting strategies are particularly effective to prevent disease transmission between cohorts in schools. Methods Using nationally representative data on adolescents in classrooms in four European countries, we simulate how four different cohorting strategies can mitigate the spread of SARS-CoV-2 in high schools. We model the effect of forming two cohorts randomly, splitting cohorts by gender, optimizing cohorts by minimizing students' out-of-school cross-cohort contacts, and approximating this optimization strategy by network chains. The rationale of all non-random cohorting strategies is to prevent the spread of SARS-CoV-2 from one cohort to the other by reducing cross-cohort out-of-school contact. We also compare the overall effect of cohorting to no cohorting and differentiate between a rota-system in which cohorts receive in-person instruction in alternating weeks and a system with separate but same-day in-person instruction for both cohorts. Data were collected between 2010 and 2011 as part of the CILS4EU project, a network panel study of 14-15-year-olds in England, Germany, the Netherlands, and Sweden. Across all four countries, we model the transmission of SARS-CoV-2 in 507 classrooms, capturing a total of 12,291 students. Findings Our simulations suggest that all four cohorting strategies reduce the spread of SARS-CoV-2 in classrooms, but vary in their effectiveness. Relative to random cohorting, all strategies that factor in out-of-school cross-cohort ties have particularly strong effects on the frequency of cross-cohort transmission but also substantively reduce the total number of infections and the share of students in quarantine when transmission dynamics are strong. Cohorting that explicitly minimizes out-of-school contact between students in different cohorts is most effective, but network-based approximation also breaks many cross-cohort ties and thus performs well. Because adolescents' out-of-school contacts tend to be strongly segregated by gender, dividing classrooms by gender also outperforms random cohorting but is less effective than directly using network information. For all cohorting strategies, rota-systems with instruction in alternating weeks contain outbreaks more effectively than same-day in-person instruction. Interpretation Cohorting of school classes as a social distancing measure can help to effectively curb SARS-CoV-2 outbreaks in the school context. If schools consider splitting up classes into two smaller cohorts, factoring in out-of-school contacts can help achieve a more effective separation of cohorts. The paper proposes effective cohorting strategies that outperform naive random cohorting in preventing the spread of SARS-CoV-2. These strategies may limit outbreaks to one cohort, keep the size of infection clusters low, and reduce the number of students in quarantine if an index case occurs in the student body. Our findings thus suggest that if schools consider cohorting, they should assign students who meet after school to the same cohort. In particular, cohorting on the basis of gender or network chains is effective and may be successfully implemented within the constraints posed by the classroom context.

Schools have long been identified as drivers of influenza and other respiratory-spread epidemics (Cauchemez et al. 2009; Glass and Glass 2008; Luca et al. 2018; Markel et al. 2007; Wallinga, Teunis, and Kretzschmar 2006) . When the novel coronavirus SARS-CoV-2 resulted in a pandemic in early 2020, many countries reacted by closing schools. For various reasons, however, school closures are controversial in both epidemiological and broader public debates.

While epidemiological studies indicate that school closures may mitigate the spread of SARS-CoV-2, their impact still is contested, partly because the consequences of synchronized measures such as banning public gatherings or introducing mandatory masks are hard to disentangle (Aleta et al. 2020; Auger et al. 2020; Chang et al. 2020; Davies, Klepac, et al. 2020; Davies, Kucharski, et al. 2020; Flaxman et al. 2020; Y. Li et al. 2020; Panovska-Griffiths et al. 2020; Prem et al. 2020; Zhang et al. 2020 ) Moreover, the infectiousness and frequency of asymptomatic infections in children and adolescents are still unclear. Emergent evidence on a lower force of infection in children under the age of ten suggests that primary schools and childcare facilities are at comparatively lower risk. 1 However, secondary schools may be at higher risk because transmission appears to be more dynamic in adolescents than in children, possibly approximating transmission dynamics in adults Goldstein, M. Lipsitch, and Cevik 2020; Levinson, Cevik, and Lipsitch 2020; Park et al. 2020; . But even if adolescents were less susceptible than adults, their frequent contacts and exposure to transmission-facilitating contexts such as schools may amplify the spread of communicable diseases in this age group (Dattner et al. 2020 ).

In broader public debates, school closures are seen as a last resort because of the high social and economic costs they involve (Bayham and Fenichel 2020; Lempel, Epstein, and Hammond 1 It has been argued that comparatively lower numbers of infected children may be a result of shielding practices of cautious parents and proactive school closure policies rather than a physical mechanism (Hyde 2020). Registered incidence numbers are also treacherous because many children may not get tested, e.g., a study from Germany recently showed that prevalence of SARS-CoV-2 antibodies in children may be up to six times higher than official numbers indicate (Hippich et al. 2020) . 5 2009; Psacharopoulos et al. 2020) . For example, distance learning is feared to be insufficient to compensate for the loss in learning that occurs when in-person classes are cancelled, which is supported by evidence from the Netherlands and Germany (Engzell, Frey, and Verhagen 2020; Grewenig et al. 2020) .

As a consequence, pediatricians and public health experts have called for schools to stay open or re-open whenever possible (Munro and Faust 2020; Sharfstein and Morphew 2020; Walger et al. 2020) . Many countries cautiously re-opened schools when incidence rates dropped in the summer of 2020. Early reports of large-scale school outbreaks in Israel soon after reopening (Stein-Zamir et al. 2020) were followed by outbreaks in Chile )and

France . Other observations suggest that outbreaks at schools are rare and that index cases in schools scarcely led to onward transmissions. Yet most of these data were collected under conditions of low incidence and strict social distancing measures in schools, including hybrid in-person and online learning as well as reduced class sizes. Therefore, the role of schools as drivers of transmission and large-scale outbreaks is not yet settled.

Cohorting as an alternative to school closures to prevent the spread of SARS-CoV-2

As transmission rates rise again, intervention strategies that keep the number of infected students and teachers as small as possible become more urgent. To avoid school closures, strategies that reduce the probability of in-school infections are of key interest. In many countries, strategies that do not radically interfere with schools' daily routines-such as wearing masks and ensuring sufficient ventilation-have already been implemented.

However, these measures are feared to be insufficient to prevent infection clusters when community transmission rises, raising calls for social distancing measures in schools, which may be more intrusive but further help avoiding the transmission of SARS-CoV-2. 2 Yet, there is hardly any evidence on social distancing measures and their effectiveness in schools, as 2 For the public discussion in Germany, see, for example https://www.sueddeutsche.de/politik/coronadeutschland-laschet-soeder-1.5119886, https://www.faz.net/aktuell/politik/inland/spahn-schlaegt-neuescorona-konzept-fuer-schulen-vor-17065077.html or https://www.zeit.de/gesellschaft/schule/2020-11/coronamassnahmen-schule-hybridunterricht-infektionsschutz-regeln-faq evidenced by Viner et al. (2020:401) , who identified "a remarkable dearth of policy-relevant data on the implementation of school social distancing during coronavirus outbreaks."

One social distancing measure that is more intrusive than some other precautions but still allows for in-person instruction is splitting larger clusters of students into smaller groups, which are kept separate to prevent cross-group infections. In line with the definition of the Centre for Disease Control and Prevention (CDC 2020), we term the corresponding process of splitting and isolating these clusters cohorting and the resulting separate groups cohorts. In the European context that we focus on, in-school instruction is typically organized in classrooms of 20-40 students. Most courses are taught in these classrooms to the same set of students. 3 In this context, cohorting refers to splitting classrooms into smaller units and instructing these units separately. There is frequent discussion of splitting classrooms in half (see footnote 2), which, unlike dividing classrooms into three or more cohorts, may still be compatible with schools' resources. Cohorts may either both be instructed in-person on the same day (using different classrooms or with one cohort present in the morning and one in the afternoon, for example) or in a rota-system with each cohort in turn being instructed in school and online in consecutive weeks.

Cohorting classrooms has several benefits. First, it facilitates social distancing within the classroom because there is more space per student. Second, it reduces the number of students who are exposed to an infection within the classroom, which can moderate both the size and the reach of an initial outbreak. When cohorting separates groups effectively, infections in the other cohort can be avoided and quarantines can be restricted to a single cohort. Results from a modeling study based on UK data suggest that cohorting of school classes in a two-week rota-system may reduce the proportion of patients that need to be 3 In the US, by contrast, instruction is largely organized at the grade level or even across grades. Comparable strategies that result in similarly small cohorts thus may be harder to implement, though the idea of learning pods (CDC 2020) captures a similar concept. One study from the US ) considers breaking up larger school interaction patterns into fixed classrooms and instructing different parts of the resulting classrooms at different points in time. However, Cohen et al. only consider randomly splitting classrooms and do not use actual student (network) data, which we focus on to compare the effects of different cohorting strategies. 7 successfully isolated through test-trace-isolate tasks in order to prevent an epidemic rebound from 75% to 65% ).

How to make cohorting effective?

While cohorting strategies were implemented in a number of places in the spring of 2020 and are still widely discussed, there is little guidance on how effective cohorting is in preventing SARS-Cov-2 transmission in schools and whether specific cohorting strategies are preferable.

Our paper fills this gap by using simulation models to assess how cohorting affects transmission dynamics within classrooms. We consider cohorting strategies that split full classrooms (i.e., about 20-40 students) into two cohorts of approximately equal size.

We first investigate the general efficacy of cohorting by simulating the transmission of SARS-CoV-2 in classrooms without cohorting and in classrooms where students are randomly separated into two distinct cohorts. This random cohorting is comparable to dividing students by alphabetical order, which has the benefit of being easy to implement. However, the effectiveness of cohorting depends on preventing transmission between cohorts. While any kind of cohorting strategy-including random cohorting-precludes interaction with members of other cohorts within school, it does not necessarily prevent such interaction outside of school. If students have out-of-school contacts, and if these contacts span cohorts, they may serve as transmission channels between cohorts even if in-school contacts are separated. Cohorting strategies that take into account out-of-school contacts and try to minimize corresponding cross-cohort contacts are therefore likely to be more effective in containing outbreaks. Therefore, we also assess whether three more sophisticated cohorting strategies that consider students' out-of-school contacts with their classmates are better in containing SARS-CoV-2 than random allocation. These strategies are based on the well-established fact that social networks mostly consist of clusters that are well-connected internally but are more loosely connected to other clusters. Such clusters can form because people tend to associate with others who are similar to them and less with others who are dissimilar (McPherson, Smith-Lovin, and Cook 2001) or due to more structural processes such as transitive closurewhen friends of friends also become friends (Lewis 2015; Rivera, Soderstrom, and Uzzi 2010) . All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi. org/10.1101 org/10. /2020 When social networks are organized in clusters with few connections between them, social distancing measures that successfully sever between-group ties may be much more efficient than strategies that do not take social network structure into account (Block et al. 2020; Manzo and Rijt 2020) . Our network-based cohorting strategies apply this insight to the school context.

The first network-based cohorting strategy splits classrooms by gender. While not explicitly considering students' out-of-school contact, this strategy takes advantage of the strong gender segregation in adolescents' friendship networks (McDougall and Hymel 2007; Rose and Rudolph 2006) . Splitting classes by gender therefore should considerably reduce out-ofschool contact between members of different cohorts. This strategy is easy to implement, but an elevated transmission risk in cross-gender romantic relationships may undermine its efficiency. The second strategy is optimization that explicitly uses students' self-reported outof-school contact with classmates to form cohorts in a way that minimizes the number of cross-cohort contacts. By definition, this strategy produces the cleanest separation of cohorts and should thus be most effective in preventing cross-cohort infection. However, this strategy is hard to implement as teachers need to know their students' out-of-school contacts with classmates and optimize cohorts accordingly. Therefore, we also propose a network chain cohorting strategy that approximates optimization based on the out-of-school contact network and is much easier to implement. In this strategy, an initial student who is wellconnected-such as a class representative or a student known to be popular-names all of her classmates with whom she has out-of-school contact, and the resulting set of students forms the basis for the first cohort. Subsequently, the listed out-of-school contacts name their out-of-school contacts, who also become members of the first cohort. The process continues until half of the classroom is allocated to the first cohort, 4 and the remaining students form the second cohort.

4 If the chain breaks, another random student can be allocated to the group and selected to nominate her outof-school contacts.

All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10.1101/2020.11.30.20241166 doi: medRxiv preprint Figure 1 : Cross-cohort out-of-school ties for different cohorting strategies in an example classroom from the CILS4EU data. Figure 1 shows an exemplary classroom in the dataset and demonstrates the success of different cohorting strategies in separating cohorts for this set of students. Nodes represent students and connections among nodes represent out-of-school contacts. Colors indicate the cohort to which students have been allocated. While there are many cross-cohort contacts under random cohorting, gender-split and network chain cohorting produce fewer crosscohort connections and optimized cohorting succeeds in perfectly separating cohorts. Of course, the effects of cohorting strategies may vary across classrooms. In the following, we use simulation models to investigate whether these cohorting strategies can help to prevent the spread of SARS-CoV-2 outbreaks across cohorts, limit infections among students, and avoid quarantines.

The CILS4EU data To assess the effectiveness of different cohorting strategies, we use student data from the first wave of the Children of Immigrants Longitudinal Study in Four European Countries (CILS4EU) project (Kalter et al. 2016) . The data provide information on 14-15-year-old students from England, Germany, the Netherlands, and Sweden. Data was collected in 2010-11 in randomly selected schools within each country, oversampling schools with a high share of immigrant students. In most schools, two ninth-grade classrooms were surveyed in full, providing individual student information as well as data on social relations between surveyed students within a classroom. The response rate at the student level was 81% in England and All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10. 1101 /2020 Germany, 91% in the Netherlands and 86% in Sweden (CILS4EU 2016). 5 Because cohorting is less likely to be an issue in small classes, our analysis considers all classrooms with information on 20 or more students. To compare the gender-split strategy with other strategies, we only consider classrooms with full information on students' gender. In total, our sample consists of 507 classrooms populated by 12,291 students.

We focus on students' social relations outside of the school context because these relations can serve as an additional channel of transmission for SARS-CoV-2 next to transmission within cohorts. Out-of-school interaction is captured by an indicator assessing the classmates a student "often spend[s] time with outside school". Students could nominate as many of their classmates as they wanted. 6 Whenever one student named another, we code an out-of-school contact between this pair of students, independent of whether the second student confirmed the nomination because contact necessarily goes both ways. The median student has out-ofschool contact with three classmates, with a mean of 3.58 contacts.

We investigate how effective different cohorting strategies are in preventing cross-cohort disease transmission through out-of-school contact with classmates. All cohorting strategies are summarized in Table 1 . Throughout our analysis, we consider outcomes when in-person instruction of cohorts takes place in alternating weeks or in different sessions on the same day.

As a baseline, we compare a scenario without cohorting (i.e., full classrooms) with random cohorting. In random cohorting, the entire classroom is randomly split into two equally-sized cohorts that are taught separately. For this comparison, we simulate the transmission of SARS-CoV-2 within classrooms (explained in more detail below) to examine the total proportion of students infected without cohorting and with random cohorting. 5 The total response rate was lower because schools or (in very rare instances) classes could also decide to not participate in the survey. Factoring in all sources of non-response, response rates are 53% in England, 79% in Germany and the Netherlands, and 65% in Sweden. (CILS4EU 2016). 6 In many classrooms, however, students were not allowed to nominate students who did not participate in the survey. Therefore, our networks are limited to students who participated in the student and/or network questionnaire.

All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10.1101/2020.11.30.20241166 doi: medRxiv preprint Optimized cohorting Two equally-sized cohorts are formed to minimize the number of cross-cohort out-of-school contacts.

Network chain cohorting An initial student names all of her out-of-school contacts, who themselves name their out-of-school contacts, etc., until the resulting set of students comprises half of the classroom. This set of students forms the first cohort, the remainder the second cohort.

Proportion of simulations with transmission to second cohort Does SARS-CoV-2 spread from the seed node's cohort to the other cohort, such that containment fails?

Average proportion of students in quarantine at the end of the simulation How many students are quarantined and can thus (temporarily) not attend school?

Average proportion of students infected at the end of the simulation How many students in the classroom (across cohorts) have been infected (by the seed node or other students)?

Subsequently, we compare random cohorting and three non-random cohorting strategies in terms of their effectiveness in avoiding transmission across cohorts, reducing quarantines, and limiting infections. In all cohorting strategies, we split classrooms into two equally sized cohorts, but according to different criteria. In the gender-split strategy, cohorts are separated All rights reserved. No reuse allowed without permission.

perpetuity.

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

The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10.1101/2020.11.30.20241166 doi: medRxiv preprint by gender. If a strict separation of boys from girls leads to unequal cohort sizes because of an uneven gender composition, members of the larger cohort (the overrepresented gender) are reallocated to the smaller cohort until cohort sizes equalize. 7 In the optimized cohorting strategy, students are allocated to cohorts in order to minimize the number of cross-cohort out-of-school contacts with classmates. 8 Network chain cohorting approximates this strategy by using chains of out-of-school contact nominations. In this strategy, a random wellconnected student names all of her out-of-school contacts and this set of students forms the core of the first cohort. 9 The nominated out-of-school contacts themselves subsequently nominate their (not yet nominated) out-of-school contacts, continuing this process until the set of nominated students comprises half of the classroom. 10 This set forms one cohort and the remaining students are pooled in the second cohort.

We compare the effectiveness of the different cohorting strategies in terms of three key indicators, which are also summarized in Table 1 . First, we consider the proportion of simulations in which SARS-CoV-2 is transmitted to the second cohort, meaning that containment of an initial outbreak fails. Second, we assess the proportion of students who are quarantined and thus temporarily cannot participate in school activities in person. Third, we evaluate the overall proportion of students in the classroom who become infected to assess whether cohorting strategies can help to reduce the severity of outbreaks. 7 There are other ways to split by gender, two of which we explore in the appendix. As explained there, from the perspective of preventing infections and quarantine, ensuring equal group sizes is preferable. 8 We use brute-force optimization, considering all possible allocations to equally-sized cohorts to find the minimum number of cross-cohort out-of-school contacts. In eight classrooms with more than 32 students, optimization fails due to computational constraints. For these classrooms, we randomly sample 1,000,000 allocations and report results for the allocation that minimizes the number of cross-cohort out-of-school contacts. 9 In the simulations, we draw the initial student from the observed out-of-school relations, with the probability of being the initial student proportional to the number of out-of-school contacts. Therefore, better-connected students are more likely to be initial students. This simplifies allocation because the algorithm is less likely to break down. 10 If there are no additional nominations at a certain nomination step and the set does not yet comprise half of the students, a random student is added to the set and can subsequently nominate her contacts. If, during the nomination process, the number of students in the group exceeds half of the class size, a random subset of the newly-nominated students is added to the set. These regulations, which ensure that the algorithm always ends up with an allocation, can also be easily implemented in practice.

All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10.1101/2020.11.30.20241166 doi: medRxiv preprint

We simulate the transmission of SARS-CoV-2 separately for each classroom and each cohorting strategy. The simulation model is summarized in Figure 2 . Each simulation starts with one randomly infected seed node (i.e., student) in the classroom and investigates how SARS-CoV-2 spreads within the class. In the simulation model, infections originate from interaction with an infectious student. Interaction is modelled on a daily basis and takes place between all students within a cohort in the classroom. In addition, interaction can happen outside of school between classmates with out-of-school contact.

Within-cohort interaction occurs only on school days (i.e., Monday to Friday) and only when the cohort is instructed in-person, i.e., every other week when cohorts are instructed in a rotasystem. The infection risk of individual students within classrooms is likely to vary, in particular with respect to physical closeness. To capture differential risks for infection, we randomly draw 25% of the within-cohort pairs of students to be at high relative risk of infecting one another. A cohort usually consists of 10-16 students, such that 2-4 other students are highrisk contacts. 11 For the remaining students, we assume a lower risk (20% of the high-risk contact) due to aerosol diffusion and unmodelled interaction patterns.

Next to within-cohort in-school interaction, out-of-school interaction with classmates can also cause infection. Out-of-school interaction is assumed to have the same risk of infection as high-risk contact within the classroom. Out-of-school contact can take place on any day of the week, but only with a given probability, as discussed below.

Irrespective of whether interaction occurs in school or outside school, it does not always lead to infection, but only with a given probability, as discussed below. Once a student is 11 We report results for 12.5% and 50% of high-risk contacts in the appendix, but substantive conclusions are unchanged. We fix high-risk interactions for pairs of students (rather than at the daily interaction) because they tend to be stable over time due to the physical organization of the classroom. All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10.1101/2020.11.30.20241166 doi: medRxiv preprint Random seed student is exposed at random day of the week. Seed student's disease trajectory is drawn.

The simulation ends if either of the following conditions holds: • All students have been infected (i.e., are either exposed, infectious, or recovered; with no student susceptible) • All students have been quarantined • Seven weeks (49 days) have passed since the beginning of the simulation.

If a non-quarantined student became symptomatic on the previous day, • the student herself • the student's entire cohort • the student's interaction partners in the last 14 days are quarantined. Quarantine lasts for 14 days.

Students interact within their classroom cohort and with their out-of-school contacts (among classmates):

• Interaction within the classroom cohort takes place Monday to Friday if the cohort is instructed in-person during that week.

Students interact with all students in the cohort. Interaction can be high-risk or low-risk. • Interaction with out-of-school contacts can take place on any day of the week. On each day, each possible interaction takes place with the probability of out-of-school contact. Out-of-school Interaction is high-risk.

Interaction with an infectious students results in exposure according to a to the baseline probability of infection upon contact, weighted by infectiousness of the infectious student and the risk level of the contact.

If a student is exposed, her disease trajectory is drawn.

Every infected student's disease trajectory is incremented or initiated (if newly exposed, see 4.). Time is incremented by one day. Students recover after they are no longer infectious. There is no reinfection.

• 25% of all pairs of students are involved in high-risk interaction in the classroom (e.g. physical closeness). The pairs are fixed within each simulation run. • All other pairs are involved in low-risk contact.

• Out-of-school contact is considered high-risk contact.

• For low-risk contact, the baseline probability of infection is weighted by factor 0.2 to determine whether an infection occurs. For high-risk contact, the probability is weighted by factor 1. all students have been infected, or (ii) all students have been quarantined, or (iii) seven weeks have passed (capturing the effect of school holidays). Given that infected seed nodes, disease trajectories, interaction patterns, and, for some cohorting strategies, cohort composition, are stochastic, we run at least 300 simulations for each classroom and each cohorting strategy and report average results throughout.

For most parameters related to disease trajectories, we rely on estimates from recent studies (discussed above). However, appropriate values for three key parameters of our transmission models are still highly uncertain. First, there is little knowledge on the probability of infection conditional on contact with an infectious student, especially for adolescents and in the school context. Second, the daily probability of out-of-school interaction with any given out-of-school contact is uncertain. Third, the proportion of clinical infections with SARS-CoV-2 is highly debated, in particular among children and adolescents.

All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10. 1101 /2020 Naturally, model outcomes are sensitive to these parameters. At a higher likelihood of infection, larger outbreaks are more likely; at higher probabilities of out-of-school interaction, out-of-school contacts become more relevant; a higher share of clinical infections triggers quarantines more quickly, but symptomatic cases may also be more infectious. Lacking precise estimates for these parameters, we model results for a range of plausible values, as discussed in the three subsequent subsections.

The probability of infection upon contact is contingent both on adolescents' general susceptibility to SARS-CoV-2, which is still unclear (Goldstein, M. Lipsitch, et al. 2020; X. Li et al. 2020; , and on how conducive the school context is for transmission. To some degree, classrooms provide ideal conditions for transmission as they are small, confined indoor spaces that are shared for multiple hours. However, in the course of the SARS-CoV-2 epidemic, many schools have adapted to pandemic conditions by enforcing frequent ventilation, the usage of masks, and other precautions.

To capture this variation in conditions, we simulate model results across a range of infection probabilities. To be precise, we model a baseline probability of infection upon contact, which is the probability of becoming infected conditional on exposure to an infectious student on any given day. 12 It is a baseline probability because this probability is further reduced if it involves a low-risk contact or a subclinical (and thus less infectious) infected student. We consider probabilities of 5%, 10%, 15%, 20%, and 25%. These values correspond to inclassroom secondary attack rates for an average infectious seed node in the range from 3-4% to 23-27%, respectively. 13 This corresponds well with ranges for the estimated secondary 12 Same-day interaction within school and outside of school are considered to be to separate encounters that pose distinct infection risks. 13 Secondary attack rates are calculated as follows. The average clinical classmate is infectious and unquarantined for three days (2 pre-symptomatic days and 1 symptomatic day), yielding a probability of infection of 1 − (1 − $%&'()*' ,-.$%$)()/0) ! for high-risk contacts and 1 − (1 − 0.2 * $%&'()*' ,-.$%$)()/0) ! for low-risk infections, with / the number of days of interaction, which depends on whether these days fall on weekdays or on the weekend. The average subclinical classmate is infectious for five days (and is not quarantined), yielding a probability of infection of 1 − (1 − 0.5 * $%&'()*' ,-.$%$)()/0) ! of infection for high-risk contacts and 1 − (1 − 0.1 * $%&'()*' ,-.$%$)()/0) ! for low-risk contacts, with / the number of days of interaction. Weighting each weekday with identical probability for the first day of infectiousness yields the secondary attack rates discussed in the text. Of course, the secondary attack rate is lower when instruction is organized in a weekly rota-system and parts of the infectious period do not All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10.1101/2020.11.30.20241166 doi: medRxiv preprint attack rate among adolescents from previous studies, with Liu et al. (2020) reporting a secondary attack rate of 4% in 10 to 19-year-olds in a contact study with all contacts, and Dattner et al. (in Viner, Mytton, et al. 2020:11) reporting a household secondary attack rate of 34% among 7 to 19-year-olds. Other estimates are in between (Chu et al. 2020; Goldstein, Marc Lipsitch, and Cevik 2020; Laxminarayan et al. 2020; Sachdev et al. 2020; ).

The probability of out-of-school interaction captures students' propensity to meet some of their classmates outside school. In the model, the parameter refers to the probability of a focal student to interact with any specific out-of-school contact on a given day. Clearly, realistic values for this parameter strongly vary with momentary conditions in society. With a high incidence of SARS-CoV-2 or even a lockdown that prohibits certain contacts, probabilities for interaction will be lower than under normal circumstances. We consider probabilities of students to meet an out-of-school contact on a given day of 5%, 10%, 15%, and 20%. In our data, the median student nominates three (mean of 3.58) classmates who she has frequent out-of-school contact with. A probability of out-of-school contact of 5% thus means that the median student on average has 1.05 out-of-school interactions in a week, with a probability of 86% to not have any out-of-school interaction on any given day and a probability of 34% to not have any interaction in an entire week. A probability of out-of-school contact of 20% means that the median student has on average 4.2 out-of-school interactions per week, with a probability of 51% of having no contact on any given day and a probability of just below 1% of having no contact in an entire week.

The proportion of subclinical SARS-CoV-2 infections, in particular among children and adolescents, is highly debated and, so far, unknown. Even if the proportion of truly asymptomatic infections is low, many other infections may come with very weak symptoms and thus go undiagnosed, in particular in high-incidence situations with limited access to correspond with in-person instruction. The secondary attack rate also varies with the share of high-risk withincohort interactions. The maximum average secondary attack rate of 23%-27% discussed above is for a proportion of 50% of high-risk interactions (results reported in the appendix), the minimum of 3%-4% for a proportion of 12.5% (see appendix) and 25% (see main text), respectively. All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10.1101/2020.11.30.20241166 doi: medRxiv preprint testing. Previous studies suggest a wide range of estimates for the share of subclinical infections, ranging from 22% (Han et al. 2020) to 86.6% ) among children and adolescents with other estimates in between (Dattner et al. 2020; Davies, Klepac, et al. 2020; Jung et al. 2020; Waterfield et al. 2020) . To capture the high uncertainty in this parameter, we model transmission processes with assumed proportions of subclinical infections of 20%, 40%, 60%, and 80%.

We first assess whether cohorting is effective in containing outbreaks. In order to do so, we compare SARS-CoV-2 transmission dynamics in regular full classrooms-without cohortingto random cohorting, which divides classrooms into two cohorts but ignores out-of-school interaction. Figure 3 shows the average proportion of students across the entire classroom that is infected at the end of the simulation, comparing no cohorting and random cohorting.

It differentiates results between teaching in a weekly rota-system and same-day instruction of both cohorts across different probabilities of infection, probabilities of out-of-school contacts, and proportions of subclinical cases. According to Figure 3 , the classroom-level proportion of infections is about two times as high in the absence of cohorting compared to random cohorting. This holds not only for all parameter constellations but also irrespective of whether cohorts are instructed in-person in consecutive weeks or on the same day. 14 Throughout the analysis, instruction in a weekly rota-system always reduces infections relative to same-day instruction: the weekly rota-system ensures that transmission dynamics within cohorts can only take place every other week, inducing a "natural quarantine" that frequently prevents outbreaks early on or stops them from spreading further-at the cost of less frequent in-person instruction.

14 Note that, even when cohorting is organized in a weekly rota-system, the baseline proportion infected for the comparison scenario without cohorting will frequently be the proportion infected under same-day instruction. This is because, under normal conditions, all students are usually instructed on the same day by default and only introducing a cohorting system requires switching to a rota-system (due to constraints in capacity). All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi. org/10.1101 org/10. /2020 Figure 4 also corroborate that larger outbreaks are substantially more likely when classrooms are not cohorted, especially when the probability for infection is high.

Under this condition, cohorting frequently contains an outbreak within one cohort, while SARS-CoV-2 spreads further if there is no cohorting. Figure 4 also shows that, at same-day instruction and an infection probability of 25%, outbreaks that affect entire classrooms are frequent. Given the other parameter values, higher probabilities of infection thus seem implausible because these outbreaks should be noticed empirically in high-incidence situations. Therefore, we consider this the upper bound for plausible probabilities of infection in the classroom context.

In relative terms, Figure 3 indicates that cohorting is most effective when the probability for infection is low. When infection risk is low, a large number of contacts is required to induce a large outbreak, and this number is lowered through cohorting. The relative effectiveness of random cohorting is also higher when out-of-school interaction is unlikely. Under this condition, random cohorting effectively separates cohorts because there is little cross-cohortinteraction and infection risk. At higher probabilities for out-of-school contact, cross-cohort contact more frequently results in transmission to the other cohort, lowering the benefit of cohorting. However, outbreaks become more severe at higher probabilities for infection and more frequent out-of-school interaction. Therefore, the lower relative effectiveness of cohorting under these conditions is compensated for by higher absolute effects on the number of resulting infections. Given that effects of cohorting are substantively large under all conditions, our model thus suggests that cohorting can effectively reduce infections in the school context. 15 15 Our model should capture both main consequences of cohorting: a reduced number of within-classroom interaction partners and improved opportunities for social distancing. Improved opportunities for social distancing are represented through a constant share of high-risk within-class contacts, such that the number of high-risk contacts falls when cohorting is introduced. A reduced number of within-classroom interaction partners is immediately implied by cohorting. All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi. org/10.1101 org/10. /2020 How to improve the effectiveness of cohorting?

The degree to which cohorting strategies succeed in containing outbreaks depends on how effective they are in breaking out-of-school ties between students from different cohorts. To see whether taking into account out-of-school contact with classmates improves the effectiveness of cohorting, we compare random cohorting with alternative strategiesgender-split cohorting, optimized cohorting that minimizes out-of-school cross-cohort interaction, and network chain cohorting that approximates such an optimization.

For an initial descriptive assessment of these strategies, Figure 5 shows the distribution of the average number of cross-cohort ties across classrooms for different cohorting strategies.

Across all countries, all cohorting strategies yield a number of cross-cohort ties that is substantively smaller than the total number of ties in the out-of-school network. As expected, the optimization strategy results in the lowest number of cross-cohort ties, with an average of 3.5 cross-cohort ties per classroom-only 17% of the average of 20 cross-cohort ties resulting under random cohorting. The gender-split strategy produces an average of 11.4 cross-cohort ties, 57% of the cross-cohort ties under random cohorting. The network chain strategy results in an average of 8.4 cross-cohort ties, thus outperforming the gender-split strategy, with 42% of the cross-cohort ties under random cohorting remaining. Therefore, next to the optimization strategy, network chain and gender-split cohorting are also likely to help contain outbreaks. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10. 1101 /2020 In order to investigate the consequences of severing cross-cohort out-of-school ties, we simulate SARS-CoV-2 transmission for the random, gender-split, network chain, and optimized cohorting strategies in all classrooms. We show results for three indicators: the proportion of instances in which an initial outbreak spreads across cohorts, the proportion of infections across the entire classroom, and the proportion of students quarantined. Note that any given proportion of clinical infections implies a specific minimum share of students who are quarantined independent of the cohorting strategy used. 16 Therefore, we show results for the excess proportion quarantined, i.e., quarantines up and above the minimum share implied by clinical infections. The choice of cohorting strategy can only affect this excess proportion. We show results aggregated across the entire sample, but provide classroom-level results by country in the appendix, with comparable substantive results. Figure 6 shows our main results. We show results for same-day instruction of both cohorts in the upper portion of the graph and results for the weekly rota-system in the lower portion.

Furthermore, we show all results separately by the probability of out-of-school interaction along the x-axis, and across different probabilities of infection and proportions of subclinical cases in horizontal panels. Therefore, Figure 6 shows model outcomes across the entire parameter space. In the subsequent discussion, however, we will, whenever possible, summarize by highlighting comparable patterns across parameters.

The choice of cohorting strategy has its strongest effect for the proportion of instances in which infection spreads to the other cohort, with substantive effects visible for all parameter constellations. Gender-split, network chain and optimized cohorting all clearly outperform random cohorting. Gender-split cohorting falls about halfway in between random and optimized cohorting. Network chain cohorting is more effective than gender-split cohorting but of course still less effective than optimized cohorting. The same pattern holds for effects on the proportion of students quarantined and the total proportion infected, though effects are weaker and almost absent for the proportion infected for some parameter constellations 16 Any clinical student eventually triggers a quarantine for her cohort, independent of cohorting strategy. For example, when the proportion of clinical infections is 60%, 60% of all simulations result in a clinical infection of the seed node, causing her cohort (50% of all students) to go into quarantine. Independent of cohorting strategy, the proportion of students in quarantine across simulation runs thus cannot fall below 30%. The excess proportion quarantined is the proportion of students quarantined up and above this baseline level. All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10.1101/2020.11.30.20241166 doi: medRxiv preprint Figure 6 : Epidemiological outcomes of different cohorting strategies: Proportion of instances of infection spreading to the second cohort, proportion of students infected, and excess proportion of students quarantined. Cumulative probabilities.

(i.e., low probabilities for infection or out-of-school contacts or low proportions of subclinical cases).

Qualitatively, all cohorting strategies react to variation in parameter values in a similar way.

The effects of cohorting strategies that successfully break cross-cohort out-of-school ties are particularly strong when the probability of infection, the probability of out-of-school interaction, and/or the proportion of subclinical students is high. Effective cohorting also has stronger effects when both cohorts are instructed on the same day rather than in consecutive weeks. (Note the different scales for the respective portions of the graph). Under these conditions, the transmission of SARS-CoV-2 is pronounced and enduring because students are easily infected, have more contacts that facilitate transmission to the other cohort, are less frequently quarantined (because infection goes unnoticed), and reinforce outbreaks within cohorts because of more frequent in-person instruction. This is also visible by higher overall cross-cohort spread and higher proportions infected and quarantined for all cohorting strategies.

An effective cohorting strategy thus is most important to avoid the spread of SARS-CoV-2 when conditions facilitate disease transmission. For example, at same-day instruction, a baseline probability of infection upon contact of 15%, a probability of out-of-school interaction of 20%, and 60% subclinical infections, random cohorting on average results in 20% of outbreaks spreading to the second cohort, gender-split cohorting results in 12%, network chain cohorting in 9%, and optimized cohorting in 4%. Thus, cohorting that considers out-of-school contact between classmates can lower the frequency of spread by 39%-79% relative to random cohorting. The excess proportion of quarantined students can be reduced from 22% (random cohorting) to 20% (gender-split), 19% (network chain cohorting) and 18% (optimized cohorting), and can thus be reduced by 9-18%. The overall proportion quarantined is 20% higher because 40% of clinical infections imply a baseline proportion of quarantined students of 20%, but this baseline proportion cannot be affected by any specific cohorting strategy. The average proportion of infections at the same time falls from 11% (random cohorting) to about 10% in gender-split, network chain and optimized cohorting, with reductions of 4% (gender-split strategy), 5% (network chain strategy) and 7% (optimized All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10.1101 https://doi.org/10. /2020 strategy). At a lower probability of infection, lower probability of out-of-school interaction, lower proportions of subclinical cases, and instruction in a weekly rota-system, effects on the frequency of spread to the second cohort and the excess proportion quarantined remain substantial, but effects on the proportion infected are smaller.

An observation from Figure 6 that may be surprising at first sight is that the total proportion of infections reacts only very mildly to the type of cohorting strategy when the baseline probability for infection is low, while the frequency of failed containment and the excess proportion quarantined react considerably. This can best be understood by inspecting Figure   7 , which shows the distribution of the proportion of infected students across all classrooms for selected parameter constellations. (To see results more clearly, Figure 7 is limited to instances with at least one infection through the seed node.) At low probabilities of infection, the transmission of SARS-CoV-2 breaks down very quickly, and outbreaks are in almost all instances limited to very few infected cases. In these situations with weak dynamics in transmissions, the substitution of cross-cohort out-of-school contacts All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10.1101 https://doi.org/10. /2020 with within-cohort out-of-school contacts in the more effective cohorting strategies means that the few resulting infections occur within the cohort rather than across cohort borders. Therefore, effective cohorting improves containment of the outbreak to one cohort. However, the transmission dynamics reach a halt quickly independent of whether within-cohort students other cross-cohort students are infected and thus cohorting strategy hardly affects the total number of infections. Therefore, the number of infections does not react strongly to cohorting strategy, as seen in the left part of Figure 7 . Only when the probability of infection is higher, such that dynamics are sustained in the second cohort if containment has failed, effective cohorting (and the ensuing containment) lowers the total number of infected students. This can be seen from the right part of Figure 7 , which shows that many students in one cohort are infected in the gender-split, network chain and optimized cohorting strategies, but the disease is contained within that cohort-represented by high probability masses of 50% or fewer infected students. By contrast, the disease more frequently spreads to and within the second cohort under random cohorting, indicated by a higher probability mass at higher proportions of infected students.

Substantively, the usefulness of cohorting strategies that prevent cross-cohort out-of-school interactions thus depends both on how well SARS-CoV-2 is transmitted among students and the underlying goals of cohorting. If transmission is weak, the main advantage of effective cohorting is to reduce the frequency of quarantines, thus keeping students in school more. At higher probabilities for infection, however, effective cohorting also reduces the total burden of infections notably by frequently containing larger outbreaks to a single cohort rather than allowing them to spread to and within the second cohort. The same holds true when the probability of out-of-school interaction is high, which more frequently induces a second outbreak after containment has failed in one cohort. Similarly, a high proportion of subclinical cases means that SARS-CoV-2 can spread unnoticed to and within the second cohort, also strengthening the consequences of effective cohorting for the total number of infections.

As we approach the end of 2020, the SARS-CV-2 pandemic has been a disruptive force in many areas of society. Social distancing measures, unprecedented in scope, have helped mitigate All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10. 1101 /2020 the first wave of the epidemic, though at high costs. Among these high costs were school closures, which resulted both in missed learning opportunities for children and in considerable strain on their families. School closures therefore are now seen as a last resort when other measures have failed or were insufficient to prevent the spread of SARS-Cov-2. With increasing incidence of SARS-CoV-2, however, infections in schools become more likely, requiring effective social distancing strategies to avoid transmission and larger outbreaks in schools.

One such measure that-though intrusive-allows schools to remain open is the decomposition of larger clusters of students into smaller isolated units. In the European context, where classroom size is between 20 and 40 students, such cohorting strategies are best applicable to splitting classrooms in half. Simulating the transmission of SARS-CoV-2 in classrooms in England, Germany, the Netherlands, and Sweden, we show that splitting classrooms in half helps contain school outbreaks, substantially reducing the number of infected students. Cohorting proves particularly effective when conducted in a rota-system, with each cohort receiving in-person instruction in alternating weeks and remote instruction in the weeks they are not in the school building. The mechanism that produces this finding is straightforward: Cohorting facilitates social distancing and helps contain initial outbreaks in a single cluster. Combining cohorting with a rota-system is even more effective because infectious students cannot transmit the disease in the classroom in the remote learning weeks, halting outbreaks or preventing them in the first place.

Though effective in terms of containing outbreaks, classroom cohorting is disruptive to school routines. It requires either the introduction of hybrid learning modes (thus reducing in-person instruction) or further resources for same-day instruction (such as additional space or teacher capacities. Therefore, when cohorting is introduced, it should be designed to prevent SARS-CoV-2 outbreaks as effectively as possible. By design, cohorting prevents interaction between members of different cohorts within school. Whether this is sufficient to completely separate cohorts, however, also depends on whether there are cross-cohort out-of-school contacts.

Because these out-of-school contacts with classmates can transmit SARS-CoV-2 as well, All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10. 1101 /2020 cohorting strategies should also factor in students' out-of-school interactions in order to be effective.

Social network-based cohorting limits the spread of SARS-CoV-2 in school classes

In our simulations, we considered three cohorting strategies that account for students' outof-school contact with classmates in allocating them to cohorts. Our results show that all of these strategies outperform random cohorting by more frequently containing outbreaks to a single cohort. They also reduce the frequency of quarantines and the number of students infected, though the latter effects are weaker when transmission dynamics are limited-i.e., when instruction is organized in a rota-system, the risk of infection is low, the share of clinical cases is high (inducing early quarantine), and/or out-of-school contact is rare. Splitting cohorts by gender reduces transmission dynamics, reflecting the fact that adolescents' out-of-school contacts are mostly among students of the same gender. A cohorting strategy that explicitly minimizes the number of cross-cohort out-of-school contacts proves even more effective.

However, since this optimization strategy requires full knowledge of students' out-of-school contact with classmates, it might be difficult to implement in practice. Fortunately, a simple approximation based on the chained nomination of out-of-school contacts among students themselves also performs substantially better than random allocation and gender-split cohorting. In this strategy, an initial student names all classmates she meets outside of school.

The nominated students in turn indicate their within-class out-of-school contacts until the resulting nominations comprise half of the class. The resulting set of students then constitutes one cohort, the remainder of students the other. While somewhat less effective than explicit optimization, this strategy has the benefit of being much easier to implement in practice, as teachers do not need full information on contact networks but can allocate students through a simple method.

Once the decision to cohort classes has been made, choosing one of these non-random allocation strategies seems beneficial. When transmission dynamics are strong, these strategies help to reduce both the frequency of quarantines and the total proportion of infections. When transmission dynamics are less pronounced, their benefits are lower and largely concentrated on producing fewer quarantines. However, given uncertainty about All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10. 1101 /2020 transmission dynamics, a strategy that prevents cross-cohort interaction more effectively may still be preferable.

Some of our cohorting strategies may have drawbacks that are not modelled explicitly in our study. Separating cohorts by gender may be undesirable from a pedagogical perspective.

Furthermore, heterosexual romantic relationships among classmates may carry an elevated transmission risk and thus may serve as infection channels between gender-specific cohorts.

Finally, the extent of gender segregation varies across classrooms, such that splitting by gender may be less effective in some classrooms. Optimizing cohorts by minimizing out-ofschool contact across cohorts is not affected by such problems. However, this strategy is harder to implement in practice because schools need to know students' out-of-school contacts and be able to optimize cohorts based on this information. Our approximation to this optimization strategy, which builds on network chains created by subsequent nominations of out-of-school contacts among students themselves, does not face such limitations. However, this strategy may cause socially awkward situations because only some students are asked to name their out-of-school contacts, because some students may be disappointed when not nominated, or because students may not be cooperative. Still, unlike the optimization strategy, it requires less information on students' out-of-school contacts and no optimization techniques, thus offering a good compromise between effectiveness and practicability.

Our modelling approach has a number of limitations. The model assumes that there is some degree of out-of-school contact among classmates. If there is no interaction among students outside of the classroom-for example, in a very strict lockdown scenario-allocation to cohorts according to out-of-school contacts becomes irrelevant, or could, under certain conditions, even be harmful. If there is no transmission between cohorts because students cease to meet after school, the focus will shift to infection probabilities within classrooms. If the risk of infection in the classroom is elevated between students who meet after school, it may be beneficial to allocate close contacts to different rather than identical cohorts. Such potential correlated infection risks among out-of-school contacts within the classroom are not modelled in our simulations. We also do not consider teachers, who may connect cohorts, All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10. 1101 /2020 have different risks for infection, and may be both more infectious and more frequently symptomatic. However, because these teacher characteristics are constant across cohorting strategies, they are unlikely to affect the relative evaluation of the different strategies.

Similarly, the out-of-school contacts we consider are limited to classmates and do not extend to parents, siblings, or other interaction partners. In reality, if an infected student induces a symptomatic infection in such an interaction partner, this may trigger a (delayed) quarantine in the classroom when the student is tested belatedly. However, this is also unlikely to change conclusions about the relative effectiveness of the different cohorting strategies.

Our data also comes with a number of limitations. The data is from 2010-2011, and interaction patterns among students may have changed since then. Interaction patterns are particularly uncertain during a pandemic, in which students are likely to rapidly adapt their behavior to prevailing conditions. We have tried to capture such uncertainty by modelling a wide range of probabilities for interaction, but we have no recent empirical data on actual interaction patterns that could be used to calibrate our models. Finally, our network data is incomplete, and like other survey data, is likely to contain errors.

However, there is no immediate reason to expect these limitations affect our qualitative conclusions. In sum, our study shows that cohorting can decrease the transmission of SARS-CoV-2 in the classroom, thus lowering the number of quarantines and infections. We have demonstrated that simple and easily implementable strategies can further improve the effectiveness of cohorting by reducing cross-cohort out-of-school interaction with classmates.

The ensuing separation between cohorts limits the spread of SARS-CoV-2 across cohorts and can further reduce quarantines and infections, especially in situations with strong transmission dynamics. Therefore, we hope that this study demonstrates how real-world data on social networks can help to evaluate and improve social distancing strategies. All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10. 1101 /2020 

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The authors acknowledge support by the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grant INST 35/1134-1 FUGG. CILS4EU research project funded in the NORFACE ERA NET Plus Migration in Europe-programme. We also thank Rosemary McGunnigle-Gonzales for comments and editing.All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted December 2, 2020. ; https://doi.org/10. 1101 /2020