key: cord-0691932-l2i3aa3k
authors: Van Egeren, D.; Stoddard, M.; Malakar, A.; Ghosh, D.; Acharya, A.; Mainuddin, S.; Majumdar, B.; Luo, D.; Nolan, R.; Joseph-McCarthy, D.; White, L. F.; Hochberg, N. S.; Basu, S.; Chakravarty, A.
title: No magic bullet: limiting in-school transmission in the face of variable SARS-CoV-2 viral loads
date: 2022-03-31
journal: nan
DOI: 10.1101/2022.03.25.22272956
sha: a7cd9d065b8b87e0960ae903e4046a49a5c5d127
doc_id: 691932
cord_uid: l2i3aa3k

In the face of a long-running pandemic, understanding the drivers of ongoing SARS-CoV-2 transmission is crucial for the rational management of COVID-19 disease burden. Keeping schools open has emerged as a vital societal imperative during the pandemic, but in-school transmission of SARS-CoV-2 can contribute to further prolonging the pandemic. In this context, the role of schools in driving SARS-CoV-2 transmission acquires critical importance. Here we model in-school transmission from first principles to investigate the effectiveness of layered mitigation strategies on limiting in-school spread. We examine the effect of masks and air quality (ventilation, filtration and ionizers) on steady-state viral load in classrooms, as well as on the number of particles inhaled by an uninfected person. The effectiveness of these measures in limiting viral transmission is assessed for variants with different levels of mean viral load (Wuhan, Delta, Omicron). Our results suggest that a layered mitigation strategy can be used effectively to limit in-school transmission, with certain limitations. First, poorly designed strategies (insufficient ventilation, no masks, staying open under high levels of community transmission) will permit in-school spread even if some level of mitigation is ostensibly present. Second, for viral variants that are sufficiently contagious, it may be difficult to construct any set of interventions capable of blocking transmission once an infected individual is present, underscoring the importance of other measures. Our findings provide several practical recommendations: the use of a layered mitigation strategy that is designed to limit transmission, with other measures such as frequent surveillance testing and smaller class sizes (such as by offering remote schooling options to those who prefer it) as needed.

Continued high levels of SARS-CoV-2 transmission strain healthcare systems and accelerate viral evolution, which undermines vaccinal efficacy 1,2 , and generates new variants with unpredictable epidemiological characteristics. For example, the transmissibility of SARS-CoV-2 has increased over time, with the Delta variant being around twice as transmissible 3 as the original Wuhan strain, which had an R0 (reproduction number) of between 3.3 4 and 5.7 5 . The incubation time of the disease has also demonstrated evolutionary change, going from 5-21 days for the original Wuhan strain 6,7 to 2-4 days for the Omicron variant 8 .

In order to slow viral evolution by limiting transmission, it is thus important to understand the role of schools in facilitating SARS-CoV-2 spread. In most countries, a significant fraction of the population consists of K-12 students, staff and first-degree household contacts of students and staff. In the US, 40% of all households have a child at home under 18 years of age 9 , and 23% of the US population is enrolled in school 10 . Thus, in-school transmission of SARS-CoV-2 will substantially impact transmission dynamics in the whole population.

A number of studies during the early part of the pandemic led to the perception that SARS-CoV-2 did not spread in schools, based on the similarity in case counts between schools and their surrounding communities and a lack of observed transmission chains among children in schools.

However, the methodological validity of these conclusions is debatable, as the metrics being used to infer a lack of spread are themselves vulnerable to a "false negative" problem (absence of evidence is not evidence of absence) (see Supplementary Text S2, and 11 for a detailed critique on the limitations of current research arguing that SARS-CoV-2 does not spread in schools). In fact, there is now a robust body of evidence supporting the contention that SARS-CoV-2 spreads efficiently in schools that lack adequate infection control measures. Empirical analyses using county-level panel data in the United States have demonstrated that counties with fully open K-12 schools with in-person learning had a 5% increase in the growth rate of COVID-19 cases during April-December 2020 (a period of time when US schools were largely closed for the first five months) 12 . Consistent with this finding, COVID-19 symptom reporting was more common in areas where schools were open compared to areas with remote learning, an effect that was attenuated in communities using multiple mitigation measures 13 . In-school transmission is apparent when systematic surveillance testing methods are used [14] [15] [16] [17] , and dramatic increases in . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 31, 2022. ;  https://doi.org/10.1101/2022.03.25.22272956 doi: medRxiv preprint case detection rates have also been observed in studies that relied on surveillance as opposed to symptomatic testing 18 .

A layered mitigation strategy is one way to limit transmission in the school setting under conditions of widespread community transmission. There are multiple potential interventions available at this point: vaccines, masks, air quality improvements, surveillance testing, contact tracing/ isolation and podding. For reasons of cost and practicality, it is rational to seek a minimal set of infection-control measures. The challenge in this regard is that ongoing viral evolution can yield further changes to the characteristics of the virus, and a set of infection control measures that works well for one viral variant may readily be defeated by the next.

An important open question then is: "What is the design of a minimal set of infection-control measures in schools that is robust to variant-to-variant differences in viral load?" In this work, we use mathematical modeling of the steps involved in viral transmission to understand the impact of infection-control measures in a range of different scenarios corresponding to variants with differing viral loads. Our intent was to address both the feasibility and robustness of strategies for limiting SARS-CoV-2 transmission in schools.

To study SARS-CoV-2 transmission and the impact of control measures in schools, we created a multistep mathematical model of viral transmission in indoor settings (Methods) occurring under the assumption of indoor aerosol spread of infectious virus (see Supplementary Text S3 for justification of this assumption). First, we estimated the concentration of virions in the air over time in a room with an infected individual present using a differential equations model. This model assumes that infected individuals emit virions into the air at a constant rate into a room that is modeled as a well-mixed container (see Supplementary Text S4 for details about this assumption). Emitted virions can be inactivated over time or filtered out via air exchanges. These viral concentration estimates were then used as an input to calculate the probability that uninfected individuals in the room will become infected with SARS-CoV-2.

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) interventions in this and subsequent sections). We further explore the effects of different types of masks on the steady-state viral concentration later in this section (Fig. 6 ). After the infected individual leaves the room, the viral concentration returns to approximately zero over a 5-hour period in a closed room (Fig. S1 ).

Filtering the air in the room decreases the steady-state concentration of virions by increasing the rate at which infectious virions are eliminated from the room (Methods). For infected individuals with a low viral emission rate (e.g., infected with the original Wuhan SARS-CoV-2 virus), air filtration and masking of the infectious individual can reduce the steady-state concentration of virus in the room to less than one virion per liter of air (Fig. 2) . However, for SARS-CoV-2 variants and individuals with higher viral loads and therefore higher viral emission rates (e.g., with Omicron-like viral loads, represented here as the 'medium' rate, or with Delta-like viral loads, represented here as the 'high' emission rate), the steady-state viral concentration in the . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted March 31, 2022. ; https://doi.org/10.1101/2022.03.25.22272956 doi: medRxiv preprint room might remain high even with high filtration efficiency and filtration rate (Fig. 2) . Notably, for schools using high efficiency filtration systems (e.g., HEPA filters), increasing the rate of air exchange across the filter is important for minimizing transmission.

Another strategy to lower the viral concentration is to use ionizers, which inactivate viral particles and remove them from the air 19 . These devices produce small ions by the corona discharge principle, according to which negatively charged ions transfer their charge to suspended particles upon collision. These charged particles then agglutinate, becoming larger until they fall out of the air under the effect of gravity 20 . Ionizers generating negatively charged ions have been shown to be efficient at removing bacteria, molds, and viruses from indoor air [21] [22] [23] . The efficiency of particle removal is dependent on the emission rate of ions within an enclosed space, as well as room volume 24 . Studies conducted with smoke particles in an enclosed room suggest a high efficiency of ionizers in removing particles from the air, that varies between 80 and 100% 20, 24, 25 . Although older ionizer technologies generate ozone, which is an undesirable byproduct, newer ionizers do not have this potential liability associated with them 26 . As with the other control measures we simulated, we found that ionizers can lower viral concentrations in a typical classroom to below one virion per liter in situations where the infected individual is emitting viral particles at a relatively low rate (masked, infected with a low viral load strain) (Fig. 3) . However, if the individual is infected with a high viral load SARS-CoV-2 variant (e.g., Delta), the viral concentration in the room will be very high even when ionizers are being used (Fig. 3) .

We used the viral concentration estimates calculated in the previous section, estimates of aerosolized particle deposition derived from computational fluid dynamics (CFD) modeling, and the minimum infectious dose of SARS-CoV-2 to estimate the probability that uninfected individuals in the room will become infected with SARS-CoV-2.

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) Infection of a new host requires the virus to be inhaled and deposited on the airway mucosa, where it can replicate. To study transmission dynamics in indoor settings with airborne SARS-CoV-2, we used a published computational fluid dynamics model of airflow in the nasopharynx to estimate the number of inhaled viral particles that are deposited in the airway mucosa 27 . This computational model of airflow in the human nasopharynx estimates the probability that an inhaled virion will reach the airway mucosa, given the size of the liquid droplet in which it is suspended. We assumed individuals breathe in virions suspended in liquid droplets with the measured steady-state size distribution of expelled respiratory droplets 28 . We used the computational fluid dynamics modeling results to compute the overall probability that an inhaled virion will hit the nasopharynx, marginalized over the empirical droplet size distribution (Methods, Table S1), and found that approximately 0.6% of virions that are inhaled during each breath are deposited in the mucosa. Thus, in this scenario, with multiple environmental control measures (masking and air filtration) in place, transmission can still occur over an 8-hour school day. Thus, in a future scenario where . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 31, 2022. ; https://doi.org/10.1101/2022.03.25.22272956 doi: medRxiv preprint 7 we are faced with a Delta-like variant with high viral loads, schools may have to rely on additional measures (e.g., rapid and widespread surveillance testing, targeted school closures) to limit in-school spread. These measures would need to supplement and not replace in-school mitigation measures in order to be effective.

High-quality masks, used correctly, have been demonstrated to reduce infection risk even in high-risk settings [29] [30] [31] . Therefore, such masks can provide an additional level of mitigation in the event that schools are faced with a high viral load (Delta-like) variant. Masks provide a double benefit, as they reduce transmission by filtering out virions emitted by infected individuals and by reducing the number of ambient virions inhaled by uninfected individuals. Masks that filter out >95% of virions increase the time to transmission by approximately 10-fold when worn on either the infected or uninfected individual (one-way masking; Fig. 6 ). If both the infected and uninfected individuals are masked (two-way masking), less-effective masks (e.g., well-fitted surgical masks) can achieve the same level of protection against transmission. Combining universal N95 masking with excellent ventilation can increase the time to transmission of even high viral load strains to longer than a typical school day (Fig. 6D ), suggesting that layered mitigation strategies featuring well-fitted and high-quality masks are critical for the control of high viral load strains in the classroom.

Up to this point, we have considered the classroom to be a well-mixed container. This simple modeling approach allows us to identify settings where the risk of in-school transmission is high.

The well-mixed container assumption is justified, particularly in settings with limited ventilation (see Supplementary Text S4 for details). However, in some settings, the assumption may not hold, particularly in well-ventilated rooms, which are thought to have a low risk of transmission overall. To better understand the risk of transmission in a well-ventilated setting, we used CFD simulations of air transport inside a classroom. To simulate a best-case scenario for ventilation, . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Figure 7D visually depicts the airflow mixing trends inside the room and the virion-bearing streamline patterns for emitters located at different parts of the auditorium-style classroom. As the room is well-ventilated, the aerosolized virions emitted by the teacher (Fig. 7E ) standing on the podium at the front of the room (with a door located proximally on the side), as well as those from the student seated in the front row (Fig. 7F) , would escape through the door quite readily.

The situation is, however, different if the infected individual were seated in the middle of the room (Fig. 7G) , and even worse if the infected individual were seated at the rear (Fig. 7H ). In these two situations, the infected individual would be efficiently spreading aerosolized pathogens, via exhaled respiratory ejecta, through the entire room. Thus, for a well-ventilated room (where the well-mixed container assumption cannot be expected to apply) total viral load in the room depends strongly on the position of the infected individual in the room. Additionally, local virion concentrations may be sufficiently high to enable efficient viral spread in the absence of other countermeasures (such as masking). The chaotic airflow patterns (invisible to the naked eye) underscore the unpredictable downside of infection risk in a closed setting (Fig. 7H) . Thus, while it is possible to use model-based approaches to identify settings with a high risk of transmission, model-based approaches that rely on the well-mixed container assumption cannot definitively identify indoor settings with a low risk of transmission. This finding further underscores the need for multiple layers of intervention, and a robust ability to detect outbreaks before they spread.

Environmental control measures (masking, air filtration, ionizers) can all have an impact on limiting transmission in the in-school setting. However, our work suggests that these measuresboth individually and in concert-are all vulnerable to defeat by a sufficiently high burden of . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 

In this study, we have used mathematical modeling to demonstrate the strengths and limitations of a layered mitigation strategy in limiting in-school transmission while keeping schools open.

We examined the impact of risk mitigation measures (masking, ionizers, ventilation, filtration) on limiting spread within a classroom when an infected person is present. Our findings underscore the critical importance of layered mitigation strategies in limiting in-school transmission. With that said, all of the examined measures can be readily defeated by sufficiently high viral loads, a biological change that has already been observed during the pandemic (for example between the Wuhan strain and the Delta variant). This is a crucial point: minimal effective measures for the disease as it is at present may have an increased risk of failure in the face of new variants of SARS-CoV-2. Our findings also indicate that the risk of transmission in schools may be hard to predict in certain settings (such as in the turbulent airflow patterns of a well-ventilated room). As a corollary, our work points to the central importance of relying on . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. This work has a number of assumptions and key limitations. We have assumed for most of the work that the classroom is a well-mixed container, and then demonstrated (Figure 7 ) that the failure of this assumption leads to higher risk than could be estimated from a well-mixed container assumption. We also assume that children are equally susceptible and infectious as adults (see Supplementary Text S1 for an in-depth discussion of this assumption). It assumes perfect compliance with mask-wearing, which is not likely to be true in practice 37 the discussion by pointing out the need to take evolutionary-driven epidemiological changes (for example due to viral load from one variant to the next) into account. In his book The Black Swan (now a classic in the risk-management community), author Nasim Nicholas Taleb argues that the key to risk management lies not so much in predicting the worst thing that could happen, but in making plans that are robust to that outcome.

Our modeling demonstrates that a layered mitigation strategy, implemented properly, can curtail viral transmission under many circumstances. With that said, there are ways to implement infection-control measures that are ineffective, and measures that are effective in the presence of . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 31, 2022. ; https://doi.org/10.1101/2022.03.25.22272956 doi: medRxiv preprint one viral variant can be readily rendered ineffective in the presence of another. Because many of the interventions have a nonlinear effect on risk mitigation, cutting corners on risk mitigation steps can degrade their utility very quickly, turning them into "hygiene theater". For example, we found that two-way masking with N95 masks increases the time until SARS-CoV-2 transmission by multiple orders of magnitude, as compared to one-way masking with cloth or surgical masks (Fig. 6) . As a corollary, it is crucial for schools to have controls in place to ensure that measures taken for mitigation are working as intended. For example, air quality can be monitored using carbon dioxide monitors, and mandatory (as opposed to opt-in) testing can be used to monitor the functional outcomes of in-school mitigation. Mitigation strategies should be pressure-tested using simple mathematical modeling approaches such as the one described in this paper. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 31, 2022 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 31, 2022 

To estimate the risk of viral transmission we built a multistep model comprising emission of viral particles into the air and subsequent inhalation and deposition into the nasopharynx of uninfected individuals (Fig. S2) .

We designed a mathematical model of viral concentrations in a room in which an infected individual emits airborne SARS-CoV-2 particles at a constant rate. We assumed that the airborne virions are distributed uniformly throughout the room (i.e., the air in the room is well-mixed).

Virions are inactivated at a constant rate and have an average lifetime of approximately 1.6 hours indoors at 73°F, 55% humidity 47 . For a closed room with no air exchange or filtration, the change in virion concentration over time is given by the differential equation

where α is the viral emission rate, δ is the viral inactivation rate, V is the volume of the room, and t is the amount of time elapsed since the infected individual entered the room. Therefore, the viral concentration in the air over time C(t) is

Air filtration removes virions in the room by filtering out a certain fraction of viruses that pass through the filter. Assuming the room is well-mixed, this adds an additional first-order virus removal process with rate εβ/V where ε is the fraction of virions that are eliminated while passing through the filter (between 0 and 1), β is the rate at which room air is passed through the filter (in units of air exchanges per hour), and V is the room volume. With filtration, the concentration in the air is

Ionizers simply increase the virus inactivation rate δ.

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) For the original Wuhan virus strain, we estimated the rate at which infected individuals emit viral particles into the air using published data which show that rooms with infected individuals have approximately 2-6 viral copies per L of air 48 . Assuming these measurements were taken when the virus was at a steady-state concentration in the room and that these individuals were in a reasonably sized room (4000 cubic feet of air), we estimated that the viral emission rate would be approximately 2319-6937 virions/h using our equation for the viral steady-state contribution shown in the Results. Therefore, we used a value within this range (5000 virions/h) for the emission rate for the low viral load simulations in this study. See

Supplemental Materials for details about parameter estimates for various interventions.

To cause a new infection, virions must be inhaled by an uninfected host and be deposited in the airway mucosa, where they can replicate and cause disease. The rate at which virions are inhaled is the product of the respiratory tidal volume (volume of air inhaled during each breath) and the respiratory rate. However, computational fluid dynamics suggests that a minority of It is noted that when wind encounters a blocking effect on its path owing to a building, the velocity pressure is converted into static pressure. Consequently, on the windward side, the pressure would increase, with consequent pressure reduction on the leeward side. The static pressure gradient generated by wind on the building surface can be estimated by:

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) where ∆ , is the static pressure difference, -is the wind pressure coefficient, . is the density of outside air, and is the mean wind velocity. Thevalue, which is independent of wind speed but depends upon wind direction and incident angle, can be calculated for low storied buildings (for up to 3 stories) 51 . The other physical parameters, namely . and are taken as 1.139 Kg/ / and 2.7 m/s, from earlier studies 52 and meteorological data 53 , respectively. Air dynamic viscosity was assumed to be 1.9065 x 10 %0 kg/m.

However, there will be another pressure difference -one between the classroom and the outside corridor (next to the door). From established building code standards 54 , we can conclude that pressure differential ∆ *12..3 between the classroom and corridor will be 5 -20 Pa, resulting in the net pressure gradient from window-to-door (i.e., main inlet to main outlet) to be ∆ &.&45 = ∆ , + ∆ *12..3

From the above assessments, the individual pressure gradients approximate to ∆ , = 3.5 Pa and ∆ *12..3 = 5 Pa. In the simulations: windows were taken as pressure inlets with 0-gauge pressure, nostrils were taken as velocity inlets with volumetric flux at 15 L/min 27,55-58 to replicate gentle steady breathing, and at the door, we imposed a pressure outlet condition with negative gauge pressure of ∆ &.&45 , i.e., -8.5 Pa. Wall boundary condition was mimicked as a stationary wall with no slip condition.

The probability that at least one student in a classroom arrives infected with SARS-CoV-2 was estimated from the overall prevalence of the infection using Poisson statistics. This probability P was estimated as

where n is the number of students in the classroom and f is the fraction of infected individuals in the overall population. To estimate the expected number of exposed individuals in a school with N total students, the entire student population was broken into classes with i students each. The . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted March 31, 2022. ; https://doi.org/10.1101/2022.03.25.22272956 doi: medRxiv preprint probability that at least one student in each class Pi was estimated using the above equation, and the expected number of students exposed to an infected individual was then calculated as NPi.

Scripts implementing the viral concentration and transmission models are available on GitHub at https://github.com/dvanegeren/covid-indoor-transmission. Input data used to estimate the fraction of inhaled droplets that hit the nasopharynx are also available in the same GitHub repository.

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