Seasonal variation in airborne infection risk in schools due to changes in ventilation inferred from monitored carbon dioxide

Abstract The year 2020 has seen the world gripped by the effects of the COVID‐19 pandemic. It is not the first time, nor will it be last, that our increasingly globalized world has been significantly affected by the emergence of a new disease. In much of the Northern Hemisphere, the academic year begins in September, and for many countries, September 2020 marked the return to full schooling after some period of enforced closure due to COVID‐19. In this paper, we focus on the airborne spread of disease and investigate the likelihood of transmission in school environments. It is crucial to understand the risk airborne infection from COVID‐19 might pose to pupils, teachers, and their wider social groups. We use monitored CO2 data from 45 classrooms in 11 different schools from within the UK to estimate the likelihood of infection occurring within classrooms regularly attended by the same staff and pupils. We determine estimates of the number of secondary infections arising via the airborne route over pre/asymptomatic periods on a rolling basis. Results show that, assuming relatively quiet desk‐based work, the number of secondary infections is likely to remain reassuringly below unity; however, it can vary widely between classrooms of the same school even when the same ventilation system is present. Crucially, the data highlight significant variation with the seasons with January being nearly twice as risky as July. We show that such seasonal variations in risk due to changes in ventilation rates are robust and our results hold for wide variations in disease parameterizations, suggesting our results may be applied to a number of different airborne diseases.


| INTRODUC TI ON
Since the outbreak began in December 2019, the novel coronavirus disease  caused by the SARS-CoV-2 virus has spread around the world resulting in a global pandemic that has become the predominant feature of 2020-2021. In an effort to contain the spread of the virus, the UK, along with many other nations, implemented lockdown measures including a "stay at home" order from March 2020, causing most public spaces to close. This led to schools closing, for all but children of key workers and children within certain These, along with their respective mitigation strategies, are detailed and discussed in a number of papers including. 3 We focus herein on relatively far-field (>2 m) airborne transmission where small infected respiratory droplets and aerosols remain suspended in the air where they can be transported by indoor air currents and inhaled. It is thought that this route may be significant in the spread of COVID-19, with growing evidence showing that it played a role in well-documented outbreaks: for example in the case of the Skagit Valley Chorale event for which Miller et al. 4 show that aerosol transmission has to be considered in order to account for the extensive resulting number of cases. While measures such as social distancing and appropriate cleaning procedures will reduce the risk linked to the droplet and contact routes, infections due to airborne transmission may be harder to contain. They require, along with other measures, 5 the appropriate use of ventilation as recommended by the WHO 6 but, as described by Bhagat et al., 7 the physics of indoor flows are complex, and thus, the consistent provision of clean air at appropriate rates to all desired locations within indoor spaces remains a scientific and engineering challenge.
To analyze the likelihood of airborne infection from COVID-19 caused by attendance at schools, this paper adopts the approach described in the seminal works of Riley et al. 8 and Rudnick & Milton,9 which were successfully applied to the transmission of measles, rhinovirus, and influenza. 10 We predict the absolute risk of airborne infection over a period of time in a given space, by taking CO 2 measurements and occupancy profiles and calculating the number of secondary infections, or the number of infections due to a single originally infected individual. We assume, as is currently required, that anyone showing any symptoms ceases attending school. We further assume that staff and students spend the vast majority of their school day in the classroom and thus infer that the risk of airborne infection is dominated by their time in the classroom which is currently likely to be the case in most schools, with break times being taken outdoors (minimal airborne infection risk) or within the classroom in inclement weather. We therefore calculate the likely consequences of airborne infection, the number of secondary infections, for pre/asymptomatic periods where the original infector continues to regularly attend the space, which, in this instance, is a classroom. The pre/asymptomatic period may be especially significant, as it corresponds to the period where infectivity is thought to reach a peak. 11 This approach is particularly applicable to primary schools (5-11 years old) where the same group of students can be assumed to attend the same classroom every day. It also remains suitable for secondary schools in which strategies have been implemented to reduce mixing between students, introducing, for example, fixed bubbles or groups of students in one classroom.
In addition, this methodology is useful because it does not require measurements nor estimates of ventilation flow rates, which are seldom recorded. Instead, it relies on the use of monitored CO 2 which is becoming increasingly widespread in newly built schools in the UK in an effort to assess the performance of ventilation systems along with showing compliance to regulations (e.g. the Department for Education 12 guidance). The focus on airborne transmission makes this study relevant to a wider range of airborne diseases, such as measles, influenza, or SARS, their spread being closely linked to ventilation as evidenced by Li et al. 13 Finally, this work provides a tool to assess Indoor Air Quality in schools more generally and thus ensure that pupils are provided with a suitable healthy learning environment.
We describe the application of the methodology to monitored CO 2 data in section 2 and how a measure of virus emission, the quanta generation rate, was chosen in section 3. Section 4 shows how the infection risk is estimated for a range of UK schools and how the resulting number of secondary infections varies between classroom and with the seasons. Finally, we draw conclusions in section 5.

| LIKELIHOOD OF AIRBORNE INFEC TION FOR CL A SS ROOMS AND N UMB ER OF S ECONDARY INFEC TI ON S ARIS ING VIA THE AIRBORNE ROUTE
Focusing only on transmission via the airborne route, we wish to determine the likely number of secondary infections that might arise within a given space should an infected individual attend the space. We assume here that all infections originate from a single infected individual. For many indoor spaces, a wide variety of people come and go with varying frequency, which makes the prediction of risk challenging. However, other indoor spaces are attended on a regular basis, day-in-day-out, and for significant durations each day by the same (or similar) group of people. We term these spaces "regularly attended spaces." Examples of these spaces include open-plan offices and school classrooms, the latter being the focus of this study.
In order to determine the likely number of secondary infections, S I , from a classroom we must first calculate the likelihood that airborne infection occurs when an infected individual regularly attends the classroom for some duration. This duration is often arbitrary, and a number of reasonable choices could be made, each leading to a different likelihood. However, for regularly attended spaces, for example, a school classroom, it is reasonable to assume that once an infected individual exhibits symptoms they cease attending. Thus, one can assess the likelihood of infection occurring during the pre/ asymptomatic infectious period, assuming the infector ceases to attended once symptoms develop. For COVID-19, the pre/ asymptomatic infectious period is estimated to be 5-7 days, therefore allowing us to examine the likelihood of infection occurring over five consecutive working days; that is, a duration T A = 5 weekdays.
For infection to occur via the airborne route, a susceptible occupant of the classroom must breathe in infectious particles that are being carried on the indoor air currents. The infectious particles, or aerosols, originate in the exhaled breath of an infector, and we assume that they remain in the "rebreathed air," that is, air that has already been breathed by another individual, which we take as a suitable surrogate for estimating airborne infection risk. The resulting probability of infection P A was described by Riley et al. 8 and then extended by Rudnick & Milton,9 as where λ is the infectivity rate, I is the number of infectors, n is the number of occupants in the space, q is the emission rate of infectious doses, known as the quanta generation rate (see section 3), and f is the fraction of rebreathed air. This, in turn, is defined as with C the monitored CO 2 within the space, C 0 indicating the outdoor ambient level and C a the concentration of CO 2 added through exhaled breath. This assumes that the main source of CO 2 is occupants, which is pertinent in classrooms and other spaces without combustion sources. In both the original formulation of Riley et al. 8 and latter form of Rudnick & Milton, 9 Wells-Riley models calculate the complement of the probability that no-one becomes infected within the duration (in our case T_ A ; hence, the exponential terms in (1) are by no means representative of the response to a cumulative dose. The method presented here assumes implicitly that susceptible people attending the classroom will not become infected elsewhere or by another route than the airborne one. This assumption allows us to determine the contribution of a specific setting (in our case the classroom) and transmission route (in our case airborne) to the spread of the disease, and thus quantify how it might vary with environmental factors as the seasons change.
Equation (1) cannot be directly applied to durations which cover varied occupancy, such as a classroom with students coming in during the day and empty at night. Herein, we assumed that each classroom was regularly attended by N = 33 people (typical in UK state schools) and the occupancy n varied between zero and N according to the school timetable (which was typically accessed through online records). The number of infectious individuals I was set to be a constant fraction of the current occupants; that is, I = n/N, which led to I/n = 1 when the space was fully occupied and I = 0 when empty. This rendered the fraction of occupants infected constant as I/n = 1=N. With these assumptions, the likelihood P A of airborne infection occurring due to a single infectious person attending school during a pre/asymptomatic period can then be calculated using: where the term σ(t) is a Heaviside operator based on whether the space is occupied or unoccupied, defined by This likelihood then provides the expected number of secondary infections that might arise via the airborne route from an infectious pre/asymptomatic person regularly attending school, via The rebreathed fraction f was found from data sets of CO 2 levels monitored in schools, described in section 4, where we took C 0 to be the average CO 2 within the space between the hours of 05:00 and 06:00 each day (which corresponds to an unoccupied space which has had time to reach a baseline CO 2 level after occupation of the previous day) and C a = 37,500 ppm. 9 The choice of a suitable value for the quanta generation rate q is discussed in section 3. We note that no assumption as to the distribution of CO 2 within the classroom has been made in deriving (2). As such, (2) provides the probability of airborne infection occurring at the sensor location, under the above assumptions, with the additional assumption that infectious airborne material is uniformly mixed but only within the rebreathed air. Hence, in order to employ (2) it is not required to assume that all of the air within the classroom is well-mixed but it is required to assume that the infected breath is evenly mixed within the rebreathed air.
Highlighting this emphasizes that when multiple CO 2 sensors are present within a single indoor space then differing CO 2 levels (1) an assumption which is valid in the case that all the air within the space is well-mixed.
Both P A and S I were calculated for all 5 weekday periods within the data gathered on a rolling basis (a total of 15,000 pre/asymptomatic periods in total). In the rolling absolute number of secondary infections reported herein, we exclude all values for which the 5 weekday period contained any unoccupied days, for example, those that included weekdays that fell in the school holidays.
The use of the measurement of CO 2 to infer the risk of airborne infection introduces several uncertainties caused by the choice of sensor location and the sensor itself. The sensors used in this study have been installed to control the ventilation provision, their location is fixed throughout the measuring period, and it can be assumed that they have been designed and maintained appropriately.
In addition, in order to limit the impact of these uncertainties as well as the ones introduced by a choice of quanta generation rate (see section 3), this paper focuses on reporting relative risks rather than absolute numbers.

| RE SULTS FOR THE S E A SONAL VARIATION IN AIRBORNE INFEC TION IN SCHOOL S
The data presented originate from recently built or renovated school classrooms, since these are more likely to have existing CO 2 monitoring provision installed. Overall, 45 spaces are monitored from within 11 different schools (8 primary and 3 secondary) and span the period November 2015 to March 2020 (when schools were subject to a UK wide lockdown), as detailed in Table 1. The data originate from schools in England, as far north as Yorkshire, as far south and west as Somerset, and as far east as Kent; the data are sourced from schools in a mix of urban and rural settings. The ventilation system varied between classrooms but was, in all cases, a hybrid ventilation system which switched between naturally driven ventilation and mechanically driven ventilation modes depending on the conditions.
The ventilation provision was controlled by automatic operation of louvers, vents, and fans. These classrooms were also typically fitted with additional windows or vents that could be opened manually by the classroom occupants.

| The airborne infection risk in one classroom in January and in July
To determine the airborne infection risk in schools, we begin by taking one classroom (selected at random) from within our set (45  Figure 1. We note that the absolute probabilities of airborne infection remain low in both cases, implying that changes in the likelihood will exhibit an approximately linear response to changes in the underlying parameters, for example, the quanta generation rate, and that results for the relative risk will be robust while the probabilities remain low.  Figure 2).

| The variation in airborne infection risk within one school
However, the variation between classrooms even within the same month is significant. The coefficient of variation of S I for the classrooms shown is 21% in January and 26% in July, with the most risky classroom being as least twice as risky as the least in both January and in July, in spite of the fact that all 17 classrooms from this particular school have the same ventilation system installed by the same contractor.
Finally, we note that variations are also observed between school years: a risky classroom one year being relatively less risky in the following academic year for instance. This could be explained by changes in the ventilation system, with adjustments being made to the building management system or interventions from the occupants, or could be due to a difference use of the space in a different academic year, for example, a class with either more or less pupils or different activities, or could be the result of differing weather conditions.

| The seasonal variation in predicted airborne secondary infections in schools
The variation in the absolute number of predicted secondary infections with season is presented in Figure 3A. We tested the seasonal variation in number of airborne secondary infections in schools for varied levels of quanta generations rate. For quanta generation rates 0.1 ≤ q ≤ 5 quanta/h, the quantitative predictions illustrated in Figure 3B remain almost unchanged.
For much high quanta generation rates 20 ≤ q ≤ 100 quanta/h, the qualitative trends in the data remain but the precise values change somewhat, for example, for q = 100 quanta/h S I in January would be predicted to be 41% higher than the July value. should be encouraged to review their behaviors to ensure that these are appropriate. These differences may be even more exaggerated in schools with no controlled ventilation provision, which represents a significant portion of the existing school stock within the UK.
Moreover, our assessment of the airborne infection risk as low may not apply to these schools but monitoring of CO 2 would provide the evidence required to make an assessment.
The seasonal variations in airborne infection risk described in this study account only for those due to changes in the indoor environment which arise due to ventilation and/or occupants' behaviors due to varying outdoor temperatures during the year. It is likely that the virus and the human response to it will also exhibit some seasonal variations and, as such, the risks of airborne infection might be compounded. If these increase the risk of transmission, for example, due to a weaker immune response in winter shown by Dopico et al. 19 or to changes in humidity as described for example by Marr Finally, the method we present is applicable to all airborne diseases. We have shown that, in particular, the variations in relative airborne infection risk with ventilation conditions, and hence season, hold over a wide range of quanta generation rates and so should be directly applicable to a wide range of diseases.

ACK N OWLED G EM ENTS
The data presented here were provided by

CO N FLI C T O F I NTE R E S T
The authors declare no competing interests.

AUTH O R CO NTR I B UTI O N S
CVMV led the study, analyzed the data, created the figures, and wrote parts of the manuscript. HCB conceived the study, secured the data, and wrote parts of the manuscript. CJN and PFL oversaw the study, supplied guidance, and advice throughout and edited the manuscript.

PEER R E V I E W
The peer review history for this article is available at https://publo ns.com/publo n/10.1111/ina.12818.

O RCI D
Henry C. Burridge https://orcid.org/0000-0002-0719-355X F I G U R E 3 The variation in (A) the absolute, and (B) the relative, average monthly number of secondary infections across the seasons. Data shown represent CO 2 monitoring in 45 classrooms (within 11 different schools) and spans the period November 2015 to March 2020 (see Table 1)