key: cord-0699686-7xhuq3wg
authors: Hou, D.; Katal, A.; Wang, L.
title: Bayesian Calibration of Using CO2 Sensors to Assess Ventilation Conditions and Associated COVID-19 Airborne Aerosol Transmission Risk in Schools
date: 2021-02-03
journal: nan
DOI: 10.1101/2021.01.29.21250791
sha: 7434024507a1f167d6a283757a5ef4ec5c257773
doc_id: 699686
cord_uid: 7xhuq3wg

Ventilation rate plays a significant role in preventing the airborne transmission of diseases in indoor spaces. Classrooms are a considerable challenge during the COVID-19 pandemic because of large occupancy density and mainly poor ventilation conditions. The indoor CO2 level may be used as an index for estimating the ventilation rate and airborne infection risk. In this work, we analyzed a one-day measurement of CO2 levels in three schools to estimate the ventilation rate and airborne infection risk. Sensitivity analysis and Bayesian calibration methods were applied to identify uncertainties and calibrate key parameters. The outdoor ventilation rate with a 95% confidence was 1.96{+/-}0.31ACH for Room 1 with mechanical ventilation and fully open window, 0.40{+/-}0.08 ACH for Rooms 2, and 0.79{+/-}0.06 ACH for Room 3 with only windows open. A time-averaged CO2 level < 450 ppm is equivalent to a ventilation rate > 10 ACH in all three rooms. We also defined the probability of the COVID-19 airborne infection risk associated with ventilation uncertainties. The outdoor ventilation threshold to prevent classroom COVID-19 aerosol spreading is between 3-8 ACH, and the CO2 threshold is around 500 ppm of a school day (< 8 hr) for the three schools.

analyzed a one-day measurement of CO2 levels in three schools to estimate the ventilation rate and 1 airborne infection risk. Sensitivity analysis and Bayesian calibration methods were applied to 2 identify uncertainties and calibrate key parameters. The outdoor ventilation rate with a 95% 3 confidence was 1.96 ± 0.31ACH for Room 1 with mechanical ventilation and fully open window, 4 0.40 ± 0.08 ACH for Rooms 2, and 0.79 ± 0.06 ACH for Room 3 with only windows open. A 5 time-averaged CO2 level < 450 ppm is equivalent to a ventilation rate > 10 ACH in all three rooms. 6

We also defined the probability of the COVID-19 airborne infection risk associated with 7 ventilation uncertainties. The outdoor ventilation threshold to prevent classroom COVID-19 8 aerosol spreading is between 3 -8 ACH, and the CO2 threshold is around 500 ppm of a school day 9 (< 8 hr) for the three schools. 10

The actual outdoor ventilation rate in a room cannot be easily measured, but it can be calculated 12 by measuring the transient indoor CO2 level. Uncertainty in input parameters can result in 13 uncertainty in the calculated ventilation rate. Our three classrooms study shows that the estimated 14 ventilation rate considering various input parameters' uncertainties is between ± 8-20 %. As a result, 15 the uncertainty of the ventilation rate contributes to the estimated COVID-19 airborne aerosol 16 infection risk's uncertainty up to ± 10 %. Other studies can apply the proposed Bayesian and 17 MCMC method to estimating building ventilation rates and airborne aerosol infection risks based 18 on actual measurement data such as CO2 levels with uncertainties and sensitivity of input 19 parameters identified. The outdoor ventilation rate and CO2 threshold values as functions of 20 exposure times could be used as the baseline models to develop correlations to be implemented by 21 cheap/portable sensors to be applied in similar situations to monitor ventilation conditions and 22 airborne risk levels. 23

The Harvard-CU Boulder Portable Air Cleaner Calculator 9 suggests a total of five air changes per 1 hour as a good ventilation condition for reducing airborne transmission risk in classrooms. 2

While recommendations are mainly based on the ventilation rate, it has been a challenge to 3 quantify the outdoor air ventilation rate in a room. Indoor air CO2 concentration is often considered 4 a surrogate/indicator for the ventilation rate. For example, the Montreal school board (Centre de 5 services scolaire de Montreal) stated in an open letter on December 14, 2020: "Establishments 6 without a mechanical ventilation system should apply the window opening guidelines to ensure 7 frequent air changes in our premises"; "Always in order to ensure good indoor air quality, we have 8 also started measuring carbon dioxide (CO2) in our establishments since November. In addition to 9 this initiative, there are the CO2 tests that must be carried out by all school service centers in 10 Quebec before December 16. The level of CO2 is a good indicator of the supply of fresh air in a 11 room. Thus, following these tests, corrective measures will be put forward, if necessary." 13 . 12 In the literature, several studies used a transient CO2 mass balance method and measured CO2 13 levels to calculate the ventilation rate in different indoor environments such as classrooms and 14 university libraries 10-12 . Batterman (2017) 12 estimated the CO2 generation rate based on the age 15 and assumed activity level for CO2 calculation in mechanically ventilated classrooms, but the real 16 activity type was unknown, so the results were subject to uncertainties. They used the whole day 17 data to estimate ventilation rate, but they did not validate the model by calculating the CO2 at a 18 different time or day. For the estimation of ventilation rate using the transient CO2 sensor data, it 19 is essential to find dominant parameters that affect the final results, calibrate the model by 20 measurement data, quantify and report the ventilation rates and infection risks with uncertainties. 21

As a result, due to various factors affecting CO2 levels, such as variable occupant numbers and 22 outdoor conditions, and the fact the unknown uncertainties of these factors, questions have been 23 . CC-BY-NC-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) of indoor air below which airborne propagation of typical respiratory infections will not occur. 22

They modeled several hypothetical cases in their work without measurement data. Peng and 23 . CC-BY-NC-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 February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint Jimenez (2020) 31 derived an analytical expression of CO2-based risk proxies for COVID-19 and 1 used it to estimate CO2 level corresponding to an acceptable airborne risk level in different indoor 2 environments. They showed that acceptable CO2 level varies by over two orders of magnitude for 3 various rooms and activities. It also depends on other factors, such as wearing face masks. No 4 measurements and uncertainties were reported. Eykelbosh (2021) 14 reviewed several studies that 5 used indoor CO2 level in assessing the transmission risk and concluded that indoor air CO2 level 6 could only represent the ventilation condition. The infection risk does not depend only on the 7 ventilation rate, and other factors such as wearing a face mask, using portable air cleaner, and 8 exposure time can also affect the infection risk. Therefore, to estimate the required ventilation rate 9 and critical CO2 level to prevent the transmission of COVID-19 aerosols in a classroom, it is 10 crucial to know the actual room condition such as occupancy profile, activity type, and other 11 parameters that affect the estimation of infection risk. 12

Therefore, to answer the "CO2 for Risk Assessment" question, in this work, we used the calibrated 13 ventilation rate and actual room parameters to calculate the COVID-19 airborne transmission in 14 the three classrooms using the modified Wells-Riley equation. We compared the results with 15 infection risk corresponding to the Reproductive number to be one (R0 = 1), and different 16 ventilation rates, and CO2 threshold levels at various exposure durations. 17

This section presents the models of CO2 concentration, airborne infection risk, and sensitivity 19 analysis, and finally, the Bayesian calibration methods. Two fully-mixed transient mass balance 20 equations are solved to calculate indoor air CO2 and COVID-19 quanta concentration. The 21 schematic of the models is plotted in Figure 1 . 22 . CC-BY-NC-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. 

where is the room volume ( 3 ); 2 is the indoor air CO2 concentration ( 3 ⁄ ); t is the 6 time duration ( ); is the CO2 generation rate by all occupants ( ⁄ ), which depends on the 7 age and activity level; 1 is the total outdoor air ventilation rate ( 3 ⁄ ); and is the outdoor 8 air CO2 concentration ( 3 ⁄ ). 9

The transient mass balance, Eq. 1, is useful for solving arbitrary occupancy patterns and generation 10 rates such as classrooms because students leave the classroom for break and launch. The solution 11

of Eq. 1 is: 12

where 2 ,0 is the observed initial CO2 concentration at each occupancy phase. 13 . CC-BY-NC-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)

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The probability of infection (PI) of a susceptible person in the room is calculated using the Wells-2

Riley formulation 32 . The method was first used by Jimenez et al. 33 for calculating infection risk 3 in different indoor environments and is recently applied to the City Reduced Probability of 4

Infection (CityRPI) model and used for city-scale infection risk analysis 34, 35 . PI is a function of 5 the number of quanta inhaled by the susceptible person (Eq. 3). We assumed that social 6 distancing is maintained between all occupants, and the current study focuses on airborne aerosol 7 transmission only. We used five assumptions for applying this model: i) there is only one infected 8 person in the room who emits SARS-CoV-2 quanta with a constant rate, ii) the initial quanta 9 concentration is zero, iii) the latent period of the disease is longer than the duration students stay 10 in the classroom. Therefore the quanta emission rate remains constant during the day, iv) the indoor 11 environment is well-mixed, and v) the infectious quanta is removed as a first-order process by the 12 ventilation, filtration, deposition on surfaces, and airborne inactivation. The PI in Eq. 1 is based 13 on the attendance of one infected person in the room, so it calculates the probability that COVID-14 19 aerosols are transmitted from the infected person to a susceptible person in the room; therefore, 15 it is a conditional probability of infection (PIcond). 16

The number of quanta inhaled by the susceptible person at the exposure time T is calculated by 17 time-averaged quanta concentration. 18

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The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint B is the inhalation rate ( 3 ℎ ⁄ ); , is the time-average quanta concentration ( 3 ⁄ ); is the 1 exposure time (ℎ);

is the fraction of people in the room who wears the mask, and is the 2 inhalation mask efficiency. A well-mixed transient mass balance equation similar to Eq. 1 is solved 3 to calculate the room's transient quanta concentration. 4

where is the indoor quanta concentration ( 3 ⁄ ); is the net quanta emission rate (ℎ −1 ); and 5 is the first-order loss rate coefficient for quanta (ℎ −1 ) . Assuming that the initial quanta 6 concentration is zero at the beginning of the day, Eq. 5 is solved as follows: 7

Because of the change in the occupancy pattern during the day, the time-averaged quanta 8 concentration is calculated using the Trapezoidal integration. is calculated based on the number 9 of infected people in the room , the fraction of people in the room with the mask , exhalation 10 mask efficiency , and quanta emission rate by one infected individual ER . 11

The first-order loss rate coefficient reflects several mechanisms: outdoor air ventilation 1 , 12 filtration 2 , deposition on surfaces 3 , and airborne inactivation 4 . 13

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The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint 1 is the air change rate of outdoor air per hour (ℎ −1 ) by the HVAC system or opening windows. 1 2 is the in-room air filtration using portable air purifiers and/or duct filters in HVAC systems. 3 2 is the removal by gravitational settling. Finally, 4 is the inactivation/decay rate. 3

As already mentioned, the is the conditional probability of infection, assuming that there is 4 one infected person in the room. The prevalence rate of the disease in the city can be used to 5 estimate the number of possible infected individuals in the room. Therefore, the absolute 6 probability of infection is calculated using the , , and number of susceptible 7 people in the room 33 : 8

is calculated using the total number of people in the room , number of infected persons, 9 and the fraction of immune people in the community . It can be estimated using the total 10 recovered cases in the study region 36,37 . 11

The prevalence rate of the disease is estimated using the daily COVID-19 statistics reported by 12 official sources: 13

where is the daily new cases, is the population, is the fraction of unreported cases. 14 A study on ten diverse geographical sites in the US shows that the estimated number of infections 15 was much greater than the number of reported cases in all sites 36 . By doing more daily tests, the 16 . CC-BY-NC-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 February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint current unreported fraction is around 0.8. is the duration of the infectious period of SARS-CoV-1 2 of 10 days in this work 38 . 2

Both CO2 and infection risk models include uncertain parameters such as ventilation rates and 4 emission rates. Some of the uncertain parameters may impact the result's accuracy and should be 5 calibrated by measurement data. Ideally, with sufficient measurements and computer resources, 6 all the uncertain parameters should be included in the calibration parameters. In reality, limited by 7 data quality/quantity or computer resources, only a few parameters could be included. Many 8 parameters and inputs could also manifest different levels of uncertainties and significances on 9 simulation outputs. So it is impracticable and unnecessary to calibrate all parameters, but for 10 dominant parameters only. Identifying these dominant parameters cannot merely rely on arbitrary 11 parameter selections from modelers' knowledge but should be based on a scientific process, i.e., a 12 sensitivity analysis. 13

To conduct a sensitivity analysis process, prior distributions and ranges of selected unknown 14 parameters should be determined. Then Monte Carlo (MC) simulation is employed to propagate 15 simulations whose model parameters' values are randomly chosen from the predefined ranges 16 using a specific sampling method to perform simulation runs iteratively. Here, the Latin Hypercube 17 Sampling (LHS) method 39 is applied since it provides good convergence of parameter space with 18 relatively fewer samples. The obtained input-output dataset is then employed to identify the 19 dominant model parameters that strongly affect the outputs. In this study, 440 parametric 20 simulations were conducted for the sensitivity analysis of the CO2 concentration model 40 . 21 . CC-BY-NC-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.

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The importance ranking results may vary with different combinations of sensitivity methods and 1 outputs depending on the variety of fundamental algorithms and conditions of each sensitivity 2 analysis method 41 . To avoid the potential inconsistency, Lim and Zhai proposed a new sensitivity 3 analysis method, sensitivity value index (SVI), to account for the differences in sensitivity analysis 4 methods and target outputs 42 . Eq. 12 shows how SVI is applied to recognizing and comparing the 5 importance rankings from different sensitivity analysis methods through the normalization and 6 aggregation process. 7

where , is the value of a sensitivity analysis method, is a parameter, is the total number of 8 the parameters, is a sensitivity method, is the total number of sensitivity methods, is the target 9 output, and is the total number of target outputs. 10

As the footstone of all Bayesian statistics, Bayes' theorem was first proposed by Reverend Thomas 12 Bayes in his doctoral dissertation 43 and can be described as: 13

The probability of an event is inferred based on the prior knowledge of conditions related to the 14 event. Bayesian inference is one application of Bayes' theorem and can be written as: 15

. CC-BY-NC-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 reality, it is impractical to apply the Bayesian inference for analytical solutions to all problems 7 because the likelihood's integrals can be computationally expensive or sometimes impossible to 8 be calculated. MCMC is a versatile approach to solve the parameter estimation problem with two 9 components. One is the well-known Monte Carlo method. It is a computational algorithm to solve 10 statistically challenging problems relying on repeated random samplings and approximate the 11 target value (e.g., mean value) using the independent samples' results. The other is the Markov 12 Chain method for solving a sequence of possible events. The probability of each event depends 13 only on the state attained in the previous event. By combining MCMC and Bayesian inference, 14 posterior distribution can be estimated efficiently. 15 Different MCMC algorithms can be classified into either a "random walking" group or a gradient-16 based group according to the acceptance-rejection criterion's adoption. In this study, Hamiltonian 17

Monte Carlo (HMC) sampling method 44 was used for the MCMC. Five thousand steps of the 18 HMC algorithms on each of four separate chains were explored in this study to make a total of 19 20,000 samplers. We used one thousand samples during the "warming-up" stage to move chains 20 . CC-BY-NC-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 February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint toward the highest density area and tune sampler hyperparameters. For each room, the first 2/3 of 1 measurements are used for the calibration, with the remaining for the validation. 2

In this study, the Coefficient of Variance of Root Mean Squared Error (CVRMSE) (Eq. 15) and 3

Normalized Mean Bais Error (NMBE) were used as indicators to estimate the calibration and 4 validation performance. 5

where ̂ is a predicted variable value for period , is an observed value for period , ̅ is the 6 mean of the observed value, and is the sample size. 7

In this study, three typical classrooms of three different schools in Montreal, Canada, were selected 9 for calibration and infection risk analysis. Each classroom was monitored during a typical 10 pandemic day, and occupants' information (students' age and number), ventilation system status, 11 window status, and transient indoor CO2 concentration are recorded. The summary information is 12 shown in Table 1 and Figure 2 . 13 14 15 . CC-BY-NC-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 February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint Notes: MV means mechanical ventilation; NV means natural ventilation. 2 Room 1 is equipped with a mechanical ventilation system, and CO2 is between 500 ~ 1000 ppm, 3 while for Room 2 and Room 3, with no access to a mechanical ventilation system, the CO2 reaches 4 up to 1800 ppm. The CO2 in Room 1 seems to indicate an acceptable level of air quality (<1000 5 ppm), and for Rooms 2 and 3, it is higher than the acceptable level of air quality requirements. The 6 occupancy pattern of Room 1 and Room 2 was measured, but for Room 3, no occupancy pattern 7 was recorded, so we assumed that number of students is constant as recorded in the morning during 8 the day. The outdoor air temperature and pressure data are provided by Environment and Climate 9

Change Canada 45 . Other parameters required for the CO2 calculation, such as outdoor air CO2, 10 generation rate, and outdoor air ventilation rate, are not available; therefore, we calibrate them 11 using the CO2 measurements. 12 . CC-BY-NC-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.

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In this section, the sensitivity analysis finds the dominant parameters for the calculation of CO2 4 concentrations. Then we use the Bayesian MCMC calibration to estimate the daily average 5 ventilation rate using the CO2 measurement data and occupancy patterns. With the calibrated 6 model, we calculate the CO2 concentrations in the three rooms under different ventilation rates. 7

Finally, we used the calibrated ventilation rate to estimate the infection risk in each classroom. We 8 . CC-BY-NC-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 February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint also find the ventilation rate and CO2 level thresholds to avoid the airborne COVID-19 aerosol 1 spread for different exposure times. 2

Outdoor/indoor pressure, outdoor/indoor air temperature, occupancy number, room volume, 4 outdoor air ventilation rate, and CO2 exhale rate are input parameters to predict CO2 concentration. 5

For the sensitivity analysis, the range of selected model inputs/parameters were defined according 6 to the references, codes, and standards available. Table 2 shows the parameters with their 7 sensitivity importance rankings: a smaller number indicates a more important/sensitive parameter. 8 

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The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint It is concluded that, for the classroom CO2 levels, the most dominant parameters are outdoor air 1 ventilation rate, CO2 generation rate per person, number of occupants, and outdoor CO2 2 concentration. Specifically, the outdoor air ventilation rate's SVI is two times the second important 3 parameter. Some sensitive parameters are often known, such as occupant number, outdoor 4 temperature, and pressure. So they may not need to be calibrated. Therefore, we selected the 5 outdoor air ventilation rate, CO2 generation rate, outdoor CO2 concentration, and indoor air 6 temperature for the next step's model calibration. Indoor pressure was assumed to be identical to 7 the outdoor pressure. 8

For the calibration of the CO2 model, the Bayesian inference method was applied. For each 10 occupancy phase (e.g., between every two measurements), we use the new measured CO2 data as 11 the initial condition for Eq. 2. The posterior distributions are shown in Figure 3 and Table 3 . In 12 each subplot, the red dash line represents the parameter's prior distribution in Table 2 . 13 . CC-BY-NC-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 February 3, 2021. The posterior distributions with the Bayesian calibration are indicated by blue, green, and orange 2 for Rooms 1, 2, and 3, respectively. The calibrated outdoor ventilation rate is 1.96 ± 0.31 ACH 3

for Room 1, 0.40 ± 0.08 ACH for Room 2, and 0.79 ± 0.06 ACH for Room 3. Here, the 4 ventilation rate is expressed by the calibrated mean value followed by the uncertainty for a 95% 5 confidence interval. Room 1 is mechanically and naturally ventilated (i.e., open windows), so its 6 ventilation rate is significantly higher than Rooms 2 and 3, with the outdoor air only from open 7 windows. The results of Room 2 and Room 3 are closer since both are naturally ventilated with 8 the same room volumes. The span of the posterior distribution of Room 1 is larger because of its 9 wider prior distribution. For Room 3, a constant occupancy was used, so it is relatively easier for 10 the MCMC iteration to find a posterior distribution. For the outdoor CO2 level and indoor air 11 temperature, Room 2 and Room 3 are closer than Room 1 due to different ventilation modes. 12 13 14 15 . CC-BY-NC-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 February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint Using the mean value of the calibration parameters, we compared the simulation results and 3 measurements of CO2 in Figure 4 . The simulation results show similar trends well as the 4 measurements. According to the American Society of Heating, Refrigerating, and Air-5 conditioning Engineers (ASHRAE) guideline 14 49 and FEMP 50 , when the CVRMSE and NMBE 6 values are less than 30% and ±10% for transient data, the calibrated accuracy meet the 7 requirements. The (CVRMSE, NMBE) of validation of Rooms 1-3 are (15.3, 7.6), (10.5, 6.1), and 8 (12.5, 12. 3), respectively, which shows that the validated accuracy is satisfied. 9

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The copyright holder for this preprint this version posted February 3, 2021. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint Fig. 4 Comparison of simulated and measured CO2 levels in three schools. 1

The calibrated ventilation rates in all three classrooms are all less than 2 ACH. The recommended 3 ventilation rate for a safe indoor environment by Harvard-CU Boulder Portable Air Cleaner 4

Calculator for Schools is at least 5 ACH. Therefore, the ventilation rate of all three classrooms 5 seems inadequate. To relate the CO2 levels, e.g., measured from a CO2 sensor, and the ventilation 6 rates, by using the calibrated CO2 model, we calculate the outdoor air ventilation rate in ACH and 7 CFM/person as a function of CO2 levels at different exposure times in Figure 5 . It helps teachers 8 find the room ventilation rate directly based on the CO2 sensors at different school hours. For 9 example, for Room 1, when CO2 > 600 ppm, OA (outdoor air) < 5 ACH (24 CFM/person); CO2 > 10 800 ppm, OA < 2 ACH (9 CFM/person) at any time of the day. A CO2 level less than 480 ppm 11 indicates a ventilation rate greater than 10 ACH (48 CFM/person) at all times. For Room 2, the 12 same CO2 levels correspond to a lower ventilation rate than Room 1. For example, when CO2 > 13 600 ppm, OA < 2 ACH (13 CFM/person); CO2 < 440 ppm indicates that OA > 10 ACH (66 14 CFM/person). This shows that Room 1 shows a higher ventilation rate (ACH) for the same CO2 15 level because of its smaller size. The ventilation rate of Room 3 at a specific CO2 level is higher 16 than Room 2 because of its constant occupancy compared to the variable occupancy of Room 2. It 17 seems lunch breaks indeed lower CO2 levels significantly (thus infectious risk in schools). For 18 Room 3, CO2 > 600 ppm indicates OA < 3 ACH (22 CFM/person). These results show that the 19 indoor CO2 could vary significantly in different classrooms even with the same ventilation rate 20 because of different room sizes, occupants number, and occupancy schedule. All three classrooms 21

show that an indoor CO2 lower than 450 ppm indicates a ventilation rate greater than 10 ACH, 22

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The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint close to the recommended 12 ACH value to prevent airborne transmission in health-care 1 facilities 7,8 . Note that here CO2 is the exposure-time-averaged instead of the instantaneous level. 2 . CC-BY-NC-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.

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To study CO2 and airborne aerosol infectious risk relation, we first calculate the probability of 3 infection risk with the posterior distribution of the ventilation rate obtained in Section 4.3. Note, 4

Here, the Bayesian and MCMC analysis allow us to quantify the uncertainties of the ventilation 5 rates to estimate airborne infectious risk by defining the probability of the infection risk: the 6 probable range of the infection risk estimated in classrooms. Then, we evaluate different 7 ventilation rates to identify the corresponding CO2 threshold level, at which the reproductive 8 number, R0 < 1, at all exposure times. We estimate the infection risk and indoor air CO2 threshold 9 based on the actual room conditions in this work. The recommended threshold could be used for 10 other rooms under a similar condition. 11 Table 4 shows the input parameters for Eqs. 3-11 to calculate the COVID-19 airborne infection 12 risk in classrooms. Actual room conditions with age and activity levels were used to determine 13 . CC-BY-NC-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 February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint breathing and quanta emission rates 51,52 . All students wore a face mask in the classroom, and mask 1 efficiency was selected based on students' typical mask type. The prevalence rate and immunity 2 fraction were calculated based on the official reported data 53 . 3 infection risk probability with a 95% confidence interval. For Rooms 1, 2, and 3, the mean PIcond 9 at the end of the day is around 14%, 14%, and 20%, respectively. Rooms 1 and 2 show lower PIcond 10 because no students were present during the break, so the quanta generation was zero. The 11 uncertain band of PIcond in Room 1 is wider than other rooms because of the larger posterior range 12 of ventilation rate due to its mechanical ventilation system. In Rooms 1 and 2, when students leave 13 the classroom, the PIabs are zero because there is no susceptible persons in the room. Room 1 shows 14 . CC-BY-NC-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 February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint the lowest PIabs because of the lower PIcond and prevalence rate: all three measurements were 1 conducted on different days. There are more students in Room 2 than Room 3; however, because 2 of the lower PIcond, the absolute PI of Room 2 is thus lower. 3

It also shows that after three hours, PIcond of Room 1 is always smaller than two other rooms due 4 to a higher ventilation rate. More frequent leaves of students (three breaks in a day) also contribute: 5 the quanta level is generally lower than the two other rooms. PIcond of Room 3 continues to rise 6 with the exposure time, unlike other classrooms, again due to the constant occupancy assumed. 7 Therefore, to reduce infection risk, more frequent breaks and leaves could be beneficial when 8

increasing the classroom's ventilation rate is relatively harder to achieve. 9

The conditional PI is the ratio of the number of infections to susceptibles = ⁄ . The 10 reproductive number (R0), defined by Rudnick and Milton (2003), is the number of secondary 11

infections when a single infected person is introduced in the room, and everyone in the room is 12 susceptible. If R0 < 1, then the infectious agent cannot spread in the population. For these three 13 classrooms if is smaller than 5.3%, 5%, and 5.5%, it is expected that the community spread 14 in the classrooms could be stopped. Figure 6 shows that, for Rooms 1, 2, and 3, the conditional PI 15 exceeds the level of R0 = 1 at around two ~ three hours, after which mitigation measures should 16 be taken. 17

. CC-BY-NC-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. To identify the relation between the required ventilation rates, corresponding CO2 levels, and the 2 COVID-19 airborne spreading risks, we calculated them for the three classrooms as shown in 3 Figure 7 . At all exposure times, the indoor CO2 and PI decrease with an increased ventilation rate. 4

The ventilation rate threshold to prevent the spread (R0<1) at all exposure times is 8, 3, and 6 ACH 5 for Rooms 1, 2, and 3, respectively. Room 1 requires a higher ventilation rate because of the 6 smaller room size. Room 2 needs a lower ventilation rate than Room 3 because students left the 7 classroom several times. Therefore, the ventilation rate threshold depends on the occupancy 8 schedule and the size of the room. 9

. CC-BY-NC-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. to reach the level of R0 = 1. On the other hand, we can find the indoor CO2 threshold for each 3 exposure time corresponding R0 = 1, as shown in Figure 8 . The CO2 threshold level decreases with 4 exposure time for all three rooms. For Room 1 at the second hour, the acceptable level is around 5 1000 ppm and later has to be 520 ppm at the end of the day to avoid spreading. For Rooms 2 and 6 3, it decreases from 660 ppm to 505 ppm and 750 ppm to 500 ppm, respectively. Meanwhile, the 7 threshold levels for 3-7 hours exposure times are close for all three classrooms. In comparison, 8

Figures 8b and 8c illustrate that the ventilation rate threshold (to prevent the spreading) increases 9 with exposure time and is not a constant number because of the three rooms' different sizes and 10 schedules. 11

. CC-BY-NC-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-NC-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 February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint Fig. 9 a) Time-averaged indoor CO2 concentration thresholds, and b, c) outdoor ventilation rate 1 thresholds with exposure times to prevent spreading. 2

In summary, the results of all three classrooms show that the ventilation rate threshold to prevent 3 the airborne transmission of COVID-19 depends on several parameters such as room size, student 4 schedules, and exposure time. The indoor CO2 threshold seems to depend on exposure time mostly, 5 and the time-averaged level of 500 ppm seems acceptable for all three classrooms. 6

The airborne transmission of COVID-19 is a major infection route in indoor spaces, especially 8 with poor ventilation conditions, large occupancy density, and high exposure time, such as school 9

classrooms. There are some recommended ventilation rates for acceptable indoor air quality or 10 preventing airborne transmission in indoor spaces, but it is not easy to measure the actual room's 11 ventilation rate. Indoor air CO2 level can be used as an indicator for the ventilation rate, whereas 12 it depends on several parameters that must be estimated. In this work, we used the sensitivity 13 analysis, MCMC Bayesian Calibration method, measured data of indoor CO2 and occupancy 14 profile of three classrooms in Montreal, Canada, to find and calibrate the dominant parameters to 15 estimate CO2 levels indoors. The results showed that the outdoor ventilation rate is the most 16 significant parameter. The calibrated ventilation rate with a 95% confidence level is 1.96 ± 0.31 17 ACH for Room 1 with mechanical ventilation, and 0.40 ± 0.08 ACH and 0.79 ± 0.06 ACH for 18 Rooms 2 and 3 with windows open only. Based on the calibrated model, we created the correlations 19 between the CO2 levels and the ventilation rates, which help teachers to estimate the ventilation 20 rates from the CO2 sensors at any time. A time-averaged CO2 lower than 450 ppm is equivalent 21 . CC-BY-NC-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 February 3, 2021. ; https://doi.org/10.1101/2021.01.29.21250791 doi: medRxiv preprint to a ventilation rate greater than 10 ACH in all three rooms, close to the recommended 12 ACH 1 value for a safe indoor environment against airborne transmission. 2

This study also proposed the approach to calculating the probability of the infection risk based on 3 the calibrated ventilation rate, which helps quantify the uncertainty of outdoor ventilation rates. 4

Moreover, this study estimated the required ventilation rate threshold and the CO2 threshold values 5 to prevent the airborne aerosol spreading of COVID-19 as a function of exposure time in the 6 classrooms. The ventilation threshold at all hours is 8, 3, and 6 ACH for rooms 1, 2, and 3, 7 respectively, and the CO2 threshold is around 500 ppm at all exposure times (< 8 hr) of a school 8 day for all three classrooms. This threshold is significantly different from the recommended value 9 of 1000 ppm for acceptable indoor air quality conditions. Therefore, it is recommended that the 10 ventilation rate and indoor CO2 concentration thresholds be reconsidered in indoor spaces, 11 especially classrooms, in the current pandemic. 

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