Safe CO₂ threshold limits for indoor long-range airborne transmission control of COVID-19

Abstract

CO₂-based infection risk monitoring is highly recommended during the current COVID-19 pandemic. However, the CO₂ monitoring thresholds proposed in the literature are mainly for spaces with fixed occupants. Determining CO₂ threshold is challenging in spaces with changing occupancy due to the co-existence of quanta and CO₂ remaining from previous occupants. Here, we propose a new calculation framework for deriving safe excess CO₂ thresholds (above outdoor level), C_t, for various spaces with fixed/changing occupancy and analyze the uncertainty involved. We categorized common indoor spaces into three scenarios based on their occupancy conditions, e.g., fixed or varying infection ratios (infectors/occupants). We proved that the rebreathed fraction-based model can be applied directly for deriving C_t in the case of a fixed infection ratio (Scenario 1 and Scenario 2). In the case of varying infection ratios (Scenario 3), C_t derivation must follow the general calculation framework due to the existence of initial quanta/excess CO₂. Otherwise, C_t can be significantly biased (e.g., 260 ppm) when the infection ratio varies greatly. C_t can vary significantly based on specific space factors such as occupant number, physical activity, and community prevalence, e.g., 7 ppm for gym and 890 ppm for lecture hall, indicating C_t must be determined on a case-by-case basis. An uncertainty of up to 6 orders of magnitude for C_t was found for all cases due to uncertainty in emissions of quanta and CO₂, thus emphasizing the role of accurate emissions data in determining C_t.

Nomenclature

B

Breathing rate, m³/h

C_CO2,i

CO₂ concentration for occupancy stage i, ppm

C_Cin,i

Initial CO₂ concentration for occupancy stage i, ppm

C_q,i

Quanta concentration for occupancy stage i, quanta/m³

C_qin,i

Initial quanta concentration for occupancy stage i, quanta/m³

C_t

Safe excess CO₂ threshold, ppm

C_t50

Median safe excess CO₂ threshold, ppm

E_CO2

CO₂ emission rate, mL/s

E_q

Quanta emission rate, quanta/h

I_i

Infector number for occupancy stage i

N_ave

Average occupant number

N_i

Occupant number for occupancy stage i

P_i

Infection risk for occupancy stage i

P_t

Predefined infection risk threshold

P_I

Community prevalence

R_A

The average number of secondary cases caused by one infector

S_i

Occupancy stage i

T_i

Exposure time for occupancy stage i, h

V

Space volume, m³

λ_i

Air change rate for occupancy stage i, h⁻¹

1. Introduction

COVID-19, as a novel coronavirus disease, has caused a worldwide pandemic since the end of 2019 [1]. Indoor transmission control is crucial in preventing the spread of the SARs-CoV-2 due to a higher transmission risk indoors than outdoors [2]. The four main transmission routes in indoor environments are droplet-borne, fomite, short-range airborne, and long-range airborne [3,4]. While short-range airborne transmission route was inferred to be the dominant route in close contact [1], long-range airborne transmission was revealed to more likely induce outbreaks in poorly ventilated and confined indoor spaces [5]. Thus, it is of primary importance to monitor and control long-range airborne transmission in indoor environments.

The exhaled infectious aerosols contributing to long-range airborne transmission are difficult to be detected. Hence, there is an urgent need for a detectable indicator to effectively monitor long-range airborne transmission. CO₂, which can be easily monitored through low-cost sensors [6], has been recommended because it can both reflect the indoor ventilation condition and the quanta concentration [7]. Accordingly, safe CO₂ thresholds are defined as the maximum CO₂ concentration level under which the indoor space is at an acceptable infection risk level. Such information is useful in guiding the design of infection-resilient buildings.

Treating CO₂ as an indicator for indoor ventilation performance, recent studies proposed CO₂ thresholds for risk control based on prevailing ventilation standards aimed at ensuring acceptable indoor air quality (IAQ) but not infection risk [[8], [9], [10]]. Although ASHRAE does not recommend a specific value of threshold [7], other organizations have suggested specific CO₂ thresholds of 800 ppm [11,12] or 800–1000 ppm [8] to ensure a safe indoor environment. However, it is questionable whether a fixed CO₂ threshold can guarantee a low infection risk for all spaces, as factors such as occupancy level and respiratory activity can all affect the value of it [6].

Moving beyond using CO₂ as a mere indicator of indoor ventilation condition, CO₂ can also be used to directly reflect quanta concentration as CO₂ and virus-laden aerosols are co-produced and co-inhaled by human. In this context, CO₂ thresholds can be calculated backward based on a pre-defined acceptable infection risk level [6,13]. Indoor airborne transmission risk is constrained under the predefined risk level in as much indoor CO₂ concentration is maintained below the derived threshold. Occupancy level and respiratory activity for a particular indoor space can all be factored in this backward calculation process [6,13,14]. In the literature of using CO₂ to reflect quanta concentration, the derived thresholds were found to be highly sensitive to factors such as activity level and community prevalence, making CO₂ thresholds vary across different indoor spaces [6]. For example, the reference excess CO₂ threshold (above outdoor level) for a classroom amounts to only about 150 ppm, while this figure is ten-fold for a supermarket [6]. This indicates that the CO₂ thresholds should be determined case by case, instead of using a fixed value for all spaces.

In addition, most proposed thresholds are for spaces with fixed occupancy level under the assumption of no initial quanta/excess CO₂ [6,13,14]. For spaces with varying occupancy, some of quanta/CO₂ released by earlier occupants can remain in the space and become initial quanta/CO₂ when the next group occupies the space, potentially increase the infection risk. The quantity of initial quanta is essential for defining CO₂ threshold, but it is difficult to estimate, as it requires information about ventilation conditions and occupancy profile of previous occupants. Hence, how can we account for initial quanta/excess CO₂ in spaces with changing occupancy in infection risk assessment remains an unsolved question [3,15].

Finally, emissions of quanta and CO₂ are crucial in determining the CO₂ threshold. However, they both exhibit inter-individual variability and can be affected by factors such as age and gender [[16], [17], [18]]. For instance, the viral load of a super-spreader can be 10 times higher than the mean level of normal infectious subjects [19], indicating a higher quanta emission [20,21]. Different values of quanta and CO₂ emission were adopted by previous studies for CO₂ threshold derivation, e.g., from 0.37 quanta/h to 100 quanta/h for classrooms [6,13,16,22]. The effect of the uncertainty in the emissions of quanta and CO₂ on CO₂ threshold needs further investigation. The present study aims to provide a new calculation framework for deriving safe excess CO₂ thresholds (C _t) by taking into account initial quanta/excess CO₂ and changing/fixed occupancy patterns in different indoor spaces, as well as propagating the uncertainty of these input variables.

2. Methodology

2.1. General calculation framework

Our model is based on four assumptions for indoor mass balance equations for CO₂ and quanta [13]: 1) both CO₂ and quanta are well mixed and evenly distributed in the air; 2) indoor excess CO₂ is released only by human exhalation, with no other indoor sources; 3) CO₂ emission rate and quanta emission rate are both constant (i.e., not time dependent); 4) the loss of quanta is mainly due to ventilation, other elimination mechanisms such as deposition, filtration and inactivation are neglected.

In deriving C _t for spaces with changing occupants, we consider a sequence of occupancy stages, S _i (I _i, N _i , T _i). Stage i represents an indoor space (with the volume of V)being occupied by a number of occupants (N _i) with infectors (I _i) for a duration of time (T_i). i = 1 represents the start of the occupancy: N ₁ occupants (with I ₁ infectors) stay in this indoor space for a period of T ₁, with no people inside prior to N ₁ occupants. The introduction of various occupancy stages aims to consider the virus released and still present in the air from previous occupancy stages (the initial quanta). This is fundamentally different from previous studies which only considered one-off occupancy or fixed occupancy throughout the exposure period of interest.

The general calculation process of C _t for one occupancy stage of a space is given as follows.

Long-range transmission risk for occupancy stage i is modeled through a Wells-Riley model [23] amended by Gammitoni and Nucci [24] to assess infection risk through unsteady-state quanta concentration:

$$P_i=1-e^{-B{\int }_0^{T_i}{C}_{q,i}\left(t\right)dt}$$ (1)

Quanta concentration in Equation (1) is modeled through transient mass balance equation:

$$\frac{d{C}_{q,i}}{dt}=\frac{I_i{E}_q}{V}-{\lambda }_i{C}_{q,i}$$ (2)

Equation (2) can be analytically solved as:

$$C_{q,i}\left(t\right)=\left(C_{\mathrm{q}\mathrm{i}\mathrm{n},\mathrm{i}}-\frac{I_i{E}_q}{{\lambda }_{i}V}\right)e^{-{\lambda }_{i}t}+\frac{I_i{E}_q}{{\lambda }_{i}V}$$ (3)

To control transmission risk of stage i under an acceptable low level, a risk threshold of P _t needs to be initially determined. Based on P _t, a required ACH (air change rate, λ _i) can be derived by substituting Equation (3) into Equation (1), λ _i should be no less than the derived value to keep transmission risk under P _t.

Indoor excess CO₂ concentration is also dominated by ACH, hence it reflects the ventilation condition of stage i.

Indoor excess CO₂ concentration for stage i is modeled by mass balance equation (4):

$$\frac{d{C}_{CO2,i}}{dt}=\frac{N_i{E}_{CO2}}{V}-{\lambda }_i{C}_{CO2,i}$$ (4)

Equation (4) is solved as:

$$C_{CO2,i}\left(t\right)=\left(C_{Cin,i}-\frac{N_i{E}_{CO2}}{{\lambda }_{i}V}\right)e^{-{\lambda }_{i}t}+\frac{N_i{E}_{CO2}}{{\lambda }_{i}V}$$ (5)

Substituting the required ACH that is backward calculated from transmission risk threshold into Equation (5), the time-averaged indoor excess CO₂ concentration (C _CO2,i) during T _i is exactly C _t for stage i [13,22]:

$$C_t=\frac{1}{T_i}{\int }_0^{T_i}{C}_{CO2,i}\left(t\right)dt$$ (6)

When indoor excess CO₂ concentration is below the reference threshold $C_{\mathrm{t}}$, it indicates that there is sufficient ventilation to keep long-range transmission risk for occupancy stage i under the risk level of P _t.

For different occupancy stages, C _t can be derived by following the steps described above, taking into account the existence of initial quanta/excess CO₂, see Equation (3) and Equation (5). Starting with occupancy stage 1, which has no initial quanta/excess CO₂,the required ACH (λ ₁) can be easily obtained following the general calculation process. For occupancy stage 2, the initial quanta and initial excess CO₂ can be estimated based on the ACH derived for occupancy stage 1 (λ ₁), under the assumption that excess CO₂ during occupancy stage 1 has been controlled below the reference threshold, C _t for occupancy stage 2 can then be calculated according to the calculation framework. This process can be repeated for all the modeled occupancy stages by using the ACH derived from the previous occupancy stages to estimate the initial quanta/excess CO₂ for the current stage, and thereby calculating C _t iteratively.

2.1.1. Infection risk threshold P_t

The infection risk threshold, P _t, is crucial in determining the safety levels of the indoor environment. It can be defined in two ways, either by using a constant value for all environments, such as 1%, 0.1% [25] or even 0.01% [6], or by determining P _t based on the reproductive number (R _A) where R _A is the average number of secondary cases caused by one infector in a given susceptible population in indoor environment. In the latter, the value of P _t is determined by the number of occupants and can become a large and inconvincible value when occupant number is small [26,27]. In this study, we use a constant value of P _t = 0.01% as suggested by Peng and Jimenez [6], which is reasonable for most occupancy stages when the number of occupants is less than 10,000.

2.2. Designed scenarios

Three scenarios were identified to calculate C _t.

• 1)Regularly attended space with fixed occupancy level and the same group of people as occupants, so that N ₁ = N ₂ = … (e.g., a lecture room used by a certain group of students) [28,29];
• 2)Non-regularly attended space with constant infection ratio (I ₁/N ₁ = I ₂/N ₂ = … = I _i/N _i), different groups of people as occupants, and a high occupancy level (e.g., shopping center, train station);
• 3)Non-regularly attended space with changing infection ratios (I ₁/N ₁ ≠ I ₂/N ₂ ≠ … ≠ I _i/N _i) and low occupancy level (e.g., gym, train coach).

All these scenarios are widely experienced in real-life situations.

2.2.1. Scenario 1: regularly attended spaces

We determined the number of infectors I _i for Scenario 1 based on both the indoor occupancy level (N _i) and local community prevalence (P _I). The expected I _i is defined as max {1, P _I N _i}. When with a low indoor occupancy level or a low community prevalence, the value of P _I N _i can be less than 1. In such case, I _i was assumed to be equal to 1. Otherwise, I _i was assumed to be P _I N _i to reflect the real local infection condition.

Quanta concentration and excess CO₂ concentration were found to have a constant proportion throughout all the occupancy stages, as determined from the mass balance equations. This proportion was only affected by infection ratio and emissions, see Equation (7) (Full derivation details can be found in Supplementary Information). As long as the infection ratios and emissions remained unchanged during the occupancy stages, the proportion remains unchanged as well, hence:

$$\frac{C_{q,1}\left(t\right)}{C_{CO\mathrm{2,1}}\left(t\right)}=\frac{C_{q,2}\left(t\right)}{C_{CO\mathrm{2,2}}\left(t\right)}=\dots =\frac{I_i}{N_i}\frac{E_q}{E_{CO2}}$$ (7)

Under these circumstances, infection risk for stage i in Eq (1) can be revised as below:

$$P_i=1-e^{-B\frac{I_i}{N_i}\frac{E_q}{E_{CO2}}{\int }_0^{T_i}{C}_{CO2,i}\left(t\right)dt}$$ (8)

Equation (8) can be treated as the classical rebreathed fraction (RF)-based infection risk model derived by Rudnick and Milton [14], with BC _CO2,i/E _CO2 representing the rebreathed fraction. This derivation proved that rebreathed fraction (RF)-based model can account for the impact of initial quanta/excess CO₂ in risk assessments for spaces with fixed occupants.

Based on Equation (8), the time averaged value C _t for occupancy stage i can then be derived as:

$$C_t=\frac{E_{CO2}{N}_i}{E_q{T}_{i}B{I}_i}\mathit{ln}\left(\frac{1}{1-P_t}\right)$$ (9)

2.2.2. Scenario 2: Non-regularly attended spaces with constant infection ratios

In Scenario 2, we assumed that community prevalence (P _I) can directly represent indoor infection ratio due to the high occupancy level (I ₁/N ₁ = I ₂/N ₂ = … = P _I). The proportion between C _q,i and C _CO2,i also becomes constant due to the constant infection ratio among occupancy stages (Detailed derivation process can be found in Supplementary Information):

$$\frac{C_{q,1}\left(t\right)}{C_{CO\mathrm{2,1}}\left(t\right)}=\frac{C_{q,2}\left(t\right)}{C_{CO\mathrm{2,2}}\left(t\right)}=\dots =P_I\frac{E_q}{E_{CO2}}$$ (10)

Similar as Scenario 1, the infection risk and excess CO₂ threshold can then be derived as:

$$P_i=1-e^{-B{P}_I\frac{E_q}{E_{CO2}}{\int }_0^{T_i}{C}_{CO2}\left(t\right)dt}$$ (11)

$$C_t=\frac{E_{CO2}}{E_q{T}_{i}B{P}_I}\mathit{ln}\left(\frac{1}{1-P_t}\right)$$ (12)

Equation (11) can be treated as an extension of the classical RF-based infection risk model. The generality of the original model is extended from scenarios with fixed occupants (scenario 1) to scenarios with varying occupancy levels (scenario 2), taking into account initial quanta/excess CO₂. It should be noted that T _i in Scenario 2 is often difficult to monitor, as the occupancy level keeps changing. An alternative method is to predefine it based on the characteristics of different spaces. For example, T _i could be set as 35 min for check-in hall and 100 min for departure hall, based on the average dwelling times measured in an airport [30].

2.2.3. Scenario 3: Non-regularly attended spaces with changing infection ratios

In Scenario 3, the indoor infection ratio cannot be represented by P _I due to the relatively low occupancy level. To ensure a safe indoor environment, it is recommended to use the maximum value of {1, P _I N _i} to determine the number of infectors (I _i), as was done in Scenario 1. In these circumstances, the infection ratio will change among the occupancy stages, and quanta concentration cannot be represented by excess CO₂ concentration. C _t derivation must follow the general calculation process (see Part 2.1).

It should be noted that the general calculation process does not require the field measurement of ACH and instead relies on a known occupancy profile, including the number of occupants and the duration of occupancy for all the occupancy stages. Thus, this method may be more suitable for spaces in Scenario 3 where the occupancy profile (N _i and T _i) of each occupancy stage can be monitored simultaneously or obtained before the spaces being occupied, such as the rail train or theatre.

2.3. Uncertainty analysis and inputs

An uncertainty analysis was carried out considering E _q and E _CO2 have interindividual variations and can vary with gender and age, leading uncertainty to C _t. The probability density functions (PDF) of E _q for three different activities were obtained from recent research by Buonanno et al. [16], where they found the quanta emissions follow a log10-normal distribution, see Table 1 . E _CO2 was also assumed to be lognormally distributed with a standard deviation equal to 20% of its mean [31]. The mean value for the distribution was calculated as the average value of E _CO2 of female and male individuals aged 30–40 years (the most frequent age cohort), and with a specific metabolic equivalent [17]. The metabolic equivalent for E _CO2 was specified by different activity levels, specifically, 1.5 met for sedentary activity, 3 met for light activity and 9 met for heavy activity [32]. Latin Hypercube sampling (LHS) [33] was used to generate a total of 30,000 samples from emissions of quanta and CO₂, due to its advantage in accurately reflecting the underlying distribution of inputs with a smaller sample size. Monte Carlo simulations [34] were used to propagate and quantify the uncertainty in predictions.

Table 1.

Inputs for Uncertainty Analysis. Distribution mean and standard deviation in brackets.

Activity	Quanta emission PDF (quanta/h)	CO2 emission PDF (mL/s)
Sedentary - breathing	LN10 (−0.429, 0.720)	LN (5.05, 1.01)
Light activity - speaking	LN10 (0.698, 0.720)	LN (10.10, 2.02)
Heavy activity - breathing	LN10 (0.399, 0.720)	LN (34.20, 6.84)

Typical indoor environments were selected for each scenario based on factors such as occupancy level, infection ratio, etc. (Table 2, Table 3 ). Cases in Scenario 1 have a fixed but different number of occupants, recognizing that occupant number is a dominant parameter in deriving C _t in Scenario 1, see Equation (9). It should be noted that lecture hall case in Scenario 1 has 3 infectors because of its high occupancy level, while other cases have only 1 infector due to the relatively low occupancy level. In Scenario 2 a shopping center was selected as the case study with variable levels of community prevalence, which were adopted from three different COVID-19 periods in the UK in 2020 [35] to represent relatively small (0.06%), median (0.4%) and high (1%) community prevalence levels. The highest level of community prevalence was adopted for Scenario 1 and Scenario 3. Two cases with low and changing occupancy levels were selected for Scenario 3 (i.e., a train coach and a gym room). As regards occupancy stages, only one stage was included for cases in Scenario 1 and Scenario 2, whereas five occupancy stages were included for cases in Scenario 3 to take into account the variability in C _t resulting from the impact of initial quanta/excess CO₂. Different categories of activities were considered in the cases of the different scenarios. Cases in Scenario 1 were assumed to have “sedentary activity - breathing”, which is typical for people sitting or standing in office or classroom environments. Cases in Scenario 2 are assumed to have “light activity - speaking”, as people are usually walking in the shopping center and talking to each other. For scenario 3, two activities were included to explore the effects of activity level on C _t determination, specifically, “sedentary activity – breathing” for the train coach and “moderate activity – breathing” for gym. The breathing rates (B) corresponding to different physical activity levels were adopted from previous research [36].

Table 2.

Inputs of uncertainty analysis for Scenario 1 and Scenario 2.

Case	Volume (m3)	Infector number	Occupant number	Exposure time (h)	Community prevalence	Breathing rate (m3/h)
Scenario 1
Classroom	231	1	30	1	1%	0.54
Lecture classroom	270	1	65	1	1%	0.54
Lecture hall	540	3	300	1	1%	0.54
Open-plan office	594	1	20	1	1%	0.54
Scenario 2
Shopping center	2040	–	–	1	0.06%, 0.4%, 1%	1.38

Table 3.

Inputs of uncertainty analysis for Scenario 3.

Scenario 3	Stage 1	Stage 2	Stage 3	Stage 4	Stage 5
Train coach (300 m3)
Infector number	1	1	1	1	1
Occupant number	20	40	80	40	20
Exposure time (h)	1	1	1	1	1
Community prevalence	1%	1%	1%	1%	1%
Breathing rate (m3/h)	0.54	0.54	0.54	0.54	0.54
Gym (600 m3)
Infector number	1	1	1	1	1
Occupant number	5	10	20	10	5
Exposure time (h)	1	1	1	1	1
Community prevalence	1%	1%	1%	1%	1%
Breathing rate (m3/h)	3.30	3.30	3.30	3.30	3.30

3. Results

3.1. Safety excess CO₂ threshold varies in different scenarios

For Scenario 1, the number of occupants (N _i) is the dominant factor that affects C _t and scales with it (see Equation (9)). C _t for cases occupied by different N _i in Scenario 1 (regularly attended spaces) have substantial differences, see Fig. 1 (a). The highest C _t50 (the median value of C _t) occurs in lecture hall (890 ppm), followed by lecture classroom (580 ppm), classroom (270 ppm), the lowest one is in office environment (180 ppm), although significant overlaps exist in the output distributions (Fig. 1(a)).

Fig. 1.

Safe excess CO₂ thresholds for 3 scenarios: (a) Scenario 1(with fixed occupancy); (b) Scenario 2 (with changing occupancy but fixed infection ratios); (c) Scenario 3 (with changing occupancy and changing infection ratios).

For Scenario 2, instead of N _i, C _t is dominated by community prevalence (P _I), as C _t is inversely proportional to P _I (see Equation (12)). Three different values of P _I (i.e., 0.06%. 0.4% and 1%) were adopted to derive C _t and the results are showed in Fig. 1(b). The highest C _t50 of 870 ppm refers to the lowest P _I of 0.06%, and the lowest C _t50 of 50 ppm to the highest P _I of 1%.

For Scenario 3, the changing infection ratios lead to different values of C _t for different occupancy stages. For train coach, C _t50 are approx. 180 ppm, 320 ppm, 650 ppm, 410 ppm and 200 ppm corresponding to infection ratios of 1/20, 1/40, 1/80, 1/40 and 1/20 for the five stages in sequence, while they are 7 ppm, 15 ppm, 30 ppm, 15 ppm and 7 ppm for gym environment corresponding to infection ratios of 1/5, 1/10, 1/20, 1/10 and 1/5. The changing infection ratios can lead to different C _t values in different stages mainly because the existence of initial quanta/excess CO₂. For instance, if initial quanta/excess CO₂ is not considered, C _t50 for Stage 2 and Stage 4 of train coach with the same occupant number should be same, but the difference of C _t50 between the two occupancy stages reaches approx. 80 ppm due to the impact of initial quanta/excess CO₂.

Furthermore, the general cases in Scenario 3 also demonstrate that the activity level is another major factor that can affect the derived thresholds, see Fig. 1(c). C _t for gym with a high activity level is much lower than that for train coach with a sedentary activity level due to the relative high activity level in gym environment (hence, high emission rate for quanta). This agrees with previous studies [37,38] that there should be much higher restrictions in spaces with high activities such as gym to control airborne infection risk.

Apart from the substantially different C _t among different cases, large uncertainty of C _t was also observed in each case, spanning up to six orders of magnitude on a log scale (see Fig. 1). Fig. 1 shows that cases with a large median value contain more uncertainty, as seen in the right-shifted log-scaled distribution of C _t, indicating that C _t can be more affected by the uncertainty of emission settings considered in our study. Given the large uncertainty of C _t and the non-normal distribution when transformed to a linear scale, the median safe excess CO₂ threshold (C _t50) is an appropriate descriptive statistic for excess CO₂ threshold, due to its high probability density [39].

3.2. Effect of infection risk threshold (P_t)

As discussed before, the infection risk threshold (P _t) plays a role in deriving C _t. Different P _t have been adopted in different research in the range of 0.01%–1% [6,25,40,41]. Here we explore how P _t will affect C _t with results shown in Fig. 2 . The base case is the classroom in Scenario 1 (see Table 2). C _t50 is found to be approximately linearly related to P _t with approx. 270 ppm for P _t = 0.01% to 27000 ppm for P _t = 1%, which reveals the high sensitivity of C _t50 to P _t.

Fig. 2.

Excess ${CO}_2$ thresholds for the classroom (see Table 2) under different infection risk thresholds.

3.3. Effect of initial conditions

We have shown that initial condition of quanta and excess CO₂ can affect the derived safe excess CO₂ threshold when infection ratio varies among the occupancy stages. However, most previous studies have neglected the consideration of the initial conditions of quanta and excess CO₂ in C _t derivation [6,13,14]. To further quantify the impact of initial condition of quanta/excess CO₂ on C _t when infection ratio varies, we compared two cases: 1) with initial quanta/excess CO₂; 2) without them. We assumed the two cases with the same indoor volume of 300 m³, both occupied with two stages. The occupants in both cases were assumed to have “sedentary activity – breathing”, and only one infector is included.

In case 1, 20 occupants were assumed to be present in Stage 1, and the number of occupants in Stage 2 changes to 5, 10, 20, 40, 80 respectively. This means the infection ratio will change from 1/20 (Stage 1) to 1/5, 1/10, 1/20, 1/40, 1/80 (Stage 2) accordingly, and initial quanta and excess CO₂ can affect C _t in Stage 2 to varying degrees. In case 2, no occupants were assumed to be present in Stage 1 (hence, no initial quanta/excess CO₂), and 5 different occupancy levels were assumed for Stage 2, similar to case 1. Our aim is to derive C _t for Stage 2 for both cases with consideration of the impacts of initial quanta and excess CO₂ from Stage 1 (case 1) and without (case 2). The differences between the results of two cases can be used to quantify the impact of initial quanta and excess CO₂ on C _t . It is straightforward to derive C _t for case 2, as there are no initial quanta and excess CO₂, while for case 1, an estimation of initial quanta/excess CO₂ is necessary. Considering the excess CO₂ concentration is affected by different factors such as exposure time and ACH during Stage 1, we assumed a constant value for initial excess CO₂ concentration for Stage 2 in case 1, namely 1000 ppm. The initial quanta can then be derived based on this value and the infection ratio of Stage 1 (see Eq. S4 in Supplementary).

Fig. 3 shows the derived C _t value of case 1 with initial quanta/excess CO₂ has distinct difference from that of case 2 without initial quanta/excess CO₂, when its infection ratio in Stage 2 deviates from Stage 1 (1/20). This suggests that the initial condition of quanta/excess CO₂ shouldn't be ignored in C _t derivation when infection ratio varies among occupancy stages. As the infection ratio for case 1 changes (either increase or decrease from 1/20 in Stage 1), the derived C _t for Stage 2 in case 1 will be larger or smaller, respectively, compared to the derived C _t for Stage 2 in case 2. The difference between the two cases becomes more pronounced as the infection ratio of case 1 deviates further from 1/20. When the infection ratio increases from 1/20 (Stage 1) to 1/5 (Stage 2), C _t50 of case 1 with initial quanta/excess CO₂ increases by 60 ppm compared to case 2 without initial quanta/excess CO₂. Conversely, when the infection ratio decreases from 1/20 to 1/80 in Stage 2, C _t50 of case 1 becomes 260 ppm lower.

Fig. 3.

Excess CO₂ threshold of the second occupancy stage of an indoor space (300 m³) under different infection ratios considering with and without initial quanta/excess CO₂.

4. Discussion

4.1. New understanding of rebreathed-fraction model

RF-based Wells-Riley model, proposed by Rudnick and Milton's [14], uses CO₂ as a maker for exhaled-breath exposure. This model does not require any knowledge about ACH, hence it has been widely used in assessing airborne infection risk [[42], [43], [44], [45], [46]]. However, we proved that RF-based model should only be adopted in spaces with fixed occupancy, otherwise initial quanta will cause bias (see Part 3.3), which is largely overlooked by many other studies. For spaces with varying occupancy, the initial quanta/excess CO₂ generated by previous occupants but remaining in the air can be very important in determining the overall quanta/excess CO₂ concentration for next-stage occupancy. The mechanism of RF-based model in dealing with initial quanta/excess CO₂ in spaces with changing occupancy has not been adequately discussed before. In this article, we provide an analytical derivation to explain its mechanism and show that initial quanta/excess CO₂ can be considered within the RF-based method in C _t derivation for Scenario 1 (with fixed occupancy) and Scenario 2 (with changing occupancy but fixed infection ratios). This extends the generalization of RF-based model from spaces with fixed occupancy to spaces with changing occupancy. It should be noted that other recent studies [28,29] resonate with our study in that they apply RF-based model to spaces with varying occupancy levels to assess infection risk. However, only two occupancy modes were considered in these studies, occupied and non-occupied, which are both included in our Scenario 1. In this contribution, we have proved that for spaces with both occupied and non-occupied modes, the non-occupied period does not affect the proportion of quanta concentration to excess CO₂ concentration in future occupied period if infection ratios remain unchanged given only ventilation is considered here (see Supplementary Information).

4.2. Implications for C_t determination

Great uncertainty in C _t can be caused by the uncertainties in emissions of E _q and E _CO2 (see Fig. 1). E _CO2 and E _q contain uncertainty because they have interindividual variation and can be affected by factors such as age, gender [[16], [17], [18]]. The value of E _q can vary by up to 3 orders of magnitude (e.g., 0.32–240 quanta/h for speaking under light activity) [16] while E _CO2 varies within only one order of magnitude (e.g., 2.88–43.2 L/h) [17]. Different studies adopted very different values of E _q and therefore lead to very different values of C _t. For example, in the classroom setting under the same activity level, the median value of E _q in our study is 0.37 quanta/h [16], while it was in the range of 27.55 quanta/h to 100 quanta/h in other studies [6,13,22], leading several hundred times lower C _t compared to our results.

The choices of P _t and I _i also impact the value of C _t. Theoretically, a lower P _t can promise a safer indoor environment, but this would come at the cost of very low C _t practically impossible to achieve in real-world scenarios. E.g., a low level of C _t may require a very high ACH, which is unfeasible and prone to cause large energy cost due to the diminishing return phenomenon of ventilation [47]. Additionally, the method to determine infector number I _i is also important, as it is related to the total quanta emission. Our study defined I _i as the maximum value of {1, P _I N _i} as the worst-case scenario. On the contrary, Bazant et al. [22] considered I _i to be the minimum value of {1, P _I N _i}, which resulted in a dramatically large value of C _t (even larger than 10000 ppm) when P _I is small.

4.3. Implications for infection risk monitoring and control

Our model has practical implications for indoor transmission monitoring and control. For Scenario 1 and Scenario 2, the safe excess CO₂ threshold can be determined based on variables such as occupancy level, duration and risk threshold through simple equations (see Equation (9) and Equation (12)), making it possible to apply our model for infection risk monitoring in Scenario 1 and 2 for public individuals. For instance, when entering a space like as a shopping center (as in our Scenario 2), individuals can easily measure the indoor excess CO₂ level first using a portable low-cost CO₂ sensor. Then, by replacing C _t in Equation (12) with the measured data, they can estimate a safe exposure duration based on their acceptable risk threshold to guide them on how long they should stay in the shopping center. Additionally, taking into account the impact of initial quanta/excess CO₂ on risk estimation and C _t derivation, our model can be adopted to further develop different ventilation control strategies, such as CO₂-based demand-controlled ventilation [48] or intermittent ventilation strategy [49,50], aimed at reducing indoor transmission risk by treating indoor excess CO₂ as a control variable.

Further applying our calculation framework into real-world scenarios, some insights can be gained by comparing derived C _t with measurement data/standard limits. In Scenario 1, the occupant numbers can largely affect C _t level, making it necessary to consider both CO₂ level and occupant level in transmission risk assessment. For example, in classrooms of Scenario 1, the measured excess CO₂ levels were found to be in the range of 300–2500 ppm (with an outdoor level of 420 ppm) dependent on the number of occupants [7,29,51]. According to our framework, 300 ppm can represent an unsafe environment if the occupant number is less than 33, and 2500 ppm can still be a safe level if occupants is larger than 278. Therefore, C _t threshold should be used in conjunction with occupant number. In scenario 2, community prevalence can dominate C _t and can be used as a reference for lockdown policy implementation. It was found that the 1-h average CO₂ level of 40% shopping mall in Hong Kong exceeded 1000 ppm [52]. To keep infection risk no more than 0.01% for shopping malls, a community prevalence of less than 0.09% is needed according to our calculation framework, otherwise, such places should be locked down. In Scenario 3, taking a restaurant (∼350 m³) with two occupancy stage (N ₁ = 20 for Stage 1 and N ₂ = 80 for Stage 2) as an example [53], according to ASHRAE 62.1 [54], the maximum excess CO₂ limits (the steady-state excess CO₂ concentration under the required ventilation rate) for the first two occupancy stages are 540 ppm (Stage 1) and 790 ppm (Stage 2) respectively. But C _t calculated from our framework amounts to 180 ppm and 610 ppm, respectively. The difference indicates the target of infection risk control should be integrated into present ventilation standards to promise both a high level of IAQ and a low infection risk.

4.4. Limitation of the study

Our study is based on the assumption that outdoor ventilation is the only loss mechanism for quanta in Scenario 1 and Scenario 2, which results in a constant proportion between quanta concentration and excess CO₂ concentration, hence making RF-based model suitable for deriving C _t in these scenarios. However, surface deposition, filtration and virus deactivation can also significantly reduce quanta concentration [[55], [56], [57]]. Neglecting these loss mechanisms may overestimate indoor quanta concentration and result in a lower C _t than needed. However, the reliability of the derived C _t for a safe indoor environment would not be affected.

The thresholds we derived are based on the assumption of a well-mixed room air. Thus, the location of CO₂ sensors need to be carefully selected to adequately reflect indoor CO₂ conditions [58,59]. Additionally, our results only account for long-range airborne transmission neglecting the contribution of short-range transmission [4,37,60]. Relying solely on C _t to monitor infection risk may not be sufficient, other measures such as wearing masks and social distancing should be implemented together to control indoor airborne transmission [[61], [62], [63]].

Another limitation lies in the application of community prevalence (P _I) in our study. In scenario 1 and scenario 3, P _I is used to determine the indoor infector number, which would cause bias because: 1) P _I might be lower than the real value due to the asymptomatic characteristic of SARS-CoV-2 [64,65]; and 2) positive individuals may not be present in public spaces due to mandatory quarantine policy which would lead to a lower indoor infection ratio than P _I. In scenario 2, simply using P _I to represent the indoor infection ratio can lead to an underestimation of the actual ratio when the number of occupants is low. Conducting field measurement to estimate the average occupancy level (N _ave) and selecting the maximum value of {1, P _I N _ave} could be an alternative method for defining a convincible infection ratio for scenario 2. In addition, considering P _I is changing during different time periods of pandemic, the indoor infection ratio would need to be updated accordingly.

In addition, the uncertainty of C _t estimated by our study may be limited as we only considered the uncertainty in emission settings (i.e., quanta emission rate, CO₂ emission rate). Community prevalence ($P_{\mathrm{I}}$) may also contain uncertainty due to the reasons described earlier. This uncertainty may increase the uncertainty of C _t for Scenario 2, where P _I is a dominating input in C _t derivation. However, it may not obviously affect C _t for Scenario 1 and 3, because P _I is only adopted in C _t derivation when P _I N _i > 1 but the occupancy level (N _i) in Scenario 1 and 3 is usually low and hence P _I N _i < 1. Similar as emission settings, breathing rate can also contain uncertainty due to interindividual variation and factors such as age and gender. In addition, quanta emission rate, CO₂ emission rate and breathing rate may all be correlated to each other [18]. In our study, we simply adopted constant breathing rates for different physical activity levels based on the study of Buonanno et al. [16]. Quanta emission rate and CO₂ emission rate are also inter-related through physical activity level (See Table 1). In future, based on more accurate data, the uncertainty and correlation of those parameters may be better interpreted, and the uncertainty of C _t can be therefore further estimated.

5. Conclusion

A new calculation framework was proposed in this study for deriving safe excess CO₂ threshold (C _t) for different spaces with consideration of initial quanta/excess CO₂ and fixed/changing occupancy levels. From our derivation process we found that the proportion of indoor excess CO₂ concentration to quanta concentration remains constant when the infection ratio (infectors/occupants) of an indoor space remains constant. Based on this relationship, the RF-based (rebreathed fraction-based) model can be applied directly for infection risk assessment and C _t derivation, but not applicable in the cases with varying infection ratios.

Affected by factors such as occupant number (N _i), community prevalence (P _I) and activity level, the median value C _t50 derived by our framework varies significantly among the selected cases, with a minimum value of 7 ppm for a gym to a maximum value of 890 ppm for a lecture hall, with long-tailed distributions. Initial quanta/excess CO₂ is found to largely affect C _t, especially when the infection ratio varies greatly during different occupancy stages. A bias of several hundred ppm (e.g., 260 ppm for a space of 300 m³ and with sedentary activity level) could be occur if the initial quanta is not well considered in C _t derivation. Our finding illustrates that different CO₂ thresholds should be derived for different spaces and different occupancy stages, rather than using a fixed value for all spaces.

Large uncertainty was also found in derived thresholds for all cases, spanning approximately 6 orders of magnitude, mainly influenced by quanta emission rate (E _q) and CO₂ emission rate (E _CO2). For a better control of indoor infection risk through CO₂ monitoring, more accurate input parameters would be necessary.

CRediT authorship contribution statement

Xiaowei Lyu: Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Zhiwen Luo: Writing – review & editing, Writing – original draft, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Conceptualization. Li Shao: Writing – review & editing, Supervision, Investigation. Hazim Awbi: Writing – review & editing, Investigation. Samuele Lo Piano: Writing – review & editing, Methodology.

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Xiaowei Lyu reports financial support was provided by China Scholarship Council.

Appendix A. Supplementary data

The following is the Supplementary data to this article.

Data availability

Data will be made available on request.