Predicting the infection probability distribution of airborne and droplet transmissions

Abstract

Herein, a method is proposed to predict the infection probability distribution rather than the room-averaged value. The infection probability is predicted by considering both airborne and droplet transmissions based on CO₂ concentration and the position of the occupants in a room. The proposed method was used in an actual office setting, and the results confirmed that it could provide a quantitative prediction of the infection probability by integrating the ventilation efficiency and the distance between occupants (i.e. social distancing). We verified the ability of the method to analyse the relative effectiveness of countermeasures for airborne and droplet transmissions. The proposed strategies can be implemented by a facility manager and can enable facility users to check the infection probability distribution in real-time to select a seat with the minimum risk of infection.

Keywords

COVID-19 Wells–Riley model social distancing local positioning system ventilation

Introduction

COVID-19 has spread worldwide since 2020. The World Health Organization (WHO)¹ assessment in March 2020 declared COVID-19 to be a pandemic. According to the WHO,² over 100 million people had been infected by January 2021, and approximately 1.75 million people had died. The damage caused by the pandemic was not only medical but also economical owing to travel restrictions and lockdowns. Infectious disease epidemics have occurred repeatedly in the history of humanity, and it is conceivable that a new virus with the same infectious power as SARS-CoV-2 might reappear in the future and cause yet another pandemic.³ Isolating infected and susceptible individuals could prevent the spread of COVID-19. However, COVID-19 is transmitted by asymptomatic patients.^4,5,6 Therefore, it is always better to assume that (an) infected persons/people are everywhere and to implement countermeasures for preventing the spread of the infection.

According to the U.S. Centers for Disease Control and Prevention (CDC),⁷ the modes of COVID-19 transmission include contact, droplet and airborne transmissions. Because airborne infectious particles remain suspended for a longer time, dilution of the air by ventilation is an effective way to prevent infection. Infection through the deposition of droplets only occurs when an infector and a susceptible individual are close to each other.⁸ Therefore, the countermeasures against airborne transmission and droplet transmission could vary. The quantitative risk of infection due to airborne transmission and droplet transmission must be evaluated separately to consider strategies for infection control.

The Wells–Riley model^9,10 is often used to provide quantitative predictions of the probability of infection by airborne transmission. Harrichandra et al.¹¹ used the Wells–Riley model to predict the probability of infection in a nail salon. Miller et al.¹² used an improved model developed by Gammaitoni and Nucci¹³ to predict the probability of infection during a choral event. However, droplet transmission differs from the airborne transmission, wherein the droplets have a high falling speed and fall to the floor in a short time. Therefore, the quantitative risk of droplet transmissions cannot be predicted using the same methods used for airborne transmissions.

To date, several examples of predicting infection probability using the Wells–Riley method and other methods based on the aforementioned method have been proposed. However, most of these methods assume that the room is perfectly mixed and that the infectious particles are uniformly dispersed. However, rooms that have non-uniform ventilation is dependent on the location of air outlets and inlets, and levels of pollution could vary from place to place.^14,15

In this study, we proposed a method for predicting the infection probability distribution rather than the room-averaged value. We predicted the probability of infection by considering both airborne transmission and droplet transmission based on CO₂ concentration and occupant positions. The term ‘airborne transmission’ refers to the transmission by airborne particles suspended in the air for a long time, and the term ‘droplet transmission’ refers to the transmission by large droplets that descend to the floor in a short time. This study is unique because it predicts (1) the distribution of the infection probability, rather than the room-averaged value, and (2) the infection probability considering both airborne transmission and droplet transmission.

Mathematical model

The equation for predicting the infection probability and number of disease cases

Airborne transmission

The term ‘airborne transmission’ refers to the transmission of airborne particles suspended in the air for a long time, such as microdroplets and droplet nuclei. This also includes larger droplets that have become small after evaporation. The Wells–Riley equation is widely used to predict the probability of infection caused by airborne transmission.^16,17,18 Rudnick and Milton extended the Wells–Riley equation^9,10 and proposed a method to determine the average number of quanta inhaled by occupants as suspended particles based on the CO₂ concentration of a target space.¹⁹ They based this method on the principle that, as exhaled air is the only source of CO₂, the CO₂ concentration can be used as a marker for particles that remain suspended in the air for a long time. CO₂ is a gaseous scalar, whereas the infectious particles are aerosols, which are multiphase mixtures. However, they can be approximated to be transported in almost a similar manner indoors.^20,21 However, the method developed by Rudnick and Milton¹⁹ assumes that the target room is in a state of perfect mixing and calculates the infection probability averaged over the entire room. Qian et al.²² proposed a method for determining the distribution of the infection probability based on computational fluid dynamics (CFD) results using the Wells–Riley equation. Guo et al.²³ proposed a method that combines the Wells–Riley equation and spatial flow impact factor (SFIF) to determine the optimal placement of occupants and air purifiers based on results of a single CFD analysis. However, these methods require CFD analysis. When performing calculations using CFD, it is necessary to assume the operating conditions of the heating, ventilation and air conditioning (HVAC) system and positions of occupants in advance. Nonetheless, the actual conditions often differ from the assumptions, and it is preferable to perform real-time monitoring, based on measurements of the actual conditions.

In this study, the target space was discretized horizontally, and equations (1–4) were proposed to predict the distribution of the infection probability, rather than the room-averaged value, based on the CO₂ concentration in each grid. By using the CO₂ concentration distribution to predict the infection probability distribution, the airflow distribution and ventilation efficiency distribution can be considered

P_{a i r} (i, j) = D_{a i r} / S = 1 - \exp (- μ_{a i r} (i, j))

(1)

μ_{a i r} (i, j) = f (i, j) B q_{a i r} t

(2)

f (i, j) = (C (i, j) - C_{0}) / C_{a}

(3)

B = I / n

(4)

where

P_{a i r} (i, j)

is the probability of infection by airborne transmission at grid

i

j

(n.d.);

D_{a i r}

is the predicted number of disease cases caused by airborne transmission (people);

S

is the number of susceptible occupants (people);

μ_{a i r} (i, j)

is the number of quanta inhaled by a susceptible occupant as airborne particles at grid

i

j

(quanta);

f (i, j)

is the proportion of exhaled air at grid

i

j

(n.d.), as expressed by equation (3);

B

is the initial infection rate (n.d.), which is expressed by Equation (4);

q_{a i r}

is the quantum generation rate of airborne particles by an infector (quanta/s);

t

is the exposure time (s);

C (i, j)

is the CO₂ concentration at grid

i

j

(ppm), which should be measured at the respiratory height (e.g. 0.7–1.2 m above the floor, assuming a sitting position);

C_{0}

is the CO₂ concentration in the outdoor air (ppm), set to 400 ppm;

C_{a}

is the CO₂ concentration added to the exhaled breath during breathing (ppm), set to 38,000 ppm assuming office work¹⁹;

I

is the number of infectors (people) and

n

is the number of occupants in the room (people).

Droplet transmission

The term ‘droplet transmission’ refers to the transmission by droplets that descend to the floor in a short amount of time. The Wells–Riley equation is designed in terms of the infection probability caused by airborne particles that are suspended in the air for a long time. Thus, a separate method is required to represent the infection probability owing to the droplets that fall to the floor in a short time.

Sun and Zhai²⁴ included the effect of falling droplets in the Wells–Riley method and predicted the infection probability in various buildings and transportation systems. According to this method, the probability of infection caused by droplet transmission can be reduced by increasing the amount of outdoor air supply. However, in nature, airflow has a minor effect on the distance covered by large droplets, and the probability of infection does not change even when the outdoor air supply varies. Therefore, this method is not suitable for predicting the infection probability of droplet transmission.

Considering these facts, equations (5–8) are proposed to predict the probability of infection by droplet transmission. The assumptions of the proposed equations are: (1) the outdoor air supply is not related to the infection probability caused by droplet transmission; (2) the distribution of the infection probability, rather than the room-averaged value, can be predicted; (3) the droplets from every single occupant can be considered and (4) the probability of droplet transmission ( $P_{d}$ ) is based on experimental results rather than numerical simulations

P_{d r o} (i, j) = D_{d r o} / S = 1 - \exp (- w_{d r o} μ_{d r o} (i, j))

(5)

μ_{d r o} (i, j) = α (i, j) B q_{d r o} t

(6)

α (i, j) = \sum_{k = 1}^{n} P_{d} (i, j, k)

(7)

P_{d} (i, j, k) = 1 / (1 + \exp ((d (i, j, k) - 0.571 m) \times 10.185))

(8)

where

P_{d r o} (i, j)

is the probability of infection by droplet transmission at grid

i

j

(n.d.);

D_{d r o}

is the predicted number of disease cases caused by droplet transmission (people);

μ_{d r o} (i, j)

is the number of quanta ingested as droplets at the grid

i

j

(quanta) and

w_{d r o}

is the weight of infection because of droplet transmission (n.d.).

α (i, j)

is the sum of the probability of droplet arrival at grid

i

j

(n.d.), expressed by equation (7);

q_{d r o}

is the quantum generation rate of droplets by an infector (quanta/s);

P_{d} (i, j, k)

is the probability of the droplets generated by the occupant

k

arriving at grid

i

j

(n.d.) expressed by equation (8), and

d (i, j, k)

is the distance between the grid

i

j

and the grid where the occupant

k

is located.

w_{d r o}

is introduced because the Wells–Riley equation is a method for predicting the probability of infection by airborne transmission, which renders it necessary to correct

μ_{d r o}

by a certain weight to predict the probability of infection by droplet transmission.

Using the variables other than $w_{d r o}$ , the number of quanta reaching the susceptible occupants as droplets was calculated. However, the droplets reaching the occupant can be classified into three categories: (1) those reach the mucous membrane (e.g. mouth or nose), (2) those reach the skin surface and are ingested indirectly and (3) those reach the occupant but are not ingested. The combined ratio of (1) and (2) is $w_{d r o}$ . By considering these, the actual number of quanta ingested as droplets ( $μ_{d r o}$ ) was calculated.

We developed equation (8) to express the probability of droplet arrival ( $P_{d}$ ) based on the experimental results reported by Ogata et al.²⁵ They reported the results of using a cough machine for measuring the relationship between the distance and the percentage of arriving droplets. We developed an approximation for this measurement result. The measurement results and approximate equations are shown in Figure 1. Assuming an office environment, we adopted the experimental results at 50% relative humidity for this study. If an environment with a different relative humidity is to be targeted, we recommend using the experimental results of the relative humidity appropriate for that environment. Notably, although droplets were sprayed in the frontal direction of the face, this study did not consider the direction of the spray and treated it isotropically because the direction of the face may change in a short time, and the sensor used in this study could not measure the direction of the face. In addition, because the particles that were treated as large droplets in this study fall to the floor within a few seconds,^26,27 at an airflow of 0.1 m/s in the room, the effect of airflow on the arrival distance of the droplets was less than 1 m. Hence, for large droplets, the effect of airflow in the room other than coughing was ignored. However, the effect of airflow in a room cannot be ignored completely. Therefore, including the effect of airflow on the calculation of the probability of droplet arrival ( $P_{d}$ ) is an issue left for future research.

Figure 1.

Relationship between the horizontal distance and droplet transmission probability.

Combined airborne and droplet transmission

In this study, the probability of infection was predicted for airborne and droplet transmissions separately. Therefore, the probability of infection due to airborne and droplet transmission at the grid point $i$ , $j$ ( $P_{a i r + d r o} (i, j)$ ) is the probability that one of the two independent infections will occur; this can be expressed by equation (9)

P_{a i r + d r o} (i, j) = 1 - (1 - P_{a i r} (i, j)) (1 - P_{d r o} (i, j))

(9)

Using equations (1) and (5), equation (9) is reduced as equations (10) and (11) as follows

P_{a i r + d r o} (i, j) = 1 - (1 - (1 - \exp (- μ_{a i r} (i, j)))) (1 - (1 - \exp (- w_{d r o} μ_{d r o} (i, j))))

(10)

P_{a i r + d r o} (i, j) = 1 - \exp (- (μ_{a i r} (i, j) + w_{d r o} μ_{d r o} (i, j)))

(11)

The predicted number of disease cases is expressed by equations (12–15)

D_{a i r + d r o} = S \times \frac{1}{n} \sum_{l = 1}^{n} P_{a i r + d r o} (l)

(12)

D_{a i r} = S \times \frac{1}{n} \sum_{l = 1}^{n} P_{a i r} (l)

(13)

D_{d r o} = S \times \frac{1}{n} \sum_{l = 1}^{n} P_{d r o} (l)

(14)

S = n - I

(15)

where

D_{a i r + d r o}

is the predicted number of disease cases caused by airborne and droplet transmission (people);

S

is the number of susceptible occupants (people), expressed by equation (15); and

P_{a i r + d r o} (l)

is the probability of infection of the occupant l by airborne and droplet transmission (n.d.), which is obtained from equation (11) by calculating

P_{a i r + d r o} (i, j)

at the grid where the occupant l is located. Notably, the values of

P_{d} (i, j, k)

, which were calculated using equation (8), is zero when

k = l

. This is because the droplets generated by the occupant l do not need to be considered when calculating the infection probability of the occupant l. Similarly,

P_{a i r} (l)

is the probability of infection by airborne transmission for the occupant l (n.d.), and

P_{d r o} (l)

is the probability of infection by droplet transmission for the occupant

l

(n.d.).

Estimation of parameters required for the proposed method

Three of the parameters used in the proposed method are unknown: the quantum generation rate of airborne particles by an infector (

q_{a i r}

), the quantum generation rate of droplets by an infector (

q_{d r o}

) and the weight of infection caused by droplet transmission (

w_{d r o}

). We estimated these parameters using the COVID-19 outbreak summarized in Table 1, following the method of Sun and Zhai.²⁴ We estimated three patterns of parameters using three patterns of combinations of cases. We calculated the predicted infection probability (

P_{a i r + d r o}

) using equation (9) for the two cases, and we optimized the values of

q_{a i r}

q_{d r o}

and

w_{d r o}

to minimize the error relative to the actual infection percentage (

P_{a i r + d r o}

) given in Table 1. As we cannot estimate

q_{d r o}

and

w_{d r o}

separately using this method, we treated

q_{d r o} w_{d r o}

as a single parameter. For validation, we used an outbreak case in a restaurant in Guangzhou.²⁸ As the seating arrangement was already known, we calculated and used the probability of droplet arrival instead of the average distance between occupants.

Table 1.

Parameters of outbreak cases of COVID-19 (based on Sun and Zhai²⁴ and Li et al.²⁸).

Case	Usage	Exposure time, $t$ (h)	Distance between occupants, $d$ (m)	Ventilation volume with clean air, $Q / n$ (m³/h/person)	Total number of occupants, $n$ (persons)	Disease cases, $D_{a i r + d r o}$ (persons)	Percentage of infection, $P_{a i r + d r o}$ ,%
Bus in Hunan-1	Calibration	2	1.05	23.91	46	8	17.8
Bus in Hunan-2	Calibration	1	1.3	28	12	3	27.3
Bus in Ningbo	Calibration	4	0.7	20	68	25	37.3
Restaurant in Guangzhou	Validation	1.4	—	13.7	21	9	45.0

The parameters

q_{a i r}

and

q_{d r o} w_{d r o}

estimated for each combination and the infection probability predicted using these parameters are listed in Table 2 along with the corresponding errors. Several studies have estimated the quantum generation rate as airborne particles (

q_{a i r}

) of SARS-CoV-2, with values ranging from 10.5 to 1030 quanta/h depending on the respiratory activities.^12,39 The estimated value in this study, 223–226 quanta/h, is between values estimated in previous studies. However, the estimated values may be larger than the actual values because we estimated them using cluster cases. Also,

q_{a i r}

and

q_{d r o} w_{d r o}

cannot be compared because

q_{a i r}

and

q_{d r o}

are the quantum generation rate, whereas

w_{d r o}

is the weight of infection because of droplet transmission. Each of the estimated parameters exhibits a gap between the minimum and maximum estimates, indicating uncertainty in the parameter estimation. The maximum error between the actual infection percentage and the predicted infection probability is 4%, which confirms that the prediction accuracy of our method is sufficiently high.

Table 2.

Estimated parameters and error between the predicted and actual infection percentage.

Cases for calibration		Quantum generation rate by an infector as airborne particles, $q_{a i r}$ (quanta/h)	Quantum generation rate by an infector as droplets and weight of infection because of droplet transmission. $q_{d r o} w_{d r o}$ (quanta/h)	Predicted infection probability, $P_{a i r + d r o}$ , %	The error between predicted and actual infection percentage (percentage points)
Bus in Ningbo	Bus in Hunan-2	226	3.31	43.0	2.0
Bus in Hunan-1	Bus in Ningbo	223	3.40	42.8	2.2
Bus in Hunan-2	Bus in Hunan-1	226	1.18	41.0	4.0

Currently, the amount of ventilation and the distance between people are known for only a few outbreak cases of infection. Therefore, in this study, we used only four outbreak cases for the calibration and validation of parameters. Thus, there is large uncertainty in the estimation of parameters (i.e. $q_{a i r}$ and $q_{d r o} w_{d r o}$ ). The larger the number of outbreak cases used for calibration, the more accurate the estimated parameters. Therefore, we recommend that when outbreak cases with parameters similar to those used in this method (e.g. ventilation volume) are available, they should also be used for calibration. In addition, because we based the estimation of parameters on outbreak cases, the parameters of the quantum generation rate may be larger than the actual value. In addition, the quanta generation rate could vary greatly with individual differences (e.g. super spreader), respiratory activities and activity levels. In the future, when more outbreak cases in which these factors are known and become available, it will be desirable to estimate the quanta generation rate for each condition and use the quanta generation rate corresponding to the actual situation in the space where the infection probability is predicted.

Application in an actual space

Outline

We evaluated the feasibility of the proposed method by applying it to an actual environment. We measured the CO₂ concentration and the occupant position information in the target space and used the data as input for the proposed method. We show the procedure for predicting the infection probability distribution and the number of disease cases based on the measurement results in Figure 2. The target space for the prediction was part of the Shimizu Corporation offices in Tokyo, Japan. An outline of the air conditioning system is shown in Figure 3. The supply air from AHU-1 was provided by linear slot diffusers on the floor, and the supply air from AHU-2 was provided by seepage through the entire carpet in the target area. Figure 4 shows the arrangement of the measurement points. The measurements were carried out on March 18–19, 2021. The HVAC system was heated during the measurement period.

Figure 2.

Procedure for predicting the infection probability distribution and the number of disease cases based on the measurement results.

Figure 3.

Outline of the air conditioning system: (a) plan view and (b) sectional view.

Figure 4.

Plan view of the measurement positions.

Measurement methods

CO₂ concentration

We measured the CO₂ concentration using a ‘TR-76Ui’ thermo-hygro-CO₂ meter (T&D). We calibrated the sensor with the outdoor air before we placed the sensor. The arrangement of the sensors among the 37 measurement points is shown in Figure 4. The height of the sensors (Figure 5) was 1.2–1.5 m above the floor. In the method proposed in this study, to ensure a high spatial resolution of the CO₂ concentration distribution, as many CO₂ sensors as feasible should be installed, at least at the four corners of the targeted space. We obtained the horizontal distribution of the CO₂ concentration by spline interpolation based on the values that were obtained at the 37 measurement points. The target space was discretized using a 10 cm × 10 cm mesh. The width of the discretization should not exceed 1 m because each seating position should be resolved.

Figure 5.

Installed CO₂ sensor.

Occupant position information

The ‘Quuppa Intelligent Locating System™’ was used to measure the positions of occupants in the room. The Quuppa system measures the position of tags using locators whose positions are known. We can identify the positions of the tags through Bluetooth communication of tags with multiple locators. ‘Quuppa LD-7L’ was used as the locator. The arrangement of the eight locators is shown in Figure 4. The installation conditions of the locators are illustrated in Figure 6. We installed all locators at a height of 2.6 m above the floor. Quuppa QT1-1 tags were used, as shown in Figure 7, and the participants always carried the tags. The participants in the experiment were employees who usually work in the targeted space. We explained the experimental procedure to the participants and obtained their consent in advance. There were nine subjects, and no one except the subjects was in the targeted space. During the measurement period, the personnel in the room worked as usual, with no restrictions on movement during the experiment. Therefore, subjects entered and exited the targeted space owing to their work.

Figure 6.

Installed Quuppa locator.

Figure 7.

Quuppa tag.

Results and discussion

Measurements of CO₂ concentration and occupant positions

The CO₂ concentrations measured by each sensor are shown in Figure 8. We corrected the results to remove any bias between sensors. For the correction, the average CO₂ concentration measured by all sensors at 3:00 on 19 March was used as the true value when the CO₂ concentration in the room was considered to be uniform. The difference between the true and measured values at 3:00 on 19 March was assumed to be the sensor bias, and the corresponding correction was applied to the entire data. Before the correction, the difference between the maximum and minimum CO₂ concentrations measured at 3:00 on 19 March was 27.1 ppm. The accuracy reported by the manufacturer was 50 ppm +5% of the value, which is within the expected margin of error. Notably, the measured CO₂ concentration may contain errors of the aforementioned order of magnitude.

Figure 8.

CO₂ concentrations measured by each sensor. The biases between sensors were corrected.

The CO₂ concentration was higher during the day than that at night because there were occupants in the office during the day but not at night. The differences in the measured CO₂ concentrations between the sensors were small at night but large during the day because of the presumably non-uniform CO₂ generation by the occupants during the day. The CO₂ concentration in the afternoon of 18 March was higher than that in the morning because of a presumable decrease in the amount of outside air introduced in the afternoon of 18 March . We denoted results at 9:30 on 18 March as a ‘well-ventilated office’ and the results at 16:50 on 18 March as a ‘poorly ventilated office’.

The CO₂ concentration distributions in the ‘well-ventilated office’ and ‘poorly ventilated office’ are shown in Figure 9, where the measured positions of occupants are shown as circles. The spatial average CO₂ concentration was higher in the ‘poorly ventilated office’ than the ‘well-ventilated office’. The CO₂ concentration was lower on the right side than on the left side of the room. This may be because the amount of outdoor air supplied on the right side near the AHU-2 was higher than that on the left side far from the AHU-2 (Figure 3(a)).

Figure 9.

Distribution of measured CO₂ concentration in the (a) ‘well-ventilated office’ and (b) ‘poorly ventilated office’. Circles show the measured positions of occupants.

Prediction of infection probability

We predicted the probability of infection using the measured CO₂ concentrations and occupant positions. The probability of infection by airborne transmission ( $P_{a i r}$ ) was predicted using equations (1–4), the probability of infection by droplet transmission $P_{d r o}$ was estimated using equations (5–8). The probability of infection by airborne and droplet transmission $P_{a i r + d r o}$ was estimated using equation (11). For the parameters estimated, the median values of the three estimates were used. Thus, the quantum generation rate of airborne particles by an infector ( $q_{a i r}$ ) was set at 226 quanta/h, and the quantum generation rate of droplets by an infector and the weight of infection caused by the droplet transmission ( $q_{d r o} w_{d r o}$ ) was set to 3.31 quanta/h. The initial infection rate ( $B$ ) was set at 0.1% based on the results of the antibody possession survey conducted in Tokyo in June 2020.²⁹ An antibody possession survey was conducted to determine the percentage of people who had been infected with COVID-19 in the past. Thus, the proportion of occupants who were infectious at the time was less than 0.1%. Therefore, the setting of the initial infection rate ( $B$ ) at 0.1% would predict a higher probability of infection than the actual situation. The exposure time ( $t$ ) was set to one hour.

The predicted infection probabilities in the ‘well-ventilated office’ and the ‘poorly ventilated office’ are shown in Figures 10–12. Figure 10 shows that the probability of infection by airborne transmission $P_{a i r}$ was higher in the ‘poorly ventilated office’ than in the ‘well-ventilated office’. This is because the CO₂ concentration was higher in the ‘poorly ventilated office’ than in the ‘well-ventilated office’, as shown in Figure 9. In contrast, Figure 11 shows that the probability of infection by droplet transmission ( $P_{d r o}$ ) was the same in both conditions and was greater in the area around the occupants. This is because equations (5–8), which were used to predict the probability of infection by droplet transmission, consider the distance between occupants rather than the CO₂ concentration. However, in the ‘poorly ventilated office’, a pair of occupants was in close proximity with each other, and the probability of infection due to droplet transmission was quite high at their position. Figure 12 shows that the probability of infection by airborne and droplet transmission ( $P_{a i r + d r o}$ ) was greater in the ‘poorly ventilated office’ than the ‘well-ventilated office’. The probability of infection by airborne and droplet transmission ( $P_{a i r + d r o}$ ) was greater on the left than the right side of the room, even though more occupants were located on the right side of the room in the ‘well-ventilated office’. This may be because the amount of outdoor air supplied by the air conditioning system was higher on the right side than on the left side of the room, as described in the previous section. Thus, the method proposed in this study can provide a quantitative analysis of the probability of infection that integrates the ventilation efficiency and the distance between occupants (i.e. social distancing).

Figure 10.

Probability of infection by airborne transmission $P_{a i r}$ in the (a) ‘well-ventilated office’ and the (b) ‘poorly ventilated office’.

Figure 11.

Probability of infection by droplet transmission $P_{d r o}$ in the (a) ‘well-ventilated office’ and the (b) ‘poorly ventilated office’. Circles show the measured positions of occupants.

Figure 12.

Probability of infection by airborne and droplet transmission $P_{a i r + d r o}$ in the (a) ‘well-ventilated office’ and the (b) ‘poorly ventilated office’. Circles show the measured positions of occupants.

Prediction of the number of disease cases

From equations (12–15), the number of disease cases caused by airborne and droplet transmission $D_{a i r + d r o}$ together, the airborne transmission $D_{a i r}$ alone and droplet transmission $D_{d r o}$ alone were predicted. The time series of the predicted number of disease cases is shown in Figure 13. The variation trend in the predicted number of disease cases was observed. This is because the variation in the probability of infection caused by the airborne transmission is due to the variation in the ventilation. In addition, there were instances of sharp pulsed increases in the number of disease cases. This was due to the increase in the probability of infection by the droplet transmission when the distance between a pair of occupants was short.

Figure 13.

Time series of predicted disease cases $D_{a i r + d r o}$ . The plots are at one-minute intervals.

The predicted number of disease cases in the ‘well-ventilated office’ and ‘poorly ventilated office’ is shown in Figure 14. The number of disease cases caused by airborne transmission $D_{a i r}$ was lower in the ‘well-ventilated office’ than in the ‘poorly ventilated office’. The proposed method could quantitatively evaluate the difference in the number of infected persons due to the ventilation efficiency. The fact that the distance between a pair of occupants was short in the ‘poorly ventilated office’ was reflected in the predicted number of disease cases caused by droplet transmission $D_{d r o}$ . Thus, $D_{d r o}$ was larger in the ‘poorly ventilated office’ than in the “well-ventilated office”. $D_{d r o}$ is not related to ventilation efficiency, but only to the distance between people. In the ‘well-ventilated office,’ the predicted number of disease cases caused by droplet transmission $D_{d r o}$ was negligible compared with that caused by airborne transmission $D_{a i r}$ . This result agrees with those of previous studies.^30,31 In such an environment, the importance of ventilation as a countermeasure against airborne transmission is relatively high. Thus, the method proposed in this study can be used to predict the relative importance of countermeasures against the airborne transmission and the droplet transmission. Notably, because there is uncertainty in the estimation of parameters (i.e. $q_{a i r}$ and $q_{d r o} w_{d r o}$ ), there is also uncertainty in the prediction of the number of disease cases. The impact of uncertainty should be reduced by using a larger number of outbreak cases for parameter estimation.

Figure 14.

Predicted disease cases $D_{a i r + d r o}$ , $D_{a i r}$ and $D_{d r o}$ in the ‘well-ventilated office’ and the ‘poorly ventilated office’.

Comparison between cases with measures taken

The probability of infection was predicted for two scenarios, assuming that countermeasures were taken in a ‘poorly ventilated office’, which is called ‘without measure’ in this section. As a countermeasure, the case of ‘increased ventilation’ and a case of ‘wearing a mask’ were assumed. In the ‘increased ventilation’ case, we assumed the difference in CO₂ concentration from the outside air to be half of that of 16:50 on 18 March. In the ‘wearing a mask’ case, we altered the quantum generation rate to assume the use of surgical masks. We assumed that wearing a mask would prevent all large droplets from spreading and reduce the spread of airborne particles by 75%.³² The predicted infection probabilities in the ‘increased ventilation’ and ‘wearing a mask’ are shown in Figure 15. In the ‘increased ventilation’ case (Figure 15 (a)), the probability of infection in ‘areas away from’ people was lower than in the ‘without measures’ case (Figure 12 (b)). Ventilation cannot control the spread of large droplets, resulting in a high probability of infection in the neighbourhood of people. The probability of infection was lower in the ‘wearing a mask’ case (Figure 15 (b)), both away from people and in the neighbourhood of people, compared to the ‘without measures’ case (Figure 12 (b)).

Figure 15.

Probability of infection by airborne and droplet transmission $P_{a i r + d r o}$ in the (a) ‘increased ventilation’ and the (b) ‘wearing a mask’.

To compare these results quantitatively, the number of disease cases was predicted as shown in Figure 16. In the ‘increased ventilation’ case, the number of disease cases caused by airborne and droplet transmission $D_{a i r + d r o}$ was decreased by 36% compared to the ‘without measure’ case. In the ‘wearing a mask’ case, the number of disease cases caused by airborne and droplet transmission $D_{a i r + d r o}$ was decreased by 82% compared to the ‘without measure’ case. In both cases, the predicted number of disease cases caused by airborne transmission $D_{a i r}$ was larger than that caused by droplet transmission $D_{d r o}$ ; therefore, the importance of countermeasures against airborne transmission is relatively high.

Figure 16.

Predicted disease cases $D_{a i r + d r o}$ , $D_{a i r}$ and $D_{d r o}$ in the (a) ‘without measure’, (b) ‘increased ventilation’ and (c) ‘wearing a mask’.

Limitations

(1) The deposition of airborne particles, decrease in viral titres over time, filters for recirculated air in HVAC systems, irradiation with UV-C and individual differences in the quantum generation rate could affect infection.^33,34,35 However, the proposed method uses CO₂ concentration as a marker for airborne particles; therefore, other factors cannot be considered. If the effects of filters or UV-C can be determined in ways other than CO₂ concentration, these effects should be added to the prediction equation.

(2) In the proposed method, the fomite transmission was ignored. This is because fomite transmission via surfaces is insignificant compared to that via droplets and airborne particles of SARS-CoV-2.^36,37,38 However, the contribution of fomite transmission was not completely absent. Therefore, the proposed method cannot be applied to infectious diseases in which the contribution of fomite transmission is significant.

(3) The quanta generation rate varies significantly with individual differences (e.g. super spreader), respiratory activities and activity levels. Additionally, the value varies significantly according to the estimation method. For example, Buonanno et al. (2020)³⁹ estimated the quanta generation rate due to breathing during light exercise to be 33.9 quanta/h. If this quanta generation rate is adopted to predict the number of disease cases due to airborne transmission in a ‘well-ventilated office’, the number was predicted to be 0.0021, which is 15% of the value obtained in this study (Figure 14). Thus, the prediction of infection probability and number of disease cases involve a significant degree of uncertainty.

Conclusion

In this study, we proposed a method for predicting the distribution of the probability of infection, rather than the room-averaged value. In addition, the probability of infection was predicted by considering both airborne transmission and droplet transmission. The infection probability caused by airborne transmission, which is the transmission by airborne particles suspended in the air for a long time, was predicted based on the CO₂ concentration. The infection probability caused by droplet transmission, which is the transmission by droplets that fall to the floor in a short time, was predicted based on the occupant position information. As the prediction accuracy of the method was verified using three patterns of combinations of actual outbreak cases of COVID-19, the difference between the actual infection percentage and predicted infection probability was a maximum of 4 percentage points, which confirms that the prediction accuracy of the method employed in this study is sufficiently high. Notably, only four outbreak cases were used for calibration and validation in this study. Thus, there is a huge uncertainty in the estimation of the parameters. By applying this method to an actual office, the infection probability was confirmed to be relatively high in a zone where the amount of outdoor air supply was relatively small. Therefore, the proposed method can be used to predict the infection probability quantitatively by integrating the ventilation efficiency and the distance between occupants that is unique to each building. Furthermore, disease cases were predicted, and the predicted number of disease cases caused by droplet transmission was not negligible when a pair of occupants were at a close distance. Therefore, it was confirmed that it is possible to analyse the relative effectiveness of countermeasures against airborne transmission and droplet transmission and study infection control strategies that are unique to a given building using the proposed method.

This study is unique in that (1) a method for predicting the distribution of the infection probability, rather than the room-averaged value, was developed and (2) the predicted infection probability considers both airborne transmission and droplet transmission. Using the proposed method and monitoring the infection probability in real-time, the relative effectiveness of countermeasures against airborne transmission and droplet transmission could be analysed, and a facility manager could consider strategies for infection control. Using the proposed method, facility users can check the infection probability distribution in real time to select a seat with a minor risk of infection.

Footnotes

Acknowledgements

The authors would like to express the deepest appreciation to Dr. Hisashi Hasebe, Mr. Nobuhiro Miura, Dr. Takashi Kurihara, Dr. Kengo Tomita and Dr. Keichi Suzuki of Shimizu Corporation for their kind support and beneficial discussion in this research project.

Author contributions

Miguel Yamamoto: conceptualization (equal); formal analysis (lead); visualization (lead); investigation (equal); methodology (equal); project administration (equal); software (lead); validation (equal); writing‐original draft preparation (lead) and writing‐review and editing (equal). Akihiro Kawamura: conceptualization (equal); data curation (lead); investigation (equal); methodology (equal); project administration (equal) and writing‐review and editing (equal). Shin-ichi Tanabe: supervision (equal); investigation (supporting); project administration (equal) and writing‐review and editing (equal). Satoshi Hori: supervision (equal); investigation (supporting); project administration (equal) and writing‐review and editing (equal).

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Correction (September 2023):

The Funding section of the article has been updated since its original publication.

ORCID iDs

Miguel Yamamoto

Shin-ichi Tanabe

Satoshi Hori

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Predicting the infection probability distribution of airborne and droplet transmissions

Abstract

Keywords

Introduction

Mathematical model

The equation for predicting the infection probability and number of disease cases

Airborne transmission

Droplet transmission

Combined airborne and droplet transmission

Estimation of parameters required for the proposed method

Application in an actual space

Outline

Measurement methods

CO2 concentration

Occupant position information

Results and discussion

Measurements of CO2 concentration and occupant positions

Prediction of infection probability

Prediction of the number of disease cases

Comparison between cases with measures taken

Limitations

Conclusion

Footnotes

Acknowledgements

Author contributions

Data availability

Declaration of conflicting interests

Funding

Correction (September 2023):

ORCID iDs

References

CO₂ concentration

Measurements of CO₂ concentration and occupant positions