Infection risk in cable cars and other enclosed spaces

Abstract As virus‐laden aerosols can accumulate and remain suspended for hours in insufficiently ventilated enclosed spaces, indoor environments can heavily contribute to the spreading of airborne infections. In the COVID‐19 pandemics, the role possibly played by cable cars has attracted media attention following several outbreaks in ski resort. To assess the real risk of infection, we experimentally characterize the natural ventilation in cable cars and develop a general stochastic model of infection in an arbitrary indoor space that accounts for the epidemiological situation, the virological parameters, and the indoor characteristics (ventilation rate and occupant number density). As a results of the high air exchange rate (we measured up to 180 air changes per hour) and the relatively short duration of the journey, the infection probability in cable cars traveling with open windows is remarkably lower than in other enclosed spaces such as aircraft cabins, train cars, offices, classrooms, and dining rooms. Accounting for the typical duration of the stay, the probability of infection during a cable‐car ride is lower by two to three orders of magnitude than in the other examples considered (the highest risk being estimated in case of a private gathering in a poorly ventilated room). For most practical purposes, the infection probability can be approximated by the inhaled viral dose, which provides an upper bound and allows a simple comparison between different indoor situations once the air exchange rate and the occupant number density are known. Our approach and findings are applicable to any indoor space in which the viral transmission is predominately airborne and the air is well mixed.

classified into droplet route and airborne route. In the droplet route, the conjunctiva or the buccal and nasal mucosae of a susceptible individual are infected by large respiratory droplets that are emitted by a symptomatic individual which would otherwise rapidly settle by gravity; in contrasts, the airborne route is characterized by the presence of infectious viral copies in very small droplets and aerosols that can accumulate in enclosed spaces and remain in the air for hours. This classification often relies on rather arbitrary cutoff sizes (e.g., about 5 μm in Ref. [2]), but the involved physical processes are much more complex as intermediate droplets can evaporate into aerosols depending on ambient conditions (see, e.g., Refs [3][4][5][6]).
Although at the very beginning of the pandemic spreading of the infection through surfaces (the fomite route) and the droplet route were considered the primary mechanisms of infection (see, e.g., Refs [2,7,8]), attention has been immediately drawn to the importance of the airborne route. 9,10 With the time, the role of surfaces has been reconsidered and scaled back 11,12 and there has been mounting evidence that airborne transmission plays a significant role in the COVID-19 pandemic (see, e.g., Refs [3,[13][14][15]). This awareness about airborne transmission has led to increase the attention paid to enclosed spaces, where potentially virus-laden aerosol can accumulate and remain suspended in the air for hours, leading to an increased risk of infection if ventilation is poor. 16 Indeed, many studies report higher probability of infection indoors. For instance, Nissen et al. 17 performed an epidemiological analysis hinting at airborne infection in a hospital ward. Kasper et al. 18 reported an outbreak on an aircraft carrier and showed that the crew members working in confined areas (such as the engine room) were exposed to a higher risk of infection. By means of epidemiological data, measurements, and computational fluid dynamics, Li et al. 19 suggested that poor indoor-air management can potentially lead to airborne transmission in restaurants. Aerosol transmission of SARS-CoV-2 has also been reported in superspreading events 20 and in poorly ventilated courtrooms. 21 Public transport has also been indicated as possibly playing a major role in the spreading of COVID-19 disease. Despite the comparatively shorter stay of the passengers, train, busses, and aircraft are characterized by high passenger density and turnover. This is the case also for cable cars and cableway gondolas, that have received major attention by mass media at the beginning of the pandemic, leading to the shutdown of ski-resort operation in most countries with severe economic consequences. 22 This was fueled by several outbreaks that have been observed in European ski resorts, particularly in Austria 23,24 and Switzerland. 25 Recently, Gianfredi et al. 22  where the proportionality constant b [m3/s] accounts for the probability of infection and is proportional to the air intake rate. In epidemiology, the law of mass action is widely used for environmentally mediated diseases (see, e.g., Refs [26][27][28]) and is also at the basis of compartmental models (see, e.g., Refs [29][30][31][32]). Integrating Equation 1, we obtain the probability of a new infection,

| The G-N model and the probability of infection
The system of Equations 1 and 7 is the G-N model, 27  As long as the risk is small, the probability of infection can be approximated by the infectious dose (P inf (t) ≈ Q(t)) and exhibits three distinct regimes (see Figure 1b

| THE PROBAB ILIT Y OF INFEC TI ON WITH S TO CHA S TI C QUANTA EMISS I ON
The probability of infection that we have derived in the previous sections assumes that breathing, virus removal, and quanta emission rates are known. If we consider these parameters as stochastic where P N Q , q is the joint probability for N Q and q , which are, in general, not independent because the normalized infectious dose depends on the virus removal rate (see Equation 10).

| A stochastic model of quanta emission
In the following, we will assume that the variability of the breathing and removal rates are small compared with the variability of the quanta emission rates, which on the contrary can vary over orders of magnitude across infectious individuals; hence, we will threat only the quanta emission rate, q , as stochastic variable. The quanta emission rate is given by the sum of the emission rates of all infectious individuals in the enclosed space, where I is the number of infected individuals, and q,i is the quanta emission rate of the ith individual. Therefore, we write the probability as which is the product of the conditional probability P q |I and the binomial distribution, which describes the probability that I of the N occupants are infectious when the prevalence of the infection (i.e., the fraction of infectious individuals in the general population) is .
). Right: The probability of infection normalized by N Q and compared with Q ∕ N Q (red line) If the individual emission rates follow a log-normal distribution, 34 we can use the Fenton-Wilkinson method to approximate their sum as a log-normal random variable. 35,36 By assuming that the emission rates of all individuals in the enclosed space belong to the same log-normal distribution (i.e., log q,i ∼  , 2 ), we have that where the variance and the mean are 2 I = log e 2 − 1 I + 1 and The conditional probability of infection, P inf t|N Q , q , is given by Equation 12, whereas the conditional probability of quanta emission, P q |I , follows a log-normal distribution, and B(I; N, ) is the binomial distribution that describes the probability that I of the N individual in the enclosed space are infectious when the prevalence is and varies according to the epidemiological situation (the case I = N is not included in the sum because at least one occupants must be susceptible for an infection to occur).
Notice that Equation 17 assumes that all non-infectious individuals are susceptible (hence, there is no immunity resulting from vaccination or recovery from a previous infection). To account for preexisting immunity, in Equation 17, we should introduce the binomial probability that only S of the N − I non-infectious individuals are susceptible, that is, the infection probability would be

| Quanta emission rate of SARS-CoV-2
In case of respiratory diseases, the emission of infectious viral copies into the environment occurs through virus-laden droplets and aerosol that are produced by infectious individuals during respiratory events such as sneezing, coughing, breathing, or talking (see, e.g., Refs [37,38] 44 on the SARS-CoV-2 concentration found in the sputum, and on the copies-to-quanta conversion factor estimated by Watanabe et al. 45 for SARS-CoV-1, they performed Monte-Carlo simulations to derive the probability density functions of the quanta

| ME A SUREMENT C AMPAIG NS IN C AB LE C AR S
The viral concentration in the air and the probability of infection are strongly dictated by the ventilation rate, E, which is able to remove infectious viral copies from the air more efficiently than gravity deposition or inactivation. In case of natural ventilation, ventilation rate is generally not known with a sufficient level of precision. In

| PROBAB ILIT Y OF INFEC TI ON IN C AB LE C AR S
With the quanta emission rate model of Buonanno, Morawska, and

F I G U R E 5 Left (A)
. The probability of infection in cable cars of OMEGA3 for different air exchange rate: 180 ACPH (cyan), 120 ACPH (light blue), 60 ACPH (blue), 20 ACPH (navy) and no air exchange (black). Right (B). Probability of infection as a function of time in the three types of cable car: ROOF (black, 45 ACPH), FLAPS (red, 160 ACPH), and OMEGA3 (blue, 120 ACPH). It is assumed that the prevalence is 1%, that the cable cars travel at full capacity (i.e., 77, 80, and 8 passengers for ROOF, FLAPS, and OMEGA3, respectively), and that the breathing rate and the potential quanta emission rate of the passengers are comparable to a standing person (see Table 1)

F I G U R E 6
Effects of vocalization and different metabolic activities on the probability of infection in the three cable cars traveling at full passengers capacity for typical air exchange rates (black, ROOF with 77 passengers and 45 ACPH; red, FLAPS with 80 passengers and 160 ACPH; and blue, OMEGA3 with 8 passengers and 120 ACPH). We assume a prevalence of 1%. Breathing rate and the potential quanta emission rates of the passengers are taken form Table 1 a total of about 5 ACPH, less than 2 ACPH of which are from outdoor air, by Ref. [49]). We can envisage two situations: in the most optimistic scenario, the ventilation system comprises HEPA filters that are assumed to have close to 100% efficiency in blocking virus-laden aerosol and the total of 15 ACPH contributes to the viral removal rate; instead, if the filter has negligible efficiency in blocking virusladen aerosol, only 8 ACPH contributes to virus removal. In the case of an aircraft cabin, certification regulation requires a minimum of 5 L/s of fresh air per person at standard cabin conditions, a value that can be increased to around 8 L/s in new aircrafts for improved comfort. 50,51 Aircraft-cabin ventilation is organized by compartments, with preferential flow across seats in the same row. Notice, however, that this does not prevent the diffusion of particles across a few adjacent seat rows 51,52 and, in any case, the passenger number density, which is the decisive parameter for infection probability, is not affected by compartmentalized ventilation.
After that is more than two orders of magnitude higher.

| CON CLUS IONS
The field campaigns to measure the airflow velocity in cable cars have allowed us to estimate the air exchange rates, which is a rather elusive parameters in naturally ventilated enclosed spaces and F I G U R E 7 Comparison between the infection probability in cable cars and in other enclosed space. We plot the infection probability in the ROOF (black) and the OMEGA3 (blue) cable cars. (For illustration we extend the solution well beyond the typical journey time of 5 and 20 min for the ROOF and the OMEGA3 -dashed lines). We assume a prevalence of 1% and, if not stated otherwise, quanta emission and breathing rate correspond to a standing person (this is used as a proxy of persons with low metabolic activity but not resting  Nevertheless, Equation 25 allows us to reliably compare the probability of infection between two indoor situations also if the emission model is inaccurate or even if it is not available. For instance, assuming the same activity level and emission model, for a given prevalence the relative risk with respect to a reference indoor situation can be simply calculated from the duration of the stay, the occupant number density, and the air exchange rate.
Finally, we remark that the stochastic infection model is general and can be applied to all indoor situations in which the viral transmission is predominately airborne and the indoor air is well mixed.
Under these conditions, our approach and findings are applicable to any enclosed space and allow us to assess the effectiveness of simple mitigation measures such as increased ventilation or reduction of the maximum occupancy in shared indoor spaces.

ACK N OWLED G M ENTS
We acknowledge the technical support of Benjamin Rembold and Andrin Landolt (streamwise gmbh) who performed the air velocimetry measurements. We are also grateful to Roger Vonbank,

CO N FLI C T O F I NTE R E S T
We declare no conflict of interest.

DATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.