A comprehensive modelling approach to estimate the transmissibility of coronavirus and its variants from infected subjects in indoor environments |…

Model

The present comprehensive model combines the detailed aerosol dynamics with anovel double Poisson model to estimate the probability that at least one carrier particle containing at least one virion will be deposited in the lungs. This model recognizes not only the discreteness of virions and their fluctuations but also that of the inhaled residues/droplets which vector them and hence, introduces fluctuations in the entire size spectrum18,19,20. The aerosol dynamics accounts for evaporation, residue formation, room dispersion, settling, plate-out and deposition in the respiratory tract of the inhaling subject.

In the present work, thefalling-to-mixing-plate-out model21 is implemented, which allows a droplet's residence time () to smoothly transition from a gravity-dominated (larger particles, diameter>50m, <100s) to a turbulence-dominated (small particle, diameter<5m, >3000s) regime as shown in Fig.1. It is worth mentioning that turbulent mixing extends the particle residence time for droplets of intermediate size. The variation of droplet lifetime with RH is significant only for large particles of diameter in the range of 2080m, mainly due to evaporation and gravitational settling in this size regime. The study results show that the lifetime of virusols in the indoor environment is determined mainly by deposition; however, viral deposition in the lungs is entirely determined by viral load and aerosol physics. The reciprocal of theresidence time of virus laden droplets to reach a given risk is an important parameter used to estimate the rate of propagation/transmissibility.

Lifetime of droplets in a typical indoor environment.

The present study attempts to calculate the exposure time required to achieve a tangible single-hit risk for a given expiratory event as well as the event reproduction number (({R}_{e})) for the given input parameters. Coughing18,22,23 and sneezing18 will be specific to the sick and symptomatic patients, although breathing23 and speaking18,20,23 are normal expiratory processes relevant to all subjects. Table 1 lists the parameters of expiratory emission18,23,24, such as droplet size distribution, frequency of emission, virion concentration in emitted droplets, etc.

For each expiratory event, numerical computations are used to determine the exposure time for different risk levels (0.1%, 1%, 10%, and 50%) and AERs (0.510h1). In the exposure time calculations, it is assumed that the emissions are continuous with the given rate and the value is estimated for a given risk. The model findings (Fig.2) reveal that, up to a critical viral load, the exposure duration decreases linearly with the viral load in the loglog graph. Although the findings are not shown here, the slope of the linear component increases with emission rate (S0). The critical viral load in this case is, 1013#/mL for breathing, 1011#/mL for coughing, 1010#/mL for sneezing, 1012#/mL for speaking for a risk of 0.1%. Beyond this critical viral load, the risk becomes a constant or invariant with respect to theviral load. These results also show that the risk is strongly dependent on the emission rate; for example, if the particle emission rate of 1000#/s for speech20 is considered, then the risk will increase in that proportion against the risk value estimated for 270#/s23,24.

Exposure time as a function of viral load for a given infection risk and ventilation rate in the indoor environment.

Alternative to the exposure time estimates, the single-hit risk (double Poisson model)is estimated under the influence of all the four expiratory events occurring simultaneously at given emission rates. The joint risk probability is then given by,

$${R}^{^{prime}}=1-{P}_{0,B}times {P}_{0,Sp}times {P}_{0,C}times {P}_{0,Sn,}$$

(1)

where ({P}_{0,B}=expleft({-N}_{d}left[1-expleft(-{n}_{v}right)right]right)) is the probability of zero-hit for thebreathing expiratory process, ({N}_{d}) is the typical number of droplets inhaled by a person, ({n}_{v}) is the average number of virions contained in a droplet, the suffices Sp, C and Sn denotes speaking, coughing and sneezing events respectively. It is to be noted that the transmissibility of a virus is measured via single-hit risk probability, dominated by the aerosol route of exposure. Also, it has been argued often that the transmissibility of the virus is linked with the viral load3,25,26, and hence, the risk of transmission to a susceptible individual is estimated as a function of viral load for specified exposure times (Fig.3).

(a) Variation of single hit risk for susceptible persons as a function of viral load for different times of exposure. (b) Variation of single hit risk for susceptible persons as a function of viral load for RH and AER.

Numerical results (Fig.3a) show that the risk is less than 1% for viral loads<108 RNA copies/mL for 1-h exposure period. But the risk rapidly approaches ahigher value (ex. 50% for 1010RNAcopies/mL and 10-min exposure), which demonstrates the high transmissibility of Delta and possibly Omicron variants which are reported to give rise to higher viral loads27,28,29,30,31 (Table 2). Thus, the present study clearly demonstrates the risk dependence on the viral load irrespective of variants. The model also explores the effect of ventilation rate on indoor infection risks (Fig.3b). When the air-exchange rate is increased from 0.5 to 10h1 for a 10-min exposure time, the single-hit risk decreases approximately by an order. This is primarily due to the elimination of airborne viruses from the indoor environment via ventilation. However, when viral load increases, the effect of enhancing ventilation reduces because smaller particles contribute to the risk as well. The ambient RH has only a minor impact on the risk; higher RH leads to larger final droplet sizes, which reduces their lifetime and therefore infection risk, as seen in Fig.3b.

Another essential metric to describe infection risk is the event reproduction number (Re), which is computed by multiplying the infection risk during the exposure time of each susceptible person by the number of susceptible people exposed for a specific exposure scenario. The following three scenarios are studied in this work to demonstrate how the model can be used: (a) 25 students in a classroom with an infected subject exposed for 4h; (b) 4 employees in an office environment with an infected subject exposed for 8h; (c) outbreak in a restaurant in Guangzhou, China. In the first scenario, the Re value approaches 2 when the viral load of the infected person in the classroom exceeds 5107#/mL, as shown in the results; also, the Re value shall remain (le)1 if the viral load is less than 2.5107#/mL for the given input and environmental parameters, as shown in Fig.4. Similarly, if the viral load is (le)7107#/mL for the given exposure conditions in an office setting, the Re value will be (le)1 in the second exposure scenario.

Event reproduction number as a function of viral load for two different indoor environments and exposure conditions.

The third case is a recognised outbreak16,32 in which a patient from an epidemic site had lunch in a restaurant of volume~435m3, with a floor area of 145m2. To compare the present study results with the literature values, the input parameters for this superspreading event from Buonanno et al.16 are considered. At the time of the presence of the case patient with a viral load of 107RNAcopies/mL, around 83 diners along with 8 staff members were present at the restaurant, and later some members of three families, who had lunch at the adjacent tables were found to be infected16,32. Considering speech as the continuous expiratory process, the infection risk is estimated as a function of exposure time (Fig.5). The infection risk due to the presence of the case patient in a limited volume of~45m3 (volume encompassing the table of case patient and other adjacent tables) is estimated as 23% for 2-h exposure period, i.e.,~(23) persons would be infected through aerosol transmission route (11 persons were present in this limited volume). This risk estimate is roughly half of the stated value in the literature16, which can be attributed to the differences in modelling because the input parameters are the same. If the complete volume of 435m3 with 91 susceptible persons is considered instead of 45m3, then the infection risk is reduced to~2% and~2 people would be infected. The decrease in risk value owing to an increase in indoor volume clearly demonstrates the dependence of the input parameter selection. The findings show that the model can be used to evaluate infection risk in low and medium risk events, including superspreading events. Thus, the new double Poissonian formalism combined with the comprehensive aerosoldynamics model introducedheremakes a significant value addition to the subject of risk evaluation of airborne diseases.

Infection risk as a function of exposure time for theoutbreak at a restaurant.

From Fig.5, it is observed that if droplet evaporation is neglected, the residual size will be higher, leading to larger removal by gravity. Hence, this would lead to lesser airborne droplet concentration and infection risk; i.e., 5.5% for this wet droplet distribution case compared to~23% (if dynamic evaporation is considered in the model) at the end of 2-h exposure scenario. Another effect related to particle deposition in the respiratory system, if lung deposition model as given by ICRP15 is replaced by a lung deposition probability of 1 (i.e., 100% deposition after inhalation), the estimated infection risk increases by~3.4 times, since a larger number of droplets are deposited in the respiratory tract leading to a higher value of infection risk. The risk value in this case is,~78% when compared to the standard model study value of~23%.

These findings imply that if the viral load is less than a certain value or if the contact period is limited for the specified emission and indoor settings, the event reproduction number will remain less than one. Alternatively, the limit on the number of people can also be estimated using the present approach for a given virus variant and the exposure duration. Hence, these studies can be used as a tool to aid decision/policy making as the spread of the disease can be directly predicted based on the viral load and other physically measurable input parameters.

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A comprehensive modelling approach to estimate the transmissibility of coronavirus and its variants from infected subjects in indoor environments |...