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# Face masks effectively limit the probability of SARS-CoV-2 transmission

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## Abstract

Airborne transmission by droplets and aerosols is important for the spread of viruses. Face masks are a well-established preventive measure, but their effectiveness for mitigating SARS-CoV-2 transmission is still under debate. We show that variations in mask efficacy can be explained by different regimes of virus abundance and related to population-average infection probability and reproduction number. For SARS-CoV-2, the viral load of infectious individuals can vary by orders of magnitude. We find that most environments and contacts are under conditions of low virus abundance (virus-limited) where surgical masks are effective at preventing virus spread. More advanced masks and other protective equipment are required in potentially virus-rich indoor environments including medical centers and hospitals. Masks are particularly effective in combination with other preventive measures like ventilation and distancing.

Airborne transmission is one of the main pathways for the transmission of respiratory viruses, including the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) (1). Wearing face masks has been widely advocated to mitigate transmission. Masks are thought to protect people in two ways: source control reducing the emission and spread of respiratory viruses through airborne droplets and aerosols, and wearer protection reducing the inhalation of airborne respiratory viruses.

The effectiveness of masks, however, is still under debate. Compared to N95/FFP2 respirators which have very low particle penetration rates (around ~5%), surgical and similar masks exhibit higher and more variable penetration rates (around ~30-70%) (2, 3). Given the large number of particles emitted upon respiration and especially upon sneezing or coughing (4), the number of respiratory particles that may penetrate masks is substantial, which is one of the main reasons leading to doubts about their efficacy in preventing infections. Moreover, randomized clinical trials show inconsistent or inconclusive results, with some studies reporting only a marginal benefit or no effect of mask use (5, 6). Thus, surgical and similar masks are often considered to be ineffective. On the other hand, observational data show that regions or facilities with a higher percentage of the population wearing masks have better control of the coronavirus disease 2019 (COVID-19) (79). So how to explain these contrasting results and apparent inconsistencies?

Here, we develop a quantitative model of airborne virus exposure that can explain these contrasting results and provide a basis for quantifying the efficacy of face masks. We show that mask efficacy strongly depends on airborne virus abundance. Based on direct measurements of SARS-CoV-2 in air samples and population-level infection probabilities, we find that the virus abundance in most environments is sufficiently low for masks to be effective in reducing airborne transmission.

When evaluating the effectiveness of masks, we want to understand and quantify their effect on the infection probability, Pinf. Assuming that every inhaled single virus (virion) has the same chance to infect a person, Pinf can be calculated by a single-hit model of infection

$Pinf=1−1−PsingleNv$

(1)where Psingle represents the infection probability for a single virus and Nv represents the total number of viruses to which the person is exposed (10). For airborne transmission, the infection probability Pinf for a given time period can be plotted as a function of inhaled virus number, Nv.

Figure 1 illustrates the dependence of Pinf on Nv based on the single-hit model (Eq. 1) and scaled by the median infectious dose IDv,50 at which the probability of infection is 50% (10). It shows a highly nonlinear sensitivity of Pinf to changes of Nv. Accordingly, the same percentage change of Nv may lead to different changes in Pinf depending on the absolute level of Nv. In a virus-rich regime, where Nv is much higher than IDv,50 (Fig. 1, A and B), Pinf is close to unity and not sensitive to changes of Nv. In this case, wearing a mask may not suffice to prevent infection. In a virus-limited regime, where Nv is close to or lower than IDv,50, however, Pinf strongly varies with Nv, and reducing Nv by wearing a mask will lead to a substantial reduction in the infection probability (Fig. 1, C and D). Thus, we need to determine the regime of airborne virus abundance to understand mask efficacy.

Respiratory particles, including aerosol particles and larger droplets, can carry viruses and are often used to visualize the transmission of airborne viruses (4). Taking a representative average of respiratory activity (11), we find that a person typically emits a total number of about 3×106 particles during a 30 min period (supplementary text, section S1.1). This very large number implies that indoor environments are usually in a respiratory particle-rich regime. Surgical masks with particle collection efficiencies around ~50% cannot prevent the release of millions of particles per person and their inhalation by others (green dots in Fig. 1, B and D). In other words, the human-emitted respiratory particle number is so high that we cannot avoid inhaling particles generated by another person even when wearing a surgical mask. If every respiratory particle were to contain one or more viruses, indoor environments would often be in a virus-rich regime because the median infectious dose IDv,50 for respiratory diseases is typically of the order of a few tens to thousands of viruses (1214).

But does a respiratory particle-rich regime really imply a respiratory virus-rich regime? To answer this question, we investigated characteristic virus distributions in both exhaled air samples and indoor air samples including coronaviruses (HCoV-NL63, -OC43, -229E and -HKU1), influenza viruses (A and B), rhinoviruses and SARS-CoV-2 (supplementary text, section S1). We find that usually just a minor fraction of exhaled respiratory particles contains viruses. In contrast to the high number of emitted respiratory particles, the number of viruses in 30-min samples of exhaled air (Nv,30,ex) are typically low with mean values around ~53 for coronaviruses (HCoV-NL63, -OC43, -229E and -HKU1), ~38 for influenza viruses (A and B), and ~96 for rhinoviruses (11) (supplementary text, section S1.2, and Fig. 2). Figure 2, A and B, shows the infection probabilities obtained by inserting the number of exhaled viruses (Nv,30,ex) for the number of potentially inhaled viruses (Nv,30) assuming a characteristic infectious dose of IDv,50 = 100 or 1000, respectively (1214). For SARS-CoV-2 in various medical centers, we obtained mean values of Nv,30 in the range of ~1 to ~600 (1518) (supplementary text, section S1.3), which correspond to Pinf values in the range of ~0.1% to 10% for IDv,50 = 1000 and ~1% to 100% for IDv,50 = 100. The wide range of Nv,30/ IDv,50 and Pinf values demonstrate that both virus-limited and virus-rich conditions can occur in indoor environments.

The high variabilities of Nv,30 and Pinf shown in Fig. 2, A and B, are consistent with the wide distribution of viral load observed in respiratory tract fluids (19) and need to be considered for estimating population-average infection probabilities, Pinf,pop (supplementary text, section S4). For this purpose, we modeled Nv for SARS-CoV-2 as lognormally distributed with standard deviations (σ) in the range of ~1 to 2, based on recently reported distributions of the viral load of SARS-CoV-2 in respiratory fluids (19) (supplementary text, section S4). As shown in Fig. 2C, the population-average infection probabilities with σ > 0 are higher than in case of uniform exposure (σ = 0) in the virus-limited regime at Pinf,pop < ~50%. In other words: When the population-average infection probability is in the virus-limited regime with Pinf,pop,0 < 0.5 (Fig. 2C), a broader distribution (larger σ) implies an increase in the fraction of transmission events under virus-rich conditions (e.g., super-spreader events), leading to a reduction of overall mask efficacy.

The basic reproduction number for COVID-19 (R0 ≈ 2-4; (20)) can be related to a basic population-average infection probability, Pinf,pop,0, through R0 = Pinf,pop,0cd (21). With the average duration of infectiousness (d ≈ 10 days), and average daily numbers of human contacts (c ≈ 10 to 25 per day) (22, 23), we obtain estimates in the range of ~0.8% to ~4% for Pinf,pop,0, as indicated by the shaded area in Fig. 2C. The low levels of Pinf,pop,0 indicate a widespread prevalence of virus-limited conditions.

Different regimes of abundance are relevant not only for the distinction of respiratory particles and viruses, but also for different types of viruses. For example, viruses with higher transmissibility, i.e., with higher loads and rates of emission/exhalation, greater environmental persistence, or lower IDv,50, may result in a virus-rich regime and lead to higher basic reproduction numbers as observed for measles and other highly infectious diseases. Our analysis shows that the levels of Pinf and R0 can vary widely for different viruses. This means that aerosol transmission does not necessarily lead to a measles-like high R0, and that relatively low values of Pinf and R0 do not rule out airborne transmission. Based on the scaling with IDv,50, the curves shown in Figs. 1 to 3 can easily be applied to assess the efficacy of masks and other preventive measures against new and more infectious mutants of SARS-CoV-2 such as B.1.1.7.

Figure 3 illustrates that source control alone is more effective than wearer protection alone, but that universal masking is the most effective. This is because masks are more effective in removing larger particles (Fig. 4) and freshly generated respiratory particles are usually largest at the source, shrinking upon evaporation in indoor air (20). Note that Fig. 3 accounts only for airborne transmission of viruses. When considering other forms of transmission, the relative importance of source control can be even higher (supplementary text, section S5) (20).

Figure 4 shows the size distribution of respiratory particles emitted by different human activities (2527). Note that aerosols are physically defined as airborne solid or liquid particles with diameters smaller than 100 μm, which can remain suspended over extended periods of time. In medical studies, however, a threshold diameter of 5 μm has often been used to distinguish between a so-called “aerosol mode” and a “droplet mode”. Our analysis of measurement data from exhaled and ambient air samples indicates that the so-called “aerosol mode” (< 5 μm) contains more viruses than the so-called “droplet mode” (> 5 μm) (11), although the latter comprises a larger volume of liquid emitted from the respiratory tract (tables S1 and S2). This may be explained by the following mechanisms: a higher viral load occurring in the lower respiratory tract where the smaller aerosol particles are generated (28); or an enrichment of organic surfactants and viruses upon generation of smaller aerosol particles (29). Enrichment of viruses in the aerosol mode can enhance their transmission because smaller particles remain suspended for a longer time, leading to stronger accumulation and dispersion in the air. This may cause higher airborne virus concentrations, inhaled virus numbers, and infection risks – especially in densely occupied rooms with poor ventilation and long periods of exposure. Moreover, small aerosol particles have a higher penetration rate and higher probability of reaching the lower respiratory tract (figs. S5 and S6).

Our analysis was focused on respiratory particles and droplets with diameters smaller than 100 μm (traditional physical definition of aerosols (30)). Because of rapid gravitational settling, respiratory droplets larger than 100 μm are removed from the air within seconds, but they may still reach the upper respiratory tract of persons in close contact and cause infections by carrying large numbers of viruses in their very large liquid volume. For example, a single one-millimeter droplet may carry as many as ~50,000 viruses in case of a viral load of 108 per milliliter respiratory fluid, which is realistic and higher than the estimated infectious dose for SARS-CoV-2 (14). Such large droplets, however, are very efficiently (~100%) removed even by simple masks (Fig. 4 and supplementary text, section S3), further emphasizing the importance and efficacy of face masks for preventing infections. Because of the strong size dependence and to avoid ambiguities, we suggest that diameter range should be explicitly specified when discussing airborne transmission by fine respiratory aerosol particles or larger droplets.

## References and Notes

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2. P. van den Driessche, J. Watmough, in Mathematical Epidemiology, F. Brauer, P. van den Driessche, J. Wu, Eds. (Springer, 2008), pp. 159–178.

3. H. Su, Face masks effectively limit the probability of SARS-CoV-2 transmission [data set], version 1.0, Open Research Data Repository of the Max Planck Society (2021); .doi:10.17617/3.5d

4. W. C. Hinds, Aerosol Technology: Properties, Behavior, and Measurement of Airborne Particles (Wiley, 1999).

5. P. A. Baron, K. Willeke, Eds., Aerosol Measurement: Principles, Techniques, and Applications (Wiley, 2005).

6. J. H. Seinfeld, S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (Wiley, 2006).

7. S. A. Kemp, B. Meng, I. A. T. M. Ferriera, R. Datir, W. T. Harvey, G. Papa, S. Lytras, D. A. Collier, A. Mohamed, G. Gallo, N. Thakur, The COVID-19 Genomics UK (COG-UK) Consortium, A. M. Carabelli, J. C. Kenyon, A. M. Lever, A. De Marco, C. Saliba, K. Culap, E. Cameroni, L. Piccoli, D. Corti, L. C. James, D. Bailey, D. L. Robertson, R. K. Gupta, Recurrent emergence and transmission of a SARS-CoV-2 spike deletion H69/V70. bioRxiv 2020.12.14.422555 [Preprint]. 8 March 2021. .doi:10.1101/2020.12.14.422555

8. O. T. Price, B. Asgharian, F. J. Miller, F. R. Cassee, R. de Winter-Sorkina, “Multiple Path Particle Dosimetry model (MPPD v1.0): A model for human and rat airway particle dosimetry,” Rijksinstituut voor Volksgezondheid en Milieu (RIVM) rapport 650010030 (RIVM, 2002).

Acknowledgments: The authors would like to thank the reviewers for the very helpful feedbacks. This study was supported by the Max Planck Society (MPG). Funding: Y.C. thanks the Minerva Program of the MPG. Author contributions: Y.C. and H.S. designed and led the study. H.S., Y.C. and N.M. performed the research. U.P. and M.O.A. discussed the results. C.W., S.R. and P.W. commented on the manuscript. Y.C., H.S. and U.P. wrote the manuscript with inputs from N.M. and all coauthors. Competing interests: Authors declare no competing interests. Data and materials availability: The data and code to generate the results in the manuscript are freely available on (31). All data are available in the main text or the supplementary materials. This work is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/. This license does not apply to figures/photos/artwork or other content included in the article that is credited to a third party; obtain authorization from the rights holder before using such material.

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