key: cord-0960544-0v51a4f9 authors: Kot, A. D. title: Critical levels of mask efficiency and of mask adoption that theoretically extinguish respiratory virus epidemics date: 2020-05-15 journal: nan DOI: 10.1101/2020.05.09.20096644 sha: 074b31746f4426d995dd8544ba982de222384db3 doc_id: 960544 cord_uid: 0v51a4f9 Using a respiratory virus epidemiological model we derive equations for the critical levels of mask efficiency (fraction blocked) and mask adoption (fraction of population wearing masks) that lower the effective reproduction number to unity. The model extends a basic epidemiological model and assumes that a specified fraction of a population dons masks at a given initial number of infections. The model includes a contribution from the ocular (nasolacrimal duct) route, and does not include contributions from contact (fomite) routes. The model accommodates dose-response (probability of infection) functions that are linear or non-linear. Our motivation to study near-population-wide mask wearing arises from the concept that, between two mask wearers, the concentration of particles at inhalation should be the square of the mask penetration fraction. This combination, or team, of masks can provide a strong dose-lowering squaring effect, which enables the use of lower-efficiency, lower-cost, lower pressure-drop (easier breathing) masks. For an epidemic with basic reproduction number R0=2.5 and with a linear dose-response, the critical mask efficiency is calculated to be 0.5 for a mask adoption level of 0.8 of the population. Importantly, this efficiency is well below that of a N95 mask, and well above that of some fabric masks. Numerical solutions of the model at near-critical levels of mask efficiency and mask adoption demonstrate avoidance of epidemics. To be conservative we use mask efficiencies measured with the most-penetrating viral-particle sizes. The critical mask adoption level for surgical masks with an efficiency of 0.58 is computed to be 0.73. With surgical masks (or equally efficient substitutes) and 80% and 90% adoption levels, respiratory epidemics with R0 of about 3 and 4, respectively, would be theoretically extinguished. When a novel respiratory viral epidemic begins exponential growth in a population, its novelty implies that there is no community (herd) immunity, nor a vaccine. The population can undertake various nonpharmaceutical measures such as physical distancing, quarantining (based on symptoms, testing, contact-tracking), work and school closures, travel restrictions, careful hand-hygiene and wearing masks [1, 2] . While applying all of these measures simultaneously would be the most effective, the impact on society may become immense. Therefore it is valuable to gauge the relative benefit provided by various measures. Herein we use a mathematical epidemiological model to examine the efficacy of near population-wide adoption of masks. Our motivation to examine wide adoption of masks arises from the concept that between two mask wearers, the concentration of particles at inhalation (after exhalation and normalization to unity) should be the square of the mask penetration (fraction transmitted). (This concept is outlined in Fig 1 of the next section.) Since squaring can be a much more powerful effect than a linear effect, some significant gains may remain for less efficient masks and incomplete adoption by the population. The combination, or team, of masks may provide a strong dose-lowering squaring effect, which would enable the use of lower-efficiency, lower-cost, lower pressure-drop (easier breathing) masks. Respiratory virus transmission can occur via droplet, aerosol (including droplet nuclei formed by evaporation of small droplets) and contact routes [3] [4] [5] . High-speed photography of visible droplets during sneezing events have shown ranges of roughly 8m [6] , but such events are rare [5] , especially during a pandemic where generally people have a heightened awareness to not cough or sneeze among others. High-speed laser-aided photography of talking at close-range revealed numerous visible droplets in the range of 20-500 μm [7] . Droplets captured at close-range and smaller than 5 μm have been reported to contain about nine-fold more viral copies than larger particles [8] . Small droplets can evaporate to form droplet-nuclei that can remain airborne [3] . Talking for five minutes can generate the same number of droplet nuclei as a cough [3] . The viability of aerosol viral particles has been demonstrated for SARS-CoV-1 and SARS-CoV-2 for the three hour duration of an experiment [9] . Non-symptomatic and pre-symptomatic transmission can be significant [10, 11] . An example of this is where 40 of 60 participants at a choir practice became infected with COVID-19, despite distancing and without coughing or sneezing [10] . With SARS, community wearing of masks when going out was associated with a 70% reduction in risk compared with never wearing a mask [12] . Mask wearing can reduce exposure especially in situations where physical distancing is difficult. Taiwan, with a population of 24 million, has only had six COVID-19 deaths, had significant surgical mask use plus other measures and early action [13] . While many masks, including home made fabric masks, might be efficient at blocking droplets, we take a more conservative approach and use mask efficiencies for aerosol sized particles of about 0.02-1 μm. Thus, if such masks were stored by a population in advance of a novel respiratory virus, their rated filtering would apply to a wide range of particle sizes and regardless of whether the novel virus is transmitted predominantly by droplets or aerosols. Various prior works have investigated epidemiological modeling of respiratory viruses with masks, including [5, [14] [15] [16] [17] [18] [19] [20] . In a pair of papers [5, 14] Atkinson and Wein construct detailed concentrationbased models of aerosol transmission of influenza with an epidemic model for households. In [15] Brienen et al assumed a simplified effect from mask usage. In [16] Tracht et al use a detailed SEIR model for H1N1 influenza and demonstrated various degrees of effect, including extinguishing the epidemic in some cases. In [20] Cui et al added compartments to include asymptomatic transmission. In [19] Yan et al use a detailed concentration-based model for Influenza. Herein we determine closed-form expressions for the critical mask efficiencies and mask adoption levels that can extinguish an epidemic, including linear or non-linear probability of infection (doseresponse) functions, and including contributions from the ocular (nasolacrimal) route. We use a relatively simple model, which requires few parameters and is consequently easier to apply to a novel virus. With this simpler model, in some cases one needs only the fundamental parameter R 0 . The non-normalized concentration would depend on the event (e.g. exhalation, speaking) and on the local environment. The decrease in concentration with distance is shown in Fig 1 as a gain, g(d , t) ⩽1 , being a nonlinear function that typically decreases with distance, and depends on time, and on other factors such as the event, and humidity, temperature. Here we consider only how masks affect the concentration in a relative manner, without needing to model the complicated function g(d , t ). Note that for lowvolume events (e.g. breathing, talking) that the region of higher concentrations would be less like the cone shown in Fig 1, and a more like a cloud nearer the source [17] In Fig 1 the mask on the infectious person is modeled as a filter with a transmission gain (fraction that penetrates) during exhalation of 0< f t , exh ⩽1 . Similarly, the mask on the susceptible person has a filter transmission during inhalation of 0< f t , inh ⩽1. Not wearing a mask corresponds to a filter transmission of unity. The four combinations of mask wearing, (i.e none, either one, or both) are shown to lead to lower concentrations given by the product of the mask filter gains. The dose input to the respiratory tract may be modeled as an integral over time of the concentration at the susceptible's face, where C source is the concentration at the source, and v (t ) is the volume of air inhaled. (The contribution for the ocular route is discussed in the next section. ) We take the filter properties to be constants (either during low-flow events like breathing, or use averages for more dynamic events). Consequently, so that the filter gain f exh (f inh + f ocular )<1 reduces the original dose d 0 by a set of four constants, where the values of f exh or f inh are set to unity for those not wearing a mask. The ocular route corresponds to collection of particles by the eye which then pass through the nasolacrimal duct [21] . The ocular route for viral particles is modeled in Fig 1 as a passage of the incident concentration through a filter denoted f ocular . It is implicit in the dose integral above that f ocular was measured during breathing. We estimate f ocular based on data in Fig 2 of [21] , as follows. The control group (Group 1) in [21] had no masks or goggles, and an ocular-exposure-only group (Group 2) had a half-mask with clean air supply. These groups were equally exposed to a standardized aerosol concentration of live attenuated influenza vaccine. Afterwards they used nasal washes followed by quantitative reverse transcription polymerase chain reaction (RT-PCR) to count the resulting number of Influenza RNA copies. The control group would have had count-contributions from both inhalation, n inh , and from ocular paths, n oc , so its count of 504 consists of n inh + n oc . The ocular-path-only group count had n oc =5. Assuming linearity of the quantitative RT-PCR tests, then the ratio n oc /n inh =0.01 is the fraction of viral particles that pass through the ocular route compared to the inhaled route, which is indeed the parameter f ocular that we seek. The ocular route transmission is about 100 times lower than inhaling without a mask, so masks would be the most effective first-approach to lowering the dose. Since f ocular summarizes aerosol reception, but not droplets, direct exposure to droplets from coughs or sneezes would be sensibly reduced by eye protection (glasses, shields). In [5] , the occurrence of cough and sneezes was modeled to be rare. Moreover, in the context of a pandemic, people have a heightened awareness of the need not to cough or sneeze in close to others, so the contribution from coughing and sneezing would be reduced. With more potent viruses or in environments of higher exposure, higher efficiency masks would be used, in which case their transmission fraction lowers to nearer that of the ocular route, so that the ocular route becomes significant. In such situations the use of sealed goggles or similar should be considered. The ocular filter value estimate of 0.01 is low relative to typical mask transmission in our examples. It will be seen later in the numerical results that the contribution from the ocular route barely affects the critical mask efficiency. The contact route was not included for several reasons. First, it's been estimated to not be significant compared to the droplet/aerosol route [5] . Second, during an epidemic there is heightened awareness of not touching one's face without hand washing. Third, mask wearing among a significant proportion of the population would decrease viral deposition on fomites. Fourth, mask wearing impedes the ability to touch one's nose or mouth. Fifth, the median-effective dose for influenza via the oral route is about 500 times higher than the nasal route [5] . All of these assumptions may not be valid other viruses or for subsets of the population, such as children. For children, one would hope that children too young to learn the value of hand washing are well supervised and in clean environments, and that ones that are old enough to learn are well taught. The effect of mask filtering is to lower the dose, and the effect of that lower dose on probability of infection is given by a dose-response function, denoted p i =P i (d ). In common SIR epidemiological models recall that the parameter β is the expected number of contacts per day that cause infection [22] . It is the expected number of infections per day from those contacts, or β = n c p i where n c is the expected number of contacts per day, and p i is the probability that such a contact results in an infection. To consider how lowering the dose affects the probability of infection, let p i 0 denote the probability of infection for the no-mask situation. Since p i 0 corresponds to some dose d 0 , we can find d 0 from For a modification of the dose by multiplying it by some value m , then the resulting probability of infection is For cases where the dose response function is linear, then Consequently, for mask use with a linear dose-response function, m=f exh ( f inh + f ocular )<1 then where p i 0 is scaled down linearly by the filtering. For the case where f ocular is insignificant and the mask transmissions are identical 6 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 15, 2020. . https://doi.org/10.1101/2020.05.09.20096644 doi: medRxiv preprint p i =f t 2 p i 0 so the probability of infection is lowered by the square of the filter transmissions. This squaring effect is potentially stronger for higher efficiency filters. In addition to the linear dose-response case, we will later consider exponential, and approximate beta Poisson dose-response functions. Before describing the epidemiological model, we discuss some assumptions on mask efficiencies. In a variety of work on masks, including [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] there are variations in measured efficiencies arising from various testing procedures, particle sizes, and fit. Masks are typically much better at blocking larger droplets, therefore it is prudent to measure mask efficiency at the most-penetrating particle-size (MPS) for infective particles. Viral particles and bacteria range from about 0.02 to 1.26 μm [23] . (It was noted in [23] that the size of the influenza and SARS-CoV-1 viruses happen to coincide with the most-penetrating particle-size for N95 respirators.) If a population stocked MPS-rated masks in advance of a novel respiratory virus, then the masks would perform at least as well as those worstcase efficiencies over a wide range of particle sizes, and regardless of whether the transmission is predominantly by droplets or by aerosols. Herein we do not attempt to compare a wide variety of masks, since our principal aim is the critical mask efficiency for extinguishing an epidemic. However in some numerical examples we assume a surgical mask. Based on [23] the MPS efficiency of a surgical mask is taken to be 0.58. That efficiency is comparable to that in [28] where the reduction in quantitative RT-PCR viral counts from fine (less than 5 μm) particles, after exhalation through a surgical mask, was 2.8 fold (which corresponds to an efficiency of 0.64). (That same work also reported that the fine particles contained 8.8 fold more viral copies than larger particles.) Overall, for all particles, the effect of the surgical mask was a 3.4 fold reduction (which corresponds to a 0.7 efficiency) [28] . Some fabrics are very poor viral filters with efficiencies of roughly 0.1 [26] . Later, for some equations we will assume that the exhalation and inhalation filter transmission gains are equal. Near equal penetration for inhalation and exhalation using surgical masks was reported in [33] . Note also that the inhalation mask efficiency of 0.58 in [23] is comparable to the exhalation mask efficiency of 0.64 in [8] . Equal filtering in either direction may be reasonable for low-flow events like breathing and talking, and perhaps less reasonable for high-flow, but rare, events like coughing and sneezing, depending on the mask seal and construction. A common concern about surgical and other types of masks, including home-made masks, is leakage between the mask and face A significant and low cost method to improve fit is an overlay of a nylon stocking [32, 34] . As reported in [32] the use of a nylon stocking overlay raised the efficiency of five of ten fabric masks above a benchmark surgical mask. A basic mathematical model of an epidemic is the SIR (Susceptible, Infectious, Recovered) model [22] . Individuals in those categories are modeled as transitioning from one compartment to the next according to various rate parameters, with a resulting set of simultaneous differential equations. We will use extensions of both the SIR model to include mask use and focus on the number of peak infectious and the final cumulative infected. All recovered are assumed to have immunity. The basic SIR model is often augmented with an exposed compartment to form a SEIR model. However, note that every person accounted for through the I, or the R, compartments do so in both the SIR and in the SEIR models. Consequently, for identical corresponding parameters, both the SIR and SEIR models will have the same value of final cumulative infected. The incubation period of the exposed compartment spreads the same flow of infections over a longer time period. Since herein we are interested in filter conditions for which an epidemic is extinguished, and such conditions will be equivalent for the SIR and the SEIR model, then for purpose it is simpler, sufficient, and more general to use a SIR model. We also express the totals of Susceptible, Infectious, and Removed as s=s nm + s m , i=i nm +i m and r=r nm + r m respectively. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. . Initially none of the population is assumed to wear masks. When the beginning of an epidemic is detected by observing some specified number i 0 of infections, then a specified fraction p m of the population dons masks. These initial conditions are, The set of differential equations for the model are, . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. . β 0 is the average number of contacts per day that are sufficient to cause infection, where the subscript 0 indicates no interventions. β 0 is related to β (our case with interventions that affect the probability of infection) by γ is the rate that the infected become recovered, which is the inverse of the duration of the infectious period . f exh and f inh are the mask transmission gains during exhalation and inhalation, respectively. It is common to specify a mask by its efficiency, which is the fraction that it blocks, rather than which it transmits, so its efficiency is An alternative model would combine the removed compartments in Fig 2, if they have identical rate of entry from the infected states, but keeping them separate enables separate accounting (for example, to demonstrate that non-mask wearers are more likely to get infected). R 0 =β 0 /γ is the basic reproduction number [22] in the basic SIR model (i.e. without masks). Below we briefly summarize some key results using R 0 because they are both fundamental and will be used alongside our mask model. R 0 can be sensibly interpreted as the average number of infections produced by a single infectious person during their infectious period [22] . The rate of growth of the infectious compartment is from which we can see that for the growth to be zero or negative, then β 0 s≤γ , or β 0 /γ =R 0 ≤1. So, at the start of an epidemic, where s≈1 , there is no initial growth unless R 0 >1. For community immunity, R 0 s < 1 , or s <1/ R 0 . For example, for such immunity with R 0 =2.5 the susceptible 10 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. . https://doi.org/10.1101/2020.05.09.20096644 doi: medRxiv preprint fraction of the population must be less than 1/ R 0 =1/2.5=0.4 . Equivalently, the fraction of the population that has been through the infectious stage and has assumed immunity is 1−1/ R 0 =0.6 . It is also possible to obtain analytical expressions for the peak infectious and the final cumulative infected [22] , which are For multi-compartment models [35] gives the determination of the effective reproduction number R 0 ,e . Of fundamental importance is that R 0 ,e =1 is again a threshold, or a critical value, below which the epidemic is theoretically extinguished and above which growth will occur. In Appendix B, using the methods of [35], R 0 ,e is found for the model of which can be simplified for identical mask transmissions, Additionally, if there is no ocular contribution 11 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. In Appendix B, we find R 0 ,e for Linear, Exponential, and Approximate Beta-Poisson dose response functions. A linear dose response is sometimes used in low-dose epidemiological models [36] . An exponential dose-response has been used to model the infection probability of SARS-CoV-1 [37] . Below we list some of the results from Appendix B. From Appendix B, with a linear dose response, and identical filter transmissions f t , With no ocular contribution, One can easily confirm that in the simple case of no masks This result corresponds to our fundamental motivation described for Fig 1, where the result of transmission through two masks is the square of their individual transmissions. From Appendix B, with an exponential dose response, with identical filter transmissions f t , and with no ocular contribution, The more general cases of non-identical filters and including an ocular contribution can be found in Appendix B. From Appendix B, with an Approximate Beta-Poisson dose response, with identical filter transmissions f t and with no ocular contribution, 12 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. The more general cases of non-identical filters and including an ocular contribution can be found in Appendix B. Here we find the filter efficiency for which R 0 ,e becomes equal to unity. In this section, we will assume masks with identical filter transmissions. In Appendix C, we find the critical mask efficiencies for Linear, Exponential, and Approximate Beta-Poisson dose response functions. Below we list some of those results. With a linear dose response, from Appendix C For no ocular contribution this simplifies to values of R 0 . Estimates of R 0 for influenza is about 1.5 [38] , and for COVID-19 it is roughly 2.5 [2] . As an example, assume that the mask adoption level is 80%, then from Fig 3 the R 0 =2.5 curve indicates a mask critical efficiency of 0.5. If the mask adoption rate was 90%, the mask critical efficiency is reduced to about 0.42. Importantly, these efficiencies are well below that of a N95 mask, and well above that of some fabric masks [26] . As R 0 is increased, it can be seen that for a particular adoption level the required efficiency heads towards 100%, so after that even perfect masks would not suffice to extinguish the epidemic. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. . https://doi.org/10.1101/2020.05.09.20096644 doi: medRxiv preprint In both Fig 3 and Fig 4, the upper red curves include the effect of the ocular route with f ocular =0.01. The curves with and without the ocular contribution are barely distinguishable. Accordingly we do not include the effect of the ocular route in examples in the remainder of this paper. With an exponential dose response, and neglecting the contribution for the ocular route, from ). (This may not be correct in high-exposure situations, such as medical workers working closely with, and for several hours per day with infected patients. However, high-exposure situations are not typical of the general mixing of a SIR model for a large population.) For a given R 0 a higher number of typical contacts implies a lower p i 0 . In [37] SARS-CoV-1 was well modeled by an exponential dose-response, and the estimated doses for residents of an apartment complex where an outbreak occurred were typically less than one-half of the median-infective dose. In [39] the mean number of contacts per day was about 13 (although that contact count does not weigh the infectiveness of each contact, as is implicit in a SIR model). With an approximate Beta-Poisson dose response, and neglecting the contribution for the ocular route, from Appendix C Influenza H1N1 has been modeled as by an approximate beta-Poisson model with α =0.581 [40] , and with R 0 ,e =1.5 [38] . The values for R 0 and the infectious duration are near those for COVID-19 [2] . Recall from the section describing the model, that for identical R 0 and infectious duration, that both our SIR model or its extension to an SEIR model would have the same value of final cumulative infected. Mainly we are interested in the overall results for the population, so we plot the totals of Susceptible, Infectious, and Removed, i.e. s=s nm + s m , i=i nm +i m and r=r nm + r m respectively. Equations were derived that give critical levels of mask efficiency and of mask adoption that lower the effective reproduction number R 0 ,e of a SIR-based epidemiological model to unity. Those critical levels 16 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. . https://doi.org/10.1101/2020.05.09.20096644 doi: medRxiv preprint correspond to a "knee" in the effectiveness curves of using masks. Rather than have a population wear masks of random or non-standardized efficiencies, it would be more effective for the population to adopt masks that exceed the critical mask efficiencies derived herein. The model used assumes that a specified fraction of a population dons masks at a given initial number of infections, and that there is no further influx of infectious people from outside the population. The model does not include contributions from indirect-contact (fomite) routes, assuming that sufficient hand hygiene is achieved by the population, and assuming that broad mask adoption would decrease deposition onto fomites and would also decrease any tendency of mask wearers to touch their nose or mouth. The model includes the ocular (nasolacrimal duct) route, and excludes direct contact from droplets. In our examples the effect on the critical mask efficiency from the ocular route was estimated to be insignificant. The ocular route would become more significant in environments of high exposure, where higher efficiency masks would be more suitable. With higher efficiency masks their transmission fraction lowers to nearer that of the ocular route, so that the ocular route becomes significant. The model accommodates dose-response (probability of infection) functions that are linear or nonlinear. In our R 0 =2.5 example for an exponential dose-response, the increase in the critical mask efficiency was about 12% (assuming a probability of infection per contact being 0.5 with the number of contacts per day being 1). That increase drops to the linear case for a higher number of contacts per day. Various assumptions were made in order to model the efficacy of population-wide mask wearing on its own. These assumptions include; no vaccinations, no physical distancing, no testing for being infectious (therefore no quarantining from testing), and symptom-less transmission (therefore no self quarantining). Such assumptions also enable a very simple and general model with few parameters. Indeed the simplest of the equations derived for the critical mask efficiency and adoption need only the fundamental parameter R 0. With the above model in an epidemic with R 0 =2.5 and with a linear dose-response, the critical mask efficiency is calculated to be 0.5 for a mask adoption level of 0.8 of the population. Importantly, this efficiency is well below that of a N95 mask, but well above that of some fabric masks. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020 To be conservative we use mask efficiencies reported for the most-penetrating viral-particle-sizes. With R 0 =2.5 , surgical masks with an efficiency of 0.58 give a computed critical mask adoption level of 0.73. With surgical masks (or equally efficient substitutes) and 80% and 90% adoption levels, respiratory epidemics with R 0 of about 3, and 4, respectively, would be theoretically extinguished. Numerical solutions of the model at near-critical levels of mask efficiency and mask adoption, for a 10 -4 initially infected fraction of the population, demonstrate avoidance of epidemic growth (and so numerically confirm the equations derived tor the critical levels). Example numerical solutions that plot infections versus time demonstrate a progression from zero effect to complete epidemic avoidance with a sharp dropoff, or knee, as the mask efficiency and mask adoption levels are increased to their calculated critical levels. A fundamental reason why population-wide mask wearing should be effective is that, between two mask wearers, the concentration of particles at inhalation would be the square of the mask penetration fraction. Since squaring can be a much more powerful effect than a linear effect, some significant gains remain for less efficient masks and incomplete adoption by the population. The combination, or team, of masks can provide a strong dose-lowering squaring effect, which enable the use of lower-efficiency, lower-cost, lower pressure-drop (easier breathing) masks. Deterministic SIR models implicitly assume that the population has homogeneous mixing, with a fixed product of the number of contacts per day and of the probability of infection per contact. However, in some situations, for example at home, one would not normally expect people to wear masks. (Such limitations also apply to physical distancing.) Since the basic reproduction number R 0 is not a constant determined solely by viral characteristics, one might consider a range in R 0 in a population in order to accommodate higher-density highercontact regions. For example, regions with busy subways would be expected to have higher R 0. The critical mask efficiency calculated herein may be insufficient in environments with exposure higher than the average expected in simple SIR modeling. One such environment would be hospitals. Another such environment would be long-term-care facilities, where the duration and closeness of the contacts between care aides and residents are significant. Additionally, mask wearing by residents would not be practical, so the advantages of combined mask wearing would not exist. Finally, such residents have a high death rate once infected. Given all of these factors, higher efficiency masks worn by personal care workers deserves consideration. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020 Other situations of higher-exposure than average might include those with tight-spacing and/or longer-duration, including public transportation and air travel. Many situations may have average exposure, including shoppers in grocery stores and street-front retail shops. In the example of R 0 =2.5 epidemics, while surgical masks already surpass the critical mask efficiency, mask leakage can be improved by using an overlay of a nylon stocking [29, 31] . As reported in [29] the use of a nylon stocking overlay raised the efficiency of several fabric masks above that of a benchmark surgical mask. Mimicking that approach, perhaps by increasing elasticity near the borders of masks, might significantly improve fit and performance of alternatives to surgical masks. Re-use of masks in pandemic situations is a natural consideration [41] not just for low supply but for lowering costs. Some possible approaches include delayed re-use and disinfection. In [42] a millionfold inactivation of SARS-CoV-2 was achieved on N95 masks using 15 minutes of dry heat at 92C. In [43] a microwave-oven steam treatment for 3 minutes attained similar results. In [44] a gauze-type surgical mask retained good filtering efficiency of about 0.78, at its most-penetrating particle size, after decontamination using a 3 minute dry heat (rice-cooker) at roughly 155 C. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. . https://doi.org/10.1101/2020.05.09.20096644 doi: medRxiv preprint The largest eigenvalue of F V −1 gives R 0 ,e , Additionally, if the contribution from the ocular route is not significant, then Here for each of the linear, exponential, and approximate Beta-Poisson dose-response functions, we use their inverses to give their response after dose scaling from mask filtering, and then use that result in the general expression from Appendix A for the effective R 0 . All of the dose-response functions below are setup for normalized doses, I.e a dose d=1 is the median-effective dose for a population, so that P i (1)=0.5 . . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020 The linear dose-response is and the response after scaling a dose corresponding to an initial P i 0 is Using p i in the expression for R 0 , e from Appendix A, we obtain and for identical filter transmission gains f t then and with no ocular contribution, The exponential dose-response is Its inverse is and the response after scaling a dose corresponding to an initial P i 0 is Using p i in the expression for R 0 , e from Appendix A, we obtain . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020 and with no ocular contribution, ] R 0 . The approximate beta-Poisson dose-response is a and the response after scaling a dose corresponding to an initial P i 0 is Using p i in the expression for R 0 , e from Appendix A, we obtain With identical filter transmission gains f t , then and with no ocular contribution, 23 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. . ). With an approximate beta-Poisson dose response, and neglecting the contribution from the ocular route, setting R 0 ,e =1 from the result in Appendix B gives the condition, which can be solved to give the critical value of f t or the critical value of mask efficiency . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. . https://doi.org/10.1101/2020.05.09.20096644 doi: medRxiv preprint . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 15, 2020. Exponential Dose-Resp. p i0 =0.5 Exponential Dose-Resp. p i0 =0.25 Linear Dose-Resp. Note that the lowest perceptible non-zero red line (infectious) on this plot is for a mask adoption of 0.6 (not the critical level of 0.73, which appears as a flat-line). Infectious . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 15, 2020. Infectious . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 15, 2020. . https://doi.org/10.1101/2020.05.09.20096644 doi: medRxiv preprint Strategies for mitigating an influenza pandemic Impact of nonpharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand How far droplets can move in indoor environments, revisiting the Wells evaporation falling curve Airborne spread of infectious agents in the indoor environment Quantifying the Routes of Transmission for Pandemic Influenza Violent expiratory events: on coughing and sneezing Visualizing Speech-Generated Oral Fluid Droplets with Laser Light Scattering 2013 -Influenza Virus Aerosols in Human Exhaled Breath Aerosol and Surface Stability of SARS-CoV-2 as Compared with SARS-CoV-1 Presymptomatic Transmission of SARS-CoV-2 -Singapore COVID-19: What proportion are asymptomatic? 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Microbiology Microwave-Generated Steam Decontamination of N95 Respirators Utilizing Universally Accessible Materials. Occupational and Environmental Health Filter quality of electret masks in filtering 14.6-594 nm aerosol particles: Effects of five decontamination methods Here for each of the linear, exponential, and approximate Beta-Poisson dose-response functions, we use the corresponding R 0 ,e determined in Appendix B, set it to unity and solve for the critical value of the mask efficiency. We assume that the masks have identical filter transmission gains f t . With a linear dose response, setting R 0 ,e =1 from the result in Appendix B gives the condition,which can be solved to give the critical value of f t , or the critical mask efficiency f b ,crit =1−f t ,crit as,For no ocular contribution this simplifies to With an exponential dose response, and neglecting the contribution from the ocular route, setting R 0 ,e =1 from the result in Appendix B gives the condition,which can be solved to give the critical value of f t or the critical value of mask efficiency f b ,crit =1−f t ,crit as,