To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic
Steffen E. Eikenberry, Marina Mancuso, Enahoro Iboi, Tin Phan, Keenan Eikenberry, Yang Kuang, Eric Kostelich, Abba B. Gumel
aa r X i v : . [ q - b i o . P E ] A p r To mask or not to mask: Modeling the potential for face maskuse by the general public to curtail the COVID-19 pandemic
Steffen E. Eikenberry ∗∗ , Marina Mancuso ∗ , Enahoro Iboi ∗ , Tin Phan ∗ , Keenan Eikenberry ∗ ,Yang Kuang ∗ , Eric Kostelich ∗ , and Abba B. Gumel ∗∗ Arizona State University, School of Mathematical and Statistical Sciences, Tempe, AZ,USAApril 8, 2020
Keywords: face mask, non-pharmaceutical intervention, cloth mask, N95 respirator, surgicalmask, SARS-CoV-2, COVID-19
Abstract
Face mask use by the general public for limiting the spread of the COVID-19 pandemicis controversial, though increasingly recommended, and the potential of this interventionis not well understood. We develop a compartmental model for assessing the community-wide impact of mask use by the general, asymptomatic public, a portion of which may beasymptomatically infectious. Model simulations, using data relevant to COVID-19 dynamicsin the US states of New York and Washington, suggest that broad adoption of even rela-tively ineffective face masks may meaningfully reduce community transmission of COVID-19and decrease peak hospitalizations and deaths. Moreover, mask use decreases the effectivetransmission rate in nearly linear proportion to the product of mask effectiveness (as afraction of potentially infectious contacts blocked) and coverage rate (as a fraction of thegeneral population), while the impact on epidemiologic outcomes (death, hospitalizations) ishighly nonlinear, indicating masks could synergize with other non-pharmaceutical measures.Notably, masks are found to be useful with respect to both preventing illness in healthy per-sons and preventing asymptomatic transmission. Hypothetical mask adoption scenarios, forWashington and New York state, suggest that immediate near universal (80%) adoption ofmoderately (50%) effective masks could prevent on the order of 17–45% of projected deathsover two months in New York, while decreasing the peak daily death rate by 34–58%, ab-sent other changes in epidemic dynamics. Even very weak masks (20% effective) can stillbe useful if the underlying transmission rate is relatively low or decreasing: In Washington,where baseline transmission is much less intense, 80% adoption of such masks could reducemortality by 24–65% (and peak deaths 15–69%), compared to 2–9% mortality reduction inNew York (peak death reduction 9–18%). Our results suggest use of face masks by the gen-eral public is potentially of high value in curtailing community transmission and the burdenof the pandemic. The community-wide benefits are likely to be greatest when face masksare used in conjunction with other non-pharmaceutical practices (such as social-distancing),and when adoption is nearly universal (nation-wide) and compliance is high.
Under the ongoing COVID-19 pandemic (caused by the SARS-CoV-2 coronavirus), recommen-dations and common practices regarding face mask use by the general public have varied greatly ∗ email [email protected] especially in combination with othernon-pharmaceutical interventions that decrease community transmission rates.Whether masks can be useful, even in principle, depends on the mechanisms for transmissionfor SARS-CoV-2, which are likely a combination of droplet, contact, and possible airborne(aerosol) modes. The traditional model for respiratory disease transmission posits infectionvia infectious droplets (generally 5–10 µ m) that have a short lifetime in the air and infect theupper respiratory tract, or finer aerosols, which may remain in the air for many hours [42], withongoing uncertainties in the relative importance of these modes (and in the conceptual modelitself [41]) for SARS-CoV-2 transmission [45, 41]. The WHO [16] has stated that SARS-CoV-2transmission is primarily via coarse respiratory droplets and contact routes. An experimentalstudy [17] using a nebulizer found SARS-CoV-2 to remain viable in aerosols ( < µ m) for threehours (the study duration), but the clinical relevance of this setup is debatable [16]. One out ofthree symptomatic COVID-19 patients caused extensive environmental contamination in [18],including of air exhaust outlets, though the air itself tested negative.Face masks can protect against both coarser droplet and finer aerosol transmission, thoughN95 respirators are more effective against finer aerosols, and may be superior in preventingdroplet transmission as well [4]. Metanalysis of studies in healthy healthcare providers (inwhom most studies have been performed) indicated a strong protective value against clinicaland respiratory virus infection for both surgical masks and N95 respirators [3]. Case controldata from the 2003 SARS epidemic suggests a strong protective value to mask use by communitymembers in public spaces, on the order of 70% [1, 2].Experimental studies in both humans and manikins indicate that a range of mask provide atleast some protective value against various infectious agents [12, 9, 14, 13, 42]. Medical maskswere potentially highly effective as both source control and primary prevention under tidallybreathing and coughing conditions in manikin studies [10, 11], with higher quality masks (e.g.N95 respirator vs. surgical mask) offering greater protection [11]. It is largely unknown to whatdegree homemade masks (typically made from cotton, teacloth, or other polyesther fibers) may2rotect against droplets/aerosols and viral transmission, but experimental results by Davies etal. [9] suggest that while the homemade masks were less effective than surgical mask, they werestill markedly superior to no mask. A clinical trial in healthcare workers [5] showed relativelypoor performance for cloth masks relative to medical masks.Mathematical modeling has been influential in providing deeper understanding on the trans-mission mechanisms and burden of the ongoing COVID-19 pandemic, contributing to the devel-opment of public health policy and understanding. Most mathematical models of the COVID-19pandemic can broadly be divided into either population-based, SIR (Kermack-McKendrick)-type models, driven by (potentially stochastic) differential equations [38, 20, 34, 22, 21, 23, 31,26, 32, 24, 33], or agent-based models [39, 28, 25, 27, 30], in which individuals typically interacton a network structure and exchange infection stochastically. One difficulty of the latter ap-proach is that the network structure is time-varying and can be difficult, if not impossible, toconstruct with accuracy. Population-based models, alternatively, may risk being too coarse tocapture certain real-world complexities. Many of these models, of course, incorporate featuresfrom both paradigms, and the right combination of dynamical, stochastic, data-driven, andnetwork-based methods will always depend on the question of interest.In [38], Li et al. imposed a metapopulation structure onto an SEIR-model to account fortravel between major cities in China. Notably, they include compartments for both documentedand undocumented infections. Their model suggests that as many as 86% of all cases went un-detected in Wuhan before travel restrictions took effect on January 23, 2020. They additionallyestimated that, on a per person basis, asymptomatic individuals were only 55% as contagious,yet were responsible for 79% of new infections, given their increased prevalence. The impor-tance of accounting for asymptomatic individuals has been confirmed by other studies ([39],[21]). In their model-based assessment of case-fatality ratios, Verity et al. [40] estimated that40–50% of cases went unidentified in China, as of February 8, 2020, while in the case of thePrincess Diamond cruise ship, 46.5% of individuals who tested positive for COVID-19 wereasymptomatic [49]. Further, Calafiore et al. [21], using a modified SIR-model, estimated that,on average, cases in Italy went underreported by a factor of 63, as of March 30, 2020.Several prior mathematical models, motivated by the potential for pandemic influenza, haveexamined the utility of mask wearing by the general public. These include a relatively simplemodification of an SIR-type model by Brienen et al. [36], while Tracht et al. [37] considered amore complex SEIR model that explicitly disaggregated those that do and do not use masks.The latter concluded that, for pandemic H1N1 influenza, modestly effective masks (20%) couldhalve total infections, while if masks were just 50% effective as source control, the epidemiccould be essentially eliminated if just 25% of the population wore masks.We adapt these previously developed SEIR model frameworks for transmission dynamics toexplore the potential community-wide impact of public use of face masks, of varying efficacyand compliance, on the transmission dynamics and control of the COVID-19 pandemic. Inparticular, we develop a two-group model, which stratifies the total population into those whohabitually do and do not wear face masks in public or other settings where transmission mayoccur. This model takes the form of a deterministic system of nonlinear differential equations,and explicitly includes asymptomatically-infectious humans. We examine mask effectivenessand coverage (i.e., fraction of the population that habitually wears masks) as our two primaryparameters of interest.We explore possible nonlinearities in mask coverage and effectiveness and the interactionof these two parameters; we find that the product of mask effectiveness and coverage levelstrongly predicts the effect of mask use on epidemiologic outcomes. Thus, homemade clothmasks are best deployed en masse to benefit the population at large. There is also a potentiallystrong nonlinear effect of mask use on epidemiologic outcomes of cumulative death and peak3ospitalizations. We note a possible temporal effect: Delaying mass mask adoption too longmay undermine its efficacy. Moreover, we perform simulated case studies using mortality datafor New York and Washington state. These case studies likewise suggest a beneficial role to massadoption of even poorly effective masks, with the relative benefit likely greater in Washingtonstate, where baseline transmission is less intense. The absolute potential for saving lives is still,however, greater under the more intense transmission dynamics in New York state. Thus, earlyadoption of masks is useful regardless of transmission intensities, and should not be delayedeven if the case load/mortality seems relatively low.In summary, the benefit to routine face mask use by the general public during the COVID-19pandemic remains uncertain, but our initial mathematical modeling work suggests a possiblestrong potential benefit to near universal adoption of even weakly effective homemade masksthat may synergize with , not replace, other control and mitigation measures. We consider a baseline model without any mask use to form the foundation for parameterestimation and to estimate transmission rates in New York and Washington state; we also usethis model to determine the equivalent transmission rate reductions resulting from public maskuse in the full model.We use a deterministic susceptible, exposed, symptomatic infectious, hospitalized, asymp-tomatic infectious, and recovered modeling framework, with these classes respectively denoted S ( t ), E ( t ), I ( t ), H ( t ), A ( t ), and R ( t ); we also include D ( t ) to track cumulative deaths. Weassume that some fraction of detected infectious individuals progress to the hospitalized class, H ( t ), where they are unable to pass the disease to the general public; we suppose that somefraction of hospitalized patients ultimately require critical care (and may die) [35], but do notexplicitly disaggregate, for example, ICU and non-ICU patients. Based on these assumptionsand simplifications, the basic model for the transmission dynamics of COVID-19 is given by thefollowing deterministic system of nonlinear differential equations: dSdt = − β ( t )( I + ηA ) SN , (2.1) dEdt = β ( t )( I + ηA ) SN − σE, (2.2) dIdt = ασE − φI − γ I I, (2.3) dAdt = (1 − α ) σE − γ A A, (2.4) dHdt = φI − δH − γ H H, (2.5) dRdt = γ I I + γ A A + γ H H, (2.6) dDdt = δH, (2.7)where N = S + E + I + A + R, (2.8)4 arameter Likely range (references) Default value β (infectious contact rate) 0.5–1.5 day − [43, 44, 38], this work 0.5 day − σ (transition exposed to infectious) 1/14–1/3 day − [19, 38] 1/5.1 day − η (infectiousness factor for asymptomatic carriers) 0.4–0.6 [39, 38] 0.5 α (fraction of infections that become symptomatic) 0.15–0.7 [38, 39, 40, 49] 0.5 φ (rate of hospitalization) 0.02–0.1 [35, 39] 0.025 day − γ A (recovery rate, Asymptomatic) 1/14-1/3 day − [34, 35] 1/7 day − γ I (recovery rate, symptomatic) 1/30-1/3 day − [34, 35] 1/7 day − γ H (recovery rate, hospitalized) [34, 35] 1/30-1/3 day − − δ (death rate, hospitalized) 0.001–0.1 [39] 0.015 day − Table 1: Baseline model parameters with brief description, likely ranges based on modeling andclinical studies (see text for further details), and default value chosen for this study.is the total population in the community, and β ( t ) is the baseline infectious contact rate, whichis assumed to vary with time in general, but typically taken fixed. Additionally, η accounts forthe relative infectiousness of asymptomatic carriers (in comparison to symptomatic carriers), σ is the transition rate from the exposed to infectious class (so 1 /σ is the disease incubationperiod), α is the fraction of cases that are symptomatic, φ is the rate at which symptomaticindividuals are hospitalized, δ is the disease-induced death rate, and γ A , γ I and γ H are recoveryrates for the subscripted population.We suppose hospitalized persons are not exposed to the general population. Thus, theyare excluded from the tabulation of N , and do not contribute to infection rates in the generalcommunity. This general modeling framework is similar to a variety of SEIR-style modelsrecently employed in [38, 39], for example.For most results in this paper, we use let β ( t ) ≡ β . However, given ongoing responses tothe COVID-19 pandemic in terms of voluntary and mandated social distancing, etc., we alsoconsider the possibility that β varies with time and adopt the following functional form fromTang et al. [34], with the modification that contact rates do not begin declining from the initialcontact rate, β , until time t , β ( t ) = (cid:26) β , t < t β min + ( β − β min ) exp( − r ( t − t )) , t ≥ t (2.9)where β min is the minimum contact rate and r is the rate at which contact decreases. The incubation period for COVID-19 is estimated to average 5.1 days [19], similar to othermodel-based estimates [38], giving σ = 1 / . − . Some previous model-based estimates ofinfectious duration are on the order of several days [38, 39, 34], with [34] giving about 7 daysfor asymptomatic individuals to recover. However, the clinical course of the disease is typicallymuch longer: In a study of hospitalized patients [35], average total duration of illness untilhospital discharge or death was 21 days, and moreover, the median duration of viral sheddingwas 20 days in survivors.The effective transmission rate (as a constant), β , ranges from around 0.5 to 1.5 day − inprior modeling studies [44, 43, 38], and typically trends down with time [34, 38]. We have leftthis as a free parameter in our fits to Washington and New York state mortality data, and find β ≈ . β ≈ . − for these states, respectively, values this range.5he relative infectiousness of asymptomatic carriers, η , is not known, although Fergusonet al. [39] estimated this parameter at about 0.5, and Li et al. [38] gave values of 0.42–0.55.The fraction of cases that are symptomatic, α , is also uncertain, with Li et al. [38] suggestingan overall case reporting rate of just 14% early in the outbreak in China, but increasing to65–69% later; further, α = 2 / α = 0 . φ ≈ .
025 day − is consistent with on the order of 5–15% of symptomatic patients beinghospitalized. If about 15% of hospitalized patients die [39], then δ ≈ .
015 day − (based on γ H = 1 /
14 day − ). We assume that some fraction of the general population wears masks with uniform inwardefficiency (i.e., primary protection against catching disease) of ǫ i , and outward efficiency (i.e.,source control/protection against transmitting disease) of ǫ o . We disaggregate all populationvariables into those that typically do and do not wear masks, respectively subscripted with U and M . Based on the above assumptions and simplifications, the extended multi-group modelfor COVID-19 (where members of the general public wear masks in public) is given by: dS U dt = − β ( I U + ηA U ) S U N − β (cid:0) (1 − ǫ o ) I M + (1 − ǫ o ) ηA M (cid:1) S U N , (2.10) dE U dt = β ( I U + ηA U ) S U N + β ((1 − ǫ o ) I M + (1 − ǫ o ) ηA M ) S U N − σE U , (2.11) dI U dt = ασE U − φI U − γ I I U , (2.12) dA U dt = (1 − α ) σE U − γ A A U , (2.13) dH U dt = φI U − δH U − γ H H U , (2.14) dR U dt = γ I I U + γ A A U + γ H H U , (2.15) dD U dt = δH U , (2.16) dS M dt = − β (1 − ǫ i )( I U + ηA U ) S M N − β (1 − ǫ i )((1 − ǫ o ) I M + (1 − ǫ o ) ηA M ) S M N , (2.17) dE M dt = β (1 − ǫ i )( I U + ηA U ) S M N + β (1 − ǫ i )((1 − ǫ o ) I M + (1 − ǫ o ) ηA M ) S M N − σE M , (2.18) dI M dt = ασE M − φI M − γ I I M , (2.19) dA M dt = (1 − α ) σE M − γ A A M , (2.20) dH M dt = φI M − δH M − γ H H M , (2.21) dR M dt = γ I I M + γ A A M + γ H H M , (2.22) dD M dt = δH M , (2.23)(2.24) N = S U + E U + I U + A U + R U + S M + E M + I M + A M + R M . (2.25)While much more complex than the baseline model, most of the complexity lies in what areessentially bookkeeping terms. We also consider a reduced version of the above model (equationsnot shown), such that only symptomatically infected persons wear a mask, to compare theconsequences of the common recommendation that only those experiencing symptoms (andtheir immediate caretakers) wear masks with more general population coverage. We assume a roughly linear relationship between the overall filtering efficiency of a mask andclinical efficiency in terms of either inward efficiency (i.e., effect on ǫ i ) or outward efficiency ( ǫ o ),based on [36]. The fit factor for homemade masks averaged 2 in [9], while the fit factor averaged5 for surgical masks. When volunteers coughed into a mask, depending upon sampling method,the number of colony-forming units resulting varied from 17% to 50% for homemade masks and0–30% for surgical masks, relative to no mask [9].Surgical masks reduced P. aeruginosa infected aerosols produced by coughing by over 80%in cystic fibrosis patients in [14], while surgical masks reduced CFU count by >
90% in a similarstudy [13]. N95 masks were more effective in both studies. Homemade teacloth masks had aninward efficiency between 58 and 77% over 3 hours of wear in [12], while inward efficiency ranged72–85% and 98–99% for surgical and N95-equivalent masks. Outward efficiency was marginalfor teacloth masks, and about 50–70% for medical masks. Surgical masks worn by tuberculosispatients also reduced the infectiousness of hospital ward air in [15], and Leung et al. [42] veryrecently observed surgical masks to decrease infectious aerosol produced by individuals withseasonal coronaviruses. Manikin studies seem to recommend masks as especially valuable undercoughing conditions for both source control [11] and prevention [10].We therefore estimate that inward mask efficiency could range widely, anywhere from 20–80% for cloth masks, with ≥
50% possibly more typical (and higher values are possible forwell-made, tightly fitting masks made of optimal materials), 70–90% typical for surgical masks,and >
95% typical for properly worn N95 masks. Outward mask efficiency could range frompractically zero to over 80% for homemade masks, with 50% perhaps typical, while surgicalmasks and N95 masks are likely 50–90% and 70–100% outwardly protective, respectively.
We use state-level time series for cumulative mortality data compiled by Center for SystemsScience and Engineering at Johns Hopkins University [47], from January 22, 2020, throughApril 2, 2020, to calibrate the model initial conditions and infective contact rate, β , as well as β min when β ( t ) is taken as an explicit function of time. Other parameters are fixed at defaultvalues in Table 1. Parameter fitting was performed using nonlinear least squares algorithm im-plemented using the lsqnonlin function in MATLAB. We consider two US states in particularas case studies, New York and Washington, and total population data for each state was definedaccording to US Census data for July 1, 2019 [48].7igure 1: Relative peak hospitalizations and cumulative mortality under simulated epidemics,under either a base β = 0.5 or 1.5 day − , under different general mask coverage level andefficacies (where ǫ o = ǫ i = ǫ ). Results are relative to a base case with no mask use. The lefthalf of the figure gives these metrics as two-dimensional functions of coverage and efficacy. Theright half gives these metrics as one-dimensional functions of coverage × efficacy. Closed-form expressions for the basic reproduction number, R , for the baseline model withoutmasks and the full model with masks are given, for β ( t ) ≡ β , in Appendix A and B, respectively. We run simulated epidemics using either β = 0.5 or 1.5 day − , with other parameters set tothe defaults given in Table 1. These parameter sets give epidemic doubling times early in time(in terms of cumulative cases and deaths) of approximately seven or three days, respectively,corresponding to case and mortality doubling times observed (early in time) in Washington andNew York state, respectively. We use as initial conditions a normalized population of 1 millionpersons, all of whom are initially susceptible, except 50 initially symptomatically infected (i.e.,5 out 100,000 is the initial infection rate), not wearing masks.We choose some fraction of the population to be initially in the masked class (“mask cov-erage”), which we also denote π , and assume ǫ o = ǫ i = ǫ . The epidemic is allowed to run itscourse (18 simulated months) under constant conditions, and the outcomes of interest are peakhospitalization, cumulative deaths, and total recovered. These results are normalized againstthe counterfactual of no mask coverage, and results are presented as heat maps in Figure 1.Note that the product ǫ × π predicts quite well the effect of mask deployment: Figure 1 alsoshows (relative) peak hospitalizations and cumulative deaths as functions of this product. Thereis, however, a slight asymmetry between coverage and efficacy, such that increasing coverage ofmoderately effective masks is generally more useful than increasing the effectiveness of masksfrom a starting point of moderate coverage. 8igure 2: Equivalent β , ˜ β (infectious contact rate) under baseline model dynamics as a functionof mask coverage × efficacy, with the left panel giving the absolute value, and the right givingthe ratio of ˜ β to the true β in the simulation with masks. That is, simulated epidemics arerun with mask coverage and effectiveness ranging from 0 to 1, and the outcomes are tracked assynthetic data. The baseline model without mask dynamics is then fit to this synthetic data,with β the trainable parameter; the resulting β is the ˜ β . This is done for simulated epidemicswith a true β of 1.5, 1, or 0.5 day − . We run the simulated epidemics described, supposing the entire population is unmasked untilmass mask adoption after some discrete delay. The level of adoption is also fixed as a constant.We find that a small delay in mask adoption (without any changes in β ) has little effect onpeak hospitalized fraction or cumulative deaths, but the “point of no return” can rapidly becrossed, if mask adoption is delayed until near the time at which the epidemic otherwise crests.This general pattern holds regardless of β , but the point of no return is further in the futurefor smaller β . β reduction The relationship between mask coverage, efficacy, and metrics of epidemic severity consideredabove are highly nonlinear. The relationship between β (the infectious contact rate) and suchmetrics is similarly nonlinear. However, incremental reductions in β , due to social distancesmeasures, etc., can ultimately synergize with other reductions to yield a meaningfully effect onthe epidemic. Therefore, we numerically determine what the equivalent change in β under thebaseline would have been under mask use at different coverage/efficacy levels, and we denotethe equivalent β value as ˜ β .That is, we numerically simulate an epidemic with and without masks, with a fixed β .Then, we fit the baseline model to this (simulated) case data, yielding a new equivalent β ,˜ β . An excellent fit giving ˜ β can almost always be obtained, though occasionally results areextremely sensitive to β for high mask coverage/efficacy, yielding somewhat poorer fits. Results9igure 3: Epidemiologic outcomes and equivalent β changes as a function of mask coveragewhen masks are either much better at blocking outgoing ( ǫ o = 0 . ǫ i = 0 .
2) or incoming( ǫ = 0 . ǫ i = 0 .
8) transmission. Results are demonstrated for both mask permutations undersimulated epidemics with baseline β = 0.5 or 1.5 day − .are summarized in Figure 2, where the ˜ β values obtained and the relative changes in equivalent β (i.e., ( ˜ β ) / ( β )) are plotted as functions of efficacy times coverage, ǫ × π , under simulatedepidemics with three baseline (true) β values.From Figure 2, we see that even 50% coverage with 50% effective masks roughly halves theeffective disease transmission rate. Widespread adoption, say 80% coverage, of masks that areonly 20% effective still reduces the effective transmission rate by about one-third. Figure 3 demonstrates the effect of mask coverage on peak hospitalizations, cumulative deaths,and equivalent β values when either ǫ o = 0 . ǫ i = 0 .
8, or visa versa (and for simulatedepidemics using either β = 0.5 or 1.5 day − . These results suggest that, all else equal, theprotection masks afford against acquiring infection ( ǫ o ) is actually slightly more importantthan protection against transmitting infection ( ǫ i ), although there is overall little meaningfulasymmetry. Finally, we consider numerical experiments where masks are given to all symptomatically in-fected persons, whether they otherwise habitually wear masks or not (i.e., both I U and I M actually wear masks). We explore how universal mask use in symptomatically infected personsinteracts with mask coverage among the general population; we let ǫ Io represent the effectivenessof masks in the symptomatic, not necessarily equal to ǫ o . We again run simulated epidemicswith no masks, universal masks among the symptomatic, and then compare different levels ofmask coverage in the general (asymptomatic) population. In this section, we use equivalent β as our primary metric. Figure 4 shows how this metric varies as a function of the maskeffectiveness given to symptomatic persons, along with the coverage and effectiveness of masks10igure 4: Equivalent β under the model where all symptomatic persons wear a mask (whetherthey otherwise habitually wear a mask or not), under varying levels of effective for the masksgiven to the symptomatic ( ǫ Io ), and in combination with different degrees of coverage and effec-tiveness for masks used by the rest of the general public. Results are for simulated epidemicswith a baseline β of 1.5 day − .worn by the general public.We also explore how conclusions vary when either 25%, 50%, or 75% of infectious COVID-19 patients are asymptomatic (i.e., we vary α ). Unsurprisingly, the greater the proportion ofinfected people are asymptomatic, the more benefit there is to giving the general public masksin addition to those experiencing symptoms. Fitting to cumulative death data, we use the baseline model to determine the best fixed β and I (0) for cumulative death data for New York and Washington state. We use New Yorkstate data beginning on March 1, 2020, through April 2, 2020, and Washington state data fromFebruary 20, 2020 through April 2, 2020. For New York state, best-fit parameters are I (0) =208 (range 154–264) and β = 1.40 (1.35–1.46) day − under fixed β . For the time-varying β ( t ),we fix r = 0 .
03 day − and t = 20, yielding a best-fit β = 1.33 (1.24–1.42) day − , β min = 0.51(-0.25–1.26) day − , and I (0) = 293 (191–394).For Washington state, parameters are I (0) = 622 (571–673) and β = 0.50 (0.49–0.52) day − under fixed β . For time-varying β ( t ), we fix r = 0 .
04 day − and t = 0, to yield a best-fit β = 1.0 (0.87–1.23) day − , β min = 0.10 (0–0.19) day − , and I (0) = 238 (177–300).We fix r and t , as it is not possible to uniquely identify r , t and β min , from death orcase data alone (see e.g., [46] on identifiability problems). Figure 5 gives cumulative death andcase data versus the model predictions for the two states, and for the two choices of β ( t ). Notethat while modeled and actual cumulative deaths match well, model-predicted cases markedlyexceed reported cases in the data, consistent with the notion of broad underreporting.We then consider either fixed β or time-varying β ( t ), according to the parameters above,in combination with the following purely hypothetical scenarios in each state.11igure 5: The left half of the figure gives model predictions and data for Washington state,using either a constant (top panels) or variable β (bottom panel), as described in the test. Theright half of the figure is similar, but for New York state.Figure 6: Simulated future (cumulative) death tolls for Washington state, using either a fixed(top panels) or variable (bottom panels) transmission rate, β , and nine different permutationsof general public mask coverage and effectiveness. The y-axes are scaled differently in top andbottom panels. 12igure 7: Simulated future daily death rates for Washington state, using either a fixed (toppanels) or variable (bottom panels) transmission rate, β , and nine different permutations ofgeneral public mask coverage and effectiveness. The y-axes are scaled differently in top andbottom panels.Figure 8: Simulated future (cumulative) death tolls for New York state, using either a fixed(top panels) or variable (bottom panels) transmission rate, β , and nine different permutationsof general public mask coverage and effectiveness.13igure 9: Simulated future daily death rates for New York state, using either a fixed (toppanels) or variable (bottom panels) transmission rate, β , and nine different permutations ofgeneral public mask coverage and effectiveness.1. No masks, epidemic runs its course unaltered with either β ( t ) ≡ β fixed or β ( t ) variableas described above.2. The two β scenarios are considered in combination with: (1) weak, moderate, or strongdeployment of masks, such that π = 0.2, 0.5, or 0.8; and (2) weak, moderate, or strongmasks, such that ǫ = 0.2, 0.5, or 0.8. No masks are used up until April 2, 2020, and thenthese coverage levels are instantaneously imposed.This yields 18 scenarios in all (nine mask coverage/efficacy scenarios, plus two underlyingtrends). Following the modeled imposition of masks on April 2, 2020, the scenarios are runfor 60 additional simulated days. Figures 6 and 8 summarize the future modeled death toll ineach city under the 18 different scenarios, along with historical mortality data. Figures 7 and9 show modeled daily death rates, with deaths peaking sometime in late April in New Yorkstate under all scenarios, while deaths could peak anywhere from mid-April to later than May,for Washington state. We emphasize that these are hypothetical and exploratory results, withpossible death tolls varying dramatically based upon the future course of β ( t ). However, theresults do suggest that even modestly effective masks, if widely used, could help “bend thecurve,” with the relative benefit greater in combination with a lower baseline β or strongerunderlying trend towards smaller β ( t ) (i.e., in Washington vs. New York). This study aims to contribute to this debate by providing realistic insight into the community-wide impact of widespread use of face masks by members of the general population. We designeda mathematical model, parameterized using data relevant to COVID-19 transmission dynamicsin two US states (New York and Washington). The model suggests a nontrivial benefit to facemask use by the general public that may vary nonlinearly with mask effectiveness, coverage,14nd baseline disease transmission intensity. Face masks should be advised not just for thoseexperiencing symptoms, and likely protect both truly healthy wearers and avoid transmissionby asymptomatic carriers. The community-wide benefits are greatest when mask coverage is asnear universal as possible.There is considerable ongoing debate on whether to recommend general public face maskuse (likely mostly homemade cloth masks or other improvised face coverings) [51], and while thesituation is in flux, more authorities are recommending public mask use, though they continueto (rightly) cite appreciable uncertainty. With this study, we hope to help inform this debateby providing insight into the potential community-wide impact of widespread face mask useby members of the general population. We have designed a mathematical model, parameter-ized using data relevant to COVID-19 transmission dynamics in two US states (New York andWashington), and our model suggests nontrivial and possibly quite strong benefit to generalface mask use. The population-level benefit is greater the earlier masks are adopted, and atleast some benefit is realized across a range of epidemic intensities. Moreover, even if they have,as a sole intervention, little influence on epidemic outcomes, face masks decrease the equiva-lent effective transmission rate ( β in our model), and thus can stack with other interventions,including social distancing and hygienic measures especially, to ultimately drive nonlinear de-creases in epidemic mortality and healthcare system burden. It bears repeating that our modelresults are consistent with the idea that face masks, while no panacea, may synergize with othernon-pharmaceutical control measures and should be used in combination with and not in lieuof these.Under simulated epidemics, the effectiveness of face masks in altering the epidemiologicoutcomes of peak hospitalization and total deaths is a highly nonlinear function of both maskefficacy and coverage in the population (see Figure 1), with the product of mask efficacy andcoverage a good one-dimensional surrogate for the effect. We have determined how mask usein the full model alters the equivalent β , denoted ˜ β , under baseline model (without masks),finding this equivalent ˜ β to vary nearly linearly with efficacy × coverage (Figure 2).Masks alone, unless they are highly effective and nearly universal, may have only a smalleffect (but still nontrivial, in terms of absolute lives saved) in more severe epidemics, such asthe ongoing epidemic in New York state. However, the relative benefit to general masks usemay increase with other decreases in β , such that masks can synergize with other public healthmeasures. Thus, it is important that masks not be viewed as an alternative, but as a com-plement, to other public health control measures (including non-pharmaceutical interventions,such as social distancing, self-isolation etc.). Delaying mask adoption is also detrimental. Thesefactors together indicate that even in areas or states where the COVID-19 burden is low (e.g.the Dakotas), early aggressive action that includes face masks may pay dividends.These general conclusions are illustrated by our simulated case studies, in which we havetuned the infectious contact rate, β (either as fixed β or time-varying β ( t )), to cumulativemortality data for Washington and New York state through April 2, 2020, and imposed hypo-thetical mask adoption scenarios. The estimated range for β is much smaller in Washingtonstate, consistent with this state’s much slower epidemic growth rate and doubling time. Modelfitting also suggests that total symptomatic cases may be dramatically undercounted in bothareas, consistent with prior conclusions on the pandemic [38]. Simulated futures for both statessuggest that broad adoption of even weak masks use could help avoid many deaths, but thegreatest relative death reductions are generally seen when the underlying transmission rate alsofalls or is low at baseline.Considering a fixed transmission rate, β , 80% adoption of 20%, 50%, and 80% effectivemasks reduces cumulative relative (absolute) mortality by 1.8% (4,419), 17% (41,317), and 55%(134,920), respectively, in New York state. In Washington state, relative (absolute) mortality15eductions are dramatic, amounting to 65% (22,262), 91% (31,157), and 95% (32,529). When β ( t ) varies with time, New York deaths reductions are 9% (21,315), 45% (103,860), and 74%(172,460), while figures for Washington are 24% (410), 41% (684), and 48% (799). In the lattercase, the epidemic peaks soon even without masks. Thus, a range of outcomes are possible,but both the absolute and relative benefit to weak masks can be quite large; when the relativebenefit is small, the absolute benefit in terms of lives is still highly nontrivial.Most of our model projected mortality numbers for New York and Washington state are quitehigh (except for variable β ( t ) in Washington), and likely represent worst-case scenarios as theyprimarily reflect β values early in time. Thus, they may be dramatic overestimates, dependingupon these states’ populations ongoing responses to the COVID-19 epidemics. Nevertheless,the estimated transmission values for the two states, under fixed and variable β ( t ) represent abroad range of possible transmission dynamics, are within the range estimated in prior studies[43, 44, 38], and so we may have some confidence in our general conclusions on the possiblerange of benefits to masks. Note also that we have restricted our parameter estimation onlyto initial conditions and transmission parameters, owing to identifiability problems with morecomplex models and larger parameter groups (see e.g. [46]). For example, the same death datamay be consistent with either a large β and low δ (death rate), or visa versa.Considering the subproblem of general public mask use in addition to mask use for sourcecontrol by any (known) symptomatic person, we find that general face mask use is still highlybeneficial (see Figure 4). Unsurprisingly, this benefit is greater if a larger proportion of infectedpeople are asymptomatic (i.e., α in the model is smaller). Moreover, it is not the case thatmasks are helpful exclusively when worn by asymptomatic infectious persons for source control,but provide benefit when worn by (genuinely) healthy people for prevention as well. Indeed, ifthere is any asymmetry in outward vs. inward mask effectiveness, inward effectiveness is actuallyslightly preferred, although the direction of this asymmetry matters little with respect to overallepidemiologic outcomes. At least one experimental study [11] does suggest that masks may besuperior at source control, especially under coughing conditions vs. normal tidal breathing andso any realized benefit of masks in the population may still be more attributable to sourcecontrol.This is somewhat surprising, given that ǫ o appears more times than ǫ i in the model termsgiving the forces of infection, which would suggest outward effectiveness to be of greater importat first glance. Our conclusion runs counter to the notion that general public masks are primarilyuseful in preventing asymptomatically wearers from transmitting disease: Masks are valuableas both source control and primary prevention. This may be important to emphasize, as somepeople who have self-isolated for prolonged periods may reasonably believe that the chance theyare asymptomatically infected is very low and therefore do not need a mask if they venture intopublic, whereas our results indicate they (and the public at large) still stand to benefit.Our theoretical results still must be interpreted with caution, owing to a combination ofpotential high rates of noncompliance with mask use in the community, uncertainty with respectto the intrinsic effectiveness of (especially homemade) masks at blocking respiratory dropletsand/or aerosols, and even surprising amounts of uncertainty regarding the basic mechanisms forrespiratory infection transmission [4, 41]. Several lines of evidence support the notion that maskscan interfere with respiratory virus transmission, including clinical trials in healthcare workers[3, 4], experimental studies as reviewed [12, 10, 9, 15, 11], and case control data from the 2003SARS epidemic [1, 2]. Given the demonstrated efficacy of medical masks in healthcare workers[3], and their likely superiority over cloth masks in [5], it is clearly essential that healthcareworks be prioritized when it comes to the most effective medical mask supply. Fortunately, ourtheoretical results suggest significant (but potentially highly variable) value even to low qualitymasks when used widely in the community. 16ith social distancing orders in place, essential service providers (such as retail workers,emergency services, law enforcement, etc.) represent a special category of concern, as theyrepresent a largely unavoidable high contact node in transmission networks: Individual public-facing workers may come into contact with hundreds or thousands of people in the course ofa day, in relatively close contact (e.g. cashiers). Such contact likely exposes the workers tomany asymptomatic carriers, and they may in turn, if asymptomatic, expose many susceptiblemembers of the general public to potential transmission. Air exposed to multiple infectiouspersons (e.g. in grocery stores) could also carry a psuedo-steady load of infectious particles,for which masks would be the only plausible prophylactic[10]. Thus, targeted, highly effectivemask use by service workers may be reasonable. We are currently extending the basic modelframework presented here to examine this hypothesis.In conclusion, our findings suggest that face mask use should be as nearly universal (i.e.,nation-wide) as possible and implemented without delay, even if most mask are homemade andof relatively low quality. This measure could contribute greatly to controlling the COVID-19pandemic, with the benefit greatest in conjunction with other non-pharmaceutical interventionsthat reduce community transmission. Despite uncertainty, the potential for benefit, the lackof obvious harm, and the precautionary principle lead us to strongly recommend as close touniversal (homemade, unless medical masks can be used without diverting healthcare supply)mask use by the general public as possible. Acknowledgements
One of the authors (ABG) acknowledge the support, in part, of the Simons Foundation (Award
Appendix A: Basic Reproduction Number for Baseline Model
The basic reproduction number for both the baseline and the full model is for the special casewhen β ( t ) ≡ β . The local stability of the DFE is explored using the next generation operatormethod [52, 53]. Using the notation in [53], it follows that the matrices F of new infectionterms and V of the remaining transfer terms associated with the version of the model are given,respectively, by F = β β η , V = σ − ασ ( φ + γ I ) 0 − (1 − α ) σ γ A . The basic reproduction number of the model, denoted by R , is given by R = β ασσ ( φ + γ I ) + β η (1 − α ) γ A . (4.1) Appendix B: Basic Reproduction Number for Full Model
The local stability of the DFE is explored using the next generation operator method [52, 53].Using the notation in [53], it follows that the matrices F of new infection terms and V of the17emaining transfer terms associated with the version of the model are given, respectively, by F = β β η β (1 − ǫ o ) β (1 − ǫ o ) η β (1 − ǫ i ) β (1 − ǫ i ) η β (1 − ǫ o )(1 − ǫ i ) β (1 − ǫ o )(1 − ǫ i ) η , V = σ − ασ ( φ + γ I ) 0 0 0 0 − (1 − α ) σ γ A σ − ασ ( φ + γ I ) 00 0 0 − (1 − α ) σ γ A . The basic reproduction number of the model, denoted by R , is given by R = β [1 + (1 − ǫ o )(1 − ǫ i )] (cid:18) ασσ ( φ + γ I ) + η (1 − α ) γ A (cid:19) . (4.2) References [1] Wu, J., Xu, F., Zhou, W., Feikin, D. R., Lin, C. Y., He, X., ... & Schuchat, A. (2004). Risk factors for SARSamong persons without known contact with SARS patients, Beijing, China. Emerging infectious diseases,10(2), 210–216.[2] Lau, J. T., Tsui, H., Lau, M., & Yang, X. (2004). SARS transmission, risk factors, and prevention in HongKong. Emerging infectious diseases, 10(4), 587–592.[3] Offeddu, V., Yung, C. F., Low, M. S. F., & Tam, C. C. (2017). Effectiveness of masks and respiratorsagainst respiratory infections in healthcare workers: a systematic review and meta-analysis. Clinical InfectiousDiseases, 65(11), 1934–1942.[4] MacIntyre, C. R., Chughtai, A. A., Rahman, B., Peng, Y., Zhang, Y., Seale, H., ... & Wang, Q. (2017). Theefficacy of medical masks and respirators against respiratory infection in healthcare workers. Influenza andother respiratory viruses, 11(6), 511–517.[5] MacIntyre, C. R., Seale, H., Dung, T. C., Hien, N. T., Nga, P. T., Chughtai, A. A., ... & Wang, Q. (2015). Acluster randomised trial of cloth masks compared with medical masks in healthcare workers. BMJ open, 5(4),e006577.[6] MacIntyre, C. R., Cauchemez, S., Dwyer, D. E., Seale, H., Cheung, P., Browne, G., ... & Ferguson, N. (2009).Face mask use and control of respiratory virus transmission in households. Emerging infectious diseases, 15(2),233.[7] Cowling, B. J., Chan, K. H., Fang, V. J., Cheng, C. K., Fung, R. O., Wai, W., ... & Chiu, B. C. (2009).Facemasks and hand hygiene to prevent influenza transmission in households: a cluster randomized trial.Annals of internal medicine, 151(7), 437–446.[8] Canini, L., Androletti, L., Ferrari, P., D’Angelo, R., Blanchon, T., Lemaitre, M., ... & Valleron, A. J. (2010).Surgical mask to prevent influenza transmission in households: a cluster randomized trial. PloS one, 5(11),e13988.[9] Davies, A., Thompson, K. A., Giri, K., Kafatos, G., Walker, J., & Bennett, A. (2013). Testing the efficacyof homemade masks: would they protect in an influenza pandemic?. Disaster medicine and public healthpreparedness, 7(4), 413–418.[10] Lai, A. C. K., Poon, C. K. M., & Cheung, A. C. T. (2012). Effectiveness of facemasks to reduce exposurehazards for airborne infections among general populations. Journal of the Royal Society Interface, 9(70), 938–948.
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