Preventing epidemics by wearing masks: An application to COVID-19
PPreventing epidemics by wearing masks: An applicationto COVID-19
Jo˜ao A. M. Gondim
Unidade Acadˆemica do Cabo de Santo Agostinho, Universidade Federal Rural dePernambuco, Cabo de Santo Agostinho, PE, Brazil
Abstract
The goal of this work is to consider widespread use of face masks as a non-pharmaceutical control strategy for the Covid-19 pandemic. A SEIR modelthat divides the population into individuals that wear masks and those thatdo not is considered. After calculating the basic reproductive number by anext generation approach, a criterion for determining when an epidemic can beprevented by the use of masks only and the critical percentage of mask users fordisease prevention in the population are derived. The results are then appliedto real world data from the United States, Brazil and Italy.
Keywords:
Epidemics, Covid-19, Face masks, Non-pharmaceutical controlstrategies, SEIR model.
1. Introduction
The Covid-19 crisis has created the biggest public health concerns of 2020.Since being first reported at the end of 2019, the disease has caused over 23million confirmed cases and 800 thousand deaths by the end of August 2020 [1].Many studies to model the pandemic spread were developed (e.g. [2, 3, 4, 5])with results influencing the policies of governments around the world.Attempts to minimize the damage of the pandemic were then implemented,such as mandatory mask use and quarantines, which improved the overall sce-nario, but so far there are no drugs or vaccines to treat or immunize people andmaintaining quarantines for longer time periods is not a viable option in some
Preprint submitted to Elsevier September 9, 2020 a r X i v : . [ q - b i o . P E ] S e p ommunities. Therefore, looking for non-pharmaceutical control strategies isessential to deal with this epidemic and others in the future.This paper addresses this issue considering the widespread use of masks asin other works such as [6, 7, 8, 9]. A SEIR model [10] with individuals dividedinto those that wear masks and those that do not is considered in Section 2, andits basic reproductive number is calculated by a next generation approach inSection 3. This leads to a criterion that determines when an epidemic outbreakcan be avoided by mask use only, and a critical percentage of mask users in thepopulation is derived.For applications of these methods, we perform the parameter fitting in Sec-tion 4 with data from the United States, Brazil and Italy and analyse whetherthe Covid-19 crisis could have been avoided in these countries with widespreadmask use from the beginning of the outbreak. In addition, numerical simula-tions are carried out to verify how the evolution of the disease is mitigated if itcannot be avoided. The conclusions are drawn in Section 5.
2. Model structure
Consider a population N that is divided into individuals that wear masks,denoted by N m and individuals that do not, denoted by N n . Let p ( t ) be thepercentage of people wearing a mask in the population at time t ≥
0, then N m ( t ) = p ( t ) N ( t ) , N n ( t ) = (1 − p ( t )) N ( t ) . (1)Both N n and N m are also divided into four epidemiological classes, consistingof susceptible, exposed, infected and removed individuals, denoted by S n and S m , E n and E m , I n and I m and R n and R m , respectively. As the model willconsider only a short time period in comparison to the demographic time frame,vital parameters will be neglected, so the total population will be assumedconstant, that is, N ( t ) = N. (2)Let r be a multiplicative factor for the transmission rate β that will take intoaccount the reduction in the probability of contagion from one person wearing2 mask in a susceptible-infected contact. We assume that this reduction is thesame whether a susceptible or an infective is wearing the mask. When onlyone individual has a mask on, we assume that the new transmission rate is rβ .In the case of both individuals with masks on, then the transmission rate isassumed to be r β . There are four ways contagions can occur, and they aredescribed in Table 1. Table 1: Transmission possibilities.
Susceptible Infected Transmission term S n I n βS n I n NS n I m rβS n I m NS m I n rβS m I n NS m I m r βS m I m N We further assume that p ( t ) = p is constant, so N m ( t ) and N n ( t ) are alsoconstant. Hence, our model can be written as3 (cid:48) n = − βS n N ( I n + rI m ) ,S (cid:48) m = − rβS m N ( I n + rI m ) ,E (cid:48) n = βS n N ( I n + rI m ) − σE n ,E (cid:48) m = rβS m N ( I n + rI m ) − σE m ,I (cid:48) n = σE n − γI n ,I (cid:48) m = σE m − γI m ,R (cid:48) n = γI n ,R (cid:48) m = γI m . (3)The parameters σ and γ denote the exit rates from the exposed and infectedclasses, respectively. It is typically assumed that σ = 1 /T e and γ = 1 /T i ,where T e and T i are the mean lengths of the latency and infectious periods,respectively.
3. The basic reproductive number and some consequences
Now, the basic reproductive number, R , of model (3) is calculated. Thiswill be done by a next generation approach (see [11, 12]). R is given by thespectral radius of K = F V − , where F = β (1 − p ) βr (1 − p )0 0 βrp βr p V = σ σ − σ γ − σ γ . Hence, K = F V − is K = β (1 − p ) γ rβ (1 − p ) γ β (1 − p ) γ rβ (1 − p ) γrβpγ r βpγ rβpγ r βpγ . Due to its block structure, the eigenvalues of K are exactly the eigenvaluesof K = β (1 − p ) γ rβ (1 − p ) γrβpγ r βpγ . It is clearly seen that the trace and the determinant of K areTr( K ) = βγ (cid:0) − p + r p (cid:1) , det( K ) = 0 , respectively, hence its eigenvalues are 0 and R = R (cid:2) − p (cid:0) − r (cid:1)(cid:3) , (4)where R = βγ , (5)which is the basic reproductive number for the standard SEIR model withoutvital dynamics (6) [13]. S (cid:48) = − β SIN ,E (cid:48) = β SIN − σE ,I (cid:48) = σE − γI ,R (cid:48) = γI. (6)5ote that (3) reduces to (6) if either p = 0 (nobody wears masks) or r = 1 (themasks offer no protection against the disease).Since both p, r ∈ [0 , R > R ≤ R , with theequality in the cases p = 0 or r = 1, when the model (3) reduces to model (6).The level set R = 1 in (4) is displayed in Figure 1 for R = 5, along with theregions of R < R > Figure 1: The level set of R = 1 for R = 5 in the rp plane. We now look for conditions the pair ( r, p ) should satisfy in order for thispoint to lie in the region of R <
1. Notice that, for fixed r ∈ [0 , R is adecreasing function of p (see Figure 2). For p = 0, we have R = R , and for p = 1, we have R = R , where R = R · r . (7)It is assumed that R >
1. Then, it is clear from Figure 2 that one can findvalues of p such that R < R <
1, that is, if and only if r < √R . (8)Moreover, there is a critical value p ∗ such that R ( p ∗ ) = 1, so R ( p ) < p > p ∗ . Solving R = 1 in (4), one sees that p ∗ = 11 − r (cid:18) − R (cid:19) . (9)6 igure 2: Plot of R as a function of p . The value of p ∗ corresponds to the critical percentage of the population thatshould wear masks in order to avoid the epidemic outbreak. In the extreme caseof r = 0, i.e., the masks are ideal and avoid contamination for users, which isthe same as immunizing the population, (9) becomes p ∗ = 1 − R , (10)which coincides with the usual threshold for herd immunity [14].
4. Data fitting and numerical results
In this Section, we collect data from the United States, Brazil and Italy touse as case studies for the results of the previous Section. The time frame inconsideration consists of the first 30 days after the cumulative number of cases ineach country reached 100, which happened in March 2, March 13 and February23 for the US, Brazil and Italy, respectively. The data, which was retrieved from[15], is displayed in Tables 2, 3 and 4.We assume that the mean latency and recovery periods are 5 . β by a minimization routine based on the leastsquares method, available in [11], in the standard SEIR model without vital7 able 2: Cumulative cases in the USA starting at the first day with more at least 100 cases. Day Cases Day Cases Day Cases Day Cases Day Cases1 100 7 541 13 2,774 19 19,608 25 86,6682 124 8 704 14 3,622 20 24,498 26 105,5843 158 9 994 15 4,611 21 33,946 27 125,2504 221 10 1,301 16 6,366 22 44,325 28 145,5265 319 11 1,631 17 9,333 23 55,579 29 168,8356 435 12 2,185 18 13,935 24 69,136 30 194,127
Table 3: Cumulative cases in Brazil starting at the first day with at least 100 cases.
Day Cases Day Cases Day Cases Day Cases Day Cases1 151 7 640 13 2,554 19 5,717 25 12,1832 151 8 970 14 2,985 20 6,880 26 14,0343 200 9 1,178 15 3,417 21 8,044 27 16,1884 234 10 1,546 16 3,904 22 9,194 28 18,1455 346 11 1,924 17 4,256 23 10,360 29 19,7896 529 12 2,247 18 4,630 24 11,254 30 20,962
Table 4: Cumulative cases in Italy starting at the first day with at least 100 cases.
Day Cases Day Cases Day Cases Day Cases Day Cases1 157 7 1,128 13 4,639 19 15,122 25 35,7322 229 8 1,702 14 5,886 20 17,670 26 41,0563 323 9 2,038 15 7,380 21 21,169 27 47,0444 470 10 2,504 16 9,179 22 24,762 28 53,5985 655 11 3,092 17 10,156 23 27,997 29 59,1586 889 12 3,861 18 12,469 24 31,524 30 63,941 dynamics (6). The routine minimizes the difference of the cumulative numberof cases, given by I ( t ) + R ( t ), and the data points.The total populations of the USA, Brazil and Italy will be rounded to 331,209 and 60 million, respectively. These numbers will be taken as the initialvalues of susceptible individuals in each country. In the first days of each data8et, the numbers of active cases (see [15]) were 85, 150 and 152 for the USA,Brazil and Italy, respective, so the initial conditions for infected and removedindividuals will be taken, respectively, as 85 and 15 for the USA, 150 and 1 forBrazil and 152 and 5 for Italy.For the initial values of exposed individuals, we use the fact that the latencyperiod is taken as 5 . Table 5: Initial conditions for the estimation of β . Country S(0) E(0) I(0) R(0)USA 331 million 335 85 15Brazil 209 million 378 150 1Italy 60 million 732 152 5
Starting with an initial guess of β = 0 .
5, the fitted values of β are, then, β US = 0 . , β BR = 0 . , β IT = 0 . . (11)Using (5), we can calculate its value for each country. The results are dis-played in Table 6. Table 6: Basic reproductive number for each country in the standard SEIR model.
Country R USA 6 . . . According to [16], when both individuals wear masks in a susceptible-infectedcontact, there is an average reduction of 82 .
18% in the transmission, so r = 1 − . . , r = 0 . R < r ≈ . . According to Table 6, the Covid-19 crisis could have been avoided in Brazil andin Italy, and according to (9), this would be possible if at least 85 .
87% and91 .
76% of all individuals wore masks, respectively.On the other hand, Table 6 indicates that Covid-19 could not have beenavoided in the USA by widespread mask use only, but the basic reproductivenumber could be lowered from R = 6 . R = 1 . R with respect to p (see [11, 17]), given byΥ R p = ∂R ∂p · pR , (12)This number provides the percentage change in R for a given percentagechange in p . For example, if Υ R p = − .
5, then a 10% increase in p produces a5% decrease in R . By (4) and (12), we haveΥ R p = − p (1 − r )1 − p (1 − r ) . (13)A plot of Υ R p as a function of p is displayed in Figure 3. It shows that R becomes very sensitive to p for bigger values of this parameter, so even if mostof a community has already become adept to wearing masks, small increases in p could still contribute greatly to epidemic control.We now assess the effect of variations in p on the numbers of infected causedby Covid-19 in the case of the USA. A comparison of the total infected curve I n ( t ) + I m ( t ) , p -5-4-3-2-10 S en s i t i v i t y i nde x o f R Figure 3: Plot of the normalized forward sensitivity index of R with respect to p as a functionof p . normalized by the total population and with initial conditions S n (0) = (1 − p ) N, S m (0) = pN, E n (0) = 0 , E m (0) = 0 ,I n (0) = 0 , I m (0) = 1 , R n (0) = 0 , R m (0) = 0 , (14)where N = 331 million, is displayed in Figure 4 for a time period of one year.The desired ”flattening of the curve“, i.e., postponing and lowering the max-imum number of cases, is achieved. A closer look at this fact is shown in Figure5, which shows that, in a period of one year, both the maximum and the timeit happens stabilize after p ≈ .
86. For these values of p , plots like the onesin Figure 4 would only reach their peak after one year, so we can say that thedisease is essentially controlled.
5. Conclusion
In this paper, a SEIR model is considered in a population that is divided intoindividuals that wear masks and individuals that do not. Parameters p and r ,which represent the (constant) percentage of the population that are mask usersand the reduction in the transmission rate due to one person wearing a mask in11
50 100 150 200 250 300 350
Time (in days) N u m be r o f c a s e s ( % o f t he popu l a t i on ) p = 0p = 0.2p = 0.4p = 0.6p = 0.8 Figure 4: Plots of the infected curves for different values of p in model (3) in the case of theUSA. p % o f t he popu l a t i on Maximum p T i m e i n da ys Time until maximum
Figure 5: Plots of the infected curves for different values of p in model (3). a susceptible-infected contact, respectively, are introduced, and their effect onthe basic reproductive number is calculated by a next generation method.This allows for the derivation of a necessary and sufficient condition forepidemic outbreaks to be prevented only by the widespread use of masks. When12his is possible, a critical percentage p ∗ of mask users in the population necessaryfor disease control is calculated.This is utterly important in dealing with public health crisis worldwide,since pharmaceutical measures such as vaccines and drugs are more laboriousand take long times to be developed while diseases spread.As case studies for the results in this paper, real world data from the Covid-19 pandemic was used, focusing on the United States, Brazil and Italy for thefirst 30 days after the total number of cases reached 100. After fitting theparameters, the results implied that the Covid-19 epidemic could have beenavoided in Brazil and Italy if at least 85.87% and 91.76% of the populations,respectively, wore masks from the beginning of the outbreak.Even though this was not possible in the case of the United States, we notedthat the basic reproductive number could have been reduced from 6.0039 to1.0699, so other control measures such as social distancing, quarantines, or evenimproving the average mask quality could help pushing this number below 1.Furthermore, numerical simulations showed that the flattening of the in-fected curve is achieved as p gets closer to 1, and that the maximum of thiscurve and the necessary time for it to happen stabilize after p ≈ .
86, i.e.,the disease is essentially controlled. Thus, simple measures such as wearingmasks can prove to be very effective in controlling, or even preventing, futureepidemics.
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