communities. Therefore, looking for non-pharmaceutical control strategies is essential to deal with this epidemic and others in the future. This paper addresses this issue considering the widespread use of masks as in other works such as [6, 7, 8, 9] . A SEIR model [10] with individuals divided into those that wear masks and those that do not is considered in Section 2, and its basic reproductive number is calculated by a next generation approach in Section 3. This leads to a criterion that determines when an epidemic outbreak can be avoided by mask use only, and a critical percentage of mask users in the population is derived.
For applications of these methods, we perform the parameter fitting in Section 4 with data from the United States, Brazil and Italy and analyse whether the Covid-19 crisis could have been avoided in these countries with widespread mask use from the beginning of the outbreak. In addition, numerical simulations are carried out to verify how the evolution of the disease is mitigated if it cannot be avoided. The conclusions are drawn in Section 5.
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 the percentage of people wearing a mask in the population at time t ≥ 0, then
(1) Both N n and N m are also divided into four epidemiological classes, consisting of 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 will consider only a short time period in comparison to the demographic time frame, vital parameters will be neglected, so the total population will be assumed constant, that is,
Let r be a multiplicative factor for the transmission rate β that will take into account the reduction in the probability of contagion from one person wearing a mask in a susceptible-infected contact. We assume that this reduction is the same whether a susceptible or an infective is wearing the mask. When only one 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 is assumed to be r 2 β. There are four ways contagions can occur, and they are described in Table 1 . We further assume that p(t) = p is constant, so N m (t) and N n (t) are also constant. Hence, our model can be written as
The parameters σ and γ denote the exit rates from the exposed and infected classes, 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.
Now, the basic reproductive number, R 0 , of model (3) is calculated. This will be done by a next generation approach (see [11, 12] ). R 0 is given by the spectral radius of
Due to its block structure, the eigenvalues of K are exactly the eigenvalues of
It is clearly seen that the trace and the determinant of K 11 are
respectively, hence its eigenvalues are 0 and
where
which is the basic reproductive number for the standard SEIR model without vital dynamics (6) [13] .
Note that (3) reduces to (6) if either p = 0 (nobody wears masks) or r = 1 (the masks offer no protection against the disease).
Since both p, r 2 ∈ [0, 1], it is clear that R 0 > 0 and that R 0 ≤ R 0 , with the equality in the cases p = 0 or r = 1, when the model (3) reduces to model (6) .
The level set R 0 = 1 in (4) is displayed in Figure 1 for R 0 = 5, along with the regions of R 0 < 1 and R 0 > 1. We now look for conditions the pair (r, p) should satisfy in order for this point to lie in the region of R 0 < 1. Notice that, for fixed r ∈ [0, 1], R 0 is a decreasing function of p (see Figure 2 ). For p = 0, we have R 0 = R 0 , and for
It is assumed that R 0 > 1. Then, it is clear from Figure 2 that one can find values of p such that R 0 < 1 if and only if R 1 < 1, that is, if and only if
Moreover, there is a critical value p * such that R 0 (p * ) = 1, so R 0 (p) < 1 if and only if p > p * . Solving R 0 = 1 in (4), one sees that The value of p * corresponds to the critical percentage of the population that should wear masks in order to avoid the epidemic outbreak. In the extreme case of r = 0, i.e., the masks are ideal and avoid contamination for users, which is the same as immunizing the population, (9) becomes
which coincides with the usual threshold for herd immunity [14] .
In this Section, we collect data from the United States, Brazil and Italy to use as case studies for the results of the previous Section. The time frame in consideration consists of the first 30 days after the cumulative number of cases in each country reached 100, which happened in March 2, March 13 and February 23 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 5.1 and 7 days, respectively, as in [6] . We fit β by a minimization routine based on the least squares method, available in [11] , in the standard SEIR model without vital dynamics (6) . The routine minimizes the difference of the cumulative number of 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 initial values of susceptible individuals in each country. In the first days of each data set, 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 removed individuals will be taken, respectively, as 85 and 15 for the USA, 150 and 1 for Brazil and 152 and 5 for Italy.
For the initial values of exposed individuals, we use the fact that the latency period is taken as 5.1 days, so we look at the number in day 6 of each data set and choose the initial number of exposed as the extra number of cases since day 1. Hence, these numbers are 335 in the USA, 378 in Brazil and 732 in Italy. A summary of the initial conditions for each country in the minimization routine is shown in Table 5 . 
Using (5), we can calculate its value for each country. The results are displayed in Table 6 . According to [16] , when both individuals wear masks in a susceptible-infected contact, there is an average reduction of 82.18% in the transmission, so r 2 = 1 − 0.8218 = 0.1782, hence r = 0.4221. Then, the result of one person wearing a mask when two individuals meet is a decrease of around 58% in the transmission coefficient.
Rewriting (8), we see that the outbreak can be avoided by a widespread use of masks is
According to Table 6 , the Covid-19 crisis could have been avoided in Brazil and in Italy, and according to (9) , this would be possible if at least 85.87% and 91.76% of all individuals wore masks, respectively.
On the other hand, Table 6 indicates that Covid-19 could not have been avoided in the USA by widespread mask use only, but the basic reproductive number could be lowered from R 0 = 6.0039 to R 1 = 1.0699, so other a combination with control measures such as social distancing, quarantines and tracking infected cases could be able to prevent the disease outbreak. This could also be done by improving the average mask protection.
Moreover, one could also calculate the normalized forward sensitivity index (or elasticity) of R 0 with respect to p (see [11, 17] ), given by
This number provides the percentage change in R 0 for a given percentage change in p. For example, if Υ R0 p = −0.5, then a 10% increase in p produces a 5% decrease in R 0 . By (4) and (12), we have
A plot of Υ R0 p as a function of p is displayed in Figure 3 . It shows that R 0 becomes very sensitive to p for bigger values of this parameter, so even if most of a community has already become adept to wearing masks, small increases in p could still contribute greatly to epidemic control. 
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 maximum number of cases, is achieved. A closer look at this fact is shown in Figure   5 , which shows that, in a period of one year, both the maximum and the time it happens stabilize after p ≈ 0.86. For these values of p, plots like the ones in Figure 4 would only reach their peak after one year, so we can say that the disease is essentially controlled.
In this paper, a SEIR model is considered in a population that is divided into individuals that wear masks and individuals that do not. Parameters p and r, which represent the (constant) percentage of the population that are mask users and the reduction in the transmission rate due to one person wearing a mask in a susceptible-infected contact, respectively, are introduced, and their effect on the basic reproductive number is calculated by a next generation method.
This allows for the derivation of a necessary and sufficient condition for epidemic outbreaks to be prevented only by the widespread use of masks. When this is possible, a critical percentage p * of mask users in the population necessary for 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 laborious and take long times to be developed while diseases spread.
As case studies for the results in this paper, real world data from the Covid- Furthermore, numerical simulations showed that the flattening of the infected curve is achieved as p gets closer to 1, and that the maximum of this curve and the necessary time for it to happen stabilize after p ≈ 0.86, i.e., the disease is essentially controlled. Thus, simple measures such as wearing masks can prove to be very effective in controlling, or even preventing, future epidemics.