Prosocial behavior of wearing a mask during an epidemic: an evolutionary explanation

In the midst of the COVID-19 pandemic, with limited or no supplies of vaccines and treatments, people and policymakers seek easy to implement and cost-effective alternatives to combat the spread of infection during the pandemic. The practice of wearing a mask, which requires change in people’s usual behavior, may reduce disease transmission by preventing the virus spread from infectious to susceptible individuals. Wearing a mask may result in a public good game structure, where an individual does not want to wear a mask but desires that others wear it. This study develops and analyzes a new intervention game model that combines the mathematical models of epidemiology with evolutionary game theory. This approach quantifies how people use mask-wearing and related protecting behaviors that directly benefit the wearer and bring some advantage to other people during an epidemic. At each time-step, a suspected susceptible individual decides whether to wear a facemask, or not, due to a social learning process that accounts for the risk of infection and mask cost. Numerical results reveal a diverse and rich social dilemma structure that is hidden behind this mask-wearing dilemma. Our results highlight the sociological dimension of mask-wearing policy.


Results
First, let us observe simple epidemic dynamics with a fixed 10%, 30%, 50%, 70%, and 90% of susceptible individuals wearing masks. In practice, we assume that the mask transmission rate is x M = 0 and the ratio of asymptomatic to symptomatic diffusing risk without wearing a mask is q = 1.0 (Figs. 1, 2(a)) and q = 0.1( Fig. 2(a)) (where, 0 ≤ q M ≤ q) . The numerical results of infected individuals at equilibrium with different mask efficiency ( η = 0.3 and η = 0.7 ) along mask benefit to others, q M /q are shown in Fig. 1. The ratio q M /q indicates the benefit of mask-wearing to neighboring people around a masked person. As a general tendency, when masks are used less and efficiency is lower, the total percentage of population that will be infected is very high. The use of higher efficiency masks significantly reduces the fraction of infected individuals when 90% of people wear masks. Figure 2(a) shows that the fraction of infected individuals is lower if the mask benefits only the wearer without any benefit to others, i.e., lower q M /q and higher efficiency. Figure 2(b) shows that the fraction of infected individuals considerably decreases when q = 0.1 , i.e., the ratio of asymptomatic to symptomatic diffusing risk is low. Figure 3 shows the fraction of total infected individuals over time where (A) shows varying mask costs (C = 0.1, 0.5, 1.0) , (B) shows changing mask efficiency (η = 0.1, 0.5, 1.0) , and (C) shows varying ratio of mask benefit to others (q M /q = 0.1, 0.5, 1.0) . In each panel, a solid green line is used for the same settings: C = 0.5 , η = 0.5 , and q M /q = 0.5 , whereas the other two colored lines (blue and red) vary depending on focal parameters. Panel A shows that compared with mask cost, the infected fraction can be considerably reduced if a mask is provided at low prices simply because it encourages an individual to use a mask to avoid infection. Meanwhile, panel B shows that even the low efficiency case (η = 0.1) can reduce and delay the peak compared with the case of not wearing a mask (black line). Now, let us observe the sensitivity of q M /q; the rate of mask benefit to others (not-wearer) is shown in panel C. We confirmed that mask-wearing benefits others by lowering q M /q ; a more significant peak reduction for all infected individuals can be achieved, which results in a larger time delay. Thus, www.nature.com/scientificreports/ besides cost, it is worth highlighting how wearing a mask is beneficial to others and directly benefits a wearer by controlling an outbreak. From a holistic perspective, Fig. 4 shows final epidemic size (FES) (Panel A), susceptible mask-wearer, S M (Panel B), average social payoff, (ASP) (Panel C), and SED (Panel D) in the 2D heat map of mask efficiency, η ; versus the ratio of mask benefit to others, q M /q (throughout this study, we presumed q = 1.0 ); for a different combination of mask cost, C; and hesitance for mask cost, δ . With an increase in the values of η , the FES is reduced because the number of mask-wearing individuals considerably increases, as expected. People are most likely to wear a face mask when the efficiency (protection capability) is high. Furthermore, with a decrease in mask wearer benefit to others, q M /q , FES is decreased ( S M increased), which indicates that mask-wearers reduce the spread The arrows that connect the compartment represent the movement of the fraction of individuals from one state to another. Non-mask wearing suspected susceptible individuals (S) can either become exposed (no-mask) (E) or mask-wearing susceptible (S M ) . Both mask-and non-mask wearing susceptible individuals (S and S M ) become exposed (E and E M ) , asymptomatic infected I A and I A M , symptomatic infected I S and I S M , and recovered (R and R M ) . (B) The mechanism of mask efficiency (η) may be apparent when the effectiveness of the mask is higher (for example, N95 and KN95 mask) that protect the wearer from viruses. (C) While an individual is asymptomatic infected and wearing a mask, the neighbor's non-mask wearer may benefit from the mask wearer. (D) Apart from the disease dynamic, the evolutionary decision-making process, an individual chooses whether to wear a mask or not, somehow depends on the relative cost of the mask and perceived risk.  Fig. 4 (D-*). If an ASP observed at NE is less than that at SO, a certain amount of social dilemma occurs (displayed by gray to black; not whiteout). Therefore, we visually show either an increase in the ratio of mask-wearing benefit to others, q M /q , or a decrease in the mask efficiency, η , under higher mask cost and higher hesitancy, which makes an individual suffer from a higher social dilemma of whether to wear a mask as a protecting provision. SED generalizes dilemma strength by measuring the potential for bettering society. If the SED is vast (deep black), there is more room for improvement. In addition, when SED is small, there is little room for improvement through cooperation, which indicates a lower incentive for an individual to wear a mask. The whiteout region shows that no dilemma ensues; society reached its social optimal situation. Figure 5 (D-*) shows that the mask efficiency (which represents the benefit to a wearer and ranges from 0 to 1) works more effectively than q M /q (which represents the benefit to others around a wearer and ranges from 0 to 1), which simplifies this social dilemma structure. This is discussed in detail below.
The heat map results [ Fig. 3(D)] confirm that SED , which quantifies the social dilemma region, will be reflected by the epidemic dynamics with EGT in the current model. Now, we evaluate how the mask efficiency and benefit behave with the different parameter perspectives on behavioral dynamics in terms of social disability. These phenomena can be explained using the line graphs in Figs The most important feature of the current model is that the benefit to the mask wearer and surrounding people can be separately formulated. The former is controlled by η (larger is more beneficial), whereas the latter is controlled by q M q (smaller is more beneficial). Thus, Fig. 4 (i.e., a set of heat maps of 2D plane of q M q and η ) In the higher η [> 0.83 (approx.)] region, owing to the absence of a gap in ASP at NE and SO, there is no social dilemma (SED = 0). It is realized by M NE = 1, which successfully suppresses FES and is quite low. The value is the same irrespective of either higher or lower q c . Referring to the knowledge of EGT 57-61 , such a situation can be described as the "cooperation (C) dominant Trivial" game.
In the middle range of η [0.49 < η < 0.74(appr.)], another non-dilemma phase appears that is confirmed by zero SED. Unlike previous situations, it can be realized by a slightly lower M NE (< 1), which results in a reasonably larger FES because of mask efficiency, which indexes the benefit of wearing a mask, is not as high as the C dominant Trivial phase. Such a relatively lower mask efficiency and nonzero cost, which occurs SO, is consistent with what can be observed at NE because allowing disease spreading to some extent owing to mask's incapability allows to realize the optimal social situation. This specific situation is called "Polymorphic Trivial" because it results in M NE < 1 (mask-wearing and non-mask wearing groups coexist); however, NE is still consistent with the optimal social situation.
Of note, between C dominant Trivial and Polymorphic Trivial, nonzero SED, i.e., a social dilemma emerges. Nevertheless, its extent is subtle because of the transitional phase between both Trivial game structures.
At lower η [ η < 0.49 (appr.)], another social dilemma differs from the above mentioned one and for which the gap of ASP is larger. Nonzero SED implies that an evolutionary equilibrium can be improved to ASP x social opt . Unlike the conventional story that is commonly observed in real social dilemma structures, what can be observed in this particular phase is quite ironic and substantially interesting in terms of EGT. In the focal region, the fact of ASP x social opt > ASP NE resulting from FES SO (= 1) > FES NE owing to the quite large gap of M SO (= 0) < M NE implies that none of the people wearing mask and allowing a full-scale epidemic is SO because mask efficiency is quite low, and the cost is nonzero. Yet, the evolutionary process backed by the behavioral dynamics that we assumed makes many people willing to wear a mask even if almost nothing minimizes the social cost, which implies that the risk of infection is overestimated by an individual, which results from the balance of I S + I S M in Eq. (12) with other remaining terms in the brackets. This dilemma extent increases when presuming higher mask cost; C = 0.8 , shown in panel (B-*). Again, by referring to EGT, let us call this social dilemma structure, "Anti-Chicken, " where NE suggests coexisting mask-wearing and non-wearing individuals. However, SO can appear at a non-wearing state, unlike the usual (pure-) Chicken game with the coexistence of cooperative and defective strategies and optimal social situation appearing at all-cooperators-state.
Let us evaluate Figs. 6 and 7. In Fig. 6 (A-*), owing to high mask efficiency and low mask cost, a dilemma-free situation is realized, except the lower region of q M q . The lower region of q M q has a slightly small SED, which originates from FES SO that is slightly larger than FES NE , which results from a small difference between M SO and According to the abovementioned terminology of "Anti-Chicken, " this specific social dilemma should be called "Anti-Prisoner's Dilemma (PD). " The occurrence of such social irony is related to how wearing a mask is beneficial to other people than to a wearer, which is quantified by q M q . If the case is highly beneficial to others (lower q M q ), wearing mask is fully justified from the SO standpoint ( M SO = 1 ), which is fairly followed www.nature.com/scientificreports/ by the evolutionary process ( M NE = 1 ). In contrast, beyond the threshold, although the SO suggests abolishing the mask, the evolutionary process is still absorbed with an exceptionally high mask-wearing fraction. In the wake of COVID-19, one of the overwhelming reactions observed in the USA 62,63 is the social controversy of whether one should wear a mask and to obey the request from the public health authority cooperatively. One of the opinions supporting "not" is that whether wearing a mask or not is the subject of an individual's liberties. Interestingly, what has been observed in Japan is entirely opposite. Such a gap between specific two countries has been explained by social compliance among people. If there are people who underestimate (or overestimate) the risk of COVID-19, question the benefits of wearing a mask to himself and others around him, and instead www.nature.com/scientificreports/ address "freedom" than "social conformity, " which is fully modeled in the brackets of Eq. (12), they may behave relatively close to what the optimal social situation shows than what the NE shows. The phase change observed at M SO from 0 (1) to 1 (0), as confirmed in Fig. 6 (B-ii), appears even at lower mask efficiency and lower cost [ Fig. 7 (A-ii)].
In Fig. 8 is a line graph for mask-wearing individuals at equilibrium, M NE , along with the conformity rate, q c , by varying mask efficiency and cost. If the mask cost is low, C = 0.1 (blue and orange), the mask-wearer fraction is relatively high M NE → 1 , irrespective of the conformity effect q c , because the price does not convey any burden to an individual. At intermediate settings for cost and efficiency at (0.5, 0.5) (green), higher M NE and lower sensitivity to the conformity effect are still observed. However, people are not interested in wearing masks owing to higher costs (C = 1.0) , which reduces the fraction of mask-wearers (purple and red). The sensitivity from the weight factor owing to conformity pressure, q c , at C = 1.0 , shows that the entire mask-wearing fraction is monotonically decreasing with q c , which is also conceivable. By deliberately observing all lines, we noted an interesting propensity that when a cheaper mask, C = 0.1 and C = 0.5 , is available, with an increase in q c , first, M NE increases and reaches its peak and then decreases even though it is not significant. Thus, social conformity pressure works both ways. This is called the bandwagon effect, which compels people to wear a mask due to it being in vogue as opposed to refusing a mask due to prevailing attitudes.

Discussion
In the COVID-19 pandemic, there is an ongoing debate on whether to endorse wearing face masks to reduce the spread of infection 64 . However, most health experts recommend mask use, which addresses the general preventive principle when the baseline risk is very high, and none of the established medical treatments, such as a vaccine and antiviral drugs, are available. An important aspect is that mask-wearing provides not only a certain extent of benefit to a wearer but also to others around him. Nevertheless, mask-wearing can feel cumbersome, unpleasant, and costly, which may be more important. Therefore, mask-wearing poses a similar structure of the "vaccination dilemma. " In each scenario, individuals benefit form actions (mask wearing and vaccination, respectively) that yield benefits to the group, but which are personally costly. Based on modeling, the effect of mask-wearing should be divided into the benefit to a wearer and to others in an explicit formulation. The model reported here deals with this point, where epidemic dynamics based on the SEIR process and behavior dynamics are deliberately quantified. Numerical results successfully highlight the focal point, as mentioned above. The benefit to a wearer dominates the benefit to other people, which emerges as a social dilemma. The structure can be quite diverse and rich and contain a variant of PD, Chicken-type dilemma, and anti-Chicken dilemma in addition to the Trivial game structure.
In conclusion, this study contains the possibility of comparative and absolute advantages of wearing a mask originating from mask-wearers. The abovementioned results show the feasibility of analyzing united multifaceted epidemics and EGT of the pandemic situation. We expect that such an outline should influence policymakers' endorsements, starting with the relevant stakeholders' involvement with further progress.

Methods
To appropriately show the disease process (Fig. 1A) observed during the COVID-19 pandemic, there are two important factor to consider: i) implementation of the exposed period in which an infected individual is not infectious and ii) infectious but has a mild symptom state (called asymptomatic infected), which should be distinguished from the usual infected stage (called symptomatic infected). Hence, we considered a population consisting of susceptible (S) , exposed (E) , asymptomatic infected (I A ) , symptomatic infected (I S ) , and recovered (R) who did not wear a mask. Within mask-wearing people, we further divided the population into mask-wearing groups, i.e., susceptible (S M ) , exposed (E M ) , asymptomatic infected (I A M ) , symptomatic infected (I S M ) , and recovered (R M ) . The system of differential equations governing this mask-wearing epidemiological model is as follows: www.nature.com/scientificreports/ where x M is the wearing mask rate at which the fraction of S individuals convert to S M , and it is governed by the behavioral dynamics in EGT. The behavioral components x M will be increased or decreased depend on expected payoff differences that allow only the fraction of suspected susceptible individuals can choose either wearing a mask or not. An infected individual, however, is not permitted to change behavior from non-masked to masked wearing person. Herein, only one connection between susceptible (non-masked) to mask-wearing susceptible is considered to avoid unnecessary complexity 18 . Regarding the epidemic dynamics, β,α , τ , and γ are disease transmission, incubation, asymptomatic to symptomatic infected, and recovery rates, respectively (see Table 1). Parameter η is the mask efficiency, which is defined by the concept of the "efficiency model" 56 , which directly shows the mask's filtering capability, i.e., the mask's ability to protect the wearer from infectious particles. For example, an N95 mask is intended to block 95% (η = 0.95) of tiny 0.3-µm particles. The parameter q represents the rate of spreading the virus from an asymptomatic individual who is not wearing a mask compared with a symptomatic one. In contrast, q M represents the case when an asymptomatic individual wears a mask. Let us call both q and q M be the "rate of asymptomatic to symptomatic diffusing risk" when not-wearing and wearing a mask, respectively. We assumed 1 > q > q M . In a nutshell, q M /q quantifies the benefit of mask-wearing to surrounding people around a focal individual who wears a mask, whereas η indicates the direct benefit of mask-wearing to the individual ( Fig. 1B and c). An important point to be confirmed in our assumption is that an individual in the state of I A M does wear a mask; for the state of symptomatic individuals ( I S M ), wearing a mask or not does not reduce the risk to others.
We impose the following constraint: Behavioral dynamics. Applying the human behavioral dynamics to the EGT concept, individuals change their strategy adoption owing to the perceived risk of infection, cost, (Fig. 1D) and conformity effect. A cooperator (self-consuming mask wearer) expects to suffer a perceived cost, C M , with the sensitivity/hesitance of mask cost δ . A defector (non-wearing individuals) has a perceived risk based on the fraction of total symptomatic infected individuals (I S + I S M ) multiplied by the perceived disease cost C I . In the following, we normalized C M with C presuming C I = 1.0 . Thus, the payoff gain depends on the difference between the perceived payoff of a mask wearer [−δ · C] and the payoff for risking infection −(I S + I S M ) × 1.0 . To obtain the conformity effect 42 (i.e., social pressure amid wearing a mask or not wearing it), we assume (M * −WM * ) M * +WM * = (M * − WM * ) . Here M * and WM * represent the total number of individuals who wear and do not wear a mask, respectively. Parameter q c is the weight factor owing to conformity pressure and m is the proportionality constant. The expected payoff gains for altering strategies can be measured as (I S + I S M ) − δC + q c (M * − WM * ) , which can be expressed (imitation dynamics) for the time evolution of x M as follows: where, www.nature.com/scientificreports/ Here, Eq. (12) represents the behavioral dynamics in which the "third brackets" give the internal equilibrium for x M other than two trivial equilibria at x M = 0 and 1 5,10,26,41,55 . The general framework of the individual behavior, which allows an individual to change decision depends mainly on the "third bracket"; either increasing x M (positive) or decreasing (negative). The first term in the "third brackets, " the total number of visible infected people works to drive individuals compliant to wear a mask. Let us keep the first term as a reference; the mask cost in the second term acts to let them hesitate to take the mask. Yet, the sensitivity of the second term to the first term should be noted because the infected fraction has a different physical dimension from that of cost. Thus, we need to introduce another parameter, δ , which implies the sensitivity of mask cost to the influence of infected fraction. Finally, the last term implies the conformity effect by referring to (M * − WM * ) , in which the parameter, q c , accounts for the weight factor to conformity pressure. ASP and SED. We evaluated the holistic social efficiency by considering both disease and mask-wearing costs. FES is defined as the sum of recovered individuals [ FES = R M (∞) + R(∞) ]. We estimated the ASP at NE (social equilibrium)for all possible values of mask cost, C , and individual fraction at equilibrium (t → ∞) for S , S M , R M , and R . In addition, we evaluated ASP at SO without a game aspect by considering the maximum ASP for each cost, C , which ranges depending on the wearing mask rate, x M , from 0 to 1. Finally, the SED (social efficiency deficit) [43][44][45] is defined as the difference between ASPs at SO and NE. To unveil the existence of a social dilemma associated with evolutionary game systems, "social efficiency deficit" has been introduced to quantify the payoff difference between social optimum (SO) (the desired state of affairs) and Nash equilibrium (NE). So that one can evaluate the SED in any context and hence predict a social dilemma. If SED = 0 implies no social dilemma, while any social dilemma causes a positive SED.
We can define ASP (NE) as, Then, we can define SED as follows,