An epidemiological model with voluntary quarantine strategies governed by evolutionary game dynamics
AAn epidemiological model with voluntary quarantine strategies governed byevolutionary game dynamics
Marco A. Amaral , ∗ Marcelo M. de Oliveira , and Marco A. Javarone Instituto de Artes, Humanidades e Ciˆencias, Universidade Federal do Sul da Bahia, Teixeira de Freitas-BA, 45996-108 Brazil Departamento de F´ısica e Matem´atica, CAP, Universidade Federal de S˜ao Jo˜ao del Rei, Ouro Branco-MG, 36420-000 Brazil, Department of Mathematics, University College London, London, UK (Dated: August 14, 2020)During pandemic events, strategies such as quarantine and social distancing can be fundamentalto curb viral spreading. Such actions can reduce the number of simultaneous infection cases andmitigate the disease spreading, which is relevant to the risk of a healthcare system collapse. Althoughthese strategies can be suggested, or even imposed, their actual implementation may depend on thepopulation perception of the risks associated with a potential infection. The current COVID-19crisis, for instance, is showing that some individuals are much more prone than others to remainisolated, avoiding unnecessary contacts, and respecting other restrictions. With this motivation,we propose an epidemiological SIR model that uses evolutionary game theory to take into accountdynamic individual quarantine strategies, intending to combine in a single process social strategies,individual risk perception, and viral spreading. The disease spreads in a population whose agents canchoose between self-isolation and a lifestyle careless of any epidemic risk. The strategy adoption isindividual and, most importantly, depends on the perceived disease risk compared to the quarantinecost. The game payoff governs the strategy adoption, while the epidemic process governs the agent’shealth state. At the same time, the infection rate depends on the agent’s strategy while the perceiveddisease risk depends on the fraction of infected agents. Our results show recurrent infection wavesphenomena, which are usually seen in previous historic epidemic scenarios with voluntary quarantine.In particular, such waves re-occur as the population reduces disease awareness. Notably, the riskperception parameter is found to be fundamental for controlling the magnitude of the infection peak,while the final infection size is mainly dictated by the infection rates. Low awareness leads to asingle and strong infection peak, while a greater disease risk perception leads to shorter, althoughmore frequent, peaks. The proposed model spontaneously captures relevant aspects of a pandemicevent, highlighting the fundamental role of social strategies.
Keywords: Epidemic Spreading ; Game Theory ; SIR model ; Voluntary Quarantine
I. INTRODUCTION
During a pandemic, quarantine and other distancingrules can constitute the only option to curb the vi-ral spreading, in particular in absence of vaccines ormedicines to control the symptoms resulting from an in-fection [1–4]. Usually, these social rules are defined byepidemiologists and other experts, however their actualimplementation can be quite challenging. For instance,the current COVID-19 crisis [5–7] is showing how somepeople are more easily prone to self-isolate under quar-antine than others, even despite evidences on the po-tential risks. By doing so, individuals that avoid anyform of restriction become an element of risk for them-selves and for their community. In these scenarios, un-derstanding how to stimulate and sustain prosocial be-haviors has a paramount relevance. In this work, we aimto study the relationship between human behavior, rep-resented by individual quarantining strategies, and theepidemic spreading of a disease. We emphasize thatthis model is not an empirical description of the cur-rent COVID-19 evolution. Instead, this is a general the- ∗ Electronic address: [email protected] oretical framework that merges evolutionary game theory(EGT) [8] and epidemiology in a single compartmentalmodel. Such framework allows rational strategy changesbetween agents and can be used to better understand thecentral aspects regarding a generic epidemic event.Usually, the approach for studying a pandemic or epi-demic process is based on compartmental models [4, 9,10], which are a ubiquitous tool in epidemiology andmodern health management systems. The SIR model isone of the most known epidemiological models [4, 9, 11].It describes the spreading of a disease, which confers im-munity against re-infection, in agents that evolve fromthe susceptible compartment, S , to the infectious, I , andeventually to the recovered (or removed) compartment R . Although simple, it has been widely used to obtainrelevant aspects of epidemic processes that present the S → I → R structure. Since its introduction in the sem-inal paper by Kermack and McKendrick [12], the modelhas been extensively studied and expanded to considerdifferent hypotheses and conditions. For example, someepidemics may require more compartments, such as theexposed and/or asymptomatic agents (known as SEIRand SEAIR models respectively) [4, 13–15]. Spread oncomplex networks was also proven useful to understandthe heterogeneity of agent contacts [16–20]. The studyof control and mitigation strategies such as vaccination[21], modeling of vector-borne diseases [22, 23], and ef- a r X i v : . [ phy s i c s . s o c - ph ] A ug fects of birth-and-death dynamics [2, 9] are other exam-ples of the wide range of applications for compartmen-tal models in epidemiology. Even rumor and corruptionspreading have found a natural framework in the SIRmodel [24–30]. Nevertheless, most of those models re-late only to the disease evolution, i.e., agents usually haveno conscious actions regarding the disease.On the other hand, many control measures for infec-tious diseases depend on individual decision making. Inthis context, the recent field of behavioral epidemiology isattracting the attention of researchers from diverse areas,applying psychology, social engineering, and game theoryapproaches to epidemiology (see [2, 11] for a review). In-stead of considering agents having static roles, behavioralepidemiology includes dynamic behavior changes. Thisis a fertile ground for the recent area of social dynam-ics, or sociophysics [31–33], which utilizes tools fromstatistical physics together with evolutionary game the-ory (and others) to better understand the complex be-havior of humans [7, 34–39]. For example, in a novelapproach, Bauch [40–42] integrated a SIR model intoan EGT framework to analyze vaccination decision dy-namics. By doing so, agents change their vaccinationstrategy dynamically, depending on their perception ofthe benefits and costs of a vaccine. This was later gen-eralized into the so-called ‘vaccination games’ framework(see [21] for a comprehensive review). Such approachled to many interesting observations and predictions invaccination protocols [11, 42–53].Recent works also investigated other mitigation strate-gies such as awareness campaigns, information spread-ing, multi-layer contact networks and dynamic contacts[45, 46, 51, 54–64]. A general overview of these investi-gations shows the presence of a cycle, where effective im-munization measures lead to a low-risk perception, whichin turn weakens said mitigation strategies, bringing thedisease back [2]. The most recent anti-vaccination move-ment is just one of a long history of such cycles [65–67].Unfortunately, vaccination is not always an option, andsocial isolation can be the only practice to prevent fur-ther disease spread [1, 2, 68, 69]. Such was the case inthe famous episodes of the Spanish flu [70, 71], SARSepidemic of 2002–2003 [72, 73] and more recently, duringthe COVID-19 pandemic [3, 6, 74–77].In the present work, we propose a ‘quarantine game’,in which agents undergo a SIR epidemic process while,at the same time, they can choose between two socialstrategies, i.e. to self-quarantine and voluntarily stay athome (Q), or continue acting normally (N). The strat-egy is constantly updated based on the individual per-ceived cost of the quarantine versus the perceived dis-ease risk. While the scope of the model is intentionallygeneral, it is mainly motivated by the recent COVID-19global pandemic and its consequences, that have shown awide spectrum of human responses to the viral spreading.For instance, countries adopted many different restrictionpolicies, from mild distancing rules to strict lock-down.However, when not mandatory, only a small fraction of individuals may decide to self-isolate, while the rest of thecommunity avoids restrictions, endangering themselvesand others. The fast scale of this phenomenon has alsoshown how collective perceptions of the disease risk haschanged in a matter of weeks (based or not on real scien-tific data) [78]. This can be seen from how individualsand policymakers across the world have so far considereda variety of options, spanning from strict lock-downs todoing nothing, with the hope of reaching some kind ofherd immunity [79–81]. The variety of social strate-gies adopted worldwide, and in particular their resultsin terms of successes and failures, constitute a relevantevidence of how important is the behavioral componentof a given strategy during pandemic events.Lastly, we emphasise that this is a theoretical model,and in no way intends to fully grasp all the social andpolitical complexities exhibited by the current pandemicscenario. On the contrary, it aims to merge two elementsof paramount relevance in these scenarios, i.e. game the-ory and epidemic spreading, on a singular time scale. II. MODEL
In the proposed model, susceptible agents (S) becomeinfected (I) with a rate β i upon contact with another in-fected agent. Then, at a constant rate γ , infected agentsget recovered (R). Besides, agents can self-impose a quar-antine (Q) and stay at home, or keep acting as in a nor-mal situation (N). In the language of game theory, theformer strategy can be interpreted as a form of cooper-ation, while the latter as a selfish strategy, i.e. a formof defection. Therefore, we shall refer interchangeably toagents adopting quarantine as cooperators, and agentsacting normally, as defectors. The main effect of strate-gies is to influence the individual infection rate β i . Weassume that quarantined agents have a lower infectionrate than normal ones, that is β Q < β N , since thoseagents reduce their interactions with other members oftheir community. We expand the usual SIR model intoa five compartment model, S Q , S N , I Q I N , and R . Asrecovered agents cannot be infected again, their strategyis irrelevant. An illustrative diagram is shown in Figure1. By using a compartmental approach, the evolutionarygame dynamics is fully integrated into the model. Thisdiffers from usual behavioral epidemiology approacheswhere the strategy fraction evolves according to a sepa-rate dynamic [2, 21, 41, 52, 55–59]. Hence, our model al-lows cross interactions (such as S Q interacting with I N ),giving rise to a rich scenario where sub-population cor-relations can be observed.Employing the game theory concept of perceived pay-off ( π ), agents base their future strategy adoption on theperceived risk of their current strategy. A cooperator(i.e. an agent self-imposing quarantine) expects to suf-fer a perceived cost Ω. This represents the difficultiesone might face in a period of quarantine, but in turn, it FIG. 1: Schematic representation of the proposed model. Weconsider five compartments where agents transition from S , I ,and R states through epidemiological dynamics. At the sametime, agents change their own strategy ( Q or N ) throughan evolutionary game dynamics. The parameter β i is theinfection rate that, depending on the strategy of an agent,is defined as β Q or β N (i.e. quarantine versus normal lifestile). The parameter γ represents the recovery rate and isindependent of the specific strategy. Φ represents the strategychange flux for each epidemic state and it is governed by theevolutionary game dynamics. strongly reduces the probability of being infected. Thisleads to a constant payoff (or perceived risk) for cooper-ators, Q , i.e.: π Q = − Ω . (1)On the other hand, defectors, i.e. agents adopting thestrategy N , have a perceived risk based on their infec-tion probability multiplied by the perceived disease costparameter δ : π N = − δβ N I. (2)We remark that the payoffs are based exclusively onthe agent’s individual perceptions. This is in accordancewith the widespread notion of individual risk perceptionbased on the number of (anecdotal) cases an agent isexposed to [63, 82–84]. The game theory dynamics con-cerns what agents perceive to be their risks and rewards,and not necessarily the actual risk of a given action.Following the usual evolutionary game dynamics, theprobability of a given agent i to adopt the strategy ofagent j is related to their payoffs π i and π j . We use thetypical Fermi rule [8]:Θ( π i , π j ) = 11 + e − ( π j − π i ) /k . (3)This allows strategy revision with a small but non zerochance of mistakes. Such irrationality is measured by the k parameter, set as k = 0 . Q and N strategies, inside each health com-partment ( S or I ), and multiply it by the strategy tran-sition probability Θ( π i , π j ) between strategies i and/or j . This is equivalent to the master equation (for eachcompartment) of an evolutionary game dynamic [8, 86]using the mean-field approximation, and leads us to thestrategy conversion rates, defined as Φ S = S Q ( S N + I N )Θ( π Q , π N ) − S N ( S Q + I Q )Θ( π N , π Q ) (4) Φ I = I Q ( S N + I N )Θ( π Q , π N ) − I N ( S Q + I Q )Θ( π N , π Q ) . (5)Here, Φ S is the rate at which S Q agents convert to S N (and conversely for Φ I ), and it is governed by the EGTpart of the model.Regarding the infection dynamics, we assume three dif-ferent infection rates, that is, β N > β a > β Q . Here, β N is the infection rate for defectors interacting with defec-tors, and similarly, β Q is the infection rate for cooper-ators. Cooperators and defectors interact through thecross-infection rate β a . For the sake of simplicity we set β a = a ( β N + β Q ) /
2, an average value of β Q and β N weighted by the external control parameter 1 > a > a = 0 . S N = − S N ( β N I N + β a I Q ) + τ Φ S (6)˙ S Q = − S Q ( β a I N + β Q I Q ) − τ Φ S (7)˙ I N = S N ( β N I N + β a I Q ) − γI N + τ Φ I (8)˙ I Q = S Q ( β a I N + β Q I Q ) − γI Q − τ Φ I (9)˙ R = γ ( I N + I Q ) , (10)where τ is the coupling parameter that controls howquickly one adopts a new strategy, in relation to the time-scale of the epidemic. Note that the current version ofthe model does not include vital dynamics, such as birthand death processes, since the model focuses on spreaddynamics that take place in a matter of months. III. RESULTS
We start by noting that the payoff structure proposedin Eqs. (1) and (2), is akin to the public goods and cli-mate change dilemma games [87–90] where each agentpayoff depends on the total number of agents in someother state. That is, the quarantine game is not a pair-wise interaction game such as the prisoner dilemma [8].In particular, in our case, the defector payoff dependson the total number of infected agents ( I ), either coop-erators or defectors, while the cooperator payoff is con-stant. In doing so, we obtain the collective equivalentof the snow-drift game (also known as chicken or hawk-dove game [8]). I.e., as long as most of the populationis healthy (susceptible or recovered), the best strategyis to defect and to continue acting normally. But assoon as most of the population chooses this strategy, theamount of infected agents grows, resulting in a change ofthe best strategy, that becomes to self-quarantine. Thispayoff structure leads to a similar case as the generalanti-coordination game class, where the best strategy isto do the opposite of what your opponents are doing. Orspecifically in our case, the opposite of what the major-ity of the population is doing [8]. However, note thatthe fraction of infected agents is not equal to the fractionof defectors, due to the epidemiology dynamics. This issimilar to the dilemma presented in vaccination games[40, 41, 50, 52, 82] where agents should vaccinate but, aslong as the majority of the population is vaccinating, theincentive to not vaccinate grows. This anti-coordinationelement is a central driver for the observed oscillatorydynamics.The numerical integration of the equations is obtainedthrough a 4th order Runge–Kutta method. For the in-terested reader, a simplified Python script for solving theequations is available at [91]. Regarding the results, un-less stated otherwise, we set Ω = 1 , τ = 1 , γ = 1 , β Q =1 , k = 0 . , a = 0 . δ , and defectors infection rate β N . As initial condition, the starting setting for the pop-ulation has only a very small fraction of infected agents,i.e. I = 0 . , S = 1 − I ., while strategies are equallydivided between C and D .Figure 2 presents the typical behavior of the popula-tion. The most evident phenomenon is the recurrent in-fection waves, even though the model has no explicit os-cillatory terms. Looking at the evolution of the epidemio-logical population, i.e. S = ( S Q + S N ), I = ( I Q + I N ), and R , we notice that susceptible agents diminish on almostdiscrete steps. The successive drops in S also coincidewith the peaks of infected agents. The inclusion of vol-untary quarantine procedures in the SIR model sponta-neously generates recurrent infection periods. This phe-nomenon can be observed for a wide range of parametersand it is a characteristic behavior of the model. Notethat such an effect is similar to the expected scenario ofreal quarantine policies [2, 3, 74], that is, re-occurring in-fection seasons. Interestingly, previous pandemics as theSpanish flu (1918) presented such infection wave behav-ior [92, 93].The cause underlying the successive infection peakscan be understood looking at the sub-population( S Q , S N , I Q , I N , R ) and the strategy distributionsthrough time. This can be seen in Figure 3. Remark-ably, the population behavior hides a complex dynamic.In particular, as the fraction of infected agents initiallygrows, the cooperator’s payoff quickly becomes advanta-geous. This is what causes the first broad peak of S Q ,as most agents start to undergo quarantine. In turn,the total fraction of infected agents begin to decline, asthe majority of the population gets quarantined, witha low value of infection rate. Nevertheless, as I tends Time P opu l a ti on F r ac ti on SIR
FIG. 2: Typical behavior of the epidemiological population, S = ( S Q + S N ) , I = ( I Q + I N ) , R . Note that recurrent infectionpeaks emerge spontaneously. Here δ = 10 , β N = 10. to 0, the payoff for agents leaving quarantine (defectorstrategy) starts to grow and eventually it becomesgreater than the cooperator’s payoff. This triggers aflux of S Q → S N , that is, people leaving quarantine.Such an event corresponds to the sharp increase in S N ,near the beginning of the second infection wave. Withmore and more agents leaving quarantine, a secondpeak of infected agents inevitably occurs. Indeed, wesee that the infection peaks are always preceded by asharp increase in the defector density. At this point, S N begins to decrease sharply because part of thembecomes infected and the others (still susceptible) startbecoming cooperators (the second and broad peak in S Q ). This process repeats itself again and again, at eachtime with less active agents. An interesting effect alsooccurs in the sub-population of infected agents, i.e. theinfection peak on defectors always precedes the peakof cooperators. We note that the number and heightof the peaks, and recurrent infection cycles, are highlydependent on δ .Next, we analyze the mixed strategy equilibrium pointto obtain the strategy inflection points. This is a similarapproach as the one used in [42] for vaccination games.Suppose a mixed strategy where an agent has a proba-bility P to cooperate. This leads to the average expectedpayoff of ¯ π = P π Q + (1 − P ) π N . We want to maximizeit in relation to P , therefore:¯ π = P ( δβ N I − Ω) − δβ N I. (11)Since all parameters are greater than zero, we obtainthe maximum expected payoff value when P = 1 (alwayscooperate) if δβ N I >
Ω. Conversely, if δβ N I <
Ω, themaximum average payoff occurs for P = 0 (always de-fect). This implies that agents will start changing strate-gies at an infection level of: I (cid:48) = Ω δβ N (12) Time P opu l a ti on F r ac ti on S N S Q I N I Q FIG. 3: Typical behavior of the sub-population, S Q , S N , I Q , I N , R . The successive infection peaks aredue to the frequent oscillations in the strategies, even if thetotal susceptible and removed individuals do not oscillate.Here, δ = 10 , β N = 10. Time P opu l a ti on F r ac ti on s ID Ω/δ β N FIG. 4: Infected agents ( I ) and defectors strategy fraction ( D )time evolution. The horizontal line represents the value I (cid:48) =Ω /δβ N . The vertical dashed lines indicate when I ( t ) = I (cid:48) .As expected, these are the strategy maximum and minimumvalues. Here δ = 10 , β N = 5. In a system composed of fully rational agents, thestrategy maximum and minimum values will coincidewith the points mentioned above. Numerical anal-ysis of the ODE integration shows good agreementwith such prediction even if we use the Fermi strat-egy probability (an approach that has inherent fluctua-tions/irrationality). This can be seen in Figure 4. Wenote that the main effect of greater irrationality, i.e.larger values of k , is to make the strategy oscillationsmore smooth around the inflection points. This analysisremained accurate for all studied values of δ, Ω, and β N .Also, note that the peak of infections always happensbetween a maximum and minimum value of D , in a wayconsistent with all studied values of parameters.To better understand the effect of the disease risk per-ceptions on the infection peak size and duration, we vary the value of δ , as this is the central parameter we areinterested in. Figure 5 shows the population dynamicswhen δ = {
0; 5; 10 } . For low-risk perceptions, agentsleave quarantine earlier and in great numbers. This cre-ates a big single infection peak, which is consistent withthe current worst-case scenarios for a pandemic [2, 3, 74].As we increase the risk perception, agents will tend to co-operate (stay in quarantine) for longer periods, leadingto the distribution of smaller infection peaks along oneor more infection cycles. We highlight that this is anemergent behavior that spontaneously appears by con-sidering the evolutionary game dynamics. In a pandemicscenario, this can be one of the most important aspectsof a quarantine policy, since the healthcare system mayhave a small capacity, and cannot take care of all infectedagents at the same time [10].The effects of different disease perception values aresummarized in Figure 6. Note that when δ = 16 thereare even five different infection peaks, all with a verysmall magnitude. Another interesting effect to observe isthat the first infection peak is not always the highest. Forlarger values of δ , the highest peak can happen after someinitial (small) infection wave. Moreover, a higher riskperception better distributes the cases over long periods.We emphasize that the infection peak magnitude canbe a very important quantity when dealing with pan-demics [10]. In Figure 7 we present the maximum si-multaneous infection size ( I max ) as a function of the per-ceived disease risk for different defector infection rates, β N . We highlight that the equations of the proposedmodel can always be normalized in relation to β Q , defin-ing a new time scale. Because of this, without loss ofgenerality, we chose to vary only β N in the presented re-sults. We see that the disease awareness, δ , can greatlyhelp diminish the maximum simultaneous infected num-ber. On the other hand, the effect of β N in I max is lesspronounced.We now analyze the infection size, measured by thefinal density of removed agents, R ∗ , shown in Figure 8.We note that the increase in δ can lead, on average, toslightly smaller R ∗ values. The decrease is more pro-nounced when β N <
2. Differently from I max however,the behavior of R ∗ is not monotonous in δ , presentingnon-periodic oscillations.Next we present the parameter space β N × δ for thefinal density of removed agents, R ∗ in Figure 9. Asexpected, increased disease risk perceptions leads to asmaller final density of removed agents. Nevertheless, itis clear that this behavior is not trivial, and different in-fection rates result in large oscillations. It is interestingto note that the valleys and peaks follow, on average, aninverse proportion with δ . For instance, for a fixed valueof R ∗ , β N ∝ /δ . Note that the value of I max is highlydependent on δ but does not change considerably with β N .As τ is the coupling constant between the epidemicand evolutionary game dynamics, it is correlated withhow quickly a population is able to respond to new in- Time P opu l a ti on F r ac ti on SIR a) Time P opu l a ti on F r ac ti on SIR b) Time P opu l a ti on F r ac ti on SIR c) FIG. 5: Typical behavior for diverse disease risk perception. In a) there is no disease risk perception, δ = 0, and the diseasebehaves according to the usual SIR dynamics, with a big and singular infection peak. In b) δ = 5 and while there are twoinfection waves, their magnitude is considerably smaller. Finally, in c) δ = 10, and we can see three shallow infection peaks.Note that as δ increases, the infection are distributed during a longer time span. In general, an increase in risk perception leadsto smaller, and more distributed, infection peaks. Here β N = 5. Time I δ=16δ=14δ=8δ=4δ=0 FIG. 6: Evolution of infected agents for different diseaseperception values, δ . This parameter plays a key role in theinfection peak magnitude, making it shorter while distribut-ing the cases over many smaller infection waves. Here β N = 5. formation regarding the current disease situation. Figure10 reports the effects of different τ in the evolution of thestrategies. Notably, increasing its value causes strategychanges to become more frequent. This in turn entailsmore oscillations in the whole population. Every peakin the defector density also leads, eventually, to a peakin the density of infected agents, I . Variations in τ donot change the final infection size considerably. We alsonote that variations in the irrationality parameter, k , didnot drastically affect the dynamics for reasonable values(0 . < k < k is tomake the strategy adoption curves sharper around theinflection points. On the other hand, a high irrationalityparameter makes the strategy changes more smoothly intime.Finally, we generalize the results of the proposed modelaccording to the evolutionary game theory framework. Itis a known result that the strategy equilibrium of a clas-sical game is invariant in relation to the multiplicationand/or sum of a constant value over all payoffs [8]. There- δ I m a x β N =2 β N =5 β N =10 β N =15 FIG. 7: Maximum simultaneous infected agents density( I max ) as a function of the perceived disease risk δ . Themagnitude of the peak decreases for greater disease aware-ness values. δ R * β N =2 β N =5 β N =10 β N =15 FIG. 8: Infection size, measured as the final value of removedagents, R ∗ . The impact of δ in R ∗ is less pronounced thanin I max . For some values of β N , the fraction R ∗ presentsoscillations. R * "phasebeta_delta.dat" using 1:2:7
0 1 2 3 4 5 6 7 8 9 10 δ β N FIG. 9: Phase space β N × δ for the final epidemic size R ∗ . Theoscillations of R ∗ in relation to both parameters are presentfor all studied values. Note that the final epidemic size de-creases with δ mainly for low β N values. fore, we can simplify the proposed payoff structure, leav-ing intact the central characteristics of the game. Thisallows us to obtain relevant information regarding thegeneral game class. We first sum Ω in both payoffs andthen divide them by β N δ . Using (cid:15) = Ω /β N δ , we get thesimplified version: π Q = 0 . (13) π N = (cid:15) − I. (14)Note that (cid:15) is the ratio between the perceived cost ofquarantine and the cost of getting infected. By defini-tion, 0 < (cid:15) < < β N δ , i.e. thecost of performing a quarantining is smaller than thatof being infected. This general payoff structure correctlypredicts the most essential feature of our model, i.e. thebest strategy is to stay on quarantine if there are manyinfected agents ( I > (cid:15) ), and leave quarantine in the op-posite case. This is very similar to the anti-coordinationgame class, where the best strategy is to do the oppo-site of your opponent. Here, however, the main factorto consider is the number of infected agents, and not ofquarantined ones.If everyone is undergoing a quarantine, one has a bigincentive to avoid such strategy. On the other hand, ifeveryone is not taking quarantine precautions, one hasa big incentive to do so. This general payoff structureis similar to the free-ride scenario obtained in vaccina-tion games [52, 82] and other models with mitigationpolicies [46, 54–59]. The inflection point where defec-tion becomes more advantageous can be clearly statedas I (cid:48) = (cid:15) . Differently from a classic game, however, I = I ( t ), that is, the number of infected agents in ourmodel is time dependent and will depend on the numberof agents using the strategy Q or N . Note however that such payoff manipulation only makes the classic gameequilibrium invariant, not its evolutionary counterpart.For the population dynamics, the payoff multiplicationhas the equivalent effect of changing the value of k in thetransition probability, (3), i.e. k (cid:48) = kβ N δ .It is also possible to show that the model is differentfrom the SIR model with two distinct infection rates.Using the definition S = S Q + S N and I = I Q + I N wesee that:˙ S = − I Q ( β a S N + β Q S Q ) − I N ( β N S N + β a S Q ) (15) ˙ I = I Q ( β a S N + β Q S Q ) + I N ( β N S N + β a S Q ) − γI (16)˙ R = γI (17)Since the flux (Φ) terms regard only transitions be-tween the same epidemiological compartment, they van-ish when we look only at the total epidemiological level ofthe population. Even so, we see that the model does notreduce to the SIR model with two infection rates. In-deed we cannot totally disappear with the sub-populationterms.Furthermore, we can also consider the population atthe level of strategy adoption dynamics. C and D repre-sent the density of cooperators and defectors respectively.For the proposed model we have C = ( S Q + I Q ) / ( S + I ),and since we only have two strategies, D = 1 − C . Therate of change in the strategies comes only from the strat-egy flux terms Φ S and Φ I . In other words, ˙ C = − Φ S − Φ I .Using Equations (4) and (5), we obtain:˙ C = ( S N + I N )( S Q + I Q )Θ( π N , π Q ) − ( S Q + I Q )( S N + I N )Θ( π Q , π N )Re-arranging the terms and noting that S Q + I Q = C ( S + I ) , S N + I N = D ( S + I ), and that S + I = 1 − R ,we finally obtain:˙ C = (1 − R ) CD [Θ( π N , π Q ) − Θ( π Q , π N )] (18)The first term, (1 − R ) , modulates the speed of thestrategy change ( ˙ C ), as it is related to the total availablepopulation allowed to vary the strategies. Most impor-tant, however, is the rest of the equation, which is pre-cisely the usual mean-field form of the master equationfor the evolution of cooperation in a two strategy game,such as the prisoner’s dilemma [8]. We can observe thatthe proposed model is self-consistent and returns the evo-lutionary game when we only look at the strategy den-sities. At the same time, (numerically) the model alsoreturns the classic SIR dynamics with two infection rateswhen we make τ = β a = 0, i.e. when we turn off thestrategy dynamics and cross infection terms. IV. CONCLUSIONS
A common approach to analyze complex systems is toisolate its essential elements and features, trying to fil-ter out less relevant components. Such is the case of
Time P opu l a ti on F r ac ti on CD a) Time P opu l a ti on F r ac ti on CD b) FIG. 10: Strategy adoption evolution for different coupling constant values τ . In a) we present a value corresponding to halfthe time-scale of the epidemics, i.e. τ = 0 .
5. Figure b) presents a time-scale twice as fast, τ = 2. The peaks in the defectorfraction always correlates to peaks in the total infected population I . Greater τ values leads to more frequent oscillations inthe strategy distribution, and consequently more infection peaks with lower heights. Here we used δ = 10 , β N = 10. social behaviors and disease spreading, two intricate pro-cesses that, mainly for the sake of simplicity, are oftenanalyzed separately. In order to describe their dynamics,identifying their essential elements and interactions, it isfundamental to define a model able to capture, as muchas possible, the observed phenomena while maintainingits simplicity. Due to the relevance of the behavioralcomponent, in particular epidemic situations such as theCOVID-19 crisis, here we proposed a theoretical frame-work devised to combine social strategies with epidemicspreading. To this end, we present a simplified version ofthe epidemiological SIR model merged with an evolution-ary game that allows agents to rationally choose betweena voluntary quarantine or a normal lifestyle during thespreading of a generic disease. Following this approach,we obtain a single compartmental model that integratesinto the same time scale the rational decision making,from game theory, and the epidemiological dynamics ofthe SIR model. The latter has been chosen as a test case,however, the proposed model can also be realized consid-ering other variations, as the SIS and SEAIR models,as well as other game theory frameworks. The infectionand recovery rates are given by the epidemiological dy-namics, while the strategy changes are controlled by theso-called strategy update rules, widely studied in evolu-tionary game theory. Nevertheless, the infection ratesdepend on the chosen strategy, whereas the risk percep-tion and payoff of each strategy depend on the numberof infected individuals.We investigate the model through numerical and ana-lytical approaches. Remarkably, the model presents indi-vidual reactions to the disease infection level, which canresult in secondary infections and the re-emergence of thedisease spreading after most of the population dismiss itsrisk. In particular, our results revealed multiple infectionpeaks for higher disease risk perceptions, very similar tothe observed behavior of past epidemic cases with vol-untary quarantine measures. The interplay between the contagion and strategy dynamics exhibited a rich behav-ior. The main parameter that we studied in the modelis the perceived disease risk, δ , i.e. a measure of howstrongly the population sees the individual cost of beinginfected. We show that while this parameter has a smalleffect on the final infection size, it is most important con-cerning the infection peak size. Notably, the maximummagnitude of the infection peak is found to be inverselyproportional to the disease perceive risk δ .It is worth to emphasize that for for no perceived dis-ease risk, agents decide to avoid quarantine and the popu-lation quickly suffers from a widespread infection, result-ing in a single and huge peak of simultaneously infectedagents. As recent events related to the global COVID-19 pandemic have shown, the total infection peak is anobservable of paramount relevance. In particular, dur-ing these critical scenarios, healthcare systems may riskto collapse, due to the possibility that the amount ofinfected individuals saturates their total capacity [10].That is one of the reasons why not only the total epi-demic size is important, but also the maximum numberof simultaneous infections. In the proposed model, theinclusion of the perceived disease risk makes individu-als prone to quarantine for longer times, resulting in asmaller infection peak. As we increase the perceived risk,multiple smaller peaks emerge. This is a direct result ofthe interconnection between two complex processes, i.e.disease spreading by the SIR model, and rational strat-egy choices by the evolutionary game dynamics. We seethat for high values of δ , the disease can stay active forlonger times and present more infection waves. Neverthe-less, those peaks are shorter and the maximum number ofsimultaneous infections is highly dependent on δ , quicklydiminishing as the disease risk perception increases.We also perform a payoff analysis to find the optimummixed strategy for a given number of infected individu-als. This allows us to analytically obtain the inflectionpoint of the strategy adoption dynamics. This may beused to understand both the dependence of the most usedstrategy as a function of the infection number, and whenthe next infection wave can emerge again. Analyzingother parameters we find that the coupling constant τ is responsible for changing the speed of the populationresponse to new infections, i.e. how fast the strategyadoption occurs, but has no strong effect on the infec-tion peak size. In the same way, the irrationality param-eter k can change the properties of the strategy adoptiondynamics without changing its inflection points or theinfection peak size. Lastly, we show that the model isself-consistent and returns the usual replicator equationwhen looking only at the strategy fractions of the popu-lation dynamics. Likewise, when we turn off the interac-tions between the populations ( τ = β a = 0) we get backtwo separated SIR populations, evolving independently.Overall, the achieved results point to the importanceof the disease perceived risk in the spreading dynamicsand how such an ingredient can be included in more re-alistic modeling. The area of behavioral epidemiology is relatively recent, and evolutionary game theory andsociophysics seem to have much to add with their ap-proaches. As examples, we cite recent works that havehighlighted how evolutionary game dynamics can be usedtogether with an epidemiology-based approach to modelsocial contact behavior such as corruption and rumorspreading [24–26, 28, 94, 95]. In this sense, we believethat this model can be used as an initial framework to un-derstand more complex phenomena regarding behavioralepidemiology, especially the integration of game theoryin compartment models. Acknowledgments
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