Game-theoretic modeling of collective decision-making during epidemics
MModelling epidemic dynamics under collective decisionmaking
Mengbin Ye ∗ , Lorenzo Zino , ∗ Alessandro Rizzo , , † & Ming Cao , † Optus–Curtin Centre of Excellence in Artificial Intelligence, Curtin University, Perth, Australia Faculty of Science and Engineering, University of Groningen, Groningen, Netherlands Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, Torino, Italy Office of Innovation, New York University Tandon School of Engineering, Brooklyn NY, USA
During the course of an epidemic, individuals constantly make decisions on how to fightagainst epidemic spreading . Collectively, these individual decisions are critical to theglobal outcome of the epidemic, especially when no pharmaceutical interventions are available .However, existing epidemic models lack the ability to capture this complex decision-makingprocess, which is shaped by an interplay of factors including government-mandated pol-icy interventions, expected socio-economic costs, perceived infection risks and social influ-ences. Here, we introduce a novel parsimonious model, grounded in evolutionary game the-ory, able to capture decision-making dynamics over heterogeneous time scales . Using realdata, we analyse three case studies in the spreading of gonorrhoea, the 1918–19 Spanishflu and COVID-19. Behavioural factors shaping the course of the epidemic are intelligi-bly mapped onto a minimal set of model parameters, and their interplay gives rise to char-acteristic phenomena, such as sustained periodic outbreaks, multiple epidemic waves, or a ∗ These authors contributed equally. † These authors contributed equally. a r X i v : . [ q - b i o . P E ] A ug uccessful eradication of the disease. Our model enables a direct assessment of the epidemi-ological and socio-economic impact of different policy interventions implemented to combatepidemic outbreaks. Besides the common-sense finding that stringent non-pharmaceuticalinterventions are essential to taming the initial phases of the outbreak, the duration of suchinterventions and the way they are phased out are key for an eradication in the medium-to-long term. Surprisingly, our findings reveal that social influence is a double-edged sword inthe control of epidemics, helping strengthen collective adoption of self-protective behavioursduring the early stages of the epidemic, but then accelerating their rejection upon lifting ofnon-pharmaceutical containment interventions.1 Introduction In the absence of pharmaceutical interventions, slowing or eradicating an epidemic in a populationcrucially depends on the actions of a sufficiently large number of individuals to adopt appropriatebehavioural responses such as physical distancing or wearing face masks. However, such humanbehaviours are typically not considered in classical mathematical epidemic models , includingthose for the ongoing COVID-19 pandemic . To bridge this gap, recent efforts have led to indi-vidual behavioural responses being incorporated into mathematical epidemic models . However,even these recent works suffer from a simplistic focus on the concept of individual awareness:while it is true that the awareness of an outbreak drives individuals to adopt self-protective be-haviours to reduce the probability of becoming infected
14, 15 , recent lessons on the complexity ofhuman behaviours from the COVID-19 outbreak have exposed inherent limitations of awareness-2ased modelling mechanisms . In particular, such mechanisms are often purely instantaneous andreactive, thus failing to capture the very factors that affect time-varying behavioural responses overthe whole duration of an epidemic, such as social influence and perceived socio-economic costs.An additional salient aspect highlighted by COVID-19 is the impact of government-mandatedpolicy interventions on individuals’ decision-making processes
7, 8, 16 . Finally, high levels of uncer-tainty cause bounded rationality to become prominent in an individual’s decision making, which isseldom included in current behavioural-epidemic models .Here, we move the focus from the instantaneous impact of reactive and fully rational be-havioural responses at the onset of an epidemic (as in awareness-based models) to a long-termoutlook where complex behavioural dynamics arise for each individual. To this end, our model isbased on an evolutionary game approach that is able to capture bounded rational decision-makingprocesses that evolve over the long term . Our approach enables the explicit inclusion of themost salient factors that each individual balances and trades off when deciding their behaviouralresponse to an epidemic outbreak, such as social influence, perceived infection risks, policy inter-ventions by authorities and the accumulation of socio-economic costs . This allows for a betterassessment of the effectiveness of different policy interventions from a healthcare point of view,while also evaluating their socio-economic impacts
19, 20 .We demonstrate our paradigm by incorporating it into two exemplary epidemic progressionmodels and applying them to three real-world case studies, including an instance of periodic out-breaks of gonorrhoea and the 1918 Spanish flu pandemic, for which it consistently reproduces the3istorical occurrence of consecutive waves of infections. We also shed light on several “what-if” scenarios of relevance to the current COVID-19 outbreak, covering the whole course of theepidemic evolution. Notably, we show that in the early stages, weak interventions can lead toa general failure to adopt self-protective behaviours, resulting in multiple overlapping infectionwaves. On the other hand, we demonstrate that in the advanced phases of the epidemic evolu-tion, the time over which a phased reduction of a lockdown occurs, rather than the severity ofthe initial lockdown itself, determines whether a second infectious wave appears, yielding severehealth and socio-economic consequences. Importantly, we identify that social influence acts as adouble-edged sword, providing benefits during the early stages of a lockdown, but accelerating thecollective rejection of self-protective behaviours during its phased reduction.
We consider a population V = { , . . . , n } of n individuals. Each individual i ∈ V is charac-terised by a two-dimensional variable ( x i ( t ) , y i ( t )) , which models their health state and the socialbehaviour adopted at the discrete time t ∈ Z ≥ , respectively.The variable x i ( t ) ∈ A takes values in a discrete set of health states A . For the sake ofsimplicity, we start by introducing our paradigm in combination with the well-known susceptible–infected–susceptible (SIS) epidemic model , which is characterised by two health states: S repre-sents susceptible individuals, who are healthy and can be infected by the disease upon interactingwith infected individuals, represented by the state I . A global observable z ( t ) measures the de- t+1 (a) IS λµ S E I R λ ν µ (b)
Figure 1: Illustration of the network model and the epidemic progression. In (a), two time-stepsin the two-layer network representation. The upper layer (green) shows influences, the lower layer(violet) contacts. In (b), schematics of the state transitions of the SIS model (above) and of theSEIR model (below). The constants λ , µ and ν indicate transition probabilities. tectable prevalence of the epidemic disease at time t : z ( t ) = n |{ i : x i ( t ) = I }| , where | · | denotesa set’s cardinality. Further health states may be added to the set A to capture specific features ofthe disease being studied .The social behaviour of individual i is captured by the binary variable y i ( t ) ∈ { , } thatexpresses whether the individual adopts self-protective behaviours ( y i ( t ) = 1 ) or continues asnormal ( y i ( t ) = 0 ). The paradigm is amenable to extensions to capture different levels of self-protection through modification of the support set of y i ( t ) .The spread of the disease and the individuals’ decision-making processes co-evolve, mutu-ally influencing one another on a two-layered network structure G = ( V , E I , E C ( t )) , illustrated inFig. 1a. The set of undirected links E I defines the static influence layer , capturing social influence between individuals during the decision-making process. The contact layer is defined through a5ime-varying set of undirected links E C ( t ) , which represent the physical contacts between pairs ofindividuals that may result in the transmission of the disease. We model the contact layer usingan activity-driven network (ADN) , which captures important features of complexity that charac-terise real-world networks. In ADNs, a constant parameter a i ∈ [0 , , called activity , expressesindividual i ’s propensity to generate m ≥ interactions with other individuals at each discretetime instant. Details are provided in the Methods. Epidemic Spread
At each discrete time-step t , every individual i that is susceptible and does notadopt self-protections (i.e. x i ( t ) = S and y i ( t ) = 0 ) may become infected. A constant parameter λ ∈ [0 , , termed infection probability , captures the probability that the disease is transmitted byan infectious individual ( x j ( t ) = I ) to a susceptible one through physical contact. We assume thatan individual adopting self-protection, y i ( t ) = 1 , is always successful in preventing contagion.Beside the contagion, at each time-step t , every infected individual i recovers with probability µ ∈ (0 , , becoming susceptible again to the disease. These transitions are illustrated in Fig. 1b,while technical details are provided in the Methods. Human Behaviour
Concurrently with the epidemic evolution, at each time-step t , each individualenacts a decision-making process on the adoption of self-protective behaviours, according to anevolutionary game-based mechanism termed logit learning . The resulting behaviour is updatedin a probabilistic fashion. We define two payoff functions π i (0) and π i (1) , which represent acombination of socio-psychological, economic and personal benefits received by individual i forenacting behaviours y i = 0 and y i = 1 , respectively. This individual then adopts self-protective6ehaviours with a probability equal to P [ y i ( t + 1) = 1] = exp { βπ i (1) } exp { βπ i (0) } + exp { βπ i (1) } . (1)The parameter β ∈ [0 , ∞ ) measures an individual’s rationality in the decision-making process;the larger the parameter β , the higher the probability that the individual chooses the behaviour thatmaximises their payoff. Payoffs are defined as π i (0) := 1 d i (cid:88) j :( i,j ) ∈E I (1 − y j ( t )) − u ( t ) , (2a) π i (1) := 1 d i (cid:88) j :( i,j ) ∈E I y j ( t ) + r ( z ( t )) − f i ( t ) , (2b)and contain the following four terms, directly related to behavioural and social factors that shapethe epidemic dynamics (more details can be found in the Methods). • Social influence.
The first summation terms in Eqs. (2a) and (2b) are inherited from co-ordination games on networks . These terms capture the social influence of neighbouringindividuals and the individual’s desire to coordinate with them, as it provides an increasedpayoff for adopting the same behaviour as that adopted by the majority of the neighbours. • Policy interventions.
The time-varying term u ( t ) ≥ in Eq. (2a) captures a broad rangeof non-pharmaceutical interventions enforced by public authorities to discourage dangerousbehaviours, e.g. by means of a fine or imprisonment or by providing benefits for workingfrom home. • Risk perception.
The risk-perception function r ( z ) : [0 , → R ≥ in Eq. (2b) is a monoton-ically non-decreasing function of the detectable prevalence z , which captures an incentive to7dopt self-protective behaviour due to the endogenous fear of becoming infected as the dis-ease spreads. Individuals may learn of z from the media coverage of the epidemic evolution.For simplicity, here we assume that all individuals share the same risk-perception function. • Cost of self-protective behaviour.
The negative impact of adopting self-protective be-haviours is captured in Eq. (2b) by the frustration function f i ( t ) = c + t (cid:88) s =1 γ s cy i ( t − s ) , (3)where c ≥ is the social, psychological and economic immediate cost per unit-time associ-ated with the adoption of self-protective behaviours, e.g. related to the inability to socialise,work from the office, enjoy public spaces, etc., and γ ∈ [0 , is its accumulation factor .Thus, f i ( t ) ≥ reflects accumulative costs for individual i up to time t . Impact of an Epidemic Outbreak
A major strength of the proposed paradigm lies in its ability tofacilitate a thorough evaluation of the immediate health and socio-economic impact of an epidemicoutbreak due to the population’s behavioural responses to prescribed policy interventions, a taskthat has been considered extremely challenging so far
18, 20 . In fact, our formalism enables thedirect computation of the health cost , which is directly related to the health state of individuals (i.e. x i ( t ) ), and the socio-economic cost , which depends on the behaviours adopted by the population(i.e. y i ( t ) ). By evaluating these costs for all individuals in the population, a cumulative healthcost H (for example, the percentage of the population infected) and socio-economic cost C for theepidemic as a whole can be determined. More details and the explicit expression of these quantitiesused in the discussion of the case studies below can be found in the Methods.8 ResultsOnset of the Epidemic
For large populations and fully connected influence layers, we use a mean-field approach to obtain insight into the behaviour of the system at the initial stages of the epi-demic through the computation of the epidemic threshold. In the absence of cumulative frustration(which is a reasonable assumption in the early stages of an outbreak) and assuming a constantintervention u ( t ) = ¯ u , the outbreak is quickly eradicated if λµ < e β (1 − ¯ u ) + (1 − β ) e − βc m ( (cid:104) a (cid:105) + (cid:112) (cid:104) a (cid:105) )( e β (1 − ¯ u ) − βe − βc ) , (4)where (cid:104) a (cid:105) and (cid:104) a (cid:105) are the average and second moment of the activity distribution, respectively (thederivation of Eq. (4) can be found in the Methods). Figure 2 shows that mild non-pharmaceuticalinterventions (small ¯ u ) have only a minor impact on the epidemic threshold, since they may notprovide a sufficient incentive to abandon the status-quo non-protective behaviours. On the otherhand, the epidemic threshold increases significantly if ¯ u is sufficiently large. The socio-economiccost c also plays an important role, whereby small increases can lower the threshold even for large ¯ u . This implies that the perception of a high socio-economic cost may hamper the outcome ofstrong non-pharmaceutical interventions.The threshold in Eq. (4) offers insight into the role of human behaviour in the early stagesof an epidemic outbreak. We now move to the medium/long-term horizon, and consider scenariosin which the threshold is exceeded, yielding a large initial outbreak across the population. To shedlight on the potential of our method, we present three case studies based on real data, viz. gonor-rhoea, Spanish flu and COVID-19. To model the spread of gonorrhoea we use the basic SIS model9 . . . . . . . . . . u c . . . . Figure 2: Ratio between the epidemic threshold computed in Eq. (4) and the corresponding quan-tity without self-protective behaviours as a function of the cost of adopting self-protective be-haviours c and of policy interventions ¯ u ( β = 2 ). The larger this ratio (i.e. in the blue region), themore resistant the system is to the epidemic outbreak.described earlier; to capture the latency period after contagion and immunity after recovery ordeath relevant to Spanish flu and COVID-19, we adopt a susceptible–exposed–infectious–removed(SEIR) model , as described in the Methods and illustrated in Fig. 1b. Parameters are also detailedin the Methods. SIS Modelling of Gonorrhoea Spreading
Despite its simplicity, the SIS model has found animportant application in the study of gonorrhoea outbreaks . We focus on revealing the role ofthe risk perception function r ( z ) in shaping the epidemic outbreak and the behavioural response.Hence, we fix all other parameters and assume that no policy interventions are implemented. InFig. 3, we consider three scenarios with progressively less cautious populations, showing that thisshift in risk perception changes not only the quantitative, but also the qualitative, characteristic10 2 4 6 8 10 . . Time (years) P r e v a l e n ce (a) . . . Time (years) (b) . . . . . Time (years) (c)
Figure 3: Prevalence of gonorrhoea infections (red curve) as a function of time as predicted byan SIS model using our co-evolution paradigm. The colour intensity of the blue vertical bandsindicates the level of adoption of self-protective behaviours, i.e. n (cid:80) ni =1 y i . In (a), a responsivepopulation ( r ( z ) = 3 √ z ) results in quick adoption of self-protective behaviours and fast eradica-tion of the disease, after the first wave. In (b), multiple waves emerge in a less cautious population( r ( z ) = 3 z ), yielding periodic waves of reinfection. In (c), in a population that underestimates therisk ( r ( z ) = 3 z ), the disease becomes endemic and a meta-stable equilibrium emerges. Parame-ters are defined in the Methods.phenomena observed. This demonstrates the power of our model, which allows to capture andreproduce a range of real-world phenomena within a unified modelling framework. In fact, bytuning just one parameter in the decision-making mechanism, we shift from fast eradication of thedisease (Fig. 3a), to periodic oscillations both in the epidemic prevalence and in the behaviouralresponse (Fig. 3b), and to the emergence of a meta-stable endemic equilibrium with a pervasiveand long-term diffusion of the disease (Fig. 3c).Notice that, as further detailed in the Methods, the risk perception function determines a11ritical prevalence which guarantees a pervasive adoption of self-protections regardless of the be-haviour of others. In our first scenario, this critical prevalence is z ∗ ≈ . However, as can beobserved in Fig. 3a, social influence causes individuals to rapidly and widely adopt self-protectivebehaviours much earlier, when the prevalence is z ( t ) ≈ , highlighting the key role played bysocial influence in shaping collective behavioural patterns and, in this instance, helping in the fasteradication of the disease. SEIR Modelling of the Spanish Flu Pandemic
We use an SEIR model combined with ourparadigm to reproduce the 1918–19 Spanish flu pandemic (see Methods). Using only model pa-rameters rooted in the actual historical context, the simulation results in Figure 4 are qualitativelyconsistent with the epidemiological data that witnessed a resurgent pandemic with three wavesincluding a massive second one
6, 26 . More importantly, key contributing factors identified from ourmodel predictions as leading to the resurgence of new and even larger outbreaks also reflect his-torical observations. In particular, a slowly increasing risk perception — associated with the initialsuppression of news about the flu — results in a delay of over a month after the initial outbreakfor the population to adopt self-protective behaviours. Meanwhile, a fast accumulating frustra-tion (corresponding for example to the emergence of anti-mask movements at the time) results inself-protective behaviours being abandoned immediately when the epidemic prevalence decreases.By doubling the duration of the interventions (from days, as reported in the literature , to days) or by relying on a more responsive population (details in the Methods), only a single waveis witnessed, consistent with the historical fact that some cities successfully stopped the Spanish12
50 100 150 200 250 30000 . . . Time (days) P r e v a l e n ce Figure 4: Prevalence of Spanish flu infections (red curve) and cumulative infections (orangecurve) as a function of time as predicted by an SEIR model in combination with the co-evolutionparadigm. The colour intensity of the blue vertical bands represents the fraction of adopters ofself-protective behaviours. Our simulation qualitatively reproduces the 1918–19 pandemic out-break with three consecutive epidemic waves. The impact of the outbreak is measured by healthcost H = 59 . and socio-economic cost C = 6 . , as detailed in the Methods.13u through longer, timely interventions . The health costs in terms of cumulative epidemic preva-lence is reduced from to and , respectively, while the socio-economic costs associatedwith the adoption of self-protective behaviours are reduced by more than (see Methods andSupplementary Fig. S1). This illustrates the power of our paradigm for predicting future scenar-ios once a particular parametrisation and model have been secured from empirical data. Existingapproaches typically estimate how infection parameters, associated with the disease dynamics, areexplicitly changed due to policy interventions . In contrast, our paradigm leaves the disease dy-namics untouched, and allows policy interventions to influence the decision-making process, thusdetermining the behavioural responses, which, in turn, shape the epidemic evolution. Modelling of the COVID-19 Pandemic
Having established that our model reproduces key fea-tures of past epidemics, we now demonstrate its power as a predictive tool by analysing severaldifferent intervention scenarios for the ongoing COVID-19 pandemic . To this end, we again inte-grate our game-theoretical framework into an SEIR model. In each of the simulations, we considerthe same cost and accumulation factor, and a population that is slow to perceive COVID-19 as athreat; by varying the intervention strategies, and the level of social influence, we reveal the globalimpact of these factors.Firstly, we observe that mild policy interventions, even if indefinite in duration, may not besufficient to ensure a timely and collective adoption of self-protective behaviours. For instance, thepopulation in Fig. 5a is overwhelmed by the epidemic because social influence acts as inertia todelay the adoption of self-protective behaviours (Supplementary Fig. S2) . Then, we consider two14
100 200 30000 . . . Time (days) P r e v a l e n ce (a) . . Time (days) (b) . . Time (days) P r e v a l e n ce (c) . . Time (days) (d)
Figure 5: Prevalence of COVID-19 infections (red curve) and cumulative infections (orangecurve) as a function of time as predicted by an SEIR model in combination with the co-evolutionparadigm. The colour intensity of the blue vertical bands represents the fraction of adopters ofself-protective behaviours. In (a), a mild policy intervention and accumulating frustration resultsin sporadic adoption of self-protection, and thus two massive epidemic waves that overlap and along tail (health cost H = 65 . , socio-economic cost C = 15 . ). In (b), a severe policy in-tervention with a long phased reduction period keeps the epidemic outbreak contained to a singlewave ( H = 5 . , C = 12 . ). In (c), a very severe policy intervention with a short phasedreduction period results in two consecutive epidemic waves ( H = 9 . , C = 23 . ). In (d), thepolicy in (b) is implemented in a scenario without social influence, showing that social influence iskey to guaranteed collective (and thus effective) behavioural responses ( H = 49 . , C = 60 . ).Parameters and costs are detailed in the Methods.15cenarios with severe but short policy interventions, followed by a linear phased reduction (Figs. 5band 5c; in the first scenario, the policy is less severe but the reduction period is longer). Comparingthe two scenarios, we conclude that, provided that the initial policy interventions are sufficientlysevere (to avoid the scenario shown in Fig. 5a), the successful eradication of the disease dependsprimarily on the phased reduction period being sufficiently long. The latter point is consistent withobservations from the recent literature on the duration of policy interventions . In fact, the diseaseis quickly eradicated in the first scenario after the first wave, with . of cumulative infectionsand even a decrease in the socio-economic cost compared to the indefinite but mild policy discussedabove, while the second (more severe, but shorter) choice of interventions yields a second wavewith a final prevalence of . and a socio-economic cost that is almost doubled.Finally, in Fig. 5d, we repeat the scenario observed in Fig. 5b but remove the factor of socialinfluence.We find that the presence of social influence during the initial severe intervention periodensures a collective population response with a adoption rate of self-protective behaviours(Fig. 5b); in contrast, the adoption rate is reduced to ≈ without social influence (Fig. 5d).However, during the phased reduction period, social influence accelerates a collective rejection ofself-protective behaviours; in Fig. 5b, individuals overwhelmingly reject self-protective behaviourcompletely two weeks before the end of the phased reduction period (Supplementary Fig. S3). Thisresembles the difficulties experienced by some countries in controlling the exit out of a lockdown inthe current COVID-19 pandemic . The crucial role played by social pressures as concluded fromour simulation results and the resonance thereof in the COVID-19 reality indicate the need forepidemics models to include time-varying behavioural responses. The model predictions highlight16he challenges policy makers experience in designing interventions, but also provide importantdirections. For example, our model results indicate that it is more effective to focus on managingthe duration and relaxation of the interventions, and our findings support the suggestion in of acloser examination of how to exploit social influence to marshal effective long-term responses toan epidemic. This analysis highlights the potential of our modelling paradigm in helping policymakers assess control actions to effectively address epidemic outbreaks. Its input parameters areclosely related to observable properties of the epidemic as well as the population; the wealth ofbehavioural data currently being gathered in the context of the COVID-19 pandemic will thereforeallow further refining of the model. Inspired by evidence of the key role played by individuals’ behavioural responses in shaping anepidemic outbreak, we have combined proven models of epidemics with an evolutionary game-based description of individual decision-making into a novel unified modelling paradigm. To ourknowledge, this is the first time an epidemic model has addressed the complex decision-makingprocess of a population responding to an epidemic. Different from existing behavioural mod-els based on reactive responses and from game theoretical models based on imitation
31, 32 , ourparadigm models how individual decisions are influenced by a large range of time-varying factors,including government interventions, risk perceptions, irrationality and social interactions through-out the duration of an epidemic outbreak, from the outbreak to eradication. When applied tocase studies of the spreading of gonorrhoea, the 1918–19 Spanish flu and COVID-19, the model17aptures a wide range of realistic outcome scenarios and specifically reveals the impact of socialinfluence and phasing out of government interventions on the persistence of an epidemic. As theframework is parsimonious and universal, with parameters rooted to the specifics of both popula-tion and epidemic, it enables well-founded predictions of the effectiveness of non-pharmaceuticalpolicy interventions in terms of health costs and socio-economic costs.18 ethodsDetails of the network structure
The two-layered network structure with a static influence layerand a time-varying physical contact layer is motivated by the observation that social relationships(e.g. between family members, friends, colleagues) typically change at a much slower pace thanthat of epidemic processes and thus can be assumed constant, while the network of physical con-tacts evolves at a time-scale comparable with the spread of the disease .The contact layer is modelled using an activity-driven network (ADN) . ADNs have emergedas a powerful paradigm to study the dynamical co-evolution of (i) the network structure, and (ii)the process unfolding at each node. ADNs have found successful application in mathematicalepidemiology . The network formation process acts as follows. At each discrete time instant t ,with probability a i , individual i activates and generates a fixed number of contacts m ≥ withother individuals chosen uniformly at random in the population. These contacts are added to thelink set E C ( t ) , contribute to the epidemic process, and are then removed before the next discretetime instant and the next activation of individuals. Despite their simplicity, which enables rigorousanalytical treatment and fast numerical simulations
22, 35 , ADNs can capture important features ofcomplexity that characterise real-world networks and are amenable to analytically-tractable exten-sions to include further features. Importantly, the proposed modelling paradigm can also use othertime-varying network models, e.g. temporal switching networks .19 etails of the SIS model According to the SIS mechanism described in the main paper, the con-tagion probability for a generic individual i ∈ V is P [ x i ( t + 1) = I | x i ( t ) = S ] = (1 − y i ( t )) (cid:0) − (1 − λ ) N i ( t ) (cid:1) , (5)where N i ( t ) = |{ j ∈ V : ( i, j ) ∈ E C ( t ) , x j ( t ) = I }| (6)is the number of infectious individuals that have a link with node i at time t on the contact layer.While Eq. (5) posits that adoption of self-protective behaviour, viz. y i ( t ) = 1 , always makesthe contagion probability equal to , the model can be generalised by introducing a parameterto reflect that self-protective behaviours have a probability in failing to prevent infection. Therecovery process described in the main paper is governed by the following probabilistic rule: P [ x i ( t + 1) = S | x i ( t ) = I ] = µ . (7) Details of the SEIR model
In the SEIR model, individuals may have four different health states:they can be susceptible to the disease ( S ); exposed ( E ), i.e. already infected but asymptomatic;infected and symptomatic ( I ); or removed ( R ), accounting for both recoveries and deaths. Thedisease spreads as follows. At each discrete time-step t , every individual i that is susceptible (i.e. x i ( t ) = S ) and does not adopt self-protective behaviours (i.e. y i ( t ) = 0 ) may become infected.Similar to the SIS model, we introduce a constant parameter λ ∈ [0 , , termed infection proba-bility , which captures the probability that the infectious disease is transmitted from an infectiousindividual to a susceptible one through a link in the contact layer. After contagion, individuals be-comes exposed ( E ). The disease propagates through contacts, each one independent of the others,20ielding P [ x i ( t + 1) = E | x i ( t ) = S ] = (1 − y i ( t )) (cid:0) − (1 − λ ) N i ( t ) (cid:1) , (8)where N i ( t ) is the number of neighbours of node i on the contact layer that are infectious at time t . Depending on the specifics of the disease, exposed (i.e. pre-symptomatic) individuals may beinfectious or not, giving N i ( t ) = |{ j ∈ V : ( i, j ) ∈ E C ( t ) , x j ( t ) ∈ { E, I }}| if exposed individuals are infectious, |{ j ∈ V : ( i, j ) ∈ E C ( t ) , x j ( t ) = I }| otherwise. (9)Beside the contagion, two other mechanisms govern the epidemic process: the emergenceof symptoms and the recovery process. Specifically, at each time-step t , every individual i that isexposed ( E ) has probability ν ∈ (0 , to become symptomatic, independent of the others, giving P [ x i ( t + 1) = I | x i ( t ) = E ] = ν . (10)Then, infected individuals have probability µ ∈ (0 , to recover, independent of the others, giving P [ x i ( t + 1) = R | x i ( t ) = I ] = µ . (11)Different from the SIS model, once removed, individuals are immunised and cannot be infectedagain. Details of the decision-making mechanism
Here, we provide with some additional details andcomments on the decision-making mechanism. The parameter β ∈ [0 , ∞ ) and the log-linear learn-ing dynamics in Eq. (16) model bounded rationality in an individual’s decision-making process.21e have assumed for simplicity that β is homogeneous among all individuals in this work, but thisis easily generalisable to be heterogeneous. Notice that if β = 0 , individuals make decisions uni-formly at random, while for β → ∞ , individuals apply perfect rationality to select the behaviourwith highest payoff. This best-response behaviour is myopic, i.e. individuals do not look forwardin time to optimise a sequence of decisions.The risk perception function is a monotonic non-decreasing function of z ( t ) , capturing thefear of being infected that rises as the number of symptomatic individuals increase. In its simplestformulation (which will be adopted in the three case studies presented in this paper), it can beassumed to be a power function r ( z ) = kz α , (12)with k > and α > . Specifically, α ∈ (0 , models cautious populations, where a small initialoutbreak causes a large increase in the risk perception. The case α = 1 captures a populationwhose reaction grows exactly proportionally to the epidemic prevalence observe. On the contrary, α > captures populations that underestimate the risk, and the epidemic prevalence must be largebefore the risk perception plays an important role in the decision-making process. The constant k is a scaling constant. In the main paper, we consider three different functions as archetypes ofthree different scenarios: a cautious population ( α < / ), a population with reaction proportionalto the epidemic prevalence ( α = 1 ), and a slow reacting population ( α = 2 ). In all three cases weset k = 3 .Note that, in the absence of the accumulation of socio-economic costs ( γ = 0 ) and without22olicy interventions u ( t ) = 0 , the risk perception function determines a critical prevalence z ∗ = min { z : r ( z ) > c } , (13)such that, z ( t ) > z ∗ implies π i (1) > π i (0) , for any individual i . In other words, if the de-tectable prevalence exceeds z ∗ , each individual will always favour the adoption of self-protectivebehaviours, regardless of the behaviour of other individuals and of the presence of policy interven-tions. We remark that the risk perception (as well as the cost of self-protection) is endogenous toeach individual, different from the policy intervention u ( t ) , which is exogenous and from the socialinfluence, which is determined by the network structure. Evaluating the socio-economic impact
The health cost H is a functional that quantifies the costsrelated to the health state of individuals and may have different definitions, depending on theepidemic model used and on the focus of the study. In our case studies, we define the heath costfor the SEIR model as the total number of infections, i.e. the cumulative prevalence. Specifically,fixed a period of observation T ≥ (the duration of the epidemic outbreak), we define H ( T ) = 1 n |{ i : x i ( T ) = R }| . (14)Note that in the SIS model, the health cost may be defined as the average epidemic prevalence overthe period of observation.We define the per capita socio-economic cost over a fixed period of observation T ≥ (the duration of the epidemic outbreak) as the average cumulative immediate costs incurred by an23ndividual in the population, that is, C := 1 n T (cid:88) t =0 n (cid:88) i =1 cy i ( t ) . (15) Derivation of the epidemic threshold for the SIS model
In the absence of cumulative frustrationand for a fully mixed influence layer, we observe that P [ y i ( t + 1) = 0] = exp { βπ i (0) } exp { βπ i (0) } + exp { βπ i (1) } (16)has the same expression for all the nodes. In fact, if we define ¯ y ( z ) as the probability that a genericnode adopts self-protective behaviours when the epidemic prevalence is equal to z , then, accordingto the strong law of large numbers, π i (0) = 1 − ¯ y ( z ) − u ( t ) = 1 − ¯ y ( z ) − ¯ u (since the controlis assumed to be a constant function) and π i (1) = ¯ y ( z ) + r ( z ) − c , which are independent of i .In a mean-field approach , we define θ ( t ) = n (cid:80) i : x i ( t )= I a i as the average activity of infectednodes and z i ( t ) = P [ x i ( t ) = I ] . Due to the strong law of large numbers, in the limit of large-scalesystems n → ∞ , z ( t ) = n (cid:80) ni =1 z i ( t ) and θ ( t ) = n (cid:80) ni =1 a i z i ( t ) . Hence, from the mean-fieldevolution of z i ( t ) , given by z i ( t + 1) = z i ( t ) − µz i ( t ) + (1 − z i ( t ))(1 − ¯ y ( z ( t ))) a i z ( t ) + (1 − z ( t ))(1 − ¯ y ( z ( t ))) λθ ( t ) , (17)we determine the following system of difference equations for the epidemic prevalence and theaverage activity of infected individuals, linearised about the disease-free equilibrium ( z = 0 , θ =0 ): z ( t + 1) = z ( t ) − µz ( t ) + mλ (cid:104) a (cid:105) z ( t )(1 − ¯ y (0)) + mλ (1 − ¯ y (0)) θ ( t ) θ ( t + 1) = θ ( t ) − µθ ( t ) + mλ (cid:104) a (cid:105) z ( t )(1 − ¯ y (0)) + mλ (cid:104) a (cid:105) (1 − ¯ y (0)) θ ( t ) . (18)24rom standard theory on the stability of discrete-time linear time-invariant systems , the origin isstable if λµ < m ( (cid:104) a (cid:105) + (cid:112) (cid:104) a (cid:105) )(1 − ¯ y (0)) . (19)In fully-mixed influence layers, the probability for an individual to adopt self-protective be-haviours ¯ y ( z ) can be derived by substituting π i (0) = 1 − ¯ y ( z ) − ¯ u and π i (1) = ¯ y ( z ) + r ( z ) − c into Eq. (16), obtaining the equilibrium equation: ¯ y = e β (¯ y − c + r ( z )) e β (¯ y − c + r ( z )) + e β (1 − ¯ y − ¯ u ) . (20)Even though it is not possible to derive a closed-form solution, we observe that at the inceptionof the epidemic outbreak, y i (0) = 0 for all individuals and, for sufficiently small values of ¯ u (i.e. ¯ u .
21 + c ), in the early stages it is verified that π i (0) > π i (1) . Hence, if the rationality β issufficiently large, the equilibrium ¯ y is close to and can be approximated by Taylor-expanding theright-hand side of the equation, obtaining ¯ y ( z ) ≈ e β ( − c + r ( z )) e β (1 − ¯ u ) + (1 − β ) e β ( − c + r ( z )) . (21) Network parameters of the simulations
In all the simulations, we consider n = 10 , indi-viduals ( of them initially infected) connected on the influence layer through a Watts–Strogatzsmall-world network, which captures many features of real-world influence networks, including aclustered structure and the presence of long-range interactions . We set an average degree andrewiring probability / , so that each node has on average long-range interaction.The contact layer is generated following an ADN with power-law distributed activities a i . , as in Aiello et al. , and lower cutoff at a min = 0 . . We set m = 1 interactionsper active node for gonorrhoea and m = 13 for Spanish flu and COVID-19, based on . Epidemic parameters of the simulations
In the three case studies we use two different classicalepidemic models: the SIS and the SEIR model. The former is used for gonorrhoea, which is asexually transmitted disease characterised by negligible protective immunity after recovery andnegligible latency period (individuals are infectious on average the day after contagion)
25, 41 . AnSEIR model is used to capture latency periods after contagion and immunity after recovery, whichhave been observed for the Spanish flu and COVID-19 . In the two SEIR case studies we relyon slightly different implementations of the epidemic model. For Spanish flu, based on establishedevidence, we assume that exposed (pre-symptomatic) individuals are not infectious . For theCOVID-19 case study, where the preliminary observations are fewer and there is no consensus onthe infectiousness of asymptomatic individuals, we assume the worst-case scenario in which allexposed individuals are infectious.The epidemic parameters are set from epidemiological data. Specifically, reliable estimationsof the time from contagion to symptoms onset τ E (for Spanish Flu and COVID-19) and the timefrom symptoms onset to recovery τ I (for all three diseases) are available
8, 25, 26 . Similar to , fromthese data we define ν = 1 − exp (cid:18) − τ E (cid:19) , and µ = 1 − exp (cid:18) − τ I (cid:19) . (22)Finally, the parameter λ is obtained from available estimations of the basic reproduction number R for the three diseases
8, 25, 26 . The basic reproduction number is defined as the average number26f secondary infections produced by an infected individual in a population where everyone issusceptible. Hence, if we define τ to be the average time that an individual is infectious ( τ = τ I + τ E for COVID-19 and τ = τ E for Spanish flu), assuming independence between the time an individualis infectious and their activity, we compute R = 1 n (cid:88) i ∈V ( a i + (cid:104) a (cid:105) ) mλτ = 2 (cid:104) a (cid:105) mλτ , (23)which implies λ = R m (cid:104) a (cid:105) τ . (24)The epidemic parameters computed using this procedure are the following. • Gonorrhoea.
We find λ = 0 . and µ = 0 . , where the time unit is a week. • Spanish Flu.
We obtain λ = 0 . , µ = 0 . and ν = 0 . , where the time unit is aday. • COVID-19.
We compute λ = 0 . , µ = 0 . and ν = 0 . , where the time unit is aday. Decision-making parameters of the simulations
We set a common level of rationality β = 6 inall simulations, which captures a moderate level of rationality so that individuals tend to maximisetheir payoff, but always have a small, non-negligible probability of adopting the behaviour withthe lower payoff. Before detailing the parameters used in the three case studies, we provide a briefdiscussion on the relative order of magnitude between the model parameters, which guided ourchoices. 27he decision-making process is based on the comparison between the two payoff functions inEqs. (2a) and (2b). The contribution of social influence to the payoff is always bounded between and . Hence, social influence is significant if the other terms do not have a higher order ofmagnitude. Consequently, policy interventions u ( t ) > can be considered severe, since theireffect is predominant with respect to social influence, while policies with u ( t ) < are milder. Thecost of self-protective behaviours consists of two terms: the immediate cost per unit-time c andthe accumulation factor γ . Small values of c become negligible in the decision making process,while, to avoid the immediate cost dominating the other terms, we should assume c < . Theaccumulation factor γ captures the cost for continued periods in which an individual adopts self-protective behaviours. To model a non-negligible effect of the accumulation of socio-economiccosts, we should guarantee that over long periods in which an individual consistently adopts self-protective behaviours, the frustration function saturates to a value comparable to the other terms.This can be achieved by imposing that lim t →∞ c + t (cid:88) s =1 γ s c = c − γ ≈ , (25)yielding c + γ ≈ (note, the above equality was obtained using the geometric series). Specifically,values of γ > − c guarantees that self-protective behaviours are eventually dismissed, afterthe complete eradication of the disease or the policy intervention is switched off. We use the riskperception function r ( t ) = kz α proposed in Eq. (12) with α = 1 / for cautious populations, α = 1 to model proportional reactions, and α = 2 for slow reacting populations. As observed in Eq. (13),the risk perception function determines a critical epidemic prevalence that triggers the adoption ofself-protective behaviours even in the absence of interventions (in the presence of accumulation,28he immediate cost c in Eq. (13) is substituted by its saturation value from Eq. (25) c/ (1 − γ ) ).We observe from Eq. (13) that risk perception becomes non-negligible if k > c − γ . To keepconsistency throughout our simulations, we set k = 3 , which is a value that verifies the conditionabove for all the choices of parameters c and γ we make in the simulations.The decision-making parameters used for the three case studies are detailed in the following. • Gonorrhoea.
We assume that the accumulation is negligible for gonorrhoea (where the useof protections has an immediate cost that typically does not accumulate, such as protectivesexual barriers). Hence, for all three simulations, we fix the immediate cost to c = 0 . andthe accumulation factor γ = 0 . No policy intervention is set, with u ( t ) = 0 for all t ≥ .In the three simulations, we test three different risk perception function. Specifically, weconsider a cautious population with r ( z ) = 3 √ z in Fig. 3a, a population with a proportionalreaction, r ( z ) = 3 z in Fig. 3b, and a population slow to react with r ( z ) = 3 z in Fig. 3c. • Spanish flu.
Self-protective behaviours involve social distancing and closures of economicactivities, which has been shown to typically yield an accumulation of psychological distressand economic losses . Hence, we assume a high accumulation factor γ = 0 . and we fix c = 0 . , in light of our discussion above. To capture the slow reaction of the populationdue to the initial suppression of information (to keep morale during World War I)
6, 42 , weset r ( z ) = 3 z . To further mirror real-world interventions by public authorities, we set aninitial intervention level equal to u (0) = 0 , which switches to u ( t ) = ¯ u = 0 . once ofthe population is infected and then remains active for days before being turned off again.29 COVID-19.
Similar as we did for the Spanish flu, we assume that accumulation is non-negligible by setting γ = 0 . . Note that we select a smaller value of γ than the one used forSpanish flu to capture how development in information and communications technologieshas helped alleviate the negative impact of extended lockdown policies during COVID-19outbreak. In view of the observations above, we fix c = 0 . . We consider a scenario inwhich individuals are slow to perceive COVID-19 as a real threat ( r ( z ) = 3 z ), as happenedin many countries including the UK and US . In Fig. 5a, we start with u (0) = 0 , andthen set the intervention as u ( t ) = 0 . when of the population is infected, which is thenturned off if consecutive days pass with z ( t ) = 0 . In Fig. 5b and Fig. 5c, we considerscenarios involving an initial severe but constant lockdown policy followed by a phasedreduction. In the first scenario (Fig. 5b), severe policies ( u ( t ) = 1 . ) are implemented for days after reaching of infections, after which u ( t ) is linearly reduced to u ( t ) = 0 over time-steps. In the second scenario (Fig. 5c), more severe policies ( u ( t ) = 1 . ) areimplemented for the same period of time-steps, after which u ( t ) is linearly reduced to u ( t ) = 0 over a shorter time-window of time-steps. Note that we select the intensity ofpolicy interventions and the duration of the phased reduction to ensure that the cumulativeintervention effort, (cid:80) t u ( t ) , over the policy intervention period, is equal in the two scenarios.Finally, in Fig. 5d, we tested the intervention scenario in Fig. 5b but in the absence of socialinfluence. Specifically, we removed the first term in each the two payoff functions in Eqs.(2a) and (2b). 30 ata Availability All data generated or analysed during this study are included in this Article(and its Supplementary Information files).
Code Availability
The code used in the simulations is available at https://github.com/lzino90/behavior.
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Acknowledgements
The work by M.Y. is partially supported by Optus Business. The work by L.Z. andM.C. is partially supported by the European Research Council (ERC-CoG-771687) and the NetherlandsOrganization for Scientific Research (NWO-vidi-14134). The work by A.R. is partially supported by Com-pagnia di San Paolo and by the Italian Ministry of Foreign Affairs and International Cooperation (Grant“Mac2Mic”).
Author Contributions
M.Y. and L.Z. contributed equally. They designed the research, performed theanalytical and numerical studies and wrote a first draft of the manuscript. A.R. and M.C. contributed equallyto the formulation of the research questions, the supervision of the research, the interpretation and analysisof the result,s and to reviewing the manuscript. All the authors contributed to the present submission.
Competing Interests
The authors declare that they have no competing interests.
Materials & Correspondence
Correspondence and requests for materials should be addressed to Meng-bin Ye. (email: [email protected]) UPPLEMENTARY INFORMATION: Modelling epidemic dynamics under collectivedecision making . . E p i d e m i c p r e v a l e n c e self-protectionscumulative infectionsepidemic prevalence 0%50%100% S e l f - p r o t e c t i o n s a d o p t i o n r a t e (a) . . E p i d e m i c p r e v a l e n c e self-protectionscumulative infectionsepidemic prevalence 0%50%100% S e l f - p r o t e c t i o n s a d o p t i o n r a t e (b) FIG. S1:
Extension of the Spanish Flu (SEIR) case study in the main article.
Prevalence of Spanish fluinfections (red curve) and cumulative infections (orange curve) as a function of time as predicted by an SEIR modelin combination with the co-evolution paradigm. The colour intensity of the blue vertical bands represents thefraction of adopters of self-protective behaviours. Consistent with the main article, we set γ = 0 . c = 0 .
1, andan initial intervention level equal to u (0) = 0, which switches to u ( t ) = ¯ u = 0 . T days before being turned off again. In Fig. S1a, we set r ( z ) = 3 z (consistent withFig. 4 in the main article), but double the intervention duration to ¯ T = 56 days. In Fig. S1b, we retain the original¯ T = 28 days from Fig. 4 in the main article, but assume a more cautious population that is better informed, with r ( z ) = 3 √ z . In both figures, we see the epidemic outbreak is successfully halted after a single wave, either by alonger policy intervention or a population that has a heightened risk perception, in contrast to Fig. 4 of the mainarticle. The impact of the outbreak is measured by health cost (a) H = 10 .
69% and (b) H = 3 . C = 2 . C = 2 .
5, as detailed in the Methods.
50 100 150 200 250 3000%25%50%75%100% Time (days) S e l f - p r o t e c t i o n s a d o p t i o n r a t e h y ( t ) i u ( t ) 00 . . . . . . . P o li c y i n t e r v e n t i o n s , u ( t ) (a) FIG. S2:
Lag in the Adoption of Self-Protective Behaviours.
This figure shows the same simulation as thatin Fig. 5a of the main article. The adoption rate (blue solid curve) of self-protective behaviours, is quantified by h y ( t ) i = n P ni =1 y i ( t ), and the intervention policy (cyan dashed curve) shows an on/off constant policy intervention u ( t ) = ¯ u = 0 . z ( t ) > .
01, and switched off after no infections are reported for 14consecutive days. Notice that although the intervention is introduced at t = 19, adoption of the self-protectivebehaviours peaks only 20 days later, at t = 39 with 73% adoption rate among the population a . This allows us toconclude that the adoption of self-protective behaviours is delayed by social influence, which has an inertia effect,and the adoption rate improves because of individuals have an increasing perceived risk of infection as the epidemicgrows. In contrast, all individuals revert back to standard behaviours at t = 215 days, almost immediately after theintervention policy ends at t = 214. a This adoption rate is well below the 99% seen in Fig. 5b and 5c of the main article, highlighting the necessity of severe interventions
10 20 30 40 50 60 70 80 900%25%50%75%100% Time (days) S e l f - p r o t e c t i o n s a d o p t i o n r a t e h y ( t ) i u ( t ) 00 . . . . P o li c y i n t e r v e n t i o n s , u ( t ) (a) S e l f - p r o t e c t i o n s a d o p t i o n r a t e h y ( t ) i u ( t ) 00 . . . . P o li c y i n t e r v e n t i o n s , u ( t ) (b) FIG. S3:
The Role of Social Influence.
Fig. S3a and Fig. S3b show the same simulation as that corresponding toFig. 5b and 5d in the main article, respectively. The adoption rate (blue solid curve) of self-protective behaviours, isquantified by h y ( t ) i = n P ni =1 y i ( t ), and the intervention policy (cyan dashed curve) shows an initial severe andconstant lockdown for 28 days after z ( t ) > .
01, followed by a linear phased reduction over 35 days. Note that forthe sake of clarity, the disease evolution is not shown here, and we only plot the time horizon up to the first policyintervention for Fig. 5d (in actuality, multiple interventions are necessary due to resurgent infection waves). Noticethat in Fig. S3a, there is a delay of 3-4 days between the start of the intervention, and all individuals adoptingself-protective behaviours, in contrast to Fig. S3b, where there is only a single day of delay. This supports the claimthat social influence has an inertia effect in the beginning, but this delay is shortened by a severe intervention.During the 28 days of constant intervention, social influence acts to ensure a collective adoption of self-protectivebehaviours, countering the effects of accumulating costs f i ( t ), with h y ( t ) i ≈ .
99 in Fig. S3a. In contrast, Fig. S3bsees an initial collective adoption driven by u ( t ), but as costs accumulate, adoption drops to a constant of h y ( t ) i ≈ .
65. However, once the phased reduction begins, we see the population with no social influence in Fig. S3bclosely mirror the phased reduction. In contrast, social influence in Fig. S3a leads to a collective rejection ofself-protective behaviours, with h y ( t ) i ≈ t = 68, over two weeks before the end of the phased reduction at tt