Games of Social Distancing during an Epidemic: Local vs Statistical Information
GGames of Social Distancing during an Epidemic: Local vsStatistical Information
A.-R. Lagos ∗ I. Kordonis ∗ G. P. Papavassilopoulos ∗ July 13, 2020
Abstract
The spontaneous behavioral changes of the agents during an epidemic can have significanteffects on the delay and the prevalence of its spread. In this work, we study a social distancinggame among the agents of a population, who determine their social interactions during thespread of an epidemic. The interconnections between the agents are modeled by a networkand local interactions are considered. The payoffs of the agents depend on their benefitsfrom their social interactions, as well as on the costs to their health due to their possiblecontamination. The information available to the agents during the decision making plays acrucial role in our model. We examine two extreme cases. In the first case, the agents knowexactly the health states of their neighbors and in the second they have statistical informationfor the global prevalence of the epidemic. The Nash equilibria of the games are studied and,interestingly, in the second case the equilibrium strategies for an agent are either full isolationor no social distancing at all. Experimental studies are presented through simulations, wherewe observe that in the first case of perfect local information the agents can affect significantlythe prevalence of the epidemic with low cost for their sociability, while in the second case theyhave to pay the burden of not being well informed. Moreover, the effects of the informationquality (fake news), the health care system capacity and the network structure are discussed ∗ National Technical University of Athens, School of Electrical and Computer Engineering, 9 Iroon Polytechnioustr.,Athens, Postal Code 157 80, Greece.E-mails: A.-R. Lagos [email protected], I. Kordonis [email protected], G.P. Papavassilopoulos [email protected] a r X i v : . [ phy s i c s . s o c - ph ] J u l nd relevant simulations are provided, which indicate that these parameters affect the size,the peak and the start of the outbreak, as well as the possibility of a second outbreak. The emergence of the Covid-19 pandemic is one of the most significant events of this era. It affectsmany sectors of human daily life, it indicates the inefficiency of many health care systems and itleads to state interventions in the functioning of the society through urgent measures, to economicdepression and to human behavioral changes. Different states followed significantly different strate-gies to contain the pandemic and achieved respectively different levels of success. However, as thepandemic progresses, the interest about its nature, its dynamics and the need to control it madeepidemiology a scientific field known to almost everyone and its terminology used daily by the mediaand included in many conversations. Humans spontaneously react to the emergence of Covid-19,following or disrespecting the state directions and legislation, adaptively adjusting their behaviorbased on their perceived risk. Thus, a question naturally arises: How this spontaneous behavioralchange of humans affects the prevalence of the disease and under what assumptions would it beeffective in reducing the spread of the outbreak?Humankind has always been haunted by epidemics, some of which have been recorded fromhistorians, such as the plague in ancient Athens (430BC) and the Black Death in medieval Europe.So, epidemiology has concerned a lot of scientists during the ages and mathematical models for thisfield were first developed in the 18 th century [1]. Nowadays, the most prevalent approach in epidemicmodeling is the compartmental models, introduced a century ago [2],[3]. These models assume thatthere exist several compartments where an agent can belong (e.g. Susceptible-Infected-Recovered)and derive ordinary differential equations for the description of the dynamics of the populationin each compartment. A main assumption for that analysis to hold is the well mixing of thepopulation. However, there is enough evidence from social and other kinds of human networks thatthis assumption does not hold in many cases.Due to that fact, novel approaches in epidemic modeling take into consideration the hetero-geneous networked structure of human interconnections [4]. A branch of these approaches usesresults from the percolation theory to estimate the spread of the epidemic [5, 6, 7, 8, 9]. Anotherbranch, that is gaining a lot of attention [10], is the agent-based models [11, 12], which consider2everal parameters of each agent profile (e.g. residence, age, mobility pattern) and run computersimulations for large populations of such agents to estimate the spread of the disease. There existalso several recent works [13, 14, 15] which take into consideration the networked structure of hu-man interconnections and they derive the -compartmental- models they use, through a mean-fieldapproach.Regardless of its derivation and its mathematical formulation, the usefulness of epidemic mod-eling is to guide states and/or individuals in taking the right protective measures to contain theepidemics. These measures, besides the efforts to develop appropriate meditation, can be roughlyorganized into two categories: vaccination [13, 14], [16, 17, 18, 19], [20, 21] and behavioral changes[22, 23, 24, 25, 26], [27, 28, 29, 30], [31, 32, 33]. In the second case, the actions taken by the agentsmay vary from usage of face masks and practice of better hygiene to voluntary quarantine, avoidanceof congregated places, application of preventive medicine and other safe social interactions.In both cases, a very important fact that determines the effectiveness of the protective measuresis that the agents make rational choices with regard to the self-protective activities they adopt bycomparing the costs and benefits of these actions. Even in the case that a central authority imposesa policy, it is often up to agents to fully comply with this or not, even if they will have to pay a highcost if they get caught. From these considerations game theory arises as a natural tool to modeland analyze the agents behavior with respect to the adoption of protective measures. Many recentstudies on this field incorporate a game theoretic analysis [16, 17, 18, 19, 27, 28, 29, 30, 34, 20,35, 21, 36, 37, 38], some of which are summarized in [39]. It should be pointed out here, that theassumption of rational agents does not always hold true, since in many cases the agents decisionsare not based on the maximization of their personal utility. Moreover, in the cases it holds it is a“double-edged sword” [20], because self-interest leads the agents to adopt strategies different thanthe ones which maximize group interest [35, 21, 36, 37, 38]. Another main characteristic of game-theoretic approaches is the crucial role of information available to the agents for their decision. Theremarkable impact of information on the epidemic outbreaks has been pointed out in [31, 32, 33],where the authors consider an extra dynamic modeling the spread of information, coupled with thecontagion dynamics. The informed agents are supposed to alter their behavior and affect this waythe disease prevalence.Following the research directions presented in the previous paragraphs, and specifically thegame-theoretic approaches for the modeling of behavioral changes [27, 28, 29, 30],[32], we propose3nd analyze a game-theoretic model for social distancing in the presence of an epidemic. Our modeldiffers from [27, 28, 29, 30],[32] since it takes under consideration the networked structure of humaninterconnections and the locality of interactions, without attempting a mean-field approach. Eachagent is considered to have her own state variables and information and choosing her action basedon these - so it could be characterized an agent-based approach. Moreover, the actions of theagents affect the intensity of their relations with their neighbors and use or do not use the availableconnections. Changes in the topology of the network have been considered as a phenomenon in[40, 41, 42], but not from a game-theoretic perspective where the agents can choose rationally whichconnections to use and induce this way an “active” topology. Furthermore, there exist several workson game-theoretic models which consider the networked structure of human interconnections, suchas [20, 33, 34, 43], where the strategy adoption is based on imitation of ones neighbors. Contrary tothat, in our model the agents do not imitate the most effective strategy of their neighborhood, butdesign their best response based on the available information. We consider two different informationpatterns: perfect local information for the states of ones neighbors and statistical information forthe global prevalence of the epidemic and investigate the different effects of these patterns.Through the analysis of the proposed model we get several results. At first, we observe thatin the case of perfect local information the agents can affect significantly the prevalence of theepidemic with low cost for their sociability, while in the case of statistical information they have topay the burden of not being well informed. Secondly, in the case the agents have only statisticalinformation, each agent’s action is either full isolation or no social distancing at all. Lastly, weinvestigate, through experimental studies, the effects of the information quality (fake or biasednews), the health care system capacity and the network structure and we conclude that theseparameters affect the size, the peak and the start of the outbreak, as well as the possibility of asecond outbreak.The rest of the paper is organized as follows. In section 2 the model for the epidemic outbreakand for the social distancing game between the agents is introduced. In section 3 we analyze thegame for the case that the agents have perfect local information for the states of their neighbors.In section 4 we analyze the game for the case that the agents have statistical information for theglobal prevalence of the epidemic. In section 5 we present simulations for the games with the twodifferent information patterns and compare the results. A discussion follows in section 6, whereseveral variations of the problem are considered, such as experimentation on various network types444]-[45], the impact of fake information and of the finite capacity of a health care system andrelated simulations are presented and annotated. We denote by G = ( V, E ) an undirected graph, where V = { , ..., n } is the set of its nodes repre-senting the agents and E ⊂ V × V is the set of its edges indicating the social relations between theagents. A = { a ij } is the adjacency matrix of the graph i.e., a ij = 1 if ( i, j ) ∈ E , otherwise a ij = 0. N i = { j : ( i, j ) ∈ E } is the neighborhood of agent i , and ¯ N i = N i ∪ { i } . d i = (cid:80) j ∈ N i a ij is thedegree of node i , that is the number of her neighbors. We consider also a matrix S = { s ij } , withthe same sparsity pattern with the adjacency matrix A , which indicates the desire of each agent tomeet with each one of her neighbors.Social distancing is one of the most effective behavioral changes that people can adopt duringan epidemic outspread. However, as mentioned in the introduction, the choice to adopt this alteredbehavior is, in many cases, up to the agents. So, we consider a social distancing game, which isrepeated at each day during the outspread of the epidemic. The actions of the agents model theintensity of the relations with each one of their neighbors they choose to have at each day. So,denoting by k the current day, the action of agent i is a vector of length equal to the number of herneighbors given by: u i ( k ) = [ u ij ( k ) ...u ij di ( k )] ∈ [0 , d i , (1)where: N i = { j , ..., j d i } . According to the strategies chosen by the agents we have an induced weighted adjacency matrix W ( k ) = [ w ij ( k )] for the network, which indicates the meeting probabilities between two neighborsat day k , where w ij ( k ) have the following form: w ij ( k ) = , if a ij = 0 u ij ( k ) u ji ( k ) , if a ij = 1 (2)We consider that each agent has a health state consisted of two variables x i ( k ), which indicatesif the agent has been infected before day k and r i ( k ), which indicates the duration of her infection5nd consequently if she has recovered. Here we assume that all the infected agents recover after R days.The vector x = [ x i ] indicates the initial conditions for the x i state of the agents. The probability p x indicates the distribution of the initial conditions, which are i.i.d. random variables: x i = , w.p. 1 − p x , w.p. p x (3)The vector r = n indicates the initial conditions for the r i state of the agents.These states evolve as follows: x i ( k + 1) = x i ( k ) , w.p. (cid:81) j ∈ N i (1 − w ij ( k ) p c x j ( k ) X { r j ( k ) In this simple model, which is a discrete analogue of the SIR model on graphs, weassume that every infected agent recovers. That is to avoid changes in the graph topology, whichwould make the analysis of the game much more difficult. We expect this to cause minor differencesin the case of an epidemic with low mortality. In order to model the probable contamination of an agent j by her neighbor agent i , we makea similar assumption with the mean field approach [15], where the authors assume that the graphtopology has no loops and there is no correlation between the states of the agents. Thus, the6ontamination probability can be expressed as a function of the well known basic reproductionnumber R : p c ( R ) = 1 − (1 − R ¯ d ) R . (6)Similar derivations for the probabilities that govern the transmission of the disease over networksof interconnected agents are existing in the relevant bibliography, such as [5].We assume that the agents choose rationally their actions, based on the available information,by maximizing their payoffs. These payoffs are considered to depend solely on the benefits fromthe social interactions between the agents and on the costs to their health due to possible contam-ination. In reality, the decision of a behavioral change depends also on socioeconomic and ethicalconsiderations, which are omitted in this first approach, for the sake of simplicity. So, in our casethe instantaneous payoffs depend on two terms. The first one indicates the satisfaction that eachagent derives by the interaction with her neighbors, these benefits differ between her neighbors.The second term shows the costs an agent suffers if she has been infected. Since the agent does notknow her health state the next day, she tries to estimate it based on the available information. Theparameters G i indicate the importance of the infection for each agent. We divide the agents intotwo groups: the vulnerable (large G i ) and the ones who are non-vulnerable (small G i ). The gameis in fact dynamic since the payoffs depend on the evolving health states of the agents. However,the agents are considered to be myopic and able to predict just a day ahead so the payoffs have thefollowing form at each day: J i ( k ) = (cid:88) j ∈ N i s ij u ij ( k ) u ji ( k ) − G i E { x i ( k + 1) | I i ( k ) }X { r i ( k +1) The strategy profile u = (cid:80) d i is a Nash equilibrium for the game with perfect localstate feedback, since it results to indifference for all the agents. However, we can observe the existence of other Nash equilibria. Proposition 2. The best response of each agent always contains a point in { , } d i , i.e. the verticesof the action space. Therefore, there is no strict Nash equilibrium in [0 , (cid:80) d i \ { , } (cid:80) d i Proof. We calculate the first and second partial derivatives of J i : ∂J i ∂u ij = u ji s ij + G i ( x i − X { r + i If agent i is infected, x i = 1 and r i < R , then J i = (cid:80) j ∈ N i s ij u ij u ji − G i and if shehas been recovered, r i = R , it is assumed that she cannot get infected again. So, in these cases,an optimal strategy for her is u ij = 1 , ∀ j ∈ N i , since if u ji = 1 = ⇒ u ij = 1 and if u ji = 0 she isindifferent so she can also choose u ij = 1 . Remark 3. If agent i and agent j are neighbors and agent i is not infected ( x i = 0 ) and agent j is not infected ( x j = 0 ) or recovered ( r j = R ) the optimal strategies for their interaction are u ij = 1 and u ji = 1 , since if u ji = 1 : J i ( u ij = 1) − J i ( u ij = 0) = s ij > and if u ij = 1 : J j ( u ji = 1) − J j ( u ji =0) = s ji > . So defining the following sets:Infected i = { j ∈ N i : x j = 1 , r j < R } (12)and | Infected i | is the number of elements of Infected i , we conclude that: J i = J i ( u ij : j ∈ Infected i ) , since the rest strategies are fixed. In this case, the computation of the equilibrium strategies is asingle objective, multi-variable, integer optimization problem for each agent, which can be solved9asily using the following algorithm for each agent in O ( | Infected i | (log( | Infected i | ) + 1)) iterations: Algorithm 1: Solution of the optimization problem for each agent Result: The optimal strategies ( u ij ) ∗ for j ∈ Infected i Sort the parameters s ij , j ∈ Infected i in decreasing order;Define the sequence of indices j ...j | Infected i | to be the j-indices of the previous ordering;Define the strategies ¯ u i = Infected i , ¯ u ik = { u ij = 1 ...u ij k = 1 , u ij k +1 = 0 ...u ij | Infected i | =0 } , k = 1 ... | Infected i | ; k = 0;∆ J i = 1; while ∆ J i > and k ≤ | Infected i | do ∆ J i = s ij k − G i p c (1 − p c ) k ; k = k + 1; end ( u ij ) ∗ = ¯ u ik − ( j k = j ); Remark 4. The strategy profile u ij = max { x i , − x j } is a Nash equilibrium for the game withperfect local state feedback (9) , if ∀ i (cid:54)∈ ∪ i Infected i : max { s ij : j ∈ Infected i } < G i p c This equilibrium shows the phenomenon that in the case the agents are highly vulnerable to thedisease and they know the state of their neighbors, they communicate with all the healthy ones inorder to maximize their payoffs and the infected try to communicate also with their neighbors forthe same reason but they are banned by them. So, this equilibrium results to higher payoffs for thenon infected agents: J i = (cid:80) j ∈ N i s ij (1 − x j X { r j 12e proceed with the calculation of the best response for each agent. If agent i has m i of herneighbors playing p ju = 1 her payoff is:˜ J i ( p iu , m i ) = a i m i p iu + (1 − bp iu ) m i . Thus: ˜ J i (0 , m i ) = 1 , ˜ J i (1 , m i ) = a i m i + (1 − b ) m i . We define the following functions: f i ( m ) = ˜ J i (1 , m ) = a i m + (1 − b ) m = a i m + e m ln(1 − b ) (21)The best response of each agent is: BR i ( m i ) = , if f i ( m i ) > , otherwise (22)So, we have the following algorithm for the computation of the strategies corresponding to a Nashequilibrium: Algorithm 2: Computation of the NE strategies for the game with information for the dis-tribution of the states Result: The optimal strategies p i ∗ u Set p iu = 1, ∀ i Compute f i ( m i ), ∀ i ( m i = d i ) while ∃ f i ( m i ) ≤ doif f i ( m i ) ≤ then Set p iu = 0 end Compute new m i , ∀ i Compute new f i ( m i ), ∀ i endProposition 3. There exists a Nash equilibrium of the game with statistical information for thedistribution of the states. Furthermore, Algorithm 2 converges to the Nash equilibrium in O ( N ) steps. roof. To prove this proposition we firstly prove the following lemma: Lemma 1. For the functions f i ( m ) , defined in (21) , there exists a unique m ∈ R + such that f ( m ) = 1 and for all m > m , m ∈ N : f ( m ) > Proof. It is easily observed that f i ( m ) is convex and f i (0) = 1 for each i . So, if f (cid:48) i (0) ≥ ⇒ f i ( m ) > ∀ m , in this case m = 0. Else if f (cid:48) i (0) ≤ ⇒ ∃ ! m ∈ R ∗ + : f ( m ) = 1 and ∀ m > m , m ∈ N : f ( m ) > f i ( m ).Due to this lemma, beginning with the maximum feasible value for m i (which is d i ) the changesin the agents strategies from 1 to 0 can result only in the decrease of their neighbors m j ’s and thusit is possible to happen only one change of strategy (1 → 0) for each agent until f i ( m i ) ≥ ∀ i , sothe algorithm converges. Moreover, due to this observation, in the worst case the ‘while-loop’ willrun N times and so the algorithm will converge in O ( N ) steps.The point that the algorithm converges is a Nash equilibrium of the game, since the agents actionsare their best responses to their active contacts numbers m i ’s and for this profile of m i ’s no agentwill be benefited from a unilateral deviation from her action.Furthermore, we should point that, since the algorithm is in fact a descent on the possible m i -profiles, i.e. it initializes with all the contacts being active ( m i = d i , ∀ i ) and each m i decreases orstays the same, the Nash equilibrium that the algorithm converges is the one corresponding to themaximum possible sociability for the agents. Remark 5. If for each agent i it holds that s i d i + G i (1 − p x )(1 − p r )[(1 − p c p x (1 − p r )) d i − > then the strategy profile p iu = 1 , ∀ i is a N.E. of that game. Proposition 4. The strategy profile p iu = 0 , ∀ i is again a N.E., since it results to indifferencebetween the unilateral changes of each agent. In this section we present several simulations for the social distancing games under the two differentinformation structures in order to compare the disease prevalence and the agents payoffs in bothcases, as well as the importance of some parameters of the model. For these simulations we considera repeated version of this game. The payoffs of the agents in this case have the form (7), indicating14he myopic behavior for the agents, who cannot predict the future consequences of their actions.The strategies considered in the following simulations are the Nash Equilibrium strategies of thestatic games of the previous sections repeated at each step of each game. The following two remarksdescribe these strategies.For the game with perfect local information we consider the following strategies: Remark 6. 1. The strategy profile u ( k ) = (cid:80) Ni =1 d i , k = 1 ...K .2. The strategy profile u ∗ ( k ) , k = 1 ...K : u ij ( k ) ∗ = , if x i ( k ) = 1 or { x i ( k ) = 0 and ( r j ( k ) = 0 or r j ( k ) = 0) } The solution of algorithm 1 , if x i ( k ) = 0 and ≤ r j ( k ) < R In the execution of algorithm 1 in this equation, the set Infected i is defined as follows:Infected i = { j ∈ N i : 1 ≤ r j ( k ) < R } 3. Any step-wise interchange of the two previous strategy profiles. and for the game with statistical information we consider the following strategies: Remark 7. 1. The strategy profile u ( k ) = (cid:80) Ni =1 d i , k = 1 ...K .2. The strategy profile u ∗ ( k ) , k = 1 ...K : The solution of Algorithm 2, where p x = p x ( k ) followsthe rule (16) .3. Any stepwise interchange of the two previous strategy profiles. In practice, since both algorithms are initialized with all the social contacts being active, theyconverge in the second category of strategies for both cases.The simulations presented here have the following parameters. The underlying graph topologyis a random graph [44] with N = 1000 agents and adjacency probability p = 1%, and thus averagedegree ¯ d = 10. The recovery period is 14 days. The sociability parameters s ij are random numbersin (0 , G i = 10000 and for the non-vulnerable G i = 1000. The percentage of the vulnerablein the community is 20%. The initial percentage of infected agents is 1%. The basic reproduction15umber of the disease R = 2, so as for the disease to be epidemic and for social distancing to benecessary.In Figure 1 we indicate the effects of the social distancing games with statistical informationand with perfect local information on the disease prevalence and on the sociability of the agents andcompare these two games. In Table 5 we present some numerical characteristics of these curves. days p e r c e n t a g e o f i n f e c t e d a g e n t s no gamestatisticalinfofull info0 50 100 150 200 250 days p e r c e n t a g e o f r e c o v e r e d a g e n t s no gamestatisticalinfofull info0 50 100 150 200 250 days p e r c e n t a g e o f a c t i v e c o n t a c t s statisticalinfofull info Figure 1: Infection, recovery and sociability curves for the game with information for the distributionof the statesCharacteristics of Infection and Sociability for the Games with Different InformationInfection Peak Total Infection Minimum SociabilityNo game 30.3% 78.7% 100%Perfect local feedback 9.1% 43.7% 94.2%Information for the distri-bution of the infected 4.2% 33.5% 22.9%Table 1: Table to compare the infection and sociability for the games with different informationWe conclude that the game with information for the distribution of the states almost extinguishthe epidemic behavior of the disease - the infection curve is similar with the infection curve of adisease with R ≤ . . 2% in the other case. This remarkable difference on the agentsbehavior affects significantly their payoffs. As we observe in Figure 2, the average payoff of the gamewith perfect local information are much higher than the average payoff of the game with statisticalinformation. Moreover, we must underline that for the vulnerable agents the difference of theirpayoffs between the two games is large, as they pay much higher the cost of not being informedabout the health state of their contacts and get infected. 1: perfect local info2: statistical info x Average payoff 1: perfect local info2: statistical info Non-vulnerable agents average payoff 1: perfect local info2: statistical info -50510 10 Vulnerable agents average payoff x x Figure 2: Comparison of the payoffs of the agents for the two gamesDue to that fact the vulnerable agents can be considered as key players for these games, sincethey tend to play conservatively and thus enhance the social distancing. So, in Figure 3 we showthe effect of the percentage of vulnerable agents in the community to the infection peak and to thetotal number of infected agents for the game with perfect local information and in Figure 4 we showthe same effects for the game with statistical information. In these figures the mean value of 30simulations is depicted at each point of the plots and the red lines are the linear regression curvesfor our experiments on different percentages.The result is the expected one, that in both games the percentage of the vulnerable agents is17 percentage of vulnerable agents p e a k o f i n f e c t e d a g e n t s peakslinear regression percentage of vulnerable agents t o t a l nu m b e r o f i n f e c t e d a g e n t s total numberslinear regression Figure 3: Correlation of the percentage of vulnerable agents with the infection outspread for theperfect local feedback information game percentage of vulnerable agents p e a k o f i n f e c t e d a g e n t s peakslinear regression percentage of vulnerable agents t o t a l nu m b e r o f i n f e c t e d a g e n t s total numberslinear regression Figure 4: Correlation of the percentage of vulnerable agents with the infection outspread for thegame with information for the distribution of the statesnegatively correlated to the disease outspread. However, in the game with statistical information(Figure 4) their role seem to be less important than in the perfect local information game, becausethe variations of the peaks and the total number of infected agents with respect to the vulnerable18gents percentage are by far smaller. In the previous sections, we have analyzed and compared the Nash equilibrium strategies of theagents for the cases of perfect local state feedback information and statistical information for thedistribution of the states. In this section, we consider several variations of the initial problem andexamine, through simulations, the effects of the varying parameters on the behavior of the agents andon the outspread of the epidemic. The first variation we study concerns the quality of the availableinformation, for the case that the agents possess statistical information for the distribution of thestates. The second variation considers the risk perception, modeled by the vulnerability parameters,to depend on the infection outspread and the capacity of health care system. And the third variationtakes under consideration the effects of the graph topology. A first modified scenario we examine is the case that the information the agents possess about thedistribution of the states is fake or biased. This is an interesting and in some cases realistic scenario,since the agents are rarely able or have the time to investigate verified data about the outspreadof the disease, but they usually get informed through mass media or social media. Consequently,the information they get is usually exaggerated or understated. The spread of fake news is anotherfactor affecting the information quality and thus the decisions of the agents. Moreover, in manycases the lack of diagnostic tests in the community makes the knowledge of the accurate infectionlevel impossible.So, we consider the following modification of the model of section 4: p fx = f p x (23)where p fx is the available fake information of the agents and f is a coefficient indicating its deviationfrom the actual information p x . So, we get the following simulations (Figure:5) indicating the effectsof an overestimation of the infection level ( f = 2) and an underestimation of the infection level( f = 0 . days p e r c e n t a g e o f i n f e c t e d a g e n t s f=0.5f=1f=2 days p e r c e n t a g e o f r e c o v e r e d a g e n t s f=0.5f=1f=2 days p e r c e n t a g e o f a c t i v e c o n t a c t s f=0.5f=1f=2 Figure 5: Infection, recovery and sociability curves for games with fake information for the distri-bution of the statesof the previous curves: Fake Information ConsequencesInfection Peak Total Infection Minimum SociabilityActual Information 3.7% 25.6% 23.9%Overestimation of Infection (2 p x ) 1.9% 11.1% 19.5%Underestimation of Infection (0 . p x ) 5.9% 36.5% 52.6%Table 2: Table to compare the infection for games with actual and fake informationWe observe that in the case of an overestimation of the infection level the agents care moreto follow social distancing and the disease prevalence is kept at low levels, while in the case ofunderestimation of the infection the agents do not care so much and the disease prevalence ishigher. It seems rational that the agents are more benefited from a low prevalence so it may beprofitable for them to receive an overestimation of the infection level. However, when applying socialdistancing they pay the costs of the effort to eradicate the epidemic, so it is interesting to examinehow their payoffs vary when they receive fake or biased information. From Figure 6, we observe20 fake info coefficient (f) Average payoff fake info coefficient (f) Non-vulnerable agents average payoff fake info coefficient (f) -1.5-1-0.500.511.52 x Vulnerable agents average payoff xx Figure 6: Payoffs of the agents in the case of fake informationthat the non vulnerable agents are slightly benefited from an underestimation of the infection level,since they are not precarious and their profits depend mostly on their social contacts. However,they have also small gains in the case of an overestimation of the infection level since they stay safeand not infected. Contrary to the non vulnerable agents, the vulnerable agents are damaged froman underestimation of the infection level, since they may get infected and suffer a lot and they arebenefited significantly from an overestimation of the infection level, which scares the whole societyand leads to strict social distancing, saving them this way from a possible infection.It is also interesting to point out the correlation of the infection outspread and the agents’ payoffswith the fake information coefficient ( f ). As we can see in Figure 7 the infection outspread andthe fake information coefficient have a definitely negative correlation, with overestimation leadingto very low infection levels and underestimation to high infection levels. Moreover, interestingly,the agents’ payoffs present a minimum average value when the information is near to the actualone and they seem to be benefited from fake or biased information. However, we should point outhere that the average payoff of all the agents is illustrated in Figure 7 and this is the reason forthe existence of that minimum, since, according to Figure 6, the non vulnerable agents - who arethe majority - are slightly benefited from a small f and the vulnerable agents - even if they areminority - are benefited significantly form a larger f , so the average payoff has this behavior.These observations could be useful for a social planer, with an aim to avoid the spreading of the21 fake info coefficient p e a k o f i n f e c t e d a g e n t s fake info coefficient t o t a l nu m b e r o f i n f e c t e d a g e n t s fake info coefficient a v e r a g e p a y o ff s x Figure 7: Correlation of the coefficient of fake information with the infection outspread and theagents’ payoffsdisease, in order to achieve social distancing without imposing it by law but by manipulating theagents strategies through the broadcasted fake or biased information. Another scenario is that the vulnerability parameters of the agents ( G i ) depend on the level ofinfection in the community. This is an interesting scenario in practice, since the health systemsworldwide have finite (and usually small) capacity, so if the number of infected agents who needhealth care pass a certain level it is not probable for the next agents who will get infected to haveaccess in the necessary facilities.We model this phenomenon considering the vulnerability parameters as functions of the infectionratio, in the model of section 4 . At first, we examine the case of linear dependence: G i = G i ( p x ) = G i αp x (24) G i are the constant vulnerability parameters used in all the previous simulations. Choosing α = p ref x we can define a reference infection level p ref x , where the agents will play as in the case of22onstant vulnerability parameters G i . Below this level, they will be indifferent for the effects of thedisease on them and care more for their social interactions and above this level they will be moreworried about the disease and follow social distancing strategies.This is confirmed by Figure: 8, where all the parameters, except the vulnerability parameters,are the same with the other simulations, as in section 5. days p e r c e n t a g e o f i n f e c t e d a g e n t s days p e r c e n t a g e o f r e c o v e r d a g e n t s days p e r c e n t a g e o f a c t i v e c o n t a c t s Figure 8: Infection, recovery and sociability curves when the vulnerability parameters have a pro-portional dependence on the infection outspreadIt is interesting to point that the agents do keep the infection level below the reference value, inthis simulation p ref x = 5%, for all the time.We also examine the case of step-function dependence, where there exists a critical infectionlevel p cr x above which the agents play with parameters G i and below which they care very littleabout the effects of the disease on them. G i = G i ( p x ) = G i /M , p x < p cr x G i , else (25)where M is a large number e.g., M = 100. In Figure 9, p cr x = 5%.It should be pointed here that in the first case (Fig:8) the peak of the infection is 4 . 3% and thetotal number of infected agents is 41 . . 1% and thetotal number of infected agents is 52 . 50 100 150 200 250 300 350 days p e r c e n t a g e o f i n f e c t e d a g e n t s days p e r c e n t a g e o f r e c o v e r e d a g e n t s days p e r c e n t a g e o f a c t i v e c o n t a c t s Figure 9: Infection, recovery and sociability curves when the vulnerability parameters have a step-wise dependence on the infection outspreadthe second case. Except of the agents strategies of social distancing, another very important factor which affectsthe outspread of the disease is the topology of the underlying network which represents the socialinteractions of the agents. So, we present here some simulations to indicate the effects of the graphtopology on the spreading of the disease and the effectiveness of the agents strategies.At first, we do not consider a social distancing game, so every two neighbors communicate freelywith each other ( w ij = a ij ). In Figure 10 we observe the infection curves for four different graphtopologies: random graph [44], stochastic block model, scale free network [47],[45] and small worldnetwork [48]. In every case we have chosen the network parameters in a way that the graphs havealmost the same average degree ( ¯ d ≈ 50 100 150 200 250 days p e r c e n t a g e o f i n f e c t e d a g e n t s random graphstochastic blockmodelscale free networksmall world network days p e r c e n t a g e o f r e c o v e r e d a g e n t s random graphstochastic blockmodelscale free networksmall world network Figure 10: Infection curves for different graph topologies for the case of no social distacningTopology Comparison (No Social Distancing)Average Degree Infection Peak Total InfectionRandom Graph 9.8 36.2% 82.8%Stochastic Block Model 10.2 22.1% 74.1%Scale Free Network 9.9 41.3% 72.3%Small World Network 10 17.9% 80.9%Table 3: Table to compare the infection for different graph topologies in the case of no socialdistancingalmost the same degrees, present a much lower but extended infection curve. However, the totalnumbers of infected agents are almost the same. Moreover, the stochastic block model - being an illconnected coalition of well connected random graphs - has a similar, in shape, reaction curve withthe random graph, but with lower peak and significantly lower total number of infected agents.Next, we proceed with the numerical study of the social distancing games with the two differentinformation structures. We begin with the game with perfect local information, illustrated in Figure11 and Table 4.From these simulations we can derive the following conclusions. Firstly, in the scale free networkthe disease spreads quickly, even in the case that the agents follow social distancing strategies. Thus,25 50 100 150 200 250 days p e r c e n t a g e o f i n f e c t e d a g e n t s random graphstochastic block modelscale free networksmall world network days p e r c e n t a g e o f r e c o v e r e d a g e n t s days p e r c e n t a g e o f a c t i v e c o n t a c t s Figure 11: Infection curves for different graph topologies for the game with perfect local informationTopology Comparison (Perfect Info Game)Infection Peak Total Infection Minimum SociabilityRandom Graph 10.2% 42.2% 94.8%Stochastic Block Model 6.1% 38.8% 96.3%Scale Free Network 21.7% 49.8% 85.7%Small World Network 4.1% 18.9% 97.5%Table 4: Table to compare the infection for different graph topologies for the case of the perfectlocal information gamewe observe the highest peak, the greatest total prevalence and consequently the lowest level on theagents sociability in their effort to flatten this curve. In the cases of small world network andstochastic block model the infection peaks are low. Thus, the agents are not so concerned aboutthe disease and relax their social distancing, resulting in second waves of the epidemic in bothcases. However, in the case of the stochastic block model the duration of the waves is larger andconsequently the total prevalence is also larger. Finally, in the case of the random graph we observea greater peak than in the stochastic block model, but due to the response of the agents and theabsence of remote cliques - which may act as sources of infection - the disease is eliminated in thiscase and we do not observe a second wave, resulting in a similar total prevalence with the stochastic26lock model.Following that, we present simulations for the game with statistical information, illustrated inFigure 12 and Table 5. days p e r c e n t a g e o f i n f e c t e d a g e n t s random graphstochastic block modelscale free networksmall world network days p e r c e n t a g e o f r e c o v e r e d a g e n t s days p e r c e n t a g e o f a c t i v e c o n t a c t s Figure 12: Infection curves for different graph topologies for the game with statistical informationTopology Comparison (Statistical Info Game)Infection Peak Total Infection Minimum SociabilityRandom Graph 3.5% 31.2% 36.1%Stochastic Block Model 4.0% 18.3% 23.4%Scale Free Network 4.6% 19.9% 17.2%Small World Network 2.9% 7.1% 52.3%Table 5: Table to compare the infection for different graph topologies for the case of the game withstatistical informationThere are several interesting observations for this case also. At first, in all graph topologies theinfection level is kept very low but with a high cost on the sociability of the agents. Secondly, theredo not exist significant differences on the peaks of the infection but there exist on the epidemic’s du-ration, affecting this way its total prevalence. In the small world network the epidemic is eliminatedquickly and with a comparatively small effort from the agents, resulting in a low total prevalence.27n the scale free network and the stochastic block model the epidemic is also eliminated but hasa relatively larger duration, resulting in similarly larger prevalence. Finally, in the random graphtopology the epidemic lasts long and has the greatest prevalence. This phenomenon is probablya result of the well mixing of the agents, that arises more in the random graph topology, whichcontributes to the persistence of the disease even in the case that the agents cut several of theircontacts. A game-theoretic approach of social distancing has been considered. In the simple model proposed,the main parameters under examination are the network describing the structure of the interactionsamong the agents, which changes according to their rationally chosen strategies and the availableinformation during the decision making. The effects of the spontaneous social distancing behavior onthe prevalence of the epidemic is investigated both analytically and numerically through simulationson artificial networks. At the current stage, the proposed model is not intended for quantitativepolicy suggestions, since on the one hand it is simplistic and on the other hand the knowledge ofrealistic values for the parameters modeling human behavior requires real observations, many dataand a proper statistical processing of them. However, it may be useful to offer a paradigm on theway the agents decide to adopt social distancing and the effects of these decisions on the prevalenceof an epidemic. References [1] D. Bernoulli, “Essai d’une nouvelle analyse de la mortalit´e caus´ee par la petite v´erole, et desavantages de l’inoculation pour la pr´evenir,” Histoire de l’Acad., Roy. Sci.(Paris) avec Mem ,pp. 1–45, 1760.[2] W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epi-demics,” Proceedings of the royal society of london. Series A, Containing papers of a mathe-matical and physical character , vol. 115, no. 772, pp. 700–721, 1927.283] R. Ross, “An application of the theory of probabilities to the study of a priori pathometry,” Proceedings of the Royal Society of London. Series A, Containing papers of a mathematicaland physical character , vol. 92, no. 638, pp. 204–230, 1916.[4] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, “Epidemic processesin complex networks,” Reviews of modern physics , vol. 87, no. 3, p. 925, 2015.[5] M. E. Newman, “Spread of epidemic disease on networks,” Physical review E , vol. 66, no. 1, p.016128, 2002.[6] C. Moore and M. E. Newman, “Epidemics and percolation in small-world networks,” PhysicalReview E , vol. 61, no. 5, p. 5678, 2000.[7] L. A. Meyers, M. Newman, and B. Pourbohloul, “Predicting epidemics on directed contactnetworks,” Journal of theoretical biology , vol. 240, no. 3, pp. 400–418, 2006.[8] G. P. Garnett and R. M. Anderson, “Sexually transmitted diseases and sexual behavior: in-sights from mathematical models,” Journal of Infectious Diseases , vol. 174, no. Supplement 2,pp. S150–S161, 1996.[9] L. Sander, C. Warren, I. Sokolov, C. Simon, and J. Koopman, “Percolation on heterogeneousnetworks as a model for epidemics,” Mathematical biosciences , vol. 180, no. 1-2, pp. 293–305,2002.[10] J. M. Epstein, “Modelling to contain pandemics,” Nature , vol. 460, no. 7256, pp. 687–687,2009.[11] J. M. Epstein, J. Parker, D. Cummings, and R. A. Hammond, “Coupled contagion dynamicsof fear and disease: mathematical and computational explorations,” PLoS One , vol. 3, no. 12,2008.[12] O. M. Cliff, N. Harding, M. Piraveenan, E. Y. Erten, M. Gambhir, and M. Prokopenko,“Investigating spatiotemporal dynamics and synchrony of influenza epidemics in australia: Anagent-based modelling approach,” Simulation Modelling Practice and Theory , vol. 87, pp. 412–431, 2018. 2913] H. Zhang, J. Zhang, C. Zhou, M. Small, and B. Wang, “Hub nodes inhibit the outbreak ofepidemic under voluntary vaccination,” New Journal of Physics , vol. 12, no. 2, p. 023015, 2010.[14] S. L. Chang, M. Piraveenan, and M. Prokopenko, “Impact of network assortativity on epidemicand vaccination behaviour,” arXiv preprint arXiv:2001.01852 , 2020.[15] F. Bagnoli, P. Lio, and L. Sguanci, “Risk perception in epidemic modeling,” Physical ReviewE , vol. 76, no. 6, p. 061904, 2007.[16] C. T. Bauch and D. J. Earn, “Vaccination and the theory of games,” Proceedings of the NationalAcademy of Sciences , vol. 101, no. 36, pp. 13 391–13 394, 2004.[17] C. T. Bauch, A. P. Galvani, and D. J. Earn, “Group interest versus self-interest in smallpoxvaccination policy,” Proceedings of the National Academy of Sciences , vol. 100, no. 18, pp.10 564–10 567, 2003.[18] T. C. Reluga, C. T. Bauch, and A. P. Galvani, “Evolving public perceptions and stability invaccine uptake,” Mathematical biosciences , vol. 204, no. 2, pp. 185–198, 2006.[19] T. C. Reluga and A. P. Galvani, “A general approach for population games with applicationto vaccination,” Mathematical biosciences , vol. 230, no. 2, pp. 67–78, 2011.[20] H. Zhang, F. Fu, W. Zhang, and B. Wang, “Rational behavior is a ‘double-edged sword’whenconsidering voluntary vaccination,” Physica A: Statistical Mechanics and its Applications , vol.391, no. 20, pp. 4807–4815, 2012.[21] P. E. M. Fine and J. A. Clarkson, “Individual versus public priorities in the determination ofoptimal vaccination policies,” American journal of epidemiology , vol. 124, no. 6, pp. 1012–1020,1986.[22] M. Kremer, “Integrating behavioral choice into epidemiological models of aids,” The QuarterlyJournal of Economics , vol. 111, no. 2, pp. 549–573, 1996.[23] R. Vardavas, R. Breban, and S. Blower, “Can influenza epidemics be prevented by voluntaryvaccination?” PLoS computational biology , vol. 3, no. 5, 2007.3024] S. Del Valle, H. Hethcote, J. M. Hyman, and C. Castillo-Chavez, “Effects of behavioral changesin a smallpox attack model,” Mathematical biosciences , vol. 195, no. 2, pp. 228–251, 2005.[25] F. H. Chen, “Rational behavioral response and the transmission of stds,” Theoretical populationbiology , vol. 66, no. 4, pp. 307–316, 2004.[26] S. Funk, M. Salath´e, and V. A. Jansen, “Modelling the influence of human behaviour on thespread of infectious diseases: a review,” Journal of the Royal Society Interface , vol. 7, no. 50,pp. 1247–1256, 2010.[27] T. C. Reluga, “Game theory of social distancing in response to an epidemic,” PLoS computa-tional biology , vol. 6, no. 5, 2010.[28] P. Poletti, M. Ajelli, and S. Merler, “The effect of risk perception on the 2009 h1n1 pandemicinfluenza dynamics,” PloS one , vol. 6, no. 2, 2011.[29] ——, “Risk perception and effectiveness of uncoordinated behavioral responses in an emergingepidemic,” Mathematical Biosciences , vol. 238, no. 2, pp. 80–89, 2012.[30] P. Poletti, B. Caprile, M. Ajelli, A. Pugliese, and S. Merler, “Spontaneous behavioural changesin response to epidemics,” Journal of theoretical biology , vol. 260, no. 1, pp. 31–40, 2009.[31] S. Funk, E. Gilad, C. Watkins, and V. A. Jansen, “The spread of awareness and its impact onepidemic outbreaks,” Proceedings of the National Academy of Sciences , vol. 106, no. 16, pp.6872–6877, 2009.[32] F. H. Chen, “Modeling the effect of information quality on risk behavior change and thetransmission of infectious diseases,” Mathematical biosciences , vol. 217, no. 2, pp. 125–133,2009.[33] A. d’Onofrio and P. Manfredi, “Information-related changes in contact patterns may triggeroscillations in the endemic prevalence of infectious diseases,” Journal of Theoretical Biology ,vol. 256, no. 3, pp. 473–478, 2009.[34] F. Fu, D. I. Rosenbloom, L. Wang, and M. A. Nowak, “Imitation dynamics of vaccinationbehaviour on social networks,” Proceedings of the Royal Society B: Biological Sciences , vol.278, no. 1702, pp. 42–49, 2011. 3135] M. van Boven, D. Klinkenberg, I. Pen, F. J. Weissing, and H. Heesterbeek, “Self-interest versusgroup-interest in antiviral control,” PLoS One , vol. 3, no. 2, 2008.[36] T. Philipson and R. Posner, Private choices and public health: The AIDS epidemic in aneconomic perspective . Harvard University Press, 1993.[37] P.-Y. Geoffard and T. Philipson, “Rational epidemics and their public control,” Internationaleconomic review , pp. 603–624, 1996.[38] ——, “Disease eradication: private versus public vaccination,” The American Economic Re-view , vol. 87, no. 1, pp. 222–230, 1997.[39] S. L. Chang, M. Piraveenan, P. Pattison, and M. Prokopenko, “Game theoretic modellingof infectious disease dynamics and intervention methods: a review,” Journal of BiologicalDynamics , vol. 14, no. 1, pp. 57–89, 2020.[40] T. Gross, C. J. D. D’Lima, and B. Blasius, “Epidemic dynamics on an adaptive network,” Physical review letters , vol. 96, no. 20, p. 208701, 2006.[41] L. B. Shaw and I. B. Schwartz, “Fluctuating epidemics on adaptive networks,” Physical ReviewE , vol. 77, no. 6, p. 066101, 2008.[42] D. H. Zanette and S. Risau-Gusm´an, “Infection spreading in a population with evolving con-tacts,” Journal of biological physics , vol. 34, no. 1-2, pp. 135–148, 2008.[43] C. E. Somarakis, G. P. Papavassilopoulos, and F. E. Udwadia, “Nonlinear dynamics of a nwecellular automata model,” ECMS Conf., Nicosia, Cyprus , June 2008.[44] P. Erd˝os and A. R´enyi, “On the evolution of random graphs,” Publ. Math. Inst. Hung. Acad.Sci , vol. 5, no. 1, pp. 17–60, 1960.[45] A.-L. Barab´asi, R. Albert, and H. Jeong, “Scale-free characteristics of random networks: thetopology of the world-wide web,” Physica A: statistical mechanics and its applications , vol.281, no. 1-4, pp. 69–77, 2000.[46] M. Renardy and R. C. Rogers, An introduction to partial differential equations . SpringerScience & Business Media, 2006, vol. 13. 3247] A.-L. Barab´asi et al. , The scale free property, Chapt. 4, in Network science . Cambridgeuniversity press, 2016.[48] D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’networks,” naturenature