Dynamics of Panic Pedestrians in Evacuation∗ Dongmei Shi† Department of Physics, Bohai University, Jinzhou Liaoning, 121000, P. R. C Wenyao Zhang Department of Physics, University of Fribourg, Chemin du Musee 3, CH-1700 Fribourg, Switzerland. Binghong Wang Department of Modern Physics, University of Science and Technology of China, 230026, Hefei Anhui, P.R.C (Dated: January 6, 2017) Amodifiedlatticegasmodelisproposedtostudypedestrianevacuationfrom asingleroom. The 7 payoffmatrixinthismodelrepresentsthecomplicated interactionsbetweenselfishindividuals,and 1 themeanforceimposedonanindividualisgivenbyconsideringtheimpactsofneighborhoodpayoff, 0 walls, and defector herding. Each passer-by moves to his selected location according to the Fermi 2 function, and the average velocity of pedestrian flow is defined as a function of the motion rule. n Two pedestrian types are included: cooperators, who adhere to the evacuation instructions; and a defectors, who ignore the rules and act individually. It is observed that the escape time increases J withthepaniclevel,andthesystemremainssmoothforalowpaniclevel,butexhibitsthreestages 5 for a high panic level. We prove that the panic level determines the dynamics of this system, and the initial density of cooperators has a negligible impact. The system experiences three phases, a ] singlephaseofcooperator,amixedtwo-phasepedestrian,andasinglephaseofdefectorsequentially G as the panic level upgrades. The phase transition has been proven basically robust to the changes C of empty site contribution, wall’s pressure, and noise amplitude in the motion rule. It is further shown that pedestrians derivethegreatest benefitfrom overall cooperation, butare trapped in the . n worst situation if they are all defectors. li PACS numbers: 87. 23. Ge, 02. 50. Le, 87. 23. Cc, 89. 90. +n n [ I. INTRODUCTION such as fear, as well as the complicated forces among 1 v individuals. Helbing used social force to simulate such 6 Evacuationofpedestriansunderpanichasbeenexten- interactions [2], and Fukui avoided force concepts by 3 sively studied [1–20], because an understanding of the applying the CA model [20]. Game theory was pro- 2 dynamic features of this phenomenon can reduce the in- posedtostudythedynamicsincrowsevacuation[32–37]. 1 cidence of injury and death. Escape panic can occur in Helio¨vaarawe presented a spatial game theoretic model 0 . different confinement sizes ranging from a rioting crowd for pedestrian behavior in situations of exit congestion 1 in a packed stadium to stunned customers in a bar. Es- [32], and Hao etal applied game theory to deal with the 0 capepanicischaracterizedbystrongcontactinteractions conflicts that arise when two pedestrians try to occupy 7 1 betweenselfishindividuals thatquicklygiverisetoherd- the same position[37]. : ing, stampede, and clogging [21–31]. Experimental ob- In this study, we modify the model [38] in which the v servationsandnumericalsimulationsaretwoapproaches effects of walls and defector herding are considered. A i X that have been widely applied to study this issue. Simu- defector herding unit is defined as a group of four indi- r lationresultshaverevealedsomeinterestingdynamicfea- viduals who are all defectors in the same neighborhood. a tures,suchaspedestrianarchformationaroundtheexit, The potential well energy or mean force imposed on an herding,andinterferencebetweenarchesinmultiple-exit individual is defined by considering three aspects. An rooms [2]. Disruptive interference, self-organized queu- individual will move to his selected location with a large ing, and scale-free escape dynamics [26, 27] have also probability if his personal energy is higher than the po- been observed. Experiments in genuine escape panic are tentialwellenergy. The velocityofa walkeris defined as difficult to set up because of ethical and legal concerns. afunctionofthemotionrule,andaveragevelocityofthe However, escape panic can be simulated by solving a pedestriansystemcanreflectdifferentdynamics,suchas set of coupled differential equations [2, 3] or by applying smoothflow,andcongestion. Wefindthatthepaniclevel the cellular automata (CA) technique [20, 26], whereby is crucial in determining the dynamics of the evacuation the movement of confined pedestrians is tracked over systemandthecompetitionbetweencooperatorsandde- time. An important task related to evacuation simula- fectors. Itisalsoobservedthatthreephasesofpedestrian tions is how to describe the effects of subjective factors, are exhibited sequentially as panic level upgrades. The remainderof the paper is organizedas follows. In Section II, we describe the model. Our numerical simu- ∗ DynamicsofPanicpedestrians lationsandanalysisarepresentedinSectionIII. Conclu- † sdm21@mail.ustc.edu.cn sions are drawn in Section IV. 2 II. MODEL Pedestrian evacuation is studied on a model of square lattice on which all the individuals are distributed uniformly. Two pedestrian types are considered: cooperators, who adhere to the evacuation instructions; and defectors, who ignore the rules and act individually. Each pedestrianonlyinteractswithhis nearestneighbors,and is deleted from the system if he arrives at the exit. We apply game theory, and the payoff matrix representing the interactions among the crowd is shown in Table 1. To be specific, a cooperatorreceives a Reward(R) when FIG. 1. Schematic presentation of a defector herding unit in he interacts with a collaborative neighbor, but suffers a which all the individuals are defectors who are circled in the Sucker(S)ifhe encountersa defectorwhogainsa payoff figure. PD is the average payoff of the defector centered in thecircle. of Temptation (T). Two defectors receive the Punish- ment (PM), respectively. In addition, a walker will gain a payoff e (0 < e < R) if there is an empty site in his a defector. Therefore, the potential well energy U will x neighborhood. Usually, T = 1+r, R = 1, S = 1−r, includeU ifadefectorherdingunitexistsinadefector’s du PM = 0, where 0 < r < 1. r is cost-to-benefit ratio neighborhood: indicating the defector-cooperator payoff divide, which describes the panic level during the escape. e = R/2 is set in this model. 1 T U = (n · )·s, (2) du du G 4 x TABLEI. Payoff matrix where n is the number of defector herding units in the du neighborhoodofx. Theparameterstakesthevalues=0 if x chooses to cooperate, and s = 1 when x decides to C D E defect. C (R, R) (S,T) (e, −) In addition, we consider the contribution of walls to D (T, S) (PM, PM) (e, −) a Ux, so Ux will include Uw if walls exist in the neighborhood of x: a Cindicatesanindividualwhocooperates, andDindicates an individualwhochooses todefect. Erepresentsanemptysite. 1 T >R>S>PM Uw = ·nw·Uw, (3) G x where n is the number of walls in the neighborhood P denotes the personal energy of individual x, and w x and U (0 < U < R) is the contribution of a wall to is the average of cumulative payoff that x receives from w w U . Combining all the quantities discussed above, the all his neighbors. Each individual stays in a potential x potential well energy U is given by wellwhichisformedbytheneighboringpeopleandwalls x (if walls exist in his neighborhood). U is the energy of x potential well x stays in, which consists of three parts. U =U +U +U , (4) Firstly, U isthe averagecumulativepayoffforallneigh- x p du w p bors of x, which represents the mean forces on x: Itshouldbenotedthatx′spersonalenergyisexcludedin U which actually reflects the pressures from x′s neigh- x 1 borhood. Up = X Pxj (1) At each time step, toward the exit’s position, each in- G x j∈Gx dividualxrandomlyselectsacell y intheneighborhood, and moves to y with probability W [38]: where G is the scale of the interacting group centered x on x, which is defined as the number of directly linked neighbors of x (excluding the empty sites). P is the 1 xj W = , (5) average of cumulative payoff that j receives from all his 1+exp(−(P −U )/κ) x x neighbors, where individual j belongs to the neighborhood of x. where κ is the noise amplitude. κ = 0 indicates deter- A defector herding unit is defined as a group in which minedoccupation,whereasκ=∞denotesstochasticoc- fourdefectorsareinthesameneighborhood(Fig. 1). We cupation. It is apparent that the larger P is, the larger x consider that a defector herding unit contributes P = W is. The equation (5) means that if x owns more per- D T/4 to the energy of the potential well for a cooperator, sonal energy than the potential well energy, x will move but contributes 0 to the energy of the potential well for to his selected location with a larger probability. If x 3 2000 r=0.1 2000 r=0.5 4800 IC=0.5 ors 1500 1500 ct Time44460000 ors \ Defe 10500000 CDoefoepcetroartor 15000000 pe erat 1 10 100 1000 1 10 100 1000 a4200 p Esc Coo 2000 r=0.65 2000 r=0.9 4000 of 1500 1500 er 1000 1000 b 3800 um 500 500 N 0.0 0.2 0.4 0.6 0.8 1.0 0 0 1 10 100 1000 1 10 100 1000 r Time step Time step FIG. 2. Escape time as a function of the panic level r at the initial density of cooperators ρIC =0.5. 2000 IC=0.1 2000 IC=0.3 ors 1500 1500 ct takes the place of y, then the individual ever occupied y Defe 1000 CDoefoepcetroartor 1000 wilSlinbceepeuqsuhaetdiobna[c5k]ctoanthreefloercitgitnhaelmlooctaitoinonveolfoxci.tyofan ors \ 5000 5000 at individual, v is defined as the system’s average velocity er 1 10 100 1000 1 10 100 1000 p and given by, Coo 2000 IC=0.7 2000 IC=0.9 of 1500 1500 ber 1000 1000 1 um 500 500 v = X Wj, (6) N NM 0 0 j∈NM 1 10 100 1000 1 10 100 1000 Time step Time step where N is the number of individuals who have the M escape ability, and W is the value of W of individual j FIG. 3. Competition of cooperators and defectors for dif- j representing its motion velocity. Different collective ferent panic levels of r at ρIC = 0.5 (UP), and Competition patterns of motion can be predicted according to v: for between cooperators and defectors for different valuesof ρIC v > 0.5, the system is realized to stay in the free flow; at r=0.65 (DOWN). when v → 0 the congestion forms. For 0 < v < 0.5, no distinct collective patterns emerge. All the individuals will update their strategies syn- chronously by studying a more successful neighbor with a probability W(s →s ), x y 1 W(s →s )= , (7) 1.0 x y 1+exp(−(P −P )/τ) y x 0.8 whereτ isthe noiselevelhavingthe identicalfunctionof 0.6 IC=0.5 κinequation[5],ands (s )isthestrategyx(y)adopts. x y C 0.4 III. SIMULATIONS AND ANALYSIS 0.2 0.0 Thesimulationsarecarriedoutonasquarelatticerep- rd rc 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 resenting a single roomwith a scale of 50×50, and2000 r pedestrians are considered. Only one exit exists, at a site range of 23 − 26 on the lattice. Initially, all the FIG.4. Pedestrian phasetransition with paniclevel r when pedestrians are distributed uniformly in the room with the initial cooperation density ρIC =0.5. ρC on the vertical a cooperation density of ρIC = 0.5. Escape is realized coordinate is cooperation density among the pedestrians in successfully when 95% of the pedestrians have escaped the room when 95% of pedestrians have escaped out of the from the room. We set Uw = R/2 and κ = τ = 0.1. In room. rd is the critical value at which defectors emerge, and all of the simulations, each data point is the average for rcisthecriticalvalueafterwhichcooperatorsbecomeextinct. 100 realizations. 4 Figure2showsthe variationinescapetime withpanic level r at ρ =0.5. It is obviously seen that the escape IC 0.4 time remains constant for small values of r (r < 0.5), but increases with r for r ≥ 0.5. The results indicate 0.3 tehscaatpree,mwahinerineagsctaolmo misuachlwfaeyasrtlheaedbsetsot satrloanteggeyr desucrainpge Pd_bound 00..12 ttwimIenee.norcdoeorpetorautonrdsearsntdanddeftehcteodrsynaaremiincvs,esitnitgeartaecdt.ioDnsiffbeer-- -Pc_bound -00..10 -0.2 ent features are shown for different values of r (see it in -0.3 r=0.1 r=0.3 Fig. 3 (UP)): (1) for r =0.1 and 0.5, cooperators domi- r=0.63 r=0.67 -0.4 r=0.8 r=0.9 nate; (2) for r = 0.65, cooperators and defectors coexist 10 100 1000 andovertimetheirfrequenciesreachalmostidenticallev- Time step els; and (3) for r = 0.9, defectors dominate the system. In Fig. 3 (DOWN), it is noted that cooperators coexist FIG.5. (Coloronline)Thedifferencesbetweenmeanpayoffs withdefectors,andevolvetoeventuallyreachalmostthe ofcooperatorsanddefectorsalongtheboundary(Pc−bound− same mutual frequency, regardless of ρIC. Combining Pd−bound) at ρIC =0.5 for different panic levels of r. this withthe resultinFig. 3(UP),wecanconcludethat thepaniclevelrremarkablyaffectsthesystemdynamics, whereas ρ has no distinct impact. IC The simulation results can be explained by analyzing the panic level. It is known from the payoff matrix that a high panic level induces a strong temptation to defect. According to the definitions of the potential well energy andmotionruleinEquations(1)−(5),alargeincrement in the percentage of defectors will lead to a relatively t1 t2 largeincreaseinpotentialwellenergyorstrongpressure, which accordingly reduces the escape velocity. Moreover,pedestrianphasetransitionwithpanic level r is discussed in Fig. 4. We observe that the system experiences three phases sequentially: a single phase of t3 t4 cooperator for r < 0.6, a mixed two-phase pedestrian of cooperator and defector for 0.6(r ) ≤ r ≤ 0.7(r ), d c and a single phase of defector for r > 0.7. It has been FIG.6. (Coloronline)Spatialevolutionsofcooperatorsand proven that as e increases from 0.1 to 0.9, rd = 0.6089, defectors at ρIC = 0.5, and r = 0.65. The black symbols the standard deviation is σ(r ) = 0.0105; r = 0.7044, represent empty sites, the brown ones indicate the defectors, d c and σ(r ) = 0.0088. As U increases from 0.1 to and thewhite symbols denotecooperators. c w 0.9, r = 0.5956, σ(r ) = 0.0133; r = 0.7111, and d d c σ(r )=0.0105. Itisobviouslyseenthatthephasetransi- c tionisbasicallyrobusttothechangesofeandU ,which itcanbe inducedthat cooperatorcluster’sboundarybe- w is consistent with the results shown in Fig. 4. Further- comes weaker as panic level r upgrades. Figure 6 shows more, noise effect κ (in equation 5) was also studied: the spatialevolutionofpedestrians at r=0.65,inwhich when κ = 0.001 (approaching determined occupation), the black symbols indicate the empty sites, the brown r = 0.58, r = 0.7; for κ = 100 (approaching stochastic ones denote the defectors, and the white symbols repre- d c occupation), r = 0.54, and r = 0.72. It is concluded sent cooperative pedestrians. It is seen that since defec- d c thatphasetransitionisalsobasicallyrobusttothenoise, tors’ boundary is slightly firmer than cooperators’ (see but the interval of mixed phase becomes broader when Fig. 5 in the phase of 0.6 ≤ r ≤ 0.7), many cooperator noise becomes very large. clusters are formedbut don’t spreadalloverthe system. To uncover the competition mechanism between the The evolution of the average velocity v for different two types of pedestrians, mean payoffs of cooperators panic levels r at ρ =0.5 is shown in Fig. 7. It is seen IC and defectors along the boundary are studied in Fig. 5. that the average velocity decreases as the panic level in- It is observed that in the phase of r < 0.6, P is creases, and declines sharply for r > 0.65. It is worthy c−bound significantly higher than P , whereas in the phase noting that v changes slowly with t for r < 0.65, but d−bound of r > 0.7, P is obviously higher. During the when r > 0.65, the velocity sharply decreases over time d−bound phase of 0.6 ≤ r ≤ 0.7, P keeps a little higher, until the system enters a uniform motion state. This d−bound but approachesto P as time evolves. Cooperator decrease in velocity is likely to be closely related to the c−bound cluster is proven crucial in cooperation spread because emergence of congestion, and the uniform-motion state of its firm boundary. According to the results in Fig. 5, corresponds to free flow (moderate or relatively high ve- 5 patterns happen. Finally,Figure8showstheescapetimeasafunctionof 0.5 tc v* ρIC for r=0.6. It is presentedthatpedestrians gainthe greatest benefit for overall cooperation (ρ = 1), while IC 0.4 the worst-case scenario occurs if all individuals defect (ρ =0). The initial density ofcooperators(0<ρ < IC IC 0.3 1) has no obvious influences on the escape time. v I IV. CONCLUSIONS 0.2 r=0.1 III r=0.5 0.1 r=0.65 Extending the model proposed in previous work, we r=0.75 II consideredtheeffectsofdefectorherdingandwallstoin- 0.0 r=0.9 vestigate pedestrian evacuation. Payoff represented the 10 100 1000 complicated interactions among individuals and was in- Time step cluded in the energy representation. We found that the FIG. 7. Evolution of average velocity v at ρIC = 0.5 for escape time increased with the panic level, and three different values of panic level r. v∗ = 0.5, I, II, and III phases of pedestrian were observedsequentially as panic represent threestages a system experiences when r>0.65. level upgraded. Analysis of the results indicated that a high panic level induced a strong temptation to defect, which leaded to the emergence of widespread de- fection. Furthermore, a large percentage of defectors in- 4600 duced a comparatively large increase in potential well 4500 energy, which reduced the average speed of pedestrian flowaccordingtothemotionrule. Payoffsofcooperators 4400 e anddefectors alongthe boundary were investigated,and m 4300 Ti theresultsrevealedthecompetitionmechanismsbetween pe 4200 thetwotypesofpedestrian. Theaveragevelocitywasde- a sc 4100 finedasthemeanofcumulativemotionprobabilityinthis E 4000 model, which revealed valuable and interesting dynamics: the system was in the free flow when the panic level 3900 was low, but exhibited three stages - congestion forma- 3800 0.0 0.2 0.4 0.6 0.8 1.0 tion, congestion, and congestion evacuation - for a high paniclevel. Weprovedthatthepaniclevelplayedanim- IC portant part in determining the dynamics for which dif- FIG. 8. Variation of escape time with the initial density of ferent behaviors were observed for different panic levels. cooperators ρIC at r=0.6. Phasetransitionexisted, andwasbasically robustto the changes of empty site contribution, wall’s pressure and noise effect. We also found that global cooperation was locity)orcongestion(verylowvelocity). Therefore,from the best strategy for the most efficient evacuation, and the phase aspect we can conclude that the system gen- the situation was worst for all defections. It was proven erally remains in a free-flow state for phase of r < 0.6 that the initial density of cooperators had a negligible when t < t (v > v∗), and then moves at a relative low c impact on the escape efficiency during evacuation. speed without distinct collective patterns. It exhibits three behavior stages for phase of r > 7: in stage I, local free flow and congestion coexist; in stage II, only ACKNOWLEDGMENTS congestionexists;andinstageIII,congestionevacuation occurs. Congestionemergesbefore the systementersthe uniform state owing to the small value for v in stage II. This work was supported by Specialized Foundation It can be predicted from this analysis that the probabil- for Theoretical Physics of China (Grant No. 11247239), ity of congestion is high for a high panic level. For the National Natural Science Foundation of China (Grants phase of coexistence (0.6≤r ≤7), no distinct collective No. 11305017,11275186and 91024026). [1] LF.Henderson,Thestatisticsofcrowdfluids.Nature229 [3] D. Helbing, Traffic and related self-driven many-particle (1971), pp.381-383. systems, Rev.Mod. Phys., 73 (2001), pp.1067-1141. [2] D.Helbing,I.Farkas,andT.Vicsek,Simulatingdynam- [4] A. Schadschneider, et al. Evacuation dynamics: Empir- icalfeaturesofescapepanic,Nature407(2000),pp.487- ical results, modeling and applications. Encyclopedia of 490. ComplexityandSystemsScience,edMeyersR(Springer, 6 Berlin), (2009), pp.3142-3176. Sociol. 20 (1995), pp. 247-269. [5] M. 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Dynamics of Panic Pedestrians in Evacuation

Similar Dynamics of Panic Pedestrians in Evacuation