Dynamics of Panic Pedestrians in Evacuation
aa r X i v : . [ n li n . C G ] J a n 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: October 15, 2018)A modified lattice gas model is proposed to study pedestrian evacuation from a single room. Thepayoff matrix in this model represents the complicated interactions between selfish individuals, andthe mean force imposed on an individual is given by considering the impacts of neighborhood payoff,walls, and defector herding. Each passer-by moves to his selected location according to the Fermifunction, and the average velocity of pedestrian flow is defined as a function of the motion rule.Two pedestrian types are included: cooperators, who adhere to the evacuation instructions; anddefectors, who ignore the rules and act individually. It is observed that the escape time increaseswith the panic level, and the system remains smooth for a low panic level, but exhibits three stagesfor a high panic level. We prove that the panic level determines the dynamics of this system, andthe initial density of cooperators has a negligible impact. The system experiences three phases, asingle phase of cooperator, a mixed two-phase pedestrian, and a single phase of defector sequentiallyas the panic level upgrades. The phase transition has been proven basically robust to the changesof empty site contribution, wall’s pressure, and noise amplitude in the motion rule. It is furthershown that pedestrians derive the greatest benefit from overall cooperation, but are trapped in theworst situation if they are all defectors.
PACS numbers:
87. 23. Ge, 02. 50. Le, 87. 23. Cc, 89. 90. +n
I. INTRODUCTION
Evacuation of pedestrians under panic has been exten-sively studied [1–20], because an understanding of thedynamic features of this phenomenon can reduce the in-cidence of injury and death. Escape panic can occur indifferent confinement sizes ranging from a rioting crowdin a packed stadium to stunned customers in a bar. Es-cape panic is characterized by strong contact interactionsbetween selfish individuals that quickly give rise to herd-ing, stampede, and clogging [21–31]. Experimental ob-servations and numerical simulations are two approachesthat have been widely applied to study this issue. Simu-lation results have revealed some interesting dynamic fea-tures, such as pedestrian arch formation around the exit,herding, and interference between arches in multiple-exitrooms [2]. Disruptive interference, self-organized queu-ing, and scale-free escape dynamics [26, 27] have alsobeen observed. Experiments in genuine escape panic aredifficult to set up because of ethical and legal concerns.However, escape panic can be simulated by solving aset of coupled differential equations [2, 3] or by applyingthe cellular automata (CA) technique [20, 26], wherebythe movement of confined pedestrians is tracked overtime. An important task related to evacuation simula-tions is how to describe the effects of subjective factors, ∗ Dynamics of Panic pedestrians † [email protected] such as fear, as well as the complicated forces amongindividuals. Helbing used social force to simulate suchinteractions [2], and Fukui avoided force concepts byapplying the CA model [20]. Game theory was pro-posed to study the dynamics in crows evacuation [32–37].Heli¨ o vaarawe presented a spatial game theoretic modelfor pedestrian behavior in situations of exit congestion[32], and Hao etal applied game theory to deal with theconflicts that arise when two pedestrians try to occupythe same position[37].In this study, we modify the model [38] in which theeffects of walls and defector herding are considered. Adefector herding unit is defined as a group of four indi-viduals who are all defectors in the same neighborhood.The potential well energy or mean force imposed on anindividual is defined by considering three aspects. Anindividual will move to his selected location with a largeprobability if his personal energy is higher than the po-tential well energy. The velocity of a walker is defined asa function of the motion rule, and average velocity of thepedestrian system can reflect different dynamics, such assmooth flow, and congestion. We find that the panic levelis crucial in determining the dynamics of the evacuationsystem and the competition between cooperators and de-fectors. It is also observed that three phases of pedestrianare exhibited sequentially as panic level upgrades.The remainder of the paper is organized as follows. InSection II, we describe the model. Our numerical simu-lations and analysis are presented in Section III. Conclu-sions are drawn in Section IV. II. MODEL
Pedestrian evacuation is studied on a model of squarelattice on which all the individuals are distributed uni-formly. Two pedestrian types are considered: coopera-tors, who adhere to the evacuation instructions; and de-fectors, who ignore the rules and act individually. Eachpedestrian only interacts with his nearest neighbors, andis deleted from the system if he arrives at the exit. Weapply game theory, and the payoff matrix representingthe interactions among the crowd is shown in Table 1.To be specific, a cooperator receives a Reward ( R ) whenhe interacts with a collaborative neighbor, but suffers aSucker ( S ) if he encounters a defector who gains a payoffof Temptation ( T ). Two defectors receive the Punish-ment ( P M ), respectively. In addition, a walker will gaina payoff e (0 < e < R ) if there is an empty site in hisneighborhood. Usually, T = 1 + r , R = 1, S = 1 − r , P M = 0, where 0 < r < r is cost-to-benefit ratioindicating the defector-cooperator payoff divide, whichdescribes the panic level during the escape. e = R/ TABLE I. Payoff matrix
C D EC (R, R) (S, T) (e, − ) D (T, S) (PM, PM) (e, − ) aa C indicates an individual who cooperates, and D indicates anindividual who chooses to defect. E represents an empty site.
T > R > S > P M P x denotes the personal energy of individual x , andis the average of cumulative payoff that x receives fromall his neighbors. Each individual stays in a potentialwell which is formed by the neighboring people and walls(if walls exist in his neighborhood). U x is the energy ofpotential well x stays in, which consists of three parts.Firstly, U p is the average cumulative payoff for all neigh-bors of x , which represents the mean forces on x : U p = 1 G x X j ∈ G x P xj (1)where G x is the scale of the interacting group centeredon x , which is defined as the number of directly linkedneighbors of x (excluding the empty sites). P xj is theaverage of cumulative payoff that j receives from all hisneighbors, where individual j belongs to the neighbor-hood of x .A defector herding unit is defined as a group in whichfour defectors are in the same neighborhood (Fig. 1). Weconsider that a defector herding unit contributes P D = T /
FIG. 1. Schematic presentation of a defector herding unit inwhich all the individuals are defectors who are circled in thefigure. P D is the average payoff of the defector centered inthe circle. a defector. Therefore, the potential well energy U x willinclude U du if a defector herding unit exists in a defector’sneighborhood: U du = 1 G x ( n du · T · s, (2)where n du is the number of defector herding units in theneighborhood of x . The parameter s takes the value s = 0if x chooses to cooperate, and s = 1 when x decides todefect.In addition, we consider the contribution of walls to U x , so U x will include U w if walls exist in the neighbor-hood of x : U w = 1 G x · n w · U w , (3)where n w is the number of walls in the neighborhoodand U w (0 < U w < R ) is the contribution of a wall to U x . Combining all the quantities discussed above, thepotential well energy U x is given by U x = U p + U du + U w , (4)It should be noted that x ′ s personal energy is excluded in U x which actually reflects the pressures from x ′ s neigh-borhood.At each time step, toward the exit’s position, each in-dividual x randomly selects a cell y in the neighborhood,and moves to y with probability W [38]: W = 11 + exp ( − ( P x − U x ) /κ ) , (5)where κ is the noise amplitude. κ = 0 indicates deter-mined occupation, whereas κ = ∞ denotes stochastic oc-cupation. It is apparent that the larger P x is, the larger W is. The equation (5) means that if x owns more per-sonal energy than the potential well energy, x will moveto his selected location with a larger probability. If x E s ca p e T i m e r IC =0.5 FIG. 2. Escape time as a function of the panic level r at theinitial density of cooperators ρ IC = 0 . takes the place of y , then the individual ever occupied y will be pushed back to the original location of x .Since equation [5] can reflect the motion velocity of anindividual, v is defined as the system’s average velocityand given by, v = 1 N M X j ∈ N M W j , (6)where N M is the number of individuals who have theescape ability, and W j is the value of W of individual j representing its motion velocity. Different collectivepatterns of motion can be predicted according to v : for v > .
5, the system is realized to stay in the free flow;when v → < v < .
5, nodistinct collective patterns emerge.All the individuals will update their strategies syn-chronously by studying a more successful neighbor witha probability W ( s x → s y ), W ( s x → s y ) = 11 + exp ( − ( P y − P x ) /τ ) , (7)where τ is the noise level having the identical function of κ in equation [5], and s x ( s y ) is the strategy x ( y ) adopts. III. SIMULATIONS AND ANALYSIS
The simulations are carried out on a square lattice rep-resenting a single room with a scale of 50 ×
50, and 2000pedestrians are considered. Only one exit exists, at asite range of 23 −
26 on the lattice. Initially, all thepedestrians are distributed uniformly in the room witha cooperation density of ρ IC = 0 .
5. Escape is realizedsuccessfully when 95% of the pedestrians have escapedfrom the room. We set U w = R/ κ = τ = 0 .
1. Inall of the simulations, each data point is the average for100 realizations. r=0.1 Cooperator Defector N u m b e r o f C oop e r a t o r s \ D e f ec t o r s r=0.5r=0.65 r=0.9 Time step Time step IC =0.1 Cooperator Defector N u m b e r o f C oop e r a t o r s \ D e f ec t o r s IC =0.3 Time step IC =0.7 Time step IC =0.9 FIG. 3. Competition of cooperators and defectors for dif-ferent panic levels of r at ρ IC = 0 . ρ IC at r = 0 .
65 (DOWN). r c C r IC =0.5 r d FIG. 4. Pedestrian phase transition with panic level r whenthe initial cooperation density ρ IC = 0 . ρ C on the verticalcoordinate is cooperation density among the pedestrians inthe room when 95% of pedestrians have escaped out of theroom. r d is the critical value at which defectors emerge, and r c is the critical value after which cooperators become extinct. Figure 2 shows the variation in escape time with paniclevel r at ρ IC = 0 .
5. It is obviously seen that the escapetime remains constant for small values of r ( r < . r for r ≥ .
5. The results indicatethat remaining calm is always the best strategy duringescape, whereas too much fear leads to a longer escapetime.In order to understand the dynamics, interactions be-tween cooperators and defectors are investigated. Differ-ent features are shown for different values of r (see it inFig. 3 (UP)): (1) for r = 0 . .
5, cooperators domi-nate; (2) for r = 0 .
65, cooperators and defectors coexistand over time their frequencies reach almost identical lev-els; and (3) for r = 0 .
9, defectors dominate the system.In Fig. 3 (DOWN), it is noted that cooperators coexistwith defectors, and evolve to eventually reach almost thesame mutual frequency, regardless of ρ IC . Combiningthis with the result in Fig. 3 (UP), we can conclude thatthe panic level r remarkably affects the system dynamics,whereas ρ IC has no distinct impact.The simulation results can be explained by analyzingthe panic level. It is known from the payoff matrix thata high panic level induces a strong temptation to defect.According to the definitions of the potential well energyand motion rule in Equations (1) − (5), a large incrementin the percentage of defectors will lead to a relativelylarge increase in potential well energy or strong pressure,which accordingly reduces the escape velocity.Moreover, pedestrian phase transition with panic level r is discussed in Fig. 4. We observe that the systemexperiences three phases sequentially: a single phase ofcooperator for r < .
6, a mixed two-phase pedestrianof cooperator and defector for 0 . r d ) ≤ r ≤ . r c ),and a single phase of defector for r > .
7. It has beenproven that as e increases from 0 . . r d = 0 . σ ( r d ) = 0 . r c = 0 . σ ( r c ) = 0 . U w increases from 0 . . r d = 0 . σ ( r d ) = 0 . r c = 0 . σ ( r c ) = 0 . e and U w , whichis consistent with the results shown in Fig. 4. Further-more, noise effect κ (in equation 5) was also studied:when κ = 0 .
001 (approaching determined occupation), r d = 0 . r c = 0 .
7; for κ = 100 (approaching stochasticoccupation), r d = 0 .
54, and r c = 0 .
72. It is concludedthat phase transition is also basically robust to the noise,but the interval of mixed phase becomes broader whennoise becomes very large.To uncover the competition mechanism between thetwo types of pedestrians, mean payoffs of cooperatorsand defectors along the boundary are studied in Fig. 5.It is observed that in the phase of r < . P c − bound issignificantly higher than P d − bound , whereas in the phaseof r > . P d − bound is obviously higher. During thephase of 0 . ≤ r ≤ . P d − bound keeps a little higher,but approaches to P c − bound as time evolves. Cooperatorcluster is proven crucial in cooperation spread becauseof its firm boundary. According to the results in Fig. 5,
10 100 1000-0.4-0.3-0.2-0.10.00.10.20.30.4 r=0.1 r=0.3 r=0.63 r=0.67 r=0.8 r=0.9 P c _bound - P d_bound Time step
FIG. 5. (Color online) The differences between mean payoffsof cooperators and defectors along the boundary ( P c − bound − P d − bound ) at ρ IC = 0 . r . t t t t FIG. 6. (Color online) Spatial evolutions of cooperators anddefectors at ρ IC = 0 .
5, and r = 0 .
65. The black symbolsrepresent empty sites, the brown ones indicate the defectors,and the white symbols denote cooperators. it can be induced that cooperator cluster’s boundary be-comes weaker as panic level r upgrades. Figure 6 showsthe spatial evolution of pedestrians at r = 0 .
65, in whichthe black symbols indicate the empty sites, the brownones denote the defectors, and the white symbols repre-sent cooperative pedestrians. It is seen that since defec-tors’ boundary is slightly firmer than cooperators’ (seeFig. 5 in the phase of 0 . ≤ r ≤ . v for differentpanic levels r at ρ IC = 0 . r > .
65. It is worthynoting that v changes slowly with t for r < .
65, butwhen r > .
65, the velocity sharply decreases over timeuntil the system enters a uniform motion state. Thisdecrease in velocity is likely to be closely related to theemergence of congestion, and the uniform-motion statecorresponds to free flow (moderate or relatively high ve-
10 100 10000.00.10.20.30.40.5 v Time step r=0.1 r=0.5 r=0.65 r=0.75 r=0.9
I II III v*t c FIG. 7. Evolution of average velocity v at ρ IC = 0 . r . v ∗ = 0 . I , II , and III represent three stages a system experiences when r > . E s ca p e T i m e IC FIG. 8. Variation of escape time with the initial density ofcooperators ρ IC at r = 0 . locity) or congestion (very low velocity). Therefore, fromthe phase aspect we can conclude that the system gen-erally remains in a free-flow state for phase of r < . t < t c ( v > v ∗ ), and then moves at a relative lowspeed without distinct collective patterns. It exhibitsthree behavior stages for phase of r >
7: in stage I,local free flow and congestion coexist; in stage II, onlycongestion exists; and in stage III, congestion evacuationoccurs. Congestion emerges before the system enters theuniform state owing to the small value for v in stage II.It can be predicted from this analysis that the probabil-ity of congestion is high for a high panic level. For thephase of coexistence (0 . ≤ r ≤ ρ IC for r = 0 .
6. It is presented that pedestrians gain thegreatest benefit for overall cooperation ( ρ IC = 1), whilethe worst-case scenario occurs if all individuals defect( ρ IC = 0). The initial density of cooperators (0 < ρ IC <
1) has no obvious influences on the escape time.
IV. CONCLUSIONS
Extending the model proposed in previous work, weconsidered the effects of defector herding and walls to in-vestigate pedestrian evacuation. Payoff represented thecomplicated interactions among individuals and was in-cluded in the energy representation. We found that theescape time increased with the panic level, and threephases of pedestrian were observed sequentially as paniclevel upgraded. Analysis of the results indicated thata high panic level induced a strong temptation to de-fect, which leaded to the emergence of widespread de-fection. Furthermore, a large percentage of defectors in-duced a comparatively large increase in potential wellenergy, which reduced the average speed of pedestrianflow according to the motion rule. Payoffs of cooperatorsand defectors along the boundary were investigated, andthe results revealed the competition mechanisms betweenthe two types of pedestrian. The average velocity was de-fined as the mean of cumulative motion probability in thismodel, which revealed valuable and interesting dynam-ics: the system was in the free flow when the panic levelwas low, but exhibited three stages - congestion forma-tion, congestion, and congestion evacuation - for a highpanic level. We proved that the panic level played an im-portant part in determining the dynamics for which dif-ferent behaviors were observed for different panic levels.Phase transition existed, and was basically robust to thechanges of empty site contribution, wall’s pressure andnoise effect. We also found that global cooperation wasthe best strategy for the most efficient evacuation, andthe situation was worst for all defections. It was proventhat the initial density of cooperators had a negligibleimpact on the escape efficiency during evacuation.
ACKNOWLEDGMENTS
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