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December  2013, 8(4): 985-1007. doi: 10.3934/nhm.2013.8.985

Structured first order conservation models for pedestrian dynamics

1. 

Siemens AG, Corporate Technology, 80200 Munich, Germany, Germany

Received  August 2012 Revised  April 2013 Published  November 2013

In this contribution, we revisit multiple first order macroscopic modelling approaches to pedestrian flows and computationally compare the results with a microscopic approach to pedestrian dynamics. We find that widely used conservation schemes show significantly different results than microscopic models. Thus, we propose to adopt on a macroscopic level a structured continuum model. The approach basically relies on fundamental diagrams - the relationship between fluxes and local densities - as well as the explicit consideration of individual velocities, thus showing similarities to generalised kinetic models. The macroscopic model is outlined in detail and shows a significantly better agreement with microscopic pedestrian models. The increased realism, important for safety relevant real life applications, is underlined considering several scenarios.
Citation: Dirk Hartmann, Isabella von Sivers. Structured first order conservation models for pedestrian dynamics. Networks & Heterogeneous Media, 2013, 8 (4) : 985-1007. doi: 10.3934/nhm.2013.8.985
References:
[1]

C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors,, Netw. Heterog. Media, 6 (2011), 351. doi: 10.3934/nhm.2011.6.351. Google Scholar

[2]

N. Bellomo and A. Bellouquid, On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms,, Netw. Heterog. Media, 6 (2011), 383. doi: 10.3934/nhm.2011.6.383. Google Scholar

[3]

N. Bellomo and V. Coscia, First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow,, CR Mecanique, 333 (2005), 843. doi: 10.1016/j.crme.2005.09.004. Google Scholar

[4]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: a survey of models, speculations, and perspectives,, SIAM Rev., 53 (2011), 409. doi: 10.1137/090746677. Google Scholar

[5]

N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular traffic flow I: Fluid dynamic and kinetic modelling,, Math. Mod. Meth. Appl. S., 12 (2002), 1801. doi: 10.1142/S0218202502002343. Google Scholar

[6]

V. Blue, M. Embrechts and J. Adler, Cellular automata modeling of pedestrian movements,, IEEE Int. Conf. on Syst., (1997). doi: 10.1109/ICSMC.1997.635272. Google Scholar

[7]

C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton,, Physica A, 295 (2001), 507. doi: 10.1016/S0378-4371(01)00141-8. Google Scholar

[8]

V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics,, Math. Mod. Meth. Appl. S., 18 (2008), 1217. doi: 10.1142/S0218202508003017. Google Scholar

[9]

C. Canuto, F. Fagnani and P. Tilli, A eulerian approach to the analysis of rendez-vous algorithms,, Proceedings of the 17th IFAC World Congress, (2008), 9039. Google Scholar

[10]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Appl. Math., 36 (2005), 1862. doi: 10.1137/S0036141004402683. Google Scholar

[11]

M. Chraibi, U. Kemloh, A. Schadschneider and A. Seyfried, Force-based models of pedestrian dynamics,, Netw. Heterog. Media, 6 (2011), 425. doi: 10.3934/nhm.2011.6.425. Google Scholar

[12]

E. Christiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with applications to crowd dynamics,, Multiscale Model. Sim., 9 (2011), 155. doi: 10.1137/100797515. Google Scholar

[13]

R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks,, Mathematical Methods in the Applied Sciences, 28 (2005), 1553. doi: 10.1002/mma.624. Google Scholar

[14]

M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.046111. Google Scholar

[15]

C. F. Daganzo, Fundamentals of Transportation and Traffic Operations,, Pergamon, (1997). Google Scholar

[16]

M. Fukui and Y. Ishibashi, Self-organized phase transitions in cellular automaton models for pedestrians,, J. Phys. Soc. Jap., 68 (1999). doi: 10.1143/JPSJ.68.2861. Google Scholar

[17]

S. Göttlich, M. Herty and A. Klar, Network models for supply chains,, Commun. Math. Sci., 3 (2005), 545. doi: 10.4310/CMS.2005.v3.n4.a5. Google Scholar

[18]

S. Göttlich, S. Kühn, J. Ohst, S. Ruzika and M. Thiemann, Evacuation dynamics influenced by spreading hazardous material,, Netw. Heterog. Media, 6 (2011), 443. doi: 10.3934/nhm.2011.6.443. Google Scholar

[19]

D. Hartmann, Adaptive pedestrian dynamics based on geodesics,, New J. Phys., 12 (2010). doi: 10.1088/1367-2630/12/4/043032. Google Scholar

[20]

D. Helbing, A fluid-dynamic model for the movement of pedestrians,, Complex Systems, 6 (1992), 391. Google Scholar

[21]

L. F. Henderson, On the fluid mechanics of human crowd motion,, Transport. Res., 8 (1974), 509. doi: 10.1016/0041-1647(74)90027-6. Google Scholar

[22]

H.-J. Huang and R.-Y. Guo, Static floor field and exit choice for pedestrian evacuation in rooms with internal obstacles and multiple exits,, Phys. Rev. E., 78 (2008). doi: 10.1103/PhysRevE.78.021131. Google Scholar

[23]

D. Helbing and A. Johansson, Pedestrian, crowd and evacuation dynamics,, Encyclopedia of Complexity and Systems Science, 16 (2010), 6476. Google Scholar

[24]

M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks,, SIAM J. Sci. Comput., 25 (2003), 1066. doi: 10.1137/S106482750241459X. Google Scholar

[25]

D. Helbing and P. Molnár, Social Force Model for Pedestrian Dynamics,, Phys. Rev. E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. Google Scholar

[26]

H. Holden and N.H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads,, SIAM J. Math. Anal., 26 (1995), 999. doi: 10.1137/S0036141093243289. Google Scholar

[27]

H. W. Hamacher and S. A. Tjandra, Mathematical modelling of evacuation problems: A state of the art,, in Pedestrian and Evacuation Dynamics (eds. M. Schreckenberg and S. D. Sharma), (2002), 227. Google Scholar

[28]

R. L. Hughes, A continuum theory for the flow of pedestrians,, Transport. Res. B - Meth., 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar

[29]

R. L. Hughes, The flow of human crowds,, Annu. Rev. Fluid Mech., 35 (2003), 169. doi: 10.1146/annurev.fluid.35.101101.161136. Google Scholar

[30]

L. Huang, S. C. Wong, M. Zhang, C.-W. Shu and W. H. K. Lam, Revisiting Hughes' dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm,, Transport. Res. B - Meth., 43 (2009), 127. doi: 10.1016/j.trb.2008.06.003. Google Scholar

[31]

P. Kachroo, S. J. Al-Nasur, S. A. Wadoo and A. Shende, Pedestrian Dynamics - Feedback Control of Crowd Evacuation,, Springer, (2008). Google Scholar

[32]

B. S. Kerner, The Physics of Traffic,, Springer, (2004). doi: 10.1007/978-3-540-40986-1. Google Scholar

[33]

G. Köster, D. Hartmann, and W. Klein, Microscopic pedestrian simulations: From passenger exchange times to regional evacuation,, in Operations Research Proceedings (eds. B. Hum, (2010), 571. Google Scholar

[34]

H. Klüpfel, A Cellular Automaton Model for Crowd Movement and Egress Simulation,, Ph.D. thesis, (2003). Google Scholar

[35]

A. Kneidl, M. Thiemann, A. Borrmann, S. Ruzika, H. W. Hamacher, G. Köster and E. Rank, Bidirectional Coupling of Macroscopic an Microscopic Approaches for Pedestrian Behavior Prediction,, in Pedestrian and Evacuation Dynamics (eds. R. D. Peacock, (2011), 459. doi: 10.1007/978-1-4419-9725-8_41. Google Scholar

[36]

A. Kneidl, M. Thiemann, D. Hartmann and A. Borrmann, Combining pedestrian simulation with a network flow optimization to support security staff in handling an evacuation of a soccer stadium,, in Proceedings of European Conference Forum 2011 (eds. E. Tobin and M. Otreba), (2011). Google Scholar

[37]

R. Löhner, On the modeling of pedestrian motion,, Appl. Math. Model., 34 (2010), 366. doi: 10.1016/j.apm.2009.04.017. Google Scholar

[38]

R. J. LeVeque, Numerical Methods for Conservation Laws,, Birkhäuser, (2005). doi: 10.1007/978-3-0348-8629-1. Google Scholar

[39]

T. I. Lakoba, D. J. Kaup and N. M. Finkelstein, Modifications of the Helbing-Molnár-Farkas-Vicsek social force model for pedestrian evolution,, Simulation, 81 (2005), 339. doi: 10.1177/0037549705052772. Google Scholar

[40]

K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider, Extended floor field CA model for evacuation dynamics,, IEICE Trans. Inf. Syst., E87D (2004), 726. Google Scholar

[41]

M. Oppenhäuser, Realisierung und Potenzialanalyse von wissenschaftlichen Konzepten zur Regionalen Evakuierung aus Polizeilicher Sicht am Beispiel des Projektes REPKA,, Master's Thesis, (2011). Google Scholar

[42]

N. Pelechano, J. M. Allbeck and N. Badler, Virtual Crowds: Methods, Simulation, and Control,, Morgan & Claypool Publishers, (2008). doi: 10.2200/S00123ED1V01Y200808CGR008. Google Scholar

[43]

D. R. Parisi, M. Gilman and H. Moldovan, A modification of the social force model can reproduce experimental data of pedestrian flows in normal conditions,, Physica A, 388 (2009), 3600. doi: 10.1016/j.physa.2009.05.027. Google Scholar

[44]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Flow,, Elsevier, (1971). Google Scholar

[45]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing,, Cambridge University Press, (2007). Google Scholar

[46]

, REPKA, Regionale Evakuierung: Planung, Kontrolle und Anpassung,, , (). Google Scholar

[47]

E. V. RiMEA, Richtlinie für Mikroskopische Entfluchtungsanalysen,, , (). Google Scholar

[48]

A. Schadschneider, W. Klingsch, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Evacuation dynamics: Empirical results, modeling and applications,, Encyclopedia of Complexity and System Science, (2009), 3142. Google Scholar

[49]

A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modeling,, Netw. Heterog. Media, 6 (2011), 545. doi: 10.3934/nhm.2011.6.545. Google Scholar

[50]

A. Varas, M. D. Cornejo, D. Mainemer, B. Toledo, J. Rogan, V. Munoz and J. A. Valdivia, Cellular automaton model for evacuation process with obstacles,, Physica A, 382 (2007), 631. doi: 10.1016/j.physa.2007.04.006. Google Scholar

[51]

U. Weidmann, Transporttechnik der Fussgänger: Transporttechnische Eigenschaften des Fussgängerverkehrs (Literaturauswertung), Schriftenreihe des IVT, (1993). Google Scholar

[52]

Y. Xia, S. C. Wong and C. W. Shu, Dynamic continuum pedestrian flow model with memory effect,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.066113. Google Scholar

[53]

W. J. Yu, R. Chen, L. Y. Dong and S. Q. Dai, Centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.026112. Google Scholar

[54]

K. Yamamoto, S. Kokubo and K. Nishinari, Simulation for pedestrian dynamics by real-coded cellular automata,, Physica A, 379 (2007). doi: 10.1016/j.physa.2007.02.040. Google Scholar

show all references

References:
[1]

C. Appert-Rolland, P. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors,, Netw. Heterog. Media, 6 (2011), 351. doi: 10.3934/nhm.2011.6.351. Google Scholar

[2]

N. Bellomo and A. Bellouquid, On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms,, Netw. Heterog. Media, 6 (2011), 383. doi: 10.3934/nhm.2011.6.383. Google Scholar

[3]

N. Bellomo and V. Coscia, First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow,, CR Mecanique, 333 (2005), 843. doi: 10.1016/j.crme.2005.09.004. Google Scholar

[4]

N. Bellomo and C. Dogbe, On the modeling of traffic and crowds: a survey of models, speculations, and perspectives,, SIAM Rev., 53 (2011), 409. doi: 10.1137/090746677. Google Scholar

[5]

N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular traffic flow I: Fluid dynamic and kinetic modelling,, Math. Mod. Meth. Appl. S., 12 (2002), 1801. doi: 10.1142/S0218202502002343. Google Scholar

[6]

V. Blue, M. Embrechts and J. Adler, Cellular automata modeling of pedestrian movements,, IEEE Int. Conf. on Syst., (1997). doi: 10.1109/ICSMC.1997.635272. Google Scholar

[7]

C. Burstedde, K. Klauck, A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton,, Physica A, 295 (2001), 507. doi: 10.1016/S0378-4371(01)00141-8. Google Scholar

[8]

V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics,, Math. Mod. Meth. Appl. S., 18 (2008), 1217. doi: 10.1142/S0218202508003017. Google Scholar

[9]

C. Canuto, F. Fagnani and P. Tilli, A eulerian approach to the analysis of rendez-vous algorithms,, Proceedings of the 17th IFAC World Congress, (2008), 9039. Google Scholar

[10]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network,, SIAM J. Appl. Math., 36 (2005), 1862. doi: 10.1137/S0036141004402683. Google Scholar

[11]

M. Chraibi, U. Kemloh, A. Schadschneider and A. Seyfried, Force-based models of pedestrian dynamics,, Netw. Heterog. Media, 6 (2011), 425. doi: 10.3934/nhm.2011.6.425. Google Scholar

[12]

E. Christiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with applications to crowd dynamics,, Multiscale Model. Sim., 9 (2011), 155. doi: 10.1137/100797515. Google Scholar

[13]

R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks,, Mathematical Methods in the Applied Sciences, 28 (2005), 1553. doi: 10.1002/mma.624. Google Scholar

[14]

M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.046111. Google Scholar

[15]

C. F. Daganzo, Fundamentals of Transportation and Traffic Operations,, Pergamon, (1997). Google Scholar

[16]

M. Fukui and Y. Ishibashi, Self-organized phase transitions in cellular automaton models for pedestrians,, J. Phys. Soc. Jap., 68 (1999). doi: 10.1143/JPSJ.68.2861. Google Scholar

[17]

S. Göttlich, M. Herty and A. Klar, Network models for supply chains,, Commun. Math. Sci., 3 (2005), 545. doi: 10.4310/CMS.2005.v3.n4.a5. Google Scholar

[18]

S. Göttlich, S. Kühn, J. Ohst, S. Ruzika and M. Thiemann, Evacuation dynamics influenced by spreading hazardous material,, Netw. Heterog. Media, 6 (2011), 443. doi: 10.3934/nhm.2011.6.443. Google Scholar

[19]

D. Hartmann, Adaptive pedestrian dynamics based on geodesics,, New J. Phys., 12 (2010). doi: 10.1088/1367-2630/12/4/043032. Google Scholar

[20]

D. Helbing, A fluid-dynamic model for the movement of pedestrians,, Complex Systems, 6 (1992), 391. Google Scholar

[21]

L. F. Henderson, On the fluid mechanics of human crowd motion,, Transport. Res., 8 (1974), 509. doi: 10.1016/0041-1647(74)90027-6. Google Scholar

[22]

H.-J. Huang and R.-Y. Guo, Static floor field and exit choice for pedestrian evacuation in rooms with internal obstacles and multiple exits,, Phys. Rev. E., 78 (2008). doi: 10.1103/PhysRevE.78.021131. Google Scholar

[23]

D. Helbing and A. Johansson, Pedestrian, crowd and evacuation dynamics,, Encyclopedia of Complexity and Systems Science, 16 (2010), 6476. Google Scholar

[24]

M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks,, SIAM J. Sci. Comput., 25 (2003), 1066. doi: 10.1137/S106482750241459X. Google Scholar

[25]

D. Helbing and P. Molnár, Social Force Model for Pedestrian Dynamics,, Phys. Rev. E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. Google Scholar

[26]

H. Holden and N.H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads,, SIAM J. Math. Anal., 26 (1995), 999. doi: 10.1137/S0036141093243289. Google Scholar

[27]

H. W. Hamacher and S. A. Tjandra, Mathematical modelling of evacuation problems: A state of the art,, in Pedestrian and Evacuation Dynamics (eds. M. Schreckenberg and S. D. Sharma), (2002), 227. Google Scholar

[28]

R. L. Hughes, A continuum theory for the flow of pedestrians,, Transport. Res. B - Meth., 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar

[29]

R. L. Hughes, The flow of human crowds,, Annu. Rev. Fluid Mech., 35 (2003), 169. doi: 10.1146/annurev.fluid.35.101101.161136. Google Scholar

[30]

L. Huang, S. C. Wong, M. Zhang, C.-W. Shu and W. H. K. Lam, Revisiting Hughes' dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm,, Transport. Res. B - Meth., 43 (2009), 127. doi: 10.1016/j.trb.2008.06.003. Google Scholar

[31]

P. Kachroo, S. J. Al-Nasur, S. A. Wadoo and A. Shende, Pedestrian Dynamics - Feedback Control of Crowd Evacuation,, Springer, (2008). Google Scholar

[32]

B. S. Kerner, The Physics of Traffic,, Springer, (2004). doi: 10.1007/978-3-540-40986-1. Google Scholar

[33]

G. Köster, D. Hartmann, and W. Klein, Microscopic pedestrian simulations: From passenger exchange times to regional evacuation,, in Operations Research Proceedings (eds. B. Hum, (2010), 571. Google Scholar

[34]

H. Klüpfel, A Cellular Automaton Model for Crowd Movement and Egress Simulation,, Ph.D. thesis, (2003). Google Scholar

[35]

A. Kneidl, M. Thiemann, A. Borrmann, S. Ruzika, H. W. Hamacher, G. Köster and E. Rank, Bidirectional Coupling of Macroscopic an Microscopic Approaches for Pedestrian Behavior Prediction,, in Pedestrian and Evacuation Dynamics (eds. R. D. Peacock, (2011), 459. doi: 10.1007/978-1-4419-9725-8_41. Google Scholar

[36]

A. Kneidl, M. Thiemann, D. Hartmann and A. Borrmann, Combining pedestrian simulation with a network flow optimization to support security staff in handling an evacuation of a soccer stadium,, in Proceedings of European Conference Forum 2011 (eds. E. Tobin and M. Otreba), (2011). Google Scholar

[37]

R. Löhner, On the modeling of pedestrian motion,, Appl. Math. Model., 34 (2010), 366. doi: 10.1016/j.apm.2009.04.017. Google Scholar

[38]

R. J. LeVeque, Numerical Methods for Conservation Laws,, Birkhäuser, (2005). doi: 10.1007/978-3-0348-8629-1. Google Scholar

[39]

T. I. Lakoba, D. J. Kaup and N. M. Finkelstein, Modifications of the Helbing-Molnár-Farkas-Vicsek social force model for pedestrian evolution,, Simulation, 81 (2005), 339. doi: 10.1177/0037549705052772. Google Scholar

[40]

K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider, Extended floor field CA model for evacuation dynamics,, IEICE Trans. Inf. Syst., E87D (2004), 726. Google Scholar

[41]

M. Oppenhäuser, Realisierung und Potenzialanalyse von wissenschaftlichen Konzepten zur Regionalen Evakuierung aus Polizeilicher Sicht am Beispiel des Projektes REPKA,, Master's Thesis, (2011). Google Scholar

[42]

N. Pelechano, J. M. Allbeck and N. Badler, Virtual Crowds: Methods, Simulation, and Control,, Morgan & Claypool Publishers, (2008). doi: 10.2200/S00123ED1V01Y200808CGR008. Google Scholar

[43]

D. R. Parisi, M. Gilman and H. Moldovan, A modification of the social force model can reproduce experimental data of pedestrian flows in normal conditions,, Physica A, 388 (2009), 3600. doi: 10.1016/j.physa.2009.05.027. Google Scholar

[44]

I. Prigogine and R. Herman, Kinetic Theory of Vehicular Flow,, Elsevier, (1971). Google Scholar

[45]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing,, Cambridge University Press, (2007). Google Scholar

[46]

, REPKA, Regionale Evakuierung: Planung, Kontrolle und Anpassung,, , (). Google Scholar

[47]

E. V. RiMEA, Richtlinie für Mikroskopische Entfluchtungsanalysen,, , (). Google Scholar

[48]

A. Schadschneider, W. Klingsch, H. Klüpfel, T. Kretz, C. Rogsch and A. Seyfried, Evacuation dynamics: Empirical results, modeling and applications,, Encyclopedia of Complexity and System Science, (2009), 3142. Google Scholar

[49]

A. Schadschneider and A. Seyfried, Empirical results for pedestrian dynamics and their implications for modeling,, Netw. Heterog. Media, 6 (2011), 545. doi: 10.3934/nhm.2011.6.545. Google Scholar

[50]

A. Varas, M. D. Cornejo, D. Mainemer, B. Toledo, J. Rogan, V. Munoz and J. A. Valdivia, Cellular automaton model for evacuation process with obstacles,, Physica A, 382 (2007), 631. doi: 10.1016/j.physa.2007.04.006. Google Scholar

[51]

U. Weidmann, Transporttechnik der Fussgänger: Transporttechnische Eigenschaften des Fussgängerverkehrs (Literaturauswertung), Schriftenreihe des IVT, (1993). Google Scholar

[52]

Y. Xia, S. C. Wong and C. W. Shu, Dynamic continuum pedestrian flow model with memory effect,, Phys. Rev. E, 79 (2009). doi: 10.1103/PhysRevE.79.066113. Google Scholar

[53]

W. J. Yu, R. Chen, L. Y. Dong and S. Q. Dai, Centrifugal force model for pedestrian dynamics,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.026112. Google Scholar

[54]

K. Yamamoto, S. Kokubo and K. Nishinari, Simulation for pedestrian dynamics by real-coded cellular automata,, Physica A, 379 (2007). doi: 10.1016/j.physa.2007.02.040. Google Scholar

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