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April  2019, 12(2): 397-409. doi: 10.3934/krm.2019017

On the interplay between behavioral dynamics and social interactions in human crowds

(1). 

Politecnico of Torino and Collegio Carlo Alberto, Torino, Italy

(2). 

School of Engineering, University of Edinburgh, Edinburgh, United Kingdom

(3). 

Mathematics and Population Dynamics Laboratory-UMMISCO, Faculty of Sciences of Semlalia of Marrakech, Cadi Ayyad Univ., Morocco

(4). 

Jacques Louis-Lions Laboratory, Pierre et Marie Curie University, Paris 6, France

Received  May 2018 Published  November 2018

This paper presents a computational modeling approach to the dynamics of human crowds, where social interactions can have an important influence on the behavioral dynamics of pedestrians. The modeling of the contagion and propagation of emotional states is carried out by looking at real physical situations where safety problems might arise in some specific circumstances. The approach is based on the methods of the kinetic theory of active particles. The evacuation of a metro station is simulated to enlighten the role of the emotional state in the overall dynamics.

Citation: Nicola Bellomo, Livio Gibelli, Nisrine Outada. On the interplay between behavioral dynamics and social interactions in human crowds. Kinetic & Related Models, 2019, 12 (2) : 397-409. doi: 10.3934/krm.2019017
References:
[1]

G. Ajmone MarsanN. Bellomo and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Math. Models Methods Appl. Sci., 26 (2016), 1051-1093.  doi: 10.1142/S0218202516500251.  Google Scholar

[2]

V.V. Aristov, Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows, Springer-Verlag, New York, 2001. doi: 10.1007/978-94-010-0866-2.  Google Scholar

[3]

P. BarbanteA. Frezzotti and L. Gibelli, A kinetic theory description of liquid menisci at the microscale, Kinet. Relat. Models, 8 (2015), 235-254.  doi: 10.3934/krm.2015.8.235.  Google Scholar

[4]

N. Bellomo and A. Bellouquid, On multiscale models of pedestrian crowds from mesoscopic to macroscopic, Comm. Math. Sciences, 13 (2015), 1649-1664.  doi: 10.4310/CMS.2015.v13.n7.a1.  Google Scholar

[5]

N. Bellomo, A. Bellouquid, L. Gibelli and N. Outada, A Quest Towards a Mathematical Theory of Living Systems, Birkhäuser, New York, 2017. doi: 10.1007/978-3-319-57436-3.  Google Scholar

[6]

N. BellomoA. Bellouquid and D. Knopoff, From the micro-scale to collective crowd dynamics, Multiscale Model. Sim., 11 (2013), 943-963.   Google Scholar

[7]

N. BellomoD. ClarkL. GibelliP. Townsend and B.J. Vreugdenhil, Vreugdenhil, Human behaviours in evacuation crowd dynamics: From modelling to ig data toward crisis management, Phys. Life Rev., 18 (2016), 1-21.   Google Scholar

[8]

N. Bellomo and L. Gibelli, Toward a behavioral-social dynamics of pedestrian crowds, Math. Models Methods Appl. Sci., 25 (2015), 395-400.  doi: 10.1142/S0218202515020017.  Google Scholar

[9]

N. Bellomo and L. Gibelli, Behavioral crowds: Modeling and Monte Carlo simulations toward validation, Comp. & Fluids, 141 (2016), 13-21.  doi: 10.1016/j.compfluid.2016.04.022.  Google Scholar

[10]

D. BuriniS. De Lillo and L. Gibelli, Stochastic differential "nonlinear" games modeling collective learning dynamics, Phys. Life Rev., 16 (2016), 123-139.   Google Scholar

[11]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Diluted Gas, Springer, Heidelberg, New York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[12]

A. CorbettaA. Mountean and K. Vafayi, Parameter estimation of social forces in pedestrian dynamics models via probabilistic method, Math. Biosci. Eng., 12 (2015), 337-356.  doi: 10.3934/mbe.2015.12.337.  Google Scholar

[13]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, Springer, 2014. doi: 10.1007/978-3-319-06620-2.  Google Scholar

[14]

P. DegondC. Appert-RollandJ. Pettré and G. Theraulaz, Vision based macroscopic pedestrian models, Kinetic Related Models, 6 (2013), 809-839.  doi: 10.3934/krm.2013.6.809.  Google Scholar

[15]

P. DegondC. Appert-RollandM. MoussaïdJ. Pettré and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068.  doi: 10.1007/s10955-013-0805-x.  Google Scholar

[16]

P. DegondJ.-G. LiuS. Merino-Aceituno and T. Tardiveau, Continuum dynamics of the intention field under weakly cohesive social interaction, Math. Models Methods Appl. Sci., 27 (2017), 159-182.  doi: 10.1142/S021820251740005X.  Google Scholar

[17]

J. M. Epstein, Modeling civil violence: An agent based computational approach, Proc. Nat. Acad. Sci., 99 (2002), 7243-7250.   Google Scholar

[18]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141.   Google Scholar

[19]

D. HelbingI. Farkas and T. Vicsek, Simulating dynamical feature of escape panic, Nature, 407 (2000), 487-490.   Google Scholar

[20]

D. Helbing and A. Johansson, Pedestrian crowd and evacuation dynamics, Enciclopedia of Complexity and System Science, (2009), 6476-6495.   Google Scholar

[21]

D. Helbing, A. Johansson and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E, 75 (2007), 046109. Google Scholar

[22]

R. L. Hughes, The flow of human crowds, Annu. Rev. Fluid Mech., 35 (2003), 169-182.  doi: 10.1146/annurev.fluid.35.101101.161136.  Google Scholar

[23]

M. Kinateder et al., Human behaviour in severe tunnel accidents: Effects of information and behavioural training, Transp. Res. Part F: Traffic Psychology and Behaviour, 17 (2013), 20–32. Google Scholar

[24]

J. Lin and T. A. Luckas, A particle swarm optimization model of emergency airplane evacuation with emotion, Net. Het. Media, 10 (2015), 631-646.  doi: 10.3934/nhm.2015.10.631.  Google Scholar

[25]

M. MoussaïdD. HelbingS. GarnierA. JohanssonM. Combe and G. Theraulaz, Experimental study of the behavioural mechanisms underlying self-organization in human crowds, Proc. Roy. Soc. B, 276 (2009), 2755-2762.   Google Scholar

[26]

M. Moussaïd and G. Theraulaz, Comment les piétons marchent dans la foule, La Recherche, 450 (2011), 56-59.   Google Scholar

[27]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2014. Google Scholar

[28]

F. RonchiF. Nieto UrizX. Criel and P. Reilly, Modelling large-scale evacuation of music festival, Fire Safety, 5 (2016), 11-19.   Google Scholar

[29]

L. WangM. Short and A. L. Bertozzi, Efficient numerical methods for multiscale crowd dynamics with emotional contagion, Math. Models Methods Appl. Sci., 27 (2017), 205-230.  doi: 10.1142/S0218202517400073.  Google Scholar

[30]

N. WijermansC. ConradoM. van SteenC. Martella and J. L. Li, A landscape of crowd management support: An integrative approach, Safety Science, 86 (2016), 142-164.   Google Scholar

[31]

N. WijermansR. JornaW. JagerT. van Vliet and O. M. J. Adang, CROSS: Modelling crowd behaviour with social-cognitive agents, Journal of Artificial Societies and Social Simulation, 16 (2013), 1-14.   Google Scholar

show all references

References:
[1]

G. Ajmone MarsanN. Bellomo and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Math. Models Methods Appl. Sci., 26 (2016), 1051-1093.  doi: 10.1142/S0218202516500251.  Google Scholar

[2]

V.V. Aristov, Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows, Springer-Verlag, New York, 2001. doi: 10.1007/978-94-010-0866-2.  Google Scholar

[3]

P. BarbanteA. Frezzotti and L. Gibelli, A kinetic theory description of liquid menisci at the microscale, Kinet. Relat. Models, 8 (2015), 235-254.  doi: 10.3934/krm.2015.8.235.  Google Scholar

[4]

N. Bellomo and A. Bellouquid, On multiscale models of pedestrian crowds from mesoscopic to macroscopic, Comm. Math. Sciences, 13 (2015), 1649-1664.  doi: 10.4310/CMS.2015.v13.n7.a1.  Google Scholar

[5]

N. Bellomo, A. Bellouquid, L. Gibelli and N. Outada, A Quest Towards a Mathematical Theory of Living Systems, Birkhäuser, New York, 2017. doi: 10.1007/978-3-319-57436-3.  Google Scholar

[6]

N. BellomoA. Bellouquid and D. Knopoff, From the micro-scale to collective crowd dynamics, Multiscale Model. Sim., 11 (2013), 943-963.   Google Scholar

[7]

N. BellomoD. ClarkL. GibelliP. Townsend and B.J. Vreugdenhil, Vreugdenhil, Human behaviours in evacuation crowd dynamics: From modelling to ig data toward crisis management, Phys. Life Rev., 18 (2016), 1-21.   Google Scholar

[8]

N. Bellomo and L. Gibelli, Toward a behavioral-social dynamics of pedestrian crowds, Math. Models Methods Appl. Sci., 25 (2015), 395-400.  doi: 10.1142/S0218202515020017.  Google Scholar

[9]

N. Bellomo and L. Gibelli, Behavioral crowds: Modeling and Monte Carlo simulations toward validation, Comp. & Fluids, 141 (2016), 13-21.  doi: 10.1016/j.compfluid.2016.04.022.  Google Scholar

[10]

D. BuriniS. De Lillo and L. Gibelli, Stochastic differential "nonlinear" games modeling collective learning dynamics, Phys. Life Rev., 16 (2016), 123-139.   Google Scholar

[11]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Diluted Gas, Springer, Heidelberg, New York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar

[12]

A. CorbettaA. Mountean and K. Vafayi, Parameter estimation of social forces in pedestrian dynamics models via probabilistic method, Math. Biosci. Eng., 12 (2015), 337-356.  doi: 10.3934/mbe.2015.12.337.  Google Scholar

[13]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, Springer, 2014. doi: 10.1007/978-3-319-06620-2.  Google Scholar

[14]

P. DegondC. Appert-RollandJ. Pettré and G. Theraulaz, Vision based macroscopic pedestrian models, Kinetic Related Models, 6 (2013), 809-839.  doi: 10.3934/krm.2013.6.809.  Google Scholar

[15]

P. DegondC. Appert-RollandM. MoussaïdJ. Pettré and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068.  doi: 10.1007/s10955-013-0805-x.  Google Scholar

[16]

P. DegondJ.-G. LiuS. Merino-Aceituno and T. Tardiveau, Continuum dynamics of the intention field under weakly cohesive social interaction, Math. Models Methods Appl. Sci., 27 (2017), 159-182.  doi: 10.1142/S021820251740005X.  Google Scholar

[17]

J. M. Epstein, Modeling civil violence: An agent based computational approach, Proc. Nat. Acad. Sci., 99 (2002), 7243-7250.   Google Scholar

[18]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141.   Google Scholar

[19]

D. HelbingI. Farkas and T. Vicsek, Simulating dynamical feature of escape panic, Nature, 407 (2000), 487-490.   Google Scholar

[20]

D. Helbing and A. Johansson, Pedestrian crowd and evacuation dynamics, Enciclopedia of Complexity and System Science, (2009), 6476-6495.   Google Scholar

[21]

D. Helbing, A. Johansson and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E, 75 (2007), 046109. Google Scholar

[22]

R. L. Hughes, The flow of human crowds, Annu. Rev. Fluid Mech., 35 (2003), 169-182.  doi: 10.1146/annurev.fluid.35.101101.161136.  Google Scholar

[23]

M. Kinateder et al., Human behaviour in severe tunnel accidents: Effects of information and behavioural training, Transp. Res. Part F: Traffic Psychology and Behaviour, 17 (2013), 20–32. Google Scholar

[24]

J. Lin and T. A. Luckas, A particle swarm optimization model of emergency airplane evacuation with emotion, Net. Het. Media, 10 (2015), 631-646.  doi: 10.3934/nhm.2015.10.631.  Google Scholar

[25]

M. MoussaïdD. HelbingS. GarnierA. JohanssonM. Combe and G. Theraulaz, Experimental study of the behavioural mechanisms underlying self-organization in human crowds, Proc. Roy. Soc. B, 276 (2009), 2755-2762.   Google Scholar

[26]

M. Moussaïd and G. Theraulaz, Comment les piétons marchent dans la foule, La Recherche, 450 (2011), 56-59.   Google Scholar

[27]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2014. Google Scholar

[28]

F. RonchiF. Nieto UrizX. Criel and P. Reilly, Modelling large-scale evacuation of music festival, Fire Safety, 5 (2016), 11-19.   Google Scholar

[29]

L. WangM. Short and A. L. Bertozzi, Efficient numerical methods for multiscale crowd dynamics with emotional contagion, Math. Models Methods Appl. Sci., 27 (2017), 205-230.  doi: 10.1142/S0218202517400073.  Google Scholar

[30]

N. WijermansC. ConradoM. van SteenC. Martella and J. L. Li, A landscape of crowd management support: An integrative approach, Safety Science, 86 (2016), 142-164.   Google Scholar

[31]

N. WijermansR. JornaW. JagerT. van Vliet and O. M. J. Adang, CROSS: Modelling crowd behaviour with social-cognitive agents, Journal of Artificial Societies and Social Simulation, 16 (2013), 1-14.   Google Scholar

Figure 1.  Geometry of the venue
Figure 2.  Density contour plots of the mean density of the emotional state, $\rho \bar{u}$, with (pedestrians on the left) and without (pedestrians on the right) social interactions at different times
Figure 3.  Density contour plots of the mean density of the emotional state, $\rho \bar{u}$, with (pedestrians on the left) and without (pedestrians on the right) social interactions at different times
Figure 4.  Averaged value of the emotional state of the crowd, $\bar{U}$, versus time for different values of the social parameter
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