Journal of Engineering Mathematics

, Volume 101, Issue 1, pp 153–173 | Cite as

From individual behaviour to an evaluation of the collective evolution of crowds along footbridges



This paper proposes a crowd dynamic macroscopic model grounded on microscopic phenomenological observations which are upscaled by means of a formal mathematical procedure. The actual applicability of the model to real-world problems is tested by considering the pedestrian traffic along footbridges, of interest for Structural and Transportation Engineering. The genuinely macroscopic quantitative description of the crowd flow directly matches the engineering need of bulk results. However, three issues beyond the sole modelling are of primary importance: the pedestrian inflow conditions, the numerical approximation of the equations for non trivial footbridge geometries and the calibration of the free parameters of the model on the basis of in situ measurements currently available. These issues are discussed, and a solution strategy is proposed.


Collective evolution Continuous crowd models Footbridges Individual behaviour 

Mathematics Subject Classification

35L65 35Q70 90B20 



The work of A. Corbetta was supported by a Lagrange Foundation PhD scholarship.


  1. 1.
    Cristiani E, Piccoli B, Tosin A (2014) Multiscale modeling of pedestrian dynamics. MS&A: modeling, simulation and applications, vol 12. Springer International Publishing, New YorkGoogle Scholar
  2. 2.
    Helbing D (2001) Traffic and related self-driven many-particle systems. Rev Mod Phys 73(4):1067–1141ADSCrossRefGoogle Scholar
  3. 3.
    Zajaca FE, Neptune RR, Kautz SA (2003) Biomechanics and muscle coordination of human walking: part ii: lessons from dynamical simulations and clinical implications. Gait Posture 17(1):1–17CrossRefGoogle Scholar
  4. 4.
    Warren WH (2006) The dynamics of perception and action. Psychol Rev 113(2):358–389MathSciNetCrossRefGoogle Scholar
  5. 5.
    Xu ML, Jiang H, Jin XG, Deng Z (2014) Crowd simulation and its applications: recent advances. J Comput Sci Technol 29(5):799–811CrossRefGoogle Scholar
  6. 6.
    Gwynne S, Galea ER, Owen M, Lawrence PJ, Filippidis L (1999) A review of the methodologies used in the computer simulation of evacuation from the built environment. Build Environ 34(6):741–749CrossRefGoogle Scholar
  7. 7.
    Zheng X, Sun J, Zhong T (2010) Study on mechanics of crowd jam based on the cusp-catastrophe model. Saf Sci 48(10):1236–1241CrossRefGoogle Scholar
  8. 8.
    Duives DC, Daamen W, Hoogendoorn SP (2013) State-of-the-art crowd motion simulation models. Transp Res Part C 37(12):193–209CrossRefGoogle Scholar
  9. 9.
    Ingólfsson ET, Georgakis CT, Jonsson J (2012) Pedestrian-induced lateral vibrations of footbridges: a literature review. Eng Struct 45:21–52CrossRefGoogle Scholar
  10. 10.
    Živanović S, Pavic A, Reynolds P (2005) Vibration serviceability of footbridges under human-induced excitation: a literature review. J Sound Vib 279:1–74ADSCrossRefGoogle Scholar
  11. 11.
    Ali S, Nishino K, Manocha D, Shah M (2013) Modeling, simulation and visual analysis of crowds: a multidisciplinary perspective. The international series in video computing, vol 11. Springer, BerlinGoogle Scholar
  12. 12.
    Venuti F, Bruno L (2009) Crowd-structure interaction in lively footbridges under synchronous lateral excitation: a literature review. Phys Life Rev 6(3):176–206ADSCrossRefGoogle Scholar
  13. 13.
    Agnelli JP, Colasuonno F, Knopoff D (2015) A kinetic theory approach to the dynamics of crowd evacuation from bounded domains. Math Models Methods Appl Sci 25(1):109–129MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Degond P, Appert-Rolland C, Pettré J, Theraulaz G (2013) Vision-based macroscopic pedestrian models. Kinet Relat Models 6(4):809–839MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Degond P, Appert-Rolland C, Moussaïd M, Pettré J, Theraulaz G (2013) A hierarchy of heuristic-based models of crowd dynamics. J Stat Phys 152(6):1033–1068ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Blue V, Adler J (1998) Emergent fundamental pedestrian flows from cellular automata microsimulation. Transp Res Board 1644:29–36CrossRefGoogle Scholar
  17. 17.
    Helbing D, Molnár P (1995) Social force models for pedestrian dynamics. Phys Rev E 51(5):4282–4286ADSCrossRefGoogle Scholar
  18. 18.
    Schadschneider A, Klingsch W, Klüpfel H, Kretz T, Rogsch C, Seyfried A (2011) Evacuation dynamics: empirical results, modeling and applications. In: Meyers RA (ed) Extreme environmental events. Springer, New York, pp 517–550CrossRefGoogle Scholar
  19. 19.
    Zheng X, Zhong T, Liu M (2009) Modeling crowd evacuation of a building based on seven methodological approaches. Build Environ 44(3):437–445MathSciNetCrossRefGoogle Scholar
  20. 20.
    Carroll SP, Owen JS, Hussein MFM (2012) Modelling crowd-bridge dynamic interaction with a discretely defined crowd. J Sound Vib 331(11):2685–2709ADSCrossRefGoogle Scholar
  21. 21.
    Haron F, Alginahi YM, Kabir MN, Mohamed AI (2012) Software evaluation for crowd evacuation—case study: Al-Masjid an-Nabawi. Int J Comput Sci Issues 9(2):128–134Google Scholar
  22. 22.
    Corbetta A, Muntean A, Vafayi K (2015) Parameter estimation of social forces in pedestrian dynamics models via a probabilistic method. Math Biosci Eng 12:337–356MathSciNetMATHGoogle Scholar
  23. 23.
    Johansson A, Helbing D, Shukla PK (2007) Specification of the social force pedestrian model by evolutionary adjustment to video tracking data. Adv Complex Syst 10(supp02):271–288MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Zanlungo F, Ikeda T, Kanda T (2011) Social force model with explicit collision prediction. Europhys Lett (EPL) 93(6):68005ADSCrossRefGoogle Scholar
  25. 25.
    Hughes RL (2000) The flow of large crowds of pedestrians. Math Comput Simul 53:367–370CrossRefGoogle Scholar
  26. 26.
    Daamen W (2004) Modelling passenger flows in public transport facilities. PhD thesis, Department of Transport and Planning, Delft University of TechnologyGoogle Scholar
  27. 27.
    Twarogowska M, Goatin P, Duvigneau R (2014) Comparative study of macroscopic pedestrian models. Transp Res Procedia 2:477–485CrossRefGoogle Scholar
  28. 28.
    Bruno L, Venuti F (2009) Crowd-structure interaction in footbridges: modelling, application to a real case-study and sensitivity analyses. J Sound Vib 323(323):475–493ADSCrossRefGoogle Scholar
  29. 29.
    Venuti F, Bruno L (2013) Mitigation of human-induced lateral vibrations on footbridges through walkway shaping. Eng Struct 56:95–104CrossRefGoogle Scholar
  30. 30.
    Fruin JJ (1987) Pedestrian planning and design. Elevator World Inc., MobileGoogle Scholar
  31. 31.
    Cristiani E, Piccoli B, Tosin A (2011) Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model Simul 9(1):155–182MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Piccoli B, Tosin A (2009) Pedestrian flows in bounded domains with obstacles. Continuum Mech Thermodyn 21(2):85–107ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    AlGadhi SAH, Mahmassani H (1990) Modelling crowd behavior and movement: application to Makkah pilgrimage. Transp Traffic Theory 59–78:1990Google Scholar
  34. 34.
    AlGadhi SAH, Mahmassani HS (1991) Simulation of crowd behavior and movement: fundamental relations and application. Transp Res Rec 1320(1320):260–268Google Scholar
  35. 35.
    Colombo RM, Goatin P, Rosini MD (2011) On the modelling and management of traffic. ESAIM Math Model Numer Anal 45(05):853–872MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Huang L, Wong S, Zhang M, Shu CW, Lam W (2009) Revisiting hughes’ dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm. Transp Res Part B 43:127–141CrossRefGoogle Scholar
  37. 37.
    Xia Y, Wong SC, Zhang M, Shu CW, Lam WHK (2008) An efficient discontinuous Galerkin method on triangular meshes for a pedestrian flow model. Int J Numer Methods Eng 76:337–350MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Piccoli B, Tosin A (2011) Time-evolving measures and macroscopic modeling of pedestrian flow. Arch Ration Mech Anal 199(3):707–738MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Lachapelle A, Wolfram MT (2011) On a mean field game approach modeling congestion and aversion in pedestrian crowds. Transp Res B 45(10):1572–1589CrossRefGoogle Scholar
  40. 40.
    Bruno L, Tosin A, Tricerri P, Venuti F (2011) Non-local first-order modelling of crowd dynamics: a multidimensional framework with applications. Appl Math Model 35(1):426–445MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Hughes TJ, Marsden JE (1976) A short course in fluid mechanics. Publish or Perish, BostonMATHGoogle Scholar
  42. 42.
    Piccoli B, Rossi F (2013) Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes. Acta Appl Math 124(1):73–105MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Tosin A, Frasca P (2011) Existence and approximation of probability measure solutions to models of collective behaviors. Netw Heterog Media 6(3):561–596MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Colombo RM, Garavello M, Lécureux-Mercier M (2012) A class of nonlocal models for pedestrian traffic. Math Models Methods Appl Sci 22(4):1150023 (34 pages)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Colombi A, Scianna M, Tosin A (2014) Differentiated cell behavior: a multiscale approach using measure theory. J Math Biol 71(5):1049–1079MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Živanović S (2012) Benchmark footbridge for vibration serviceability assessment under vertical component of pedestrian load. ASCE J Struct Eng 138(10):1193–1202CrossRefGoogle Scholar
  47. 47.
    Dallard P, Fitzpatrick T, Flint A, Le Bourva S, Low A, Ridsdill Smith RM, Willford M (2001) The London millennium footbridge. Struct Eng 79(22):17–33Google Scholar
  48. 48.
    Fujino Y, Pacheco BM, Nakamura S, Warnitchai P (1993) Synchronization of human walking observed during lateral vibration of a congested pedestrian bridge. Earthq Eng Struct Dyn 22(9):741–758CrossRefGoogle Scholar
  49. 49.
    Setareh M (2011) Study of verrazano-narrows bridge movements during a New York City marathon. J Bridge Eng 16(1):127–138CrossRefGoogle Scholar
  50. 50.
    Evers JHM, Fetecau RC, Ryzhik L (2014) Anisotropic interactions in a first-order aggregation model: a proof of concept. Nonlinearity 28(8):2847–2871ADSMathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Maury B, Roudneff-Chupin A, Santambrogio F (2010) A macroscopic crowd motion model of gradient flow type. Math Models Methods Appl Sci 20(10):1787–1821MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Rimon E, Koditschek DE (1992) Exact robot navigation using artificial potential functions. IEEE Trans Robot Autom 8(5):501–518CrossRefGoogle Scholar
  53. 53.
    Connolly CI, Burns JB, Weiss R (1990) Path planning using Laplace’s equation. IEEE Int Conf Robot 3:2102–2106Google Scholar
  54. 54.
    Doob JL (2001) Classical potential theory and its probabilistic counterpart. Springer, BerlinCrossRefMATHGoogle Scholar
  55. 55.
    Iñiguez P, Rosell J (2009) Path planning using sub- and super-harmonic functions. In: Proceedings of the 40th international symposium on robotics, Barcelona, Spain, March 2009, pp 319–324Google Scholar
  56. 56.
    Russell L (2005) Footbridge awards 2005. Bridge Des Eng 11(41):35–49Google Scholar
  57. 57.
    Bögle A (2004) Footbridges. In: Schlaich J, Bergermann R (eds) Light structures. Prestel, New York, pp 232–267Google Scholar
  58. 58.
    Caetano E, Cunha A, Magalhaes F, Moutinho C (2010) Studies for controlling human-induced vibration of the Pedro e Inês footbridge, Portugal. Part 1: assessment of dynamic behaviour. Eng Struct 32(4):1069–1081CrossRefGoogle Scholar
  59. 59.
    Faure S, Maury B (2015) Crowd motion from the granular standpoint. Math Models Methods Appl Sci 25(3):463–493MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Maury B, Roudneff-Chupin A, Santambrogio F (2011) Handling congestion in crowd motion modeling. Netw Heterog Media 6(3):485–519MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Sampoli ML (2004) An automatic procedure to compute efficiently the intersection of two triangles. Technical report, University of Siena, ItalyGoogle Scholar
  62. 62.
    O’Rourke J (1994) Computational geometry in C. Cambridge University Press, New YorkMATHGoogle Scholar
  63. 63.
    Toussaint GT (1983) Solving geometric problems with the rotating calipers. In: Proceedings of the IEEE Melecon, vol 83. p A10Google Scholar
  64. 64.
    Vázquez JL, Vitillaro E (2008) Heat equation with dynamical boundary conditions of reactive type. Commun Partial Differ Equ 33(4):561–612MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Vázquez JL, Vitillaro E (2009) On the Laplace equation with dynamical boundary conditions of reactive-diffusive type. J Math Anal Appl 354(2):674–688MathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    Colli P, Fukao T (2015) Cahn–Hilliard equation with dynamic boundary conditions and mass constraint on the boundary. J Math Anal Appl 429(2):1190–1213MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Miranville A, Zelik S (2005) Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions. Math Methods Appl Sci 28(6):709–735MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Evers JHM, Hille SC, Muntean A (2015) Mild solutions to a measure-valued mass evolution problem with flux boundary conditions. J Differ Equ 259(3):1068–1097ADSMathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    Buchmueller S, Weidmann U (2006) Parameters of pedestrians, pedestrian traffic and walking facilities. Technical Report 132, ETH, ZürichGoogle Scholar
  70. 70.
    Venuti F, Bruno L (2007) An interpretative model of the pedestrian fundamental relation. C R Mecanique 335(4):194–200ADSCrossRefMATHGoogle Scholar
  71. 71.
    Pushkarev BS, Zupan JM (1975) Urban space for pedestrians. MIT Press, CambridgeGoogle Scholar
  72. 72.
    Habicht AT, Braaksma JP (1984) Effective width of pedestrian corridors. J Transp Eng 110(1):80–93CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Architecture and DesignPolitecnico di TorinoTurinItaly
  2. 2.Department of Structural and Geotechnical EngineeringPolitecnico di TorinoTurinItaly
  3. 3.Department of Mathematics and Computer Science, CASA - Centre for Analysis, Scientific Computing and ApplicationsEindhoven University of TechnologyEindhovenThe Netherlands
  4. 4.Istituto per le Applicazioni del Calcolo “M. Picone”Consiglio Nazionale delle RicercheRomeItaly

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