# American Institute of Mathematical Sciences

March  2017, 12(1): 93-112. doi: 10.3934/nhm.2017004

## A discrete Hughes model for pedestrian flow on graphs

 1 Dip. di Scienze di Base e Applicate per l'Ingegneria, "Sapienza" Università di Roma, via Scarpa 16, 00161 Roma, Italy 2 RICAM, Austrian Academy of Sciences (ÖAW), Altenbergerstr. 69, 4040 Linz, Austria 3 Dip. di Matematica, "Sapienza" Università di Roma, P.le Aldo Moro 5, 00185 Roma, Italy

* Corresponding author:Fabio Camilli

Received  April 2016 Revised  December 2016 Published  February 2017

Fund Project: AF is supported the Austrian Academy of Sciences ÖAW via the New Frontiers Group NST-001.

In this paper, we introduce a discrete time-finite state model for pedestrian flow on a graph in the spirit of the Hughes dynamic continuum model. The pedestrians, represented by a density function, move on the graph choosing a route to minimize the instantaneous travel cost to the destination. The density is governed by a conservation law whereas the minimization principle is described by a graph eikonal equation. We show that the discrete model is well-posed and the numerical examples reported confirm the validity of the proposed model and its applicability to describe real situations.

Citation: Fabio Camilli, Adriano Festa, Silvia Tozza. A discrete Hughes model for pedestrian flow on graphs. Networks and Heterogeneous Media, 2017, 12 (1) : 93-112. doi: 10.3934/nhm.2017004
##### References:
 [1] "Jamarat: Study of Current Conditions and Means of Improvements", Hajj Research Centre, Um Al-Qura University Saudi Arabia, 1984. [2] A. Alla, M. Falcone and D. Kalise, An efficient policy iteration algorithm for dynamic programming equations, SIAM J. Sci. Comput., 37 (2015), A181-A200.  doi: 10.1137/130932284. [3] D. Amadori and M.Di Francesco, The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions, Acta Math. Sci., 32 (2012), 259-280.  doi: 10.1016/S0252-9602(12)60016-2. [4] D. Amadori, P. Goatin and M. D. Rosini, Existence results for Hughes model for pedestrian flows, J. Math. Anal. Appl., 420 (2014), 387-406.  doi: 10.1016/j.jmaa.2014.05.072. [5] M. Bardi and J. P. Maldonado Lopez, A Dijkstra-type algorithm for dynamic games, Dyn. Games Appl., (2015), 1-4.  doi: 10.1007/s13235-015-0156-0. [6] N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463.  doi: 10.1137/090746677. [7] M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: theoretical study, Netw. Heterog. Media, 9 (2014), 519-552.  doi: 10.3934/nhm.2014.9.519. [8] F. Camilli and C. Marchi, A comparison among various notions of viscosity solutions for Hamilton-Jacobi equations on networks, J. Math. Anal. Appl., 407 (2013), 112-118.  doi: 10.1016/j.jmaa.2013.05.015. [9] F. Camilli and C. Marchi, Staionary mean field games systems defined on networks, SIAM J. Cont. Optim., 54 (2016), 1085-1103.  doi: 10.1137/15M1022082. [10] F. Camilli, A. Festa and D. Schieborn, An approximation scheme for a Hamilton-Jacobi equation defined on a network, Appl. Numer. Math., 73 (2013), 33-47.  doi: 10.1016/j.apnum.2013.05.003. [11] E. Carlini, A. Festa, F. J. Silva and M. T. Wolfram, A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow, Dyn. Games Appl., (2016), 1-23.  doi: 10.1007/s13235-016-0202-6. [12] G. Costeseque, J. P. Lebacque and R. Monneau, A convergent scheme for Hamilton-Jacobi equations on a junction: Application to traffic, Numer. Math., 129 (2015), 405-447.  doi: 10.1007/s00211-014-0643-z. [13] E. Cristiani and F. S. Priuli, A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks, Netw. Heterog. Media, 10 (2015), 857-876.  doi: 10.3934/nhm.2015.10.857. [14] M. Di Francesco, P. A. Markowich, J. F. Pietschmann and M. T. Wolfram, On the Hughes model for pedestrian flow: The one-dimensional case, J. Differential Equations, 250 (2011), 1334-1362.  doi: 10.1016/j.jde.2010.10.015. [15] Z. Fang, Q. Li, Q. Li, L. D. Han and D. Wang, A proposed pedestrian waiting-time model for improving space-time use efficiency in stadium evacuation scenarios, Build. Environ., 46 (2011), 1774-1784.  doi: 10.1016/j.buildenv.2011.02.005. [16] M. Garavello and B. Piccoli, "Traffic Flow on Networks" AIMS Series on Applied Mathematics, Vol. 1, American Institute of Mathematical Sciences, 2006. [17] 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, Transportat. Res. B-Meth., 43 (2009), 127-141.  doi: 10.1016/j.trb.2008.06.003. [18] R. L. Hughes, The flow of large crowds of pedestrians, Math. Comput. Simulat., 53 (2000), 367-370.  doi: 10.1016/S0378-4754(00)00228-7. [19] R. L. Hughes, A continuum theory for the flow of pedestrians, Transport. Res. B-Meth., 36 (2002), 507-535.  doi: 10.1016/S0191-2615(01)00015-7. [20] R. L. Hughes, The flow of human crowds, Annu. rev. fluid mech., 35 (2003), 169-182.  doi: 10.1146/annurev.fluid.35.101101.161136. [21] P.-L. Lions and P. E. Souganidis, Viscosity solutions for junctions: Well posedness and stability, Rend. Lincei Mat. Appl., 27 (2016), 535-545.  doi: 10.4171/RLM/747. [22] J. Manfredi, A. Oberman and A. Sviridov, Nonlinear elliptic partial differential equations and p-harmonic functions on graphs, Differ. Integral Equ., 28 (2015), 79-102. [23] M. Puterman and S. L. Brumelle, On the convergence of policy iteration in stationary dynamic programming, Math. Oper. Res., 4 (1979), 60-69.  doi: 10.1287/moor.4.1.60. [24] J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698.  doi: 10.1137/S0036142999363668. [25] A. Treuille, S. Cooper and Z. Popovîc, Continuum crowds, ACM Trans. Graph., 25 (2006), 1160-1168.  doi: 10.1145/1179352.1142008.

show all references

##### References:
 [1] "Jamarat: Study of Current Conditions and Means of Improvements", Hajj Research Centre, Um Al-Qura University Saudi Arabia, 1984. [2] A. Alla, M. Falcone and D. Kalise, An efficient policy iteration algorithm for dynamic programming equations, SIAM J. Sci. Comput., 37 (2015), A181-A200.  doi: 10.1137/130932284. [3] D. Amadori and M.Di Francesco, The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions, Acta Math. Sci., 32 (2012), 259-280.  doi: 10.1016/S0252-9602(12)60016-2. [4] D. Amadori, P. Goatin and M. D. Rosini, Existence results for Hughes model for pedestrian flows, J. Math. Anal. Appl., 420 (2014), 387-406.  doi: 10.1016/j.jmaa.2014.05.072. [5] M. Bardi and J. P. Maldonado Lopez, A Dijkstra-type algorithm for dynamic games, Dyn. Games Appl., (2015), 1-4.  doi: 10.1007/s13235-015-0156-0. [6] N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev., 53 (2011), 409-463.  doi: 10.1137/090746677. [7] M. Briani and E. Cristiani, An easy-to-use algorithm for simulating traffic flow on networks: theoretical study, Netw. Heterog. Media, 9 (2014), 519-552.  doi: 10.3934/nhm.2014.9.519. [8] F. Camilli and C. Marchi, A comparison among various notions of viscosity solutions for Hamilton-Jacobi equations on networks, J. Math. Anal. Appl., 407 (2013), 112-118.  doi: 10.1016/j.jmaa.2013.05.015. [9] F. Camilli and C. Marchi, Staionary mean field games systems defined on networks, SIAM J. Cont. Optim., 54 (2016), 1085-1103.  doi: 10.1137/15M1022082. [10] F. Camilli, A. Festa and D. Schieborn, An approximation scheme for a Hamilton-Jacobi equation defined on a network, Appl. Numer. Math., 73 (2013), 33-47.  doi: 10.1016/j.apnum.2013.05.003. [11] E. Carlini, A. Festa, F. J. Silva and M. T. Wolfram, A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow, Dyn. Games Appl., (2016), 1-23.  doi: 10.1007/s13235-016-0202-6. [12] G. Costeseque, J. P. Lebacque and R. Monneau, A convergent scheme for Hamilton-Jacobi equations on a junction: Application to traffic, Numer. Math., 129 (2015), 405-447.  doi: 10.1007/s00211-014-0643-z. [13] E. Cristiani and F. S. Priuli, A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks, Netw. Heterog. Media, 10 (2015), 857-876.  doi: 10.3934/nhm.2015.10.857. [14] M. Di Francesco, P. A. Markowich, J. F. Pietschmann and M. T. Wolfram, On the Hughes model for pedestrian flow: The one-dimensional case, J. Differential Equations, 250 (2011), 1334-1362.  doi: 10.1016/j.jde.2010.10.015. [15] Z. Fang, Q. Li, Q. Li, L. D. Han and D. Wang, A proposed pedestrian waiting-time model for improving space-time use efficiency in stadium evacuation scenarios, Build. Environ., 46 (2011), 1774-1784.  doi: 10.1016/j.buildenv.2011.02.005. [16] M. Garavello and B. Piccoli, "Traffic Flow on Networks" AIMS Series on Applied Mathematics, Vol. 1, American Institute of Mathematical Sciences, 2006. [17] 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, Transportat. Res. B-Meth., 43 (2009), 127-141.  doi: 10.1016/j.trb.2008.06.003. [18] R. L. Hughes, The flow of large crowds of pedestrians, Math. Comput. Simulat., 53 (2000), 367-370.  doi: 10.1016/S0378-4754(00)00228-7. [19] R. L. Hughes, A continuum theory for the flow of pedestrians, Transport. Res. B-Meth., 36 (2002), 507-535.  doi: 10.1016/S0191-2615(01)00015-7. [20] R. L. Hughes, The flow of human crowds, Annu. rev. fluid mech., 35 (2003), 169-182.  doi: 10.1146/annurev.fluid.35.101101.161136. [21] P.-L. Lions and P. E. Souganidis, Viscosity solutions for junctions: Well posedness and stability, Rend. Lincei Mat. Appl., 27 (2016), 535-545.  doi: 10.4171/RLM/747. [22] J. Manfredi, A. Oberman and A. Sviridov, Nonlinear elliptic partial differential equations and p-harmonic functions on graphs, Differ. Integral Equ., 28 (2015), 79-102. [23] M. Puterman and S. L. Brumelle, On the convergence of policy iteration in stationary dynamic programming, Math. Oper. Res., 4 (1979), 60-69.  doi: 10.1287/moor.4.1.60. [24] J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698.  doi: 10.1137/S0036142999363668. [25] A. Treuille, S. Cooper and Z. Popovîc, Continuum crowds, ACM Trans. Graph., 25 (2006), 1160-1168.  doi: 10.1145/1179352.1142008.
Scheme of the network and initial density.
Test 1 (Dirichlet boundary conditions): density and potential before the first time of interaction.
Test 1 (Dirichlet boundary conditions): density and potential after the first time of interaction at $(0.2,-0.8)$.
Test 2 (No-flux boundary conditions): stable configuration obtained for $t>3.5$.
Test 3 (Dirichlet BCs with diffusion $\epsilon=1$): Density at two different time steps ($t=0.75$ and $t=1.75$).
The Wuhan Sports Centre (left) and the evacuation network considered in our study (right).
Initial distribution of density on the graph (up) and drift potential in the initial configuration (down).
Initial distribution of density on the graph (up) and drift potential in the initial configuration (down).
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