Today's Talk

• Macroscopic (hydrodynamic) models for pedestrians.
• Models derived/inspired from vehicular traffic models.
• Singular terms to enforce capacity constraints .
• Structure-preserving numerical analysis.

But There Are Other Models for Pedestrians!

• Reviews: Bellomo, Gibelli, Quaini, Reali (2022) , Bellomo, Liao, Quaini, Russo, Siettos (2023) .
• Force-based models: Reynolds (1987) , Helbing, Molnár (1995) , D'Orsogna, Chuang, Bertozzi, Chayes (2006) .
• Rational behaviour models: Moussaïd, Helbing, Theraulaz (2011) , Degond, Appert-Rolland, Moussaïd, Pettré, Theraulaz (2013) , RB, Carrillo, Degond (2018) , Salam, Tiwari, Klar, Sundar (2023) .
• Other hydrodynamic models: Hughes (2002) , Hughes (2003) , Colombo, Rosini (2005) , Carrillo, Martin, Wolfram (2016) .
• Cellular automata: Burstedde, Klauck, Schadschneider, Zittartz (2001) .
• Mean-field games: Dogbé (2010) , Lachapelle, Wolfram (2011) .
• Multi-scale models: Cristiani, Piccoli, Tosin (2011) .
• Collective learning: Liao, Ren, Yan (2023) .
• Differential games: Barreiro-Gomez, Masmoudi (2023) .

The Fundamental Diagram

$\rho$ is the density, $u$ is the velocity, $J$ is the flux.

Weidmann (1993), Seyfried, Steffen, Klingsch, Boltes (2005), Cao, Seyfried, Zhang, Holl, Song (2017).

Early Traffic Models (1950s)

$$ \pt\rho + \px(\rho u_{\text{FD}}(\rho)) = 0, $$or, defining $J_{\text{FD}}(\rho) = \rho u_{\text{FD}}(\rho)$,$$ \pt\rho + J'_{\text{FD}}(\rho) \px\rho = 0. $$Information propagates at speed $ J'_{\text{FD}}(\rho) - u_{\text{FD}}(\rho) = \rho u'_{\text{FD}}(\rho) \leq 0 .$

Lighthill, Whitham (1955), Richards (1956).

Second-Order Models (1970s)

$$ \begin{cases} \pt\rho + \px(\rho u) = 0, \\ \pt u + u \px u = -\frac{1}{\tau} \left( u - u_{\text{FD}}(\rho) + \frac{\nu}{\rho}\px\rho \right) , \quad \tau>0, \nu>0. \end{cases} $$The $\px \rho$ term on the second equation is an attempt to model congestion.

Payne (1971), Whitham (1974).

Requiem for Second-Order Models (1995)

For some increasing function $p(\rho)$, rewrite previous model as$$ \begin{cases} \pt\rho + \px(\rho u) = 0, \\ \pt u + u \px u + \frac{\nu}{\rho}\px p(\rho) = -\frac{1}{\tau} ( u - u_{\text{FD}}(\rho) ) , \quad \tau>0, \nu>0. \end{cases} $$

Characteristic speeds are $u\pm \nu\sqrt{p'(\rho)}$: information can reach drivers from behind!

Daganzo (1995).

Resurrection of Second-Order Models (2000)

A new model:$$ \begin{cases} \pt\rho + \px(\rho u) = 0, \\ (\pt + u \px) (u + p(\rho)) = 0. \end{cases} $$$p(\rho)$ is an increasing function, the pseudo-pressure (think of $p(\rho)=\rho^\gamma$ for $\gamma>0$).

Characteristic speeds are $u$ and $u - \rho p'(\rho)$.

Aw, Rascle (2000).

Rescaled Aw-Rascle Model (2008)

Make the pseudo-pressure singular, and rescale:$$ \begin{cases} \pt\rho + \px(\rho u) = 0, \\ (\pt + u \px) (u + \info{ \eps p(\rho)} ) = 0, \quad 0 < \eps \ll 1, \\ \info{p(\rho)} = ( \rho^{-1} - \rhomax^{-1} ) ^ {-\gamma},\quad \gamma > 0. \end{cases} $$The singular term enforces the capacity constraint $\rho\leq\rhomax$.

Berthelin, Degond, Delitala, Rascle (2008), Berthelin, Degond, Blanc, Moutari, Royer (2008).

The Desired Velocity $\omega$

Rewrite$$ \begin{cases} \pt\rho + \px(\rho u) = 0, \\ (\pt + u \px) (u + \eps p(\rho)) = 0, \end{cases} $$as$$ \begin{cases} \pt\rho + \px(\rho u) = 0, \\ \info{(\pt + u \px) \omega} = 0, \\ u = \omega - \eps p(\rho). \end{cases} $$

$\omega$, the desired velocity, is advected by the model.

Towards Pedestrians

The relation $u = \omega - \eps p(\rho)$ is not dimensionally consistent. A natural alternative is $u = \omega - \eps \info{\grad \phi (\rho)}$. Model becomes$$ \begin{cases} \pt\rho + \div(\rho u) = 0, \\ (\pt + u \cdot \grad) \omega = 0, \\ u = \omega - \eps \grad \phi (\rho), \end{cases} $$for a choice of increasing congestion function $\phi(\rho)$, such as$$ \phi(\rho) = ( \rho^{-1} - \rhomax^{-1} ) ^ {-\gamma},\quad \gamma > 0. $$

Properties of the Model

Rewrite model in advection-diffusion form,$$ \begin{cases} \pt\rho + \div(\rho \omega) = \eps \div(\rho \grad \phi(\rho)), \\ (\pt + u \cdot \grad) \omega = 0, \\ u = \omega - \eps \grad \phi (\rho). \end{cases} $$

The first equation enforces capacity bound.

The second equation introduces steering behaviour.

Derivation from Particle System

We formally derive the model from an agent-based model:$$ \begin{cases} \dot{X}_k(t) = V_k(t), \\ \dot{W}_k(t) = 0, \\ V_k(t) = W_k(t) - \varepsilon \grad\brk{\phi(\rho^R)}(t,X_k), \end{cases} $$where $X_k$, $V_k$, and $W_k$ are, respectively, the position, velocity, and desired velocity of the $k$^th agent.

$\rho^R$ is non-local estimator of the density with radius $R$,$$ \rho^R(t,x) = \frac{1}{NR^2} \sum_{k=1}^{N} M\prt{\frac{\abs{x - X_k(t)}}{R}}, $$where $M$ is a symmetric mollifier with unit integral.

$N\rightarrow\infty$ Limit

$$ \begin{cases} \pt f(t,x,w) + \div\prt{U_f^R f} = 0, \\ U_f^R(t,x,w) = w - \varepsilon \grad\brk{\phi(\rho_f^R)}(t, x), \\ \rho_f^R(t, x) = \frac{1}{R^2} \int_{\Rtwo\times\Rtwo} M\prt{\frac{\abs{x - y}}{R}} f(t,y,w) \dy \dw. \end{cases} $$

$R\rightarrow 0$ Limit

$$ \begin{cases} \pt f + \div\prt{U_f f} = 0, \\ U_f(t,x,w) = w - \varepsilon \grad\brk{\phi(\rho_f)}(t, x), \\ \rho_f(t, x) = \int_{\Rtwo} f(t,x,w) \dw. \end{cases} $$

Oligokinetic Ansatz

The local kinetic model admits a measure-valued solution$$ \mu(t,x,w) = \sum_{i=1}^{P} \rho\i(t,x) \delta(w - \omega\i(t,x)) $$if $(\rho_i,\omega_i)$ satisfy the multi-fluid model$$ \begin{cases} \pt \rho\i + \div\prt{\rho\i u\i} = 0, \\ \pt \omega\i + \prt{u\i\cdot\grad}\omega\i = 0, \\ u\i = \omega\i - \eps \grad\phi(\rho), \\ \rho = \sum_{i=1}^{P} \rho\i. \end{cases} $$

Stiffness Issues (CFL)

Model in conservative variables $(\rho,q = \rho\omega)$ (1D):$$ \begin{cases} \pt\rho + \px q = \eps \px(\rho \px \phi(\rho)), \\ \pt q + \px (\rho^{-1}q^2) = \eps \px(q \px \phi(\rho)). \end{cases} $$

Diffusion coefficient in the first equation is $\eps\rho\phi'(\rho)$. Near congestion, when $\rhomax-\rho\sim\eps$, it is$$ \eps\rho\phi'(\rho) = \eps\gamma\rho^\gamma\left( 1 - \frac{\rho}{\rhomax} \right)^{-(\gamma+1)} \sim \eps^{-\gamma}. $$

A naïve numerical treatment will require a CFL condition, $\info{\Dt \sim \eps^{\gamma} \Dx^2}$.

Stiffness Issues (Congestion)

Model in conservative variables (1D):$$ \begin{cases} \pt\rho + \px q = \eps \px(\rho \px \phi(\rho)), \\ \pt q + \px (\rho^{-1}q^2) = \eps \px(q \px \phi(\rho)). \end{cases} $$

Recall $q = \rho \omega$ and the equation for $\omega$,$$ \pt \omega + \omega \px \omega = \eps \px \phi(\rho) \px\omega. $$Ignoring diffusive effects, $\omega$ satisfies Burgers' equation. It tends to develop shocks, which correspond to delta-shock waves on the density equation.

Without specialised schemes, the bound $\rho\leq\rhomax$ will not hold.

Semi-Discrete & Semi-Implicit Scheme

$$ \begin{align} & \frac{\rho\np - \rho\n}{\Dt} + \px \prt{\rho\n \omega\n} = \varepsilon\px \prt{\rho\n \info{\px\phi\np}}, \\& \frac{q\np - q\n}{\Dt} + \px \prt{q\n \omega\n} = \varepsilon\px \prt{q\n \info{\px\phi\np}}, \\& q\n = \rho\n \omega\n, \\& \phi\n = \phi\prt{\rho\n}, \\& \phi\prt{\rho} = \prt{\rho^{-1} - \rho_{\text{max}}^{-1}}^{-\gamma}. \end{align} $$

Elliptic Problem on $\phi\np$

The equation for the density becomes$$ -\varepsilon\Dt\px \prt{\rho\n \px\phi\np} + \rho\prt{\phi\np} = \rho\n - \Dt\px \prt{\rho\n \omega\n}, $$where $\rho(\phi)$ is the inverse of $\phi(\rho)$. Existence and well-posedness of $\phi\np$ holds provided the RHS is positive. This will hold for sufficiently small $\Dt$, independently of $\eps$.

Once $\phi\np$ is found, the update of $q\np$ is explicit.

Fully Discrete Schemes

Finite-volume discretisation:$$ \begin{cases} \pt\rho + \px q = \eps \px(\rho \px \phi(\rho)), \\ \pt q + \px (\rho^{-1}q^2) = \eps \px(q \px \phi(\rho)), \end{cases} $$becomes$$ \begin{align} & \frac{\rho\i\np - \rho\i\n}{\Dt} + \frac{F\ih\n - F\imh\n}{\Dx} = \eps \frac{D\ih\np - D\imh\np}{\Dx}, \\& \frac{q\i\np - q\i\n}{\Dt} + \frac{G\ih\n - G\imh\n}{\Dx} = \eps \frac{C\ih\np - C\imh\np}{\Dx}. \end{align} $$

Hyperbolic Transport

$$ \begin{align} & \frac{\rho\i\np - \rho\i\n}{\Dt} + \info{\frac{F\ih\n - F\imh\n}{\Dx}} = \eps \frac{D\ih\np - D\imh\np}{\Dx}, \\& \frac{q\i\np - q\i\n}{\Dt} + \info{\frac{G\ih\n - G\imh\n}{\Dx}} = \eps \frac{C\ih\np - C\imh\np}{\Dx}. \end{align} $$The first-order scheme uses an upwind discretisation for the transport terms:$$ \begin{align} & F\ih\n = \rho\i\n \pos{\omega\ih\n} + \rho\ip\n \neg{\omega\ih\n}, \\& G\ih\n = q\i\n \pos{\omega\ih\n} + q\ip\n \neg{\omega\ih\n}, \\& \omega\ih\n = \frac{\omega\i\n+\omega\ip\n}{2}. \end{align} $$We also construct a second-order flux using a minmod limiter.

Parabolic Terms

$$ \begin{align} & \frac{\rho\i\np - \rho\i\n}{\Dt} + \frac{F\ih\n - F\imh\n}{\Dx} = \eps \info{\frac{D\ih\np - D\imh\np}{\Dx}}, \\& \frac{q\i\np - q\i\n}{\Dt} + \frac{G\ih\n - G\imh\n}{\Dx} = \eps \info{\frac{C\ih\np - C\imh\np}{\Dx}}. \end{align} $$The schemes use a (second-order) centred discretisation for the diffusion:$$ \begin{align} & D\ih\np = \frac{\prt{\rho\i\n+\rho\ip\n}\prt{\phi\ip\np - \phi\i\np}}{2\Dx}, & C\ih\np = \frac{\prt{q\i\n+q\ip\n}\prt{\phi\ip\np - \phi\i\np}}{2\Dx}. \end{align} $$

Elliptic Problem on $\phi\i\np$

The equation for the density becomes$$ -\varepsilon\Dt \frac{D\ih\np - D\imh\np}{\Dx} + \rho\prt{\phi\i\np} = \rho\i\n - \Dt \frac{F\ih\n - F\imh\n}{\Dx}, $$where $\rho(\phi)$ is the inverse of $\phi(\rho)$. Existence and well-posedness of $\phi\np$ holds provided the RHS is positive, which only requires a hyperbolic-like CFL, $\Dt\sim\Dx$, independent of $\eps$.

Once $\phi\i\np$ is found, the update of $q\i\np$ is explicit.

Validation of the Schemes

A simple test case that develops congestion:$$ \begin{cases} \rho_0(x) = 0.7, \\ \omega_0(x) = 0.5 - 0.4\sin(2\pi x). \end{cases} $$

Comparison of First and Second-Order Schemes

Same test case. $M=2^{10}$ points.

Singular Limit Behaviour

Same test case. $M=2^{10}$ points.

Corridor Experiments

Corridor Experiments

Corridor Experiments

Crowd Collision and Model Instability

Perhaps a Saffman–Taylor instability, not clear yet.

Conclusion

• A hydrodynamic model for pedestrians with a capacity bound.
• The model captures the fundamental diagram through the congestion effects.
• A structure-preserving numerical scheme.
• Work now published in M3AS: Aceves-Sánchez, RB, Degond, Mercier (2024) .

Outlook

• Congestion function $\phi$ is insensitive to crowd speed.
• Empirical calibration (e.g. match corridor simulations to experimental data).
• Learning the form of the congestion function from data.
• Resolve the model instability, likely through a multi-fluid or kinetic extension to the model.

References I

• P. Aceves-Sánchez, R. Bailo, P. Degond, and Z. Mercier. Pedestrian models with congestion effects. Math. Models Methods Appl. Sci., 34(06):1001–1041, 2024.
• A. Aw and M. Rascle. Resurrection of "Second Order" Models of Traffic Flow. SIAM J. Appl. Math., 60(3):916–938, 2000.
• R. Bailo, J.A. Carrillo, and P. Degond. In Crowd Dynamics, Volume 1, 259–292. Springer International Publishing, 2018.
• J. Barreiro-Gomez and N. Masmoudi. Differential games for crowd dynamics and applications. Math. Models Methods Appl. Sci., 33(13):2703–2742, 2023.

References II

• N. Bellomo, L. Gibelli, A. Quaini, and A. Reali. Towards a mathematical theory of behavioral human crowds. Math. Models Methods Appl. Sci., 32(02):321–358, 2022.
• N. Bellomo, J. Liao, A. Quaini, L. Russo, and C. Siettos. Human behavioral crowds review, critical analysis and research perspectives. Math. Models Methods Appl. Sci., 33(08):1611–1659, 2023.
• F. Berthelin, P. Degond, V.L. Blanc, S. Moutari, and M.R.A.J. Royer. A traffic-flow model with constraints for the modeling of traffic jams. Math. Models Method Appl. Sci., 18(supp01):1269–1298, 2008.
• F. Berthelin, P. Degond, M. Delitala, and M. Rascle. A Model for the Formation and Evolution of Traffic Jams. Arch. Ration. Mech. An., 187(2):185–220, 2008.

References III

• C. Burstedde, K. Klauck, A. Schadschneider, and J. Zittartz. Simulation of Pedestrian Dynamics Using a Two-Dimensional Cellular Automaton. Phys. A, 295(3-4):507–525, 2001.
• S. Cao, A. Seyfried, J. Zhang, S. Holl, and W. Song. Fundamental diagrams for multidirectional pedestrian flows. J. Stat. Mech: Theory Exp., 2017(3):033404, 2017.
• J.A. Carrillo, S. Martin, and M. Wolfram. An Improved Version of the Hughes Model for Pedestrian Flow. Math. Models Methods Appl. Sci., 26(04):671–697, 2016.
• R.M. Colombo and M.D. Rosini. Pedestrian Flows and Non-Classical Shocks. Math. Meth. Appl. Sci., 28(13):1553–1567, 2005.

References IV

• E. Cristiani, B. Piccoli, and A. Tosin. Multiscale Modeling of Granular Flows with Application to Crowd Dynamics. Multiscale Model. Simul., 9(1):155–182, 2011.
• M.R. D'Orsogna, Y.L. Chuang, A.L. Bertozzi, and L.S. Chayes. Self-Propelled Particles with Soft-Core Interactions: Patterns, Stability, and Collapse. Phys. Rev. Lett., 96(10):104302, 2006.
• C.F. Daganzo. Requiem for Second-Order Fluid Approximations of Traffic Flow. Transportation Research Part B: Methodological, 29(4):277–286, 1995.
• P. Degond, C. Appert-Rolland, M. Moussaïd, J. Pettré, and G. Theraulaz. A Hierarchy of Heuristic-Based Models of Crowd Dynamics. J. Stat. Phys., 152(6):1033–1068, 2013.

References V

• C. Dogbé. Modeling Crowd Dynamics by the Mean-Field Limit Approach. Math. Comput. Model., 52(9-10):1506–1520, 2010.
• D. Helbing and P. Molnár. Social Force Model for Pedestrian Dynamics. Phys. Rev. E, 51(5):4282–4286, 1995.
• R.L. Hughes. A Continuum Theory for the Flow of Pedestrians. Transportation Research Part B: Methodological, 36(6):507–535, 2002.
• R.L. Hughes. The Flow of Human Crowds. Annu. Rev. Fluid Mech., 35(1):169–182, 2003.

References VI

• A. Lachapelle and M. Wolfram. On a Mean Field Game Approach Modeling Congestion and Aversion in Pedestrian Crowds. Transportation Research Part B: Methodological, 45(10):1572–1589, 2011.
• J. Liao, Y. Ren, and W. Yan. Kinetic modeling of a leader-follower system in crowd evacuation with collective learning. Math. Models Methods Appl. Sci., 33(05):1099–1117, 2023.
• M.J. Lighthill and G.B. Whitham. On Kinematic Waves II - A Theory of Traffic Flow on Long Crowded Roads. Proc. R. Soc. A Math. Phys. Eng. Sci., 229(1178):317–345, 1955.
• M. Moussaïd, D. Helbing, and G. Theraulaz. How simple rules determine pedestrian behavior and crowd disasters. Proc. Natl. Acad. Sci., 108(17):6884–6888, 2011.

References VII

• H.J. Payne. Model of Freeway Traffic and Control. Mathematical Model of Public System, 28:51–61, 1971.
• C.W. Reynolds. Flocks, Herds and Schools: A Distributed Behavioral Model. ACM SIGGRAPH Comput. Graph., 21(4):25–34, 1987.
• P.I. Richards. Shock Waves on the Highway. Oper. Res., 4(1):42–51, 1956.
• P.S.A. Salam, S. Tiwari, A. Klar, and S. Sundar. A numerical investigation of pedestrian dynamics based on rational behaviour in different density scenarios. Phys. A, 624:128933, 2023.

References VIII

• A. Seyfried, B. Steffen, W. Klingsch, and M. Boltes. The fundamental diagram of pedestrian movement revisited. J. Stat. Mech: Theory Exp., 2005(10):P10002–P10002, 2005.
• U. Weidmann. Transporttechnik Der Fussgänger, Transporttechnische Eigenschaften Des Fussgängerverkehrs (Literturauswertung). Institut für Verkehrsplanung, Transporttechnik, Strassen und Eisenbahnbau (IVT), ETH Zürich, 1993.
• G.B. Whitham. Linear and Nonlinear Waves. Wiley, 1974.