# American Institute of Mathematical Sciences

December  2018, 15(6): 1271-1290. doi: 10.3934/mbe.2018059

## Modeling crowd dynamics through coarse-grained data analysis

 1 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, USA 2 Adaptive Behavior and Cognition Group, Max Planck Institut for Human Development, Berlin, Germany 3 Centre de Recherches sur la Cognition Animale, Centre de Biologie Intégrative (CBI), Centre National de la Recherche Scientifique (CNRS) & Université de Toulouse 3 Paul Sabatier, 31062 Toulouse, France 4 INRIA Rennes-Bretagne Atlantique, Campus de Beaulieu, Rennes, France 5 CNRS, Laboratoire de Physique Théorique, Orsay, France 6 Department of Mathematics, Imperial College London, London SW7 2AZ, UK

* Corresponding author: smotsch@asu.edu

Received  June 03, 2017 Revised  April 25, 2018 Published  September 2018

Understanding and predicting the collective behaviour of crowds is essential to improve the efficiency of pedestrian flows in urban areas and minimize the risks of accidents at mass events. We advocate for the development of crowd traffic management systems, whereby observations of crowds can be coupled to fast and reliable models to produce rapid predictions of the crowd movement and eventually help crowd managers choose between tailored optimization strategies. Here, we propose a Bi-directional Macroscopic (BM) model as the core of such a system. Its key input is the fundamental diagram for bi-directional flows, i.e. the relation between the pedestrian fluxes and densities. We design and run a laboratory experiments involving a total of 119 participants walking in opposite directions in a circular corridor and show that the model is able to accurately capture the experimental data in a typical crowd forecasting situation. Finally, we propose a simple segregation strategy for enhancing the traffic efficiency, and use the BM model to determine the conditions under which this strategy would be beneficial. The BM model, therefore, could serve as a building block to develop on the fly prediction of crowd movements and help deploying real-time crowd optimization strategies.

Citation: Sebastien Motsch, Mehdi Moussaïd, Elsa G. Guillot, Mathieu Moreau, Julien Pettré, Guy Theraulaz, Cécile Appert-Rolland, Pierre Degond. Modeling crowd dynamics through coarse-grained data analysis. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1271-1290. doi: 10.3934/mbe.2018059
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Experiments and data acquisition. (a) A typical experiment where bi-directional circulation is analyzed. (b) Participants equipped with reflexive markers were tracked by means of an optoelectronic motion capture system (VICON MX-40, Oxford Metrics, UK). (c) The area-weighting assignment procedure: The particle $i$ (red dot) is located in the cell ${\mathcal C}^n(i) = [\theta_k, \theta_{k+1}]$ and is assigned to the two nodes $\theta_k$ and $\theta_{k+1}$ in proportion to the area enclosing the opposite node. For instance, on the picture, the assignment to the node $\theta_k$ is in proportion to the ratio of the shaded area to the area of ${\mathcal C}^n(i)$
Characteristic speeds in the BM model. (a) Perspective view of the BFD $f(\rho_+, \rho_-)$ as a function of $\rho_+$ and $\rho_-$. (b) Sketchy fundamental diagram (in red) of a one-way flow $f(\rho)$ as a function of the single density $\rho$. The function $f(\rho)$ has the same monotony as cuts of the BFD $f(\rho_+, \rho_-)$ along lines $\rho_- = \mbox{Constant}$. The average velocity $u(\rho) = f(\rho)/\rho$ and the cluster velocity $\lambda(\rho) = f'(\rho)$ are respectively the slopes of the secant (in red) and tangent (in green) lines to this curve
Bi-directional Fundamental Diagram (BFD). (a) Estimated BFD expressing the flux $f$ in one direction as a function of the density of pedestrians moving in the same direction $\rho_+$ and in the opposite direction $\rho_-$. (b) Parametrized BFD used in the model. The values of $f$, in $(m.s)^{-1}$, are color coded according to the color bar
Model results and comparisons with experimental data. (a) Setting for the model: the density at the entry of the corridor ($x = 0$) is taken from the experiment and the model is used to predict the occupancy inside the corridor. (b-c) Clockwise and anti-clockwise (resp. b and c) pedestrian densities as functions of position (horizontal axis) and time (vertical axis running downwards), for one of the replications with $60$ pedestrians corresponding to balanced fluxes ($50 \%$ of pedestrians walking in each direction). Left picture of each panel: experiment; right picture: BM model, run with initial and boundary data and BFD estimated from the experimental data. The density is color-coded according to the lateral scale (in m$^{-2}$)
Model results and comparisons with experiment data with the $75 \%\! -\! 25\%$ flux balance: $\%75$ are moving to the left ($\rho_-$) and $\%25$ are moving to the right ($\rho_+$). See Fig. 4b, c for balanced flux
Pedestrian and cluster velocities. Comparison between the pedestrian velocity $u$ (red curve) and cluster velocity $\lambda$ (green curve) as functions of the local density of co-moving pedestrians $\rho$ from the experiments (circles) and from the BM model (solid line) with balanced flux ($50\%\! -\! 50\%$) on the left figure and with $75\%\! -\! 25\%$ flux balance on the right figure. Since $\lambda <u$, information is propagating upstream as predicted by the BM model
Efficiency segregation strategy. Estimation of the relative gain using the segregation strategy (in %) as a function of the densities $\rho_+, \, \rho_-$ (in m$^{-2}$) of the two types of pedestrians (level curve representation). The strategy is efficient when both densities $\rho_+$ and $\rho_-$ are roughly balanced and large
Cluster and cluster velocity. (a) Illustration of a cluster defined from a density distribution $\rho(x, t^n)$ for a given threshold $h$. The edges of the cluster (i.e. $X^+$ and $X^-$) described the level curves of $\rho(x, t)$. (b) Graphical representation of the cluster velocity using the level curves of $\rho$. The picture depicts the edges (in yellow) of the cluster (in red). The cluster edge velocity i.e. the slope of the cluster edge in the position-time plane is given by $f'(h)$ and is illustrated by the green segment
Level curves of the density. Level curves $X(t)$ for various levels $h$ for the replication displayed in Fig. 4 and clockwise pedestrian density $\rho_+$. Horizontal axis is space and vertical axis is time, running downwards. The color code corresponds to the level height $h$, from blue (lower levels) to red (higher levels). (a): with no cutoff and no filtering. (b): after cutoff (only level curves with life-time greater than $3$ seconds are kept) but before filtering. (c): after cutoff and filtering
Parametric estimation of the bi-directional fundamental diagram. The coefficients $a$, $b$ and $c$ refer to Eq. 9. To measure the accuracy of each regression, we estimate the coefficient of determination $R^2$ in the last column (the closer $R^2$ is to $1$, the better the estimation is)
 Sample set a b c $R^2$ $50\%-50\%$ $1.218$ $0.273$ $0.181$ 0.944 $75\%-25\%$ $1.216$ $0.087$ $0.203$ 0.972 $100\%-0\%$ $1.269$ $0.077$ $0$ 0.982
 Sample set a b c $R^2$ $50\%-50\%$ $1.218$ $0.273$ $0.181$ 0.944 $75\%-25\%$ $1.216$ $0.087$ $0.203$ 0.972 $100\%-0\%$ $1.269$ $0.077$ $0$ 0.982
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