# American Institute of Mathematical Sciences

September  2020, 15(3): 405-426. doi: 10.3934/nhm.2020025

## Micro- and macroscopic modeling of crowding and pushing in corridors

 1 Radon Institute for Applied and Computational Mathematics, Altenbergerstr. 69, 4040, Linz, Austria 2 University of Warwick, Mathematics Institute, Gibbet Hill Road, CV47AL Coventry, UK

Received  November 2019 Revised  July 2020 Published  September 2020 Early access  September 2020

Experiments with pedestrians revealed that the geometry of the domain, as well as the incentive of pedestrians to reach a target as fast as possible have a strong influence on the overall dynamics. In this paper, we propose and validate different mathematical models at the micro- and macroscopic levels to study the influence of both effects. We calibrate the models with experimental data and compare the results at the micro- as well as macroscopic levels. Our numerical simulations reproduce qualitative experimental features on both levels, and indicate how geometry and motivation level influence the observed pedestrian density. Furthermore, we discuss the dynamics of solutions for different modeling approaches and comment on the analysis of the respective equations.

Citation: Michael Fischer, Gaspard Jankowiak, Marie-Therese Wolfram. Micro- and macroscopic modeling of crowding and pushing in corridors. Networks & Heterogeneous Media, 2020, 15 (3) : 405-426. doi: 10.3934/nhm.2020025
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Left: Sketch of experimental setup at the University of Wuppertal, showing the corridor width $w_E$ in the experiments, the exit $\Gamma_E$ and the measurement area. Right: computational domain with adapted width $w_{CA}$ to ensure a consistent discretization of the exit and an increased length $l_{CA} = 9.6$m
Cellular automaton: transition rules
Discretization of the exit. In the case of an even number of cells, a central positioned agent will leave the corridor faster than in the case of three cells
(A) distribution of $n = 40$ agents at time $t = 0$ and $t = 16 \Delta t$, (B) the simulated density, (C) densities of exit times when increasing the number of Monte-Carlo runs
Influence of the scaling parameter $\beta$ on individual dynamics
Average exit time as a function of $\beta$ and $p_{ex}$, or $p_{ex}$ only
Impact of the corridor width on the maximum density. The CA approach yields comparable results for high density regimes and low motivation level
Impact of the motivation level on the maximum pedestrian density: experimental (A) vs. microscopic simulations (B)
Comparison of the potentials $\phi_E$ and $\phi_L$ for a convex obstacle
Comparison of the potentials $\phi_E$ and $\phi_L$ for a non-convex obstacle
Left: Bifurcation diagram detailing the behavior of the solution to (12)-(13). The behavior along the interface lines is identical as in the bottom right corner. Right: exit time corresponding to $L = 1$
Simulations for $n = 60$, $\beta = 3.84$, $\mu = 1$. We observe a good agreement between the CA (microscopic) and PDE (macroscopic) solutions. The effect of higher densities for wider corridors also occur on a macroscopic scale. There is a clear difference in behavior between narrow and wide corridors
Effects of pushing
Comparison of the congestion at the exit in case of pushing (bottom row) and no-pushing (top row) for $60$ individuals: We observe that people move faster towards the exit and the formation of a larger congested area in front of it
exit times for different runs and different motivations from [3]. Run 01 is used to set the desired maximum velocity $v_{max}$
 Run $\mu=1$ $\mu_0$ 01, $n=1$ $8\mathrm{s}$ 02, $n=63, w=1.2\mathrm{m}$ $53\mathrm{s}$ $64\mathrm{s}$ 03, $n=67, w=3.4\mathrm{m}$ $60\mathrm{s}$ $68\mathrm{s}$ 04, $n=57, w=5.6\mathrm{m}$ $55\mathrm{s}$ $57\mathrm{s}$
 Run $\mu=1$ $\mu_0$ 01, $n=1$ $8\mathrm{s}$ 02, $n=63, w=1.2\mathrm{m}$ $53\mathrm{s}$ $64\mathrm{s}$ 03, $n=67, w=3.4\mathrm{m}$ $60\mathrm{s}$ $68\mathrm{s}$ 04, $n=57, w=5.6\mathrm{m}$ $55\mathrm{s}$ $57\mathrm{s}$
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