A macroscopic pedestrian model with variable maximal density

TL;DR

The paper introduces a macroscopic pedestrian model in which the maximal density $\tau(x,t)$ is dynamic rather than fixed, achieved by coupling a density conservation law with a Burgers-like equation for an information field $u$ that propagates through the crowd. The key ingredients include a nonlocal average $\tau^{\text{ave}}$, a threshold $\theta$, and a nonlocal flux controlled by a triangular fundamental diagram $f(\rho,\tau)$, enabling pushing effects and density-flux tails in congested regimes. 1D and 2D numerical simulations demonstrate emergent density patterns, self-organized density near targets, and a faster-is-faster evacuation phenomenon in bottleneck scenarios, with ex post analysis showing a tail in the fundamental diagram reminiscent of real crowds. The 1D code is made publicly available, supporting reproducibility and further exploration of variable-density crowd dynamics.

Abstract

In this paper we propose a novel macroscopic (fluid dynamics) model for describing pedestrian flow in low and high density regimes. The model is characterized by the fact that the maximal density reachable by the crowd - usually a fixed model parameter - is instead a state variable. To do that, the model couples a conservation law, devised as usual for tracking the evolution of the crowd density, with a Burgers-like PDE with a nonlocal term describing the evolution of the maximal density. The variable maximal density is used here to describe the effects of the psychological/physical pushing forces which are observed in crowds during competitive or emergency situations. Specific attention is also dedicated to the fundamental diagram, i.e., the function which expresses the relationship between crowd density and flux. Although the model needs a well defined fundamental diagram as known input parameter, it is not evident a priori which relationship between density and flux will be actually observed, due to the time-varying maximal density. An a posteriori analysis shows that the observed fundamental diagram has an elongated "tail" in the congested region, thus resulting similar to the concave/concave fundamental diagram with a "double hump" observed in real crowds. The main features of the model are investigated through 1D and 2D numerical simulations. The numerical code for the 1D simulation is freely available at https://gitlab.com/cristiani77/code_arxiv_2406.14649

Figures (7)

• Figure 1: Fundamental diagram $\rho\mapsto f(\rho,\tau)$.
• Figure 2: Test 1. Screenshots of the solutions $(\rho,\tau,u)$ at four time steps. From left to right, top to bottom: Initial condition, the queue behind the closed gate begins, the queue reinforces and back-propagates (with noncostant density), both the density and the maximal density drop after the gate opened. The numerical code for this test is freely available https://gitlab.com/cristiani77/code_arxiv_2406.14649.
• Figure 3: Test 1. Point-wise fundamental diagram computed ex post by \ref{['FDexpost']}. Each color is associated to a grid cell. The green line is the Matlab best fit obtained with sums of 5 sine's. The numerical code for this test is freely available https://gitlab.com/cristiani77/code_arxiv_2406.14649.
• Figure 4: Test 2. Screenshots of the solutions $(\rho,u,\tau)$. From left to right, top to bottom: default set of parameters (Table \ref{['tab:parameters']}), $\nu$ increased to 0.2, $\nu$ decreased to 0.05, $\beta$ increased to 2.
• Figure 5: Test 3. Screenshots of the solutions $\rho$ (first column), $u$ (second column), and $\tau$ (third column), at $t=0$, 75, 160.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3