A macroscopic pedestrian model with variable maximal density

TL;DR

This paper introduces a new macroscopic pedestrian flow model where the maximum crowd density is a dynamic variable influenced by crowd behavior, providing insights into complex congestion patterns and fundamental diagrams.

Contribution

The novel aspect is modeling the maximal density as a state variable coupled with a conservation law and a nonlocal PDE, capturing psychological and physical crowd effects.

Findings

The model reproduces a fundamental diagram with a double hump in congested conditions.

Numerical simulations demonstrate the model's ability to mimic real crowd dynamics.

The variable maximal density influences the shape of the fundamental diagram.

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