Dynamic macroscopic modeling of crowd dynamics under panic conditions

Abstract

Large gatherings and natural disasters can lead to panicked crowds and associated crowd disasters, which have caused thousands of deaths worldwide in the last two decades. The high fatalities that result from such disasters and their negative social effects have generated significant research interest. Empirical studies based on actual observations of crowd dynamics have revealed three important characteristics of panicked crowds for which mathematical models can be developed: 1) high density, 2) aggregated pushing pressure and 3) turbulence. To simulate the collective behavior of pedestrians during such large-scale events, continuum models are favored for their computational efficiency and theoretical analysis of key variables and parameters. However, owing to the difficulty of quantitatively estimating the aggregated pushing pressure and its variation in crowd disasters, the existing continuum approaches fail to describe important features of panicked crowds. Moreover, as the literature has focused on the qualitative features of the model properties, simulations and evaluations based on real-world problems are lacking.

In this study, two second-order continuum models are developed that explicitly consider the crowd force and panic effects in the unidirectional pedestrian flow and multidirectional flow, respectively. The unidirectional model is composed of three constituent parts, namely continuum theory, the route strategy model and the crowd pressure model. This enables the unidirectional model to consider two main characteristics of crowd movement: individual attention, which dominates the characteristics of pedestrian movement in low-density conditions, and crowd pressure, which dominates the crowd dynamics in high-density conditions. These two characteristics are reflected in the route strategy model and the crowd pressure model, respectively. By introducing panic sentiment into the crowd pressure model, the proposed model is able to clarify the effects of both the force chains and panic sentiment. Based on the unidirectional model, the multidirectional model is further developed to describe the multiple pedestrian streams and interaction observations in multidirectional cases, in which the hydroponic law is still satisfied by appropriately setting the functions and parameters.

The continuum models are then formulated as a set of partial differential equations (PDEs) with appropriate boundary conditions. Two efficient solution algorithms, the first-order finite difference method for rectangular meshes and the first-order finite volume method for triangular meshes, are developed to solve the PDE systems, which are validated through application to designed numerical examples. The reproduction of crowd phenomena, such as congestion, lane formation and the “second peak,” indicates that the model can effectively simulate the characteristics of panicked crowds. Furthermore, the models are applied to two real-world scenarios: the 2015 Hajj crowd disaster and the 2010 Love Parade crowd disaster. Consistent with the empirical observations, the simulation results reveal three important features of panicked crowds, namely, high density, aggregated pushing pressure and turbulence. A comparative study is then conducted based on empirical data on the crowd disasters. Overall, the qualitative analysis and the quantitative comparison of the simulation results and actual observations validate the effectiveness of the models developed in this study.