Narrow Results

This paper deals with the mathematical modelling of crowd dynamics within the framework of continuum mechanics. The method uses the mass conservation equation closed by phenomenological models linking the local velocity to density and density gradients. The closures take into account movement in more than one space dimension, presence of obstacles, pedestrian strategies, and modelling of panic conditions. Numerical simulations of the initial-boundary value problems visualize the ability of the models to predict several interesting phenomena related to the complex system under consideration.

This paper, that deals with the modelling of crowd dynamics, is the first one of a project finalized to develop a mathematical theory refereing to the modelling of the complex systems constituted by several interacting individuals in bounded and unbounded domains. The first part of the paper is devoted to scaling and related representation problems, then the macroscopic scale is selected and a variety of models are proposed according to different approximations of the pedestrian strategies and interactions. The second part of the paper deals with a qualitative analysis of the models with the aim of analyzing their properties. Finally, a critical analysis is proposed in view of further development of the modelling approach. Additional reasonings are devoted to understanding the conceptual differences between crowd and swarm modelling.

We present a new class of macroscopic models for pedestrian flows. Each individual is assumed to move towards a fixed target, deviating from the best path according to the instantaneous crowd distribution. The resulting equation is a conservation law with a nonlocal flux. Each equation in this class generates a Lipschitz semigroup of solutions and is stable with respect to the functions and parameters defining it. Moreover, key qualitative properties such as the boundedness of the crowd density are proved. Specific models are presented and their qualitative properties are shown through numerical integrations. In particular, the present model accounts for the possibility of reducing the exit time from a room by carefully positioning obstacles that direct the crowd flow.

In this paper we model pedestrian flows evacuating a narrow corridor through an exit by a one-dimensional hyperbolic conservation law with a point constraint in the spirit of [Colombo and Goatin, J. Differential Equations, 2007]. We introduce a nonlocal constraint to restrict the flux at the exit to a maximum value p(ξ), where ξ is the weighted averaged instantaneous density of the crowd in an upstream vicinity of the exit. Choosing a non-increasing constraint function p(⋅), we are able to model the capacity drop phenomenon at the exit.

Existence and stability results for the Cauchy problem with Lipschitz constraint function p(⋅) are achieved by a procedure that combines the wave-front tracking algorithm with the operator splitting method. In view of the construction of explicit examples (one is provided), we discuss the Riemann problem with discretized piecewise constant constraint p(⋅). We illustrate the fact that nonlocality induces loss of self-similarity for the Riemann solver; moreover, discretization of p(⋅) may induce non-uniqueness and instability of solutions.

This paper presents a new approach to behavioral-social dynamics for pedestrian crowds by suitable development of mathematical tools of the kinetic theory. It is shown how pedestrians heterogeneity and the propagation of local unusual behaviors in the crowd can be accounted for. The proposed model is applied to the study of two groups of pedestrians walking in opposite directions in a crowded street and its predictive ability is demonstrated by showing that emerging behaviors, such as pedestrian segregation, can be depicted.

A mathematical model of the evacuation of a crowd from bounded domains is derived by a hybrid approach with kinetic and macro-features. Interactions at the micro-scale, which modify the velocity direction, are modeled by using tools of game theory and are transferred to the dynamics of collective behaviors. The velocity modulus is assumed to depend on the local density. The modeling approach considers dynamics caused by interactions of pedestrians not only with all the other pedestrians, but also with the geometry of the domain, such as walls and exits. Interactions with the boundary of the domain are non-local and described by games. Numerical simulations are developed to study evacuation time depending on the size of the exit zone, on the initial distribution of the crowd and on a parameter which weighs the unconscious attraction of the stream and the search for less crowded walking directions.

Roger Hughes proposed a macroscopic model for pedestrian dynamics, in which individuals seek to minimize their travel time but try to avoid regions of high density. One of the basic assumptions is that the overall density of the crowd is known to every agent. In this paper we present a modification of the Hughes model to include local effects, namely limited vision, and a conviction towards decision making. The modified velocity field enables smooth turning and temporary waiting behavior. We discuss the modeling in the micro- and macroscopic setting as well as the efficient numerical simulation of either description. Finally we illustrate the model with various numerical experiments and evaluate the behavior with respect to the evacuation time and the overall performance.

In this paper, we develop two efficient numerical methods for a multiscale kinetic equation in the context of crowd dynamics with emotional contagion [A. Bertozzi, J. Rosado, M. Short and L. Wang, Contagion shocks in one dimension, J. Stat. Phys.158 (2014) 647–664]. In the continuum limit, the mesoscopic kinetic equation produces a natural Eulerian limit with nonlocal interactions. However, such limit ceases to be valid when the underlying microscopic particle characteristics cross, corresponding to the blow up of the solution in the Eulerian system. One method is to couple these two situations — using Eulerian dynamics for regions without characteristic crossing and kinetic evolution for regions with characteristic crossing. For such a hybrid setting, we provide a regime indicator based on the macroscopic density and fear level, and propose an interface condition via continuity to connect these two regimes. The other method is based on a level set formulation for the continuum system. The level set equation shares similar forms as the kinetic equation, and it successfully captures the multi-valued solution in velocity, which implies that the multi-valued solution other than the viscosity solution should be the physically relevant ones for the continuum system. Numerical examples are presented to show the efficiency of these new methods.

The goal of this work is to study an infectious disease spreading in a medium size population occupying a confined environment. For this purpose, we consider a kinetic theory approach to model crowd dynamics in bounded domains and couple it to a kinetic equation to model contagion. The interactions of a person with other pedestrians and the environment are modeled by using tools of game theory. The pedestrian dynamics model allows to weight between two competing behaviors: the search for less congested areas and the tendency to follow the stream unconsciously in a panic situation. Each person in the system has a contagion level that is affected by the people in their neighborhood. For the numerical solution of the coupled problem, we propose a numerical algorithm that at every time step solves one crowd dynamics problem and one contagion problem, i.e. with no subiterations between the two. We test our coupled model on a problem involving a small crowd walking through a corridor.

This paper proposes a multiscale vision to human crowds which provides a consistent description at the three possible modeling scales, namely, microscopic, mesoscopic, and macroscopic. The proposed approach moves from interactions at the microscopic scale and shows how the same modeling principles lead to kinetic and hydrodynamic models. Hence, a unified framework is developed which permits to derive models at each scale using the same principles and similar parameters. This approach can be used to simulate crowd dynamics in complex environments composed of interconnected areas, where the most appropriate scale of description can be selected for each area. This offers a pathway to the development of a multiscale computational model which has the capability to optimize the granularity of the description depending on the pedestrian local flow conditions. An important feature of the modeling at each scale is that the complex interaction between emotional states of walkers and their motion is taken into account.

In this paper, we present a computational modeling approach for the dynamics of human crowds, where the spreading of an emotion (specifically fear) has an influence on the pedestrians’ behavior. Our approach is based on the methods of the kinetic theory of active particles. The model allows us to weight between two competing behaviors depending on fear level: the search for less congested areas and the tendency to follow the stream unconsciously (herding). The fear level of each pedestrian influences their walking speed and is influenced by the fear levels of their neighbors. Numerically, we solve our pedestrian model with emotional contagion using an operator splitting scheme. We simulate evacuation scenarios involving two groups of interacting pedestrians to assess how domain geometry and the details of fear propagation impact evacuation dynamics. Further, we reproduce the evacuation dynamics of an experimental study involving distressed ants.

The modeling of living systems composed of many interacting entities is treated in this paper with the aim of describing their collective behaviors. The mathematical approach is developed within the general framework of the kinetic theory of active particles. The presentation is in three parts. First, we derive the mathematical tools, subsequently, we show how the method can be applied to a number of case studies related to well defined living systems, and finally, we look ahead to research perspectives.

The first part of our paper presents a general survey on the modeling, analytic problems, and applications of the dynamics of human crowds, where the specific features of living systems are taken into account in the modeling approach. This critical analysis leads to the second part which is devoted to research perspectives on modeling, analytic problems, multiscale topics which are followed by hints towards possible achievements. Perspectives include the modeling of social dynamics, multiscale problems and a detailed study of the link between crowds and swarms modeling.

This paper presents a survey and critical analysis of the mathematical literature on modeling and simulation of human crowds taking into account behavioral dynamics. The main focus is on research papers published after the review [N. Bellomo and C. Dogbè, On the modeling of traffic and crowds: A survey of models, speculations, and perspectives, SIAM Rev. 53 (2011) 409–463], thus providing important research perspectives related to new, emerging trends. The presentation addresses the scaling problem corresponding to microscopic (individual-based), mesoscopic (kinetic), and macroscopic (hydrodynamic) modeling and analysis. A multiscale vision guides the overall content of the paper. The critical analysis of the overall content naturally leads to research perspectives. A selection of them is brought to the attention of the interested reader together with hints on how to deal with them.

We study crowd dynamics by means of both non-atomic and atomic differential games, which are also known as macroscopic and microscopic models, respectively; and we consider few crowd-related applications and experiments. Mainly, we study the modeling for Hajj motion (Tawaf), the most important event for Muslim pilgrims taking place in Mecca. We formally show that the proposed game-theoretical model is a potential-cost-dependent version of the well-known Hughes model, and the existing connection with the Lighthill–Whitham–Richards traffic model. We show that for a particular value in the potential cost, one obtains the same Hughes crowd model. Hence, we introduce a mild approximation that allows the computation of semi-explicit/explicit solutions for the proposed game problems. We study four main components that may be used by a central planner: (i) the effect of the clusters (delegations) over the motion, (ii) the inflow control to optimize the flux, (iii) how organized crowds evolve in comparison to disorganized ones, and (iv) how the creation of corrals with time-delays over their motion can be beneficial for the crowd evolution efficiency. We show that a faster performance is exhibited when pilgrims do Tawaf individually than when clustered groups are made, suggesting a possible direction for the policies design. Hence, we also present a simple ON/OFF control over the inflow that can be implemented by a central planner such that the flux is maximized. Related to this last mentioned policy, we have shown that delaying a group to integrate into the crowd can be beneficial for their efficient and faster motion. Also, we show that organized crowds evolve faster than disorganized ones, suggesting that education programs to perform Tawaf may potentially improve the flow. Finally, we show that other type of behaviors could be captured by means of the appropriate design of the cost functional. For example, the consideration of stress during an evacuation can be captured by means of the suitable modification of the cost functional, and as another example, we model the drafting tactics in bicycling where small densities can be beneficial for the performance. We present numerical and simulation results for all the applications using both the macroscopic and microscopic models, which certify the suitability of these models to capture the main features of the real behavior.

Social behavior in crowds, such as herding or increased interpersonal spacing, is driven by the psychological states of pedestrians. Current macroscopic crowd models assume that these are static, limiting the ability of models to capture the complex interplay between evolving psychology and collective crowd dynamics that defines a “social crowd”. This paper introduces a novel approach by explicitly incorporating an “activity” variable into the modeling framework, which represents the evolving psychological states of pedestrians and is linked to crowd dynamics. To demonstrate the role of activity, we model pedestrian egress when this variable captures stress and awareness of contagion. In addition, to highlight the importance of dynamic changes in activity, we examine a scenario in which an unexpected incident necessitates alternative exits. These case studies demonstrate that activity plays a pivotal role in shaping crowd behavior. The proposed modeling approach thus opens avenues for more realistic macroscopic crowd descriptions with practical implications for crowd management.

The relation between speed and density is connected with every self-organization phenomenon of pedestrian dynamics and offers the opportunity to analyze them quantitatively. But even for the simplest systems, like pedestrian streams in corridors, this fundamental relation is not completely understood. A comparison of data from literature shows that specifications in text books as well as measurements under various experimental conditions differ significantly. In this contribution it is studied whether cultural influences and length of the corridor can be the causes for these deviations. To reduce as much as possible unintentional effects, a system is chosen with reduced degrees of freedom and thus the most simple system, namely the movement of pedestrians along a line under closed boundary conditions. It is found that the speed of Indian test persons is less dependent on density than the speed of German test persons. Surprisingly the more unordered behavior of the Indians is more effective than the ordered behavior of the Germans. This may be due to differences in their self-organization behavior. Without any statistical measure one cannot conclude about whether there are differences or not. By hypothesis test it is found quantitatively that these differences exist, suggesting cultural differences in the fundamental diagram of pedestrians.