First-Order Macroscopic Modelling of Human Crowd Dynamics

Abstract

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.

Pedestrian flows in bounded domains with obstacles

“…As a consequence, the classical maximum principle for nonlinear hyperbolic equations, stating that the solution ρ(t, x) remains confined within the same lower and upper bounds of the initial datum for all x ∈ R and all t > 0, no longer holds true and the model is able to describe the transition of pedestrians to panic even starting from an initial density entirely bounded below the standard maximum R. The resulting fundamental diagram, i.e., the mapping ρ → f (ρ), agrees well with experimental observations reported by Helbing et al in [12]. Bellomo and Dogbé [3], Coscia and Canavesio [8] refer instead to a two-dimensional setting, in which the walking area is represented by a bounded domain Ω ⊂ R 2 with possible inlet and outlet regions along the boundary ∂Ω. In [3] the motion of pedestrians is described by a system of two partial differential equations invoking the conservation of mass and the balance of linear momentum:…”

Pedestrian flows in bounded domains with obstacles

“…As a consequence, the classical maximum principle for nonlinear hyperbolic equations, stating that the solution ρ(t, x) remains confined within the same lower and upper bounds of the initial datum for all x ∈ R and all t > 0, no longer holds true and the model is able to describe the transition of pedestrians to panic even starting from an initial density entirely bounded below the standard maximum R. The resulting fundamental diagram, i.e., the mapping ρ → f (ρ), agrees well with experimental observations reported by Helbing et al in [12]. Bellomo and Dogbé [3], Coscia and Canavesio [8] refer instead to a two-dimensional setting, in which the walking area is represented by a bounded domain Ω ⊂ R 2 with possible inlet and outlet regions along the boundary ∂Ω. In [3] the motion of pedestrians is described by a system of two partial differential equations invoking the conservation of mass and the balance of linear momentum:…”

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

“…[3,4,21,23]. In many aspects these models are quite similar to those of traffic flows [8] and pedestrian flows [7,9,10].…”

“…As a generalization of [14] (in which λ = λ(W (t))), here we assume that the velocity λ varies also with respect to the local position x, as can be naturally encountered in practice. These phenomena also appear in some biologic models (modeling the development of ovarian follicles, see [17,18]) and pedestrian flow models (see [8,9,15]). …”

PDE-Free Mass-Constrained Learning of Complex Systems with Hidden States

“…Although the local density-velocity relationship appears explicitly in the conservation law, the coupling with the Eikonal equation imparts a global character to the dynamics. This allows the model to reproduce complex behavioral features such as obstacle avoidance and re-routing in congested environments [92]. Both theoretical investigations and empirical studies support the notion that pedestrian speed and direction are influenced by the full spatial distribution of the crowd, rather than just local surroundings [17].…”

Pedestrian flows in bounded domains with obstacles

“…As a consequence, the classical maximum principle for nonlinear hyperbolic equations, stating that the solution ρ(t, x) remains confined within the same lower and upper bounds of the initial datum for all x ∈ R and all t > 0, no longer holds true and the model is able to describe the transition of pedestrians to panic even starting from an initial density entirely bounded below the standard maximum R. The resulting fundamental diagram, i.e., the mapping ρ → f (ρ), agrees well with experimental observations reported by Helbing et al in [12]. Bellomo and Dogbé [3], Coscia and Canavesio [8] refer instead to a two-dimensional setting, in which the walking area is represented by a bounded domain Ω ⊂ R 2 with possible inlet and outlet regions along the boundary ∂Ω. In [3] the motion of pedestrians is described by a system of two partial differential equations invoking the conservation of mass and the balance of linear momentum:…”

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

“…[3,4,21,23]. In many aspects these models are quite similar to those of traffic flows [8] and pedestrian flows [7,9,10].…”

“…As a generalization of [14] (in which λ = λ(W (t))), here we assume that the velocity λ varies also with respect to the local position x, as can be naturally encountered in practice. These phenomena also appear in some biologic models (modeling the development of ovarian follicles, see [17,18]) and pedestrian flow models (see [8,9,15]). …”

PDE-Free Mass-Constrained Learning of Complex Systems with Hidden States

“…Although the local density-velocity relationship appears explicitly in the conservation law, the coupling with the Eikonal equation imparts a global character to the dynamics. This allows the model to reproduce complex behavioral features such as obstacle avoidance and re-routing in congested environments [92]. Both theoretical investigations and empirical studies support the notion that pedestrian speed and direction are influenced by the full spatial distribution of the crowd, rather than just local surroundings [17].…”

Pedestrian flows in bounded domains with obstacles

“…As a consequence, the classical maximum principle for nonlinear hyperbolic equations, stating that the solution ρ(t, x) remains confined within the same lower and upper bounds of the initial datum for all x ∈ R and all t > 0, no longer holds true and the model is able to describe the transition of pedestrians to panic even starting from an initial density entirely bounded below the standard maximum R. The resulting fundamental diagram, i.e., the mapping ρ → f (ρ), agrees well with experimental observations reported by Helbing et al in [12]. Bellomo and Dogbé [3], Coscia and Canavesio [8] refer instead to a two-dimensional setting, in which the walking area is represented by a bounded domain Ω ⊂ R 2 with possible inlet and outlet regions along the boundary ∂Ω. In [3] the motion of pedestrians is described by a system of two partial differential equations invoking the conservation of mass and the balance of linear momentum:…”

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

“…[3,4,21,23]. In many aspects these models are quite similar to those of traffic flows [8] and pedestrian flows [7,9,10].…”

“…As a generalization of [14] (in which λ = λ(W (t))), here we assume that the velocity λ varies also with respect to the local position x, as can be naturally encountered in practice. These phenomena also appear in some biologic models (modeling the development of ovarian follicles, see [17,18]) and pedestrian flow models (see [8,9,15]). …”

PDE-Free Mass-Constrained Learning of Complex Systems with Hidden States

“…Although the local density-velocity relationship appears explicitly in the conservation law, the coupling with the Eikonal equation imparts a global character to the dynamics. This allows the model to reproduce complex behavioral features such as obstacle avoidance and re-routing in congested environments [92]. Both theoretical investigations and empirical studies support the notion that pedestrian speed and direction are influenced by the full spatial distribution of the crowd, rather than just local surroundings [17].…”