A hierarchy of heuristic-based models of crowd dynamics

TL;DR

This paper develops a hierarchy of models for crowd dynamics based on a noisy behavioral individual-based model, deriving kinetic and macroscopic equations with different closure relations, and linking equilibrium states to game theory.

Contribution

It introduces a new hierarchy of kinetic and macroscopic crowd models derived from a noisy IBMs, including novel closure relations and a game-theoretic interpretation of equilibria.

Findings

Derived kinetic model for pedestrian probability distribution.

Proposed three macroscopic closure relations: delta, von Mises-Fisher, and hydrodynamic equilibrium.

Discussed model features and practical applicability.

Abstract

We derive a hierarchy of kinetic and macroscopic models from a noisy variant of the heuristic behavioral Individual-Based Model of Moussaid et al, PNAS 2011, where the pedestrians are supposed to have constant speeds. This IBM supposes that the pedestrians seek the best compromise between navigation towards their target and collisions avoidance. We first propose a kinetic model for the probability distribution function of the pedestrians. Then, we derive fluid models and propose three different closure relations. The first two closures assume that the velocity distribution functions are either a Dirac delta or a von Mises-Fisher distribution respectively. The third closure results from a hydrodynamic limit associated to a Local Thermodynamical Equilibrium. We develop an analogy between this equilibrium and Nash equilibia in a game theoretic framework. In each case, we discuss the features of the models and their suitability for practical use.