Existence of solutions to a class of one-dimensional models for pedestrian evacuations

TL;DR

This paper proves the existence of solutions for a class of one-dimensional pedestrian evacuation models inspired by Hughes' framework, using topological fixed point methods and illustrating with various examples including panic and inertial dynamics.

Contribution

It introduces a fixed point approach to establish solutions for generalized Hughes-type models with discontinuous flux and complex boundary conditions.

Findings

Existence of solutions for models with discontinuous flux.

Application to models with panic behavior and capacity drops.

Extension to models with inertial turning dynamics.

Abstract

In the framework inspired by R. L. Hughes model (Transp. Res. B, 2002) for pedestrian evacuation in a corridor, we establish existence of a solution by a topological fixed point argument. This argument applies to a class of models where the dynamics of the pedestrian density $\rho$ (governed by a discontinuous-flux Lighthill,Whitham and Richards model $\rho$t + (sign(x -- $\xi$(t))$\rho$v($\rho$)) x = 0) is coupled via an abstract operator to the computation of a Lipschitz continuous "turning curve" $\xi$. We illustrate this construction by several examples, including the standard Hughes' model with affine cost, and either with open-end conditions or with conditions corresponding to panic behaviour with capacity drop at exits. Other examples put forward versions of the Hughes model with inertial dynamics of the turning curve and general costs.