A master equation-based framework for the modeling of pedestrian dynamics
This paper is devoted to the derivation of a master equation-based framework for the modeling of the pedestrian dynamics into a bounded domain of the plane. The framework is based on the domain decomposition into squares with length side $\epsilon$ and on the definition of the transition rates in the admissible directions that define the mathematical operators of the master equation fulfilled by the joint probability. In particular two specific mathematical models are derived within the new framework: An isotropic model for the pedestrian dynamics in the checkout area of a supermarket where the pedestrians move towards low-density regions of the domain, and an anisotropic model describing the movement of pedestrians during an evacuation where high-density regions of the domain are reached. The macroscopic dynamics, obtained by letting $\epsilon$ go to zero, is described by reaction-diffusion equations.