Multiscale modeling by time-evolving measures

This chapter is devoted to a multiscale approach to the modeling of crowd dynamics, which is the core topic of the book. We begin by presenting, in Sect. 5.1, a general measure-based modeling framework suitable to include the basic features of pedestrian kinematics at any scale. Specifically, we assume that pedestrian motion results from the interplay between the individual will to follow a preferred travel program and the necessity to face the rest of the crowd. We discuss in Sect. 5.2 how to properly model these behavioral aspects. In Sect. 5.3 we show how discrete (microscopic) and continuous (macroscopic) models can be obtained in the proposed framework, before focusing, in Sect. 5.4, on multiscale modeling issues. We also propose a detailed dimensional analysis, which highlights the role of a few significant parameters, and a numerical scheme for the approximate solution of the equations. The scheme is obtained in two steps in Sect. 5.5. First we derive a discrete-in-time model; next we discretize the space variable as well, obtaining an algorithm (cf. Appendix B) which can be implemented on a computer to produce simulations (cf. Chap.  2). Finally, in Sect. 5.6 we extend the previous modeling structures to the case of two interacting crowds.