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This paper presents an overview of the mathematical approaches to modeling crowd dynamics by taking into account the complexity of living systems. The contents refer to the classical representation scales (microscopic, kinetic, and macroscopic) and to the mathematical frameworks that can be used for the modeling approach. The analysis focuses on the kinetic scale and, specifically, on developments of the mathematical kinetic theory of particles with heterogeneous behaviors. The existing literature is critically analyzed looking ahead to research perspectives and, more precisely, to a unified modeling strategy in view also of swarm dynamics modeling.

This paper presents a new mathematical model of vehicular traffic, based on the methods of the generalized kinetic theory, in which the space of microscopic states (position and velocity) of the vehicles is genuinely discrete. While in the recent literature discrete-velocity kinetic models of car traffic have already been successfully proposed, this is, to our knowledge, the first attempt to account for all aspects of the physical granularity of car flow within the formalism of the aforesaid mathematical theory. Thanks to a rich but handy structure, the resulting model allows one to easily implement and simulate various realistic scenarios giving rise to characteristic traffic phenomena of practical interest (e.g., queue formation due to roadworks or to a traffic light). Moreover, it is analytically tractable under quite general assumptions, whereby fundamental properties of the solutions can be rigorously proved., Comment: 22 pages, 3 figures