Modelling complex systems in applied sciences; methods and tools of the mathematical kinetic theory for active particles

This paper deals with the modelling of large systems of interacting individuals characterized by a microscopic state which includes both mechanical and socio-biological activities. The first part of the paper is devoted to the derivation and critical analysis of the modelling of microscopic equations and subsequently of the derivation of evolution equations. The second part analyzes how these mathematical structures can be properly used to model a variety of models in different fields of applied sciences: traffic flow, population dynamics, biology. Some research perspectives are analyzed in the last part of the paper.