Complexity analysis and mathematical tools towards the modelling of living systems

This paper is a review and critical analysis of the mathematical kinetic theory of active particles applied to the modelling of large living systems made up of interacting entities. The first part of the paper is focused on a general presentation of the mathematical tools of the kinetic theory of active particles. The second part provides a review of a variety of mathematical models in life sciences, namely complex social systems, opinion formation, evolution of epidemics with virus mutations, and vehicular traffic, crowds and swarms. All the applications are technically related to the mathematical structures reviewed in the first part of the paper. The overall contents are based on the concept that living systems, unlike the inert matter, have the ability to develop behaviour geared towards their survival, or simply to improve the quality of their life. In some cases, the behaviour evolves in time and generates destructive and/or proliferative events.