A decade of thermostatted kinetic theory models for complex active matter living systems

In the last decade, the thermostatted kinetic theory has been proposed as a general paradigm for the modeling of complex systems of the active matter and, in particular, in biology. Homogeneous and inhomogeneous frameworks of the thermostatted kinetic theory have been employed for modeling phenomena that are the result of interactions among the elements, called active particles, composing the system. Functional subsystems contain heterogeneous active particles that are able to perform the same task, called activity. Active matter living systems usually operate out-of-equilibrium; accordingly, a mathematical thermostat is introduced in order to regulate the fluctuations of the activity of particles. The time evolution of the functional subsystems is obtained by introducing the conservative and the nonconservative interactions which represent activity-transition, natural birth/death, induced proliferation/destruction, and mutation of the active particles. This review paper is divided in two parts: In the first part the review deals with the mathematical frameworks of the thermostatted kinetic theory that can be found in the literature of the last decade and a unified approach is proposed; the second part of the review is devoted to the specific mathematical models derived within the thermostatted kinetic theory presented in the last decade for complex biological systems, such as wound healing diseases, the recognition process and the learning dynamics of the human immune system, the hiding-learning dynamics and the immunoediting process occurring during the cancer-immune system competition. Future research perspectives are discussed from the theoretical and application viewpoints, which suggest the important interplay among the different scholars of the applied sciences and the desire of a multidisciplinary approach or rather a theory for the modeling of every active matter system.