Kinetic models for systems of interacting agents with multiple microscopic states

We propose and investigate general kinetic models with transition probabilities that can describe the simultaneous change of multiple microscopic states of the interacting agents. These models can be applied to many problems in socio-economic sciences, where individuals may change both their compartment and their characteristic microscopic variable, as for instance kinetic models for epidemic diffusion or for international trade with possible transfers of agents. Mathematical properties of the kinetic model are proved, as existence and uniqueness of a solution for the Cauchy problem in suitable Wasserstein spaces. The quasi-invariant asymptotic regime, leading to simpler kinetic Fokker-Planck-type equations, is investigated and commented on in comparison with other existing models. Some numerical tests are performed in order to show the time evolution of distribution functions and of meaningful macroscopic fields, even in case of non-constant interaction rates and transfer probabilities.