Kinetic models for populations with multiple microscopic states and individual transfers
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We propose and investigate general kinetic models 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 variables. In this work, we consider a multi-population system where agents are involved in international trades with possible transfers of individuals from a country to another, and trading interaction rules are influenced by the individual knowledge and vice versa. We derive from the kinetic level the macroscopic equations governing each population density, mean knowledge and national wealth, and we study equilibrium values for suitable choices of microscopic interaction functions. Moreover, we perform a suitable quasi-invariant asymptotic limit of the Boltzmann model, leading to a system of simpler Fokker-Planck-type equations.