A kinetic framework under the action of an external force field: Analysis and application in epidemiology

Abstract

The tools of kinetic theory allow to describe the dynamics and evolution of a system composed of stochastically interacting particles. The interaction is modeled by means of two classes of parameters, i.e. interaction rates and transition probabilities. Therefore, a system of nonlinear ordinary differential equations is derived. Nevertheless, in general, this structure does not consider the action of an external environment. This paper aims at providing a new kinetic model where an external action occurs. Specifically, this action over the system is modeled by introducing an external force field. Then, a new kinetic model is derived, and some analytical results towards the solution of the related Cauchy problem are provided, in the conservative case: existence, uniqueness, positivity and boundedness. Finally, an application in the contest of mathematical epidemiology is given; the new kinetic framework is characterized for three classical compartmental models: SIR, SEIIR and SEIIRS. Stability results and numerical simulations, in agreement with classical theory, confirm the adherence to reality of this new model.