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In this paper we introduce the optimal control of a kinetic model describing agents who migrate on a graph and interact within its nodes exchanging a physical quantity. As a prototype model, we consider the spread of an infectious disease on a graph, so that the exchanged quantity is the viral-load. The control, exerted on both the mobility and on the interactions separately, aims at minimising the average macroscopic viral-load. We prove that minimising the average viral-load weighted by the mass in each node is the most effective and convenient strategy. We consider two different interactions: in the first one the infection (gain) and the healing (loss) processes happen within the same interaction, while in the second case the infection and healing result from two different processes. With the appropriate controls, we prove that in the first case it is possible to stop the increase of the disease, but paying a very high cost in terms of control, while in the second case it is possible to eradicate the disease. We test numerically the role of each intervention and the interplay between the mobility and the interaction control strategies in each model.

Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, model of BGK type implementing a velocity-jump process. We study both a linear and a nonlinear case and describe the concentration profile. In particular, we analyse a hyperbolic (or high frequency) regime that can be interpreted both as a local (microscopic) or as a nonlocal (macroscopic) rescaling. We consider a Hopf-Cole transform and derive a Hamilton-Jacobi equation. The concentrations are then explained as a consequence of the stationary points of the Hamiltonian that is spatially heterogeneous like the velocity-jump process. After revising the classical hydrodynamic limits for the aggregate quantities and the eikonal equation that can be derived from those with a Hopf-Cole transform, we find that the Hamilton-Jacobi equation is a second order approximation of the eikonal equation in the limit of small diffusivity. For nonlinear turning kernels, the Hopf-Cole transform allows to study the stability of the possible homogeneous configurations and of patterns and the results of a linear stability analysis previously obtained are found and extended to a nonlinear regime. In particular, it is shown that instability (pattern formation) occurs when the Hamiltonian is convex-concave.