Reaction-diffusion systems derived from kinetic models for Multiple Sclerosis

Abstract

We present a mathematical study for the development of Multiple Sclerosis in which a spatio-temporal kinetic model describes, at mesoscopic level, the dynamics of a high number of interacting agents. We consider both interactions among different populations of human cells and motion of immune cells, stimulated by cytokines. Moreover, we reproduce the consumption of myelin sheath due to anomalously activated lymphocytes and its restoration by oligodendrocytes. Successively, we fix a small time parameter and assume that the considered processes occur at different scales. This allows to perform a formal limit, obtaining macroscopic reaction-diffusion equations for the number densities with a chemotaxis term. A natural step is then to study the system, inquiring about the formation of spatial patterns through a Turing instability analysis of the problem and basing the discussion on microscopic parameters of the model. In particular, we get spatial patterns oscillating in time that may reproduce brain lesions characteristic of different phases of the pathology