Reaction–diffusion systems derived from kinetic theory for Multiple Sclerosis

Author:

Oliveira João Miguel1ORCID,Travaglini Romina1ORCID

Affiliation:

1. Centre of Mathematics, University of Minho, Campus of Gualtar, Braga 4710-057, Portugal

Abstract

In this paper, we present a mathematical study for the development of Multiple Sclerosis in which a spatio-temporal kinetic theory model describes, at the mesoscopic level, the dynamics of a high number of interacting agents. We consider both interactions among different populations of human cells and the 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 us 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 the 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.

Funder

Cost Action

Portuguese FCT/MCTES

Publisher

World Scientific Pub Co Pte Ltd

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