  We present a mathematical study for the development of multiple sclerosis
based on a reaction-diffusion system. The model describes interactions among
different populations of human cells, motion of immune cells stimulated by
cytokines, consumption of myelin sheath due to anomalously activated
lymphocytes and its restoration by oligodendrocytes. Successively, we introduce
a therapy term representing injection of low-dose IL-2 interleukine. A natural
step is then to study the system, investigating the formation of spatial
patterns by means of a Turing instability analysis of the problem. In
particular, we get spatial patterns oscillating in time that may reproduce
brain lesions characteristic of the early stage of the pathology, in both
non-treatment and treatment scenarios.
