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.

在本文中，我们提出了一项关于多发性硬化症发展的数学研究，其中时空动力学理论模型在介观水平上描述了大量相互作用的因素的动力学。我们考虑不同人类细胞群之间的相互作用以及细胞因子刺激下的免疫细胞的运动。此外，我们再现了由于淋巴细胞异常激活而导致的髓鞘消耗及其由少突胶质细胞的恢复。接下来，我们固定一个小的时间参数，并假设所考虑的过程发生在不同的尺度上。这使我们能够执行形式限制，获得具有趋化项的数密度的宏观反应扩散方程。然后自然的步骤是研究系统，通过问题的图灵不稳定性分析来探究空间模式的形成，并基于模型的微观参数进行讨论。特别是，我们得到了随时间振荡的空间模式，这可能会再现病理学不同阶段特征的脑损伤。