Crohn’s disease is an inflammatory bowel disease (IBD) that is not well understood. In particular, unlike other IBDs, the inflamed parts of the intestine compromise deep layers of the tissue and are not continuous but separated and distributed through the whole gastrointestinal tract, displaying a patchy inflammatory pattern. In the present paper, we introduce a toy-model which might explain the appearance of such patterns. We consider a reaction-diffusion system involving bacteria and phagocyte and prove that, under certain conditions, this system might reproduce an activator-inhibitor dynamic leading to the occurrence of Turing-type instabilities. In other words, we prove the existence of stable stationary solutions that are spatially periodic and do not vanish in time. We also propose a set of parameters for which the system exhibits such phenomena and compare it with realistic parameters found in the literature. This is the first time, as far as we know, that a Turing pattern is investigated in inflammatory models.

克罗恩病是一种尚未完全了解的炎症性肠病 (IBD)。特别是，与其他 IBD 不同的是，肠道的发炎部分会损害组织的深层，并且不是连续的，而是在整个胃肠道中分开和分布，显示出斑片状的炎症模式。在本文中，我们介绍了一个玩具模型，它可以解释这种模式的出现。我们考虑了一个涉及细菌和吞噬细胞的反应扩散系统，并证明在某些条件下，该系统可能会重现激活剂-抑制剂的动态，导致图灵型不稳定性的发生。换句话说，我们证明了空间周期性且不会随时间消失的稳定平稳解的存在。我们还提出了系统表现出这种现象的一组参数，并将其与文献中的实际参数进行比较。据我们所知，这是第一次在炎症模型中研究图灵模式。