Diabetes mellitus, at the forefront of the diseases of our age, is a type of disease that plays a leading role in the formation of many deadly diseases and is very common all over the world. In this work, based on the system in Shabestari et al. (Chaos Solitons Fractals 112:44–51, 2018), we give the fractional glucose–insulin regulatory system for the first time by Caputo–Fabrizio derivative. We present special solutions, stability analysis, uniqueness of the solution for this fractional system with the aid of Banach fixed point theory. Taking various values belonging to the fractional order into consideration, we present some numerical results according to the relation between glucose, insulin and \(\beta \)-cells.

糖尿病是当今时代疾病的最前沿，是一种在许多致命疾病的形成中起着主导作用的疾病，在世界范围内非常普遍。在这项工作中，基于Shabestari等人的系统。（Chaos Solitons Fractals 112：44–51，2018），我们首次使用Caputo-Fabrizio衍生物给出了葡萄糖-胰岛素分数调节系统。我们借助Banach不动点理论为该分数阶系统提供特殊的解决方案，稳定性分析，解决方案的唯一性。考虑到属于分数阶的各种值，我们根据葡萄糖，胰岛素和\（\ beta \）细胞之间的关系给出了一些数值结果。