This article generalizes the existing minimal model of the hypothalamic-pituitary-adrenal (HPA) axis in a realistic way, by including memory terms: distributed time delays, on one hand and fractional-order derivatives, on the other hand. The existence of a unique equilibrium point of the mathematical models is proved and a local stability analysis is undertaken for the system with general distributed delays. A thorough bifurcation analysis for the distributed delay model with several types of delay kernels is provided. Numerical simulations are carried out for the distributed delays models and for the fractional-order model with discrete delays, which substantiate the theoretical findings. It is shown that these models are able to capture the vital mechanisms of the HPA system.

中文翻译：

---

具有记忆的下丘脑-垂体-肾上腺轴模型的稳定性和Hopf分叉分析。

本文通过包括存储项：一方面是分布式时延，另一方面是分数阶导数，以一种现实的方式概括了下丘脑-垂体-肾上腺（HPA）轴的现有最小模型。证明了数学模型唯一平衡点的存在，并对具有一般分布时滞的系统进行了局部稳定性分析。提供了具有几种类型的延迟内核的分布式延迟模型的彻底分叉分析。对分布的时滞模型和具有离散时滞的分数阶模型进行了数值模拟，这证实了理论上的发现。结果表明，这些模型能够捕获HPA系统的重要机制。