Stability and Hopf bifurcation analysis for the hypothalamic-pituitary-adrenal axis model with memory

TL;DR

This paper extends the minimal hypothalamic-pituitary-adrenal (HPA) axis model by incorporating memory effects through distributed delays and fractional derivatives, analyzing stability and bifurcations to better understand system dynamics.

Contribution

It introduces a generalized HPA axis model with memory terms, proves the existence of a unique equilibrium, and conducts stability and bifurcation analysis with numerical validation.

Findings

Models capture key HPA system mechanisms.

Distributed delays affect stability and bifurcation.

Fractional derivatives influence system dynamics.

Abstract

This paper 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.