# Existence and stability analysis of solutions for a ultradian glucocorticoid rhythmicity and acute stress model

## TL;DR

This paper analyzes the existence and stability of solutions in mathematical models of the HPA axis, focusing on ultradian glucocorticoid rhythms and stress response, including effects of system delays.

## Contribution

It introduces a mathematical framework for analyzing stability and periodic solutions in HPA axis models, incorporating delays and nonlinear dynamics.

## Key findings

- Existence of periodic solutions under specific parameter conditions
- Stability of solutions depends on the delay parameter
- Linear and nonlinear stability analyses conducted

## Abstract

The hypothalamic pituitary adrenal (HPA) axis responds to physical and mental challenge to maintain homeostasis in part by controlling the body's cortisol level. Dysregulation of the HPA axis is implicated in numerous stress-related diseases. For a structured model of the HPA axis that includes the glucocorticoid receptor but does not take into account the system response delay, we analyze linear and non-linear stability of stationary solutions. For a second mathematical model that describes the mechanism of the HPA axis self-regulatory activities and takes into account a delay of system response, we prove existence of periodic solutions under certain assumptions on ranges of parameter values and analyze stability of these solutions with respect to the time delay value.