Stability and Hopf bifurcation analysis of a four-dimensional hypothalamic-pituitary-adrenal axis model with distributed delays

TL;DR

This paper analyzes a four-dimensional HPA axis model with distributed delays, revealing stability conditions, bifurcations, and oscillatory behaviors that model physiological rhythms and disease states.

Contribution

It introduces a novel four-dimensional model with distributed delays and provides detailed stability and bifurcation analysis, including coexistence of equilibria and limit cycles.

Findings

Existence of at least one equilibrium state.

Large delays lead to stable limit cycles modeling ultradian rhythms.

Parameter choices can produce coexisting stable states.

Abstract

A four-dimensional mathematical model of the hypothalamus-pituitary-adrenal (HPA) axis is investigated, incorporating the influence of the GR concentration and general feedback functions. The inclusion of distributed time delays provides a more realistic modeling approach, since the whole past history of the variables is taken into account. The positivity of the solutions and the existence of a positively invariant bounded region are proved. It is shown that the considered four-dimensional system has at least one equilibrium state and a detailed local stability and Hopf bifurcation analysis is given. Numerical results reveal the fact that an appropriate choice of the system's parameters leads to the coexistence of two asymptotically stable equilibria in the non-delayed case. When the total average time delay of the system is large enough, the coexistence of two stable limit cycles is revealed, which successfully model the ultradian rhythm of the HPA axis both in a normal disease-free situation and in a diseased hypocortisolim state, respectively. Numerical simulations reflect the importance of the theoretical results.