Stability and Hopf bifurcation analysis of a 4‐dimensional hypothalamic‐pituitary‐adrenal axis model with distributed delays

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.