  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.
