Differential flatness and Liouvillian character of two HPA axis models

In this paper, we study two existing quantitative models of the hypothalamic- pituitary-adrenal (HPA) axis from a control systems theory viewpoint, that is, we suppose that we can act on the dynamics of the HPA axis throughout some parameters, which are the system inputs. In particular, we will focus on flatness and Liouvillian properties of the considered control systems of the HPA axis. We first study the minimal three-dimensional model Bangsgaard and Ottesen [2017] which is shown to be flat, and then we consider the more involved and important model proposed in Rao and Androulakis [2019, 2020], with seven states, for which we prove that flatness no longer holds. The more involved model satisfies however a similar but weaker property than flatness: it is a Liouvillian system.
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