Stability analysis and estimate of the region of attraction of a human respiratory model

Abstract

In this paper, we complete the stability analysis of various human respiratory non-linear time delay models introduced in Vielle & Chauvet (1998, Delay equation analysis of human respiratory stability. Math. Biosci., 152, 105–122), Kollár & Turi (2004, Numerical stability analysis in respiratory control system models. Conference on Differential Equations and Applications in Mathematical Biology, Nanaimo, BC, Canada, pp. 65–78), Batzel & Tran (2000a, Stability of the human respiratory control system. Part I: analysis of two-dimensional delay state-space model. J. Math. Biol., 41, 45–79) and Batzel & Tran (2000b, Stability analysis of the human respiratory control system. Part II: analysis of three-dimensional delay state-space model. J. Math. Biol., 41, 80–102). More precisely, we present a detailed mathematical analysis of the stability of the non-linear model trivial equilibrium, an estimate of its region of attraction and exponential estimates of the solutions starting in this region. The proposed approach is constructive and it is based on the use of Lyapunov–Krasovskii functionals of complete type for time-delay systems with a cross term in the time derivative.