Stability, convergence, and limit cycles in some human physiological processes

Abstract: Mathematical models for physiological processes aid qualitative understanding of the impact of various parameters on the underlying process. We analyse two such models for human physiological processes: the Mackey-Glass and the Lasota equations, which model the change in the concentration of blood cells in the human body. We first study the local stability of these models, and derive bounds on various model parameters and the feedback delay for the concentration to equilibrate. We then deduce conditions for non-oscillatory convergence of the solutions, which could ensure that the blood cell concentration does not oscillate. Further, we define the convergence characteristics of the solutions which govern the rate at which the concentration equilibrates when the system is stable. Owing to the possibility that physiological parameters can seldom be estimated precisely, we also derive bounds for robust stability\textemdash which enable one to ensure that the blood cell concentration equilibrates despite parametric uncertainty. We also highlight that when the necessary and sufficient condition for local stability is violated, the system transits into instability via a Hopf bifurcation, leading to limit cycles in the blood cell concentration. We then outline a framework to characterise the type of the Hopf bifurcation and determine the asymptotic orbital stability of limit cycles. The analysis is complemented with numerical examples, stability charts and bifurcation diagrams. The insights into the dynamical properties of the mathematical models may serve to guide the study of dynamical diseases.