Abstract

To simulate glucose-insulin metabolism, this work proposes a mathematical model of a three-state glucose-insulin system that accounts for the interplay among blood glucose levels, plasma insulin concentration, and insulin delay dynamics. The main contribution is to portray physiological processes as components of an integrated nonlinear dynamical system, rather than as processes themselves. The mathematical approach uses Jacobian analysis to identify biologically plausible steady states and their local stability while guaranteeing the existence of an invariant set. To reduce parameter redundancy and reveal the underlying rate ratio, we employ a non-dimensionalisation procedure. We then identify oscillatory dynamics using delay-dependent spectral criteria, which also characterise Hopf-type bifurcations when oscillatory dynamics replace steady-state control. Sensitivities identify the primary regulatory factors and provide more detail on how metabolism affects equilibrium attributes. Thus, this method provides a mathematically sound foundation for analysing resilience, instability, and parameter leverage in metabolism-related diseases by connecting physiological mechanisms with qualitative dynamical-systems analysis. It offers a broad framework for expanding the study to more complex contexts, such as multi-scale, stochastic, or customised models.