A Nonlinear Ordinary Differential Equation Model for the Effects of Fasting on the Human Body System

This paper develops and analyzes a nonlinear ordinary differential equation model for the metabolic dynamics of fasting. The model tracks five interacting state variables: blood glucose G(t), glycogen reserve Y(t), lipid/fat reserve F(t), ketone bodies K(t), and labile protein reserve P(t). A fasting state vector x(t)=(G,Y,F,K,P)^T is formulated from biological assumptions on glycogenolysis, gluconeogenesis, lipolysis, ketogenesis, and protein sparing. The resulting system is proved to preserve non-negativity and to admit a positively invariant bounded region. The fasting-adapted equilibrium is characterized analytically by reducing the steady-state conditions to a scalar nonlinear equation in glucose. The full Jacobian matrix is derived and evaluated at a calibrated equilibrium E*=(0.4144, 0, 0.8617, 1.1757, 0.8397), where all eigenvalues have negative real parts. In addition to the local analysis, a sufficient contractive Lyapunov criterion for global asymptotic stability is established, and multistart numerical experiments provide strong evidence of global attraction of E* for the calibrated parameter set. Finite-difference normalized sensitivity indices identify a5 and e3 as the dominant control parameters for terminal glucose and protein levels, while c1, d1, and d2 exert the greatest influence on lipid-ketone adaptation. Time-series simulations under parameter variation reproduce key fasting features: rapid glycogen depletion, progressive ketone accumulation, declining glucose, and partial protein preservation. The paper provides a mathematically consistent template for physiological fasting studies and a platform for future extensions involving endocrine regulation, circadian forcing, and patient-specific calibration.

Keywords: fasting dynamics; nonlinear ordinary differential equations; Jacobian matrix; eigenvalues; eigenvectors; local stability; global stability; sensitivity analysis; ketogenesis; metabolic modeling.