ANALYSIS OF A MODEL OF THE GLUCOSE-INSULIN REGULATORY SYSTEM WITH TWO DELAYS ∗

Abstract. We continue a recent attempt to better understand the glucose-insulin regulatory system via a mathematical model of delay differential equations with two discrete time delays. With explicit delays, the model is more realistic in physiology, more accurate in mathematics, and more robust in applications. We study this model analytically and perform carefully designed numerical simulations by allowing two parameters to vary. Our analytical and numerical results confirm most current existing physiological observations and reveal more insightful information. The following factors are critical for ensuring the sustained oscillatory regulation and insulin secretion: (1) the time lag for insulin secretion stimulated by glucose and the newly synthesized insulin becoming “remote insulin ” (Theorem 4.2 (b) and Theorem 5.6); (2) the delayed effect of hepatic glucose production (Theorem 4.2 (c) and Theorem 5.6); (3) moderate insulin clearance rate (Theorem 5.6 and simulations in section 6.4); and (4) nonoverwhelming glucose infusion (simulations in section 6.2