This chapter proposes a delay differential model for glucose-insulin endocrine metabolic regulation model, incorporating beta-cell dynamics to regulate and maintain bloodstream insulin concentration. In the model, two time-delays are involved, namely, \(\delta _g\) and \(\delta _\iota \) , which represent delayed insulin secretion and delayed glucose reduction. A moderate hyperglycemia results in beta-cell growth (negative feedback), while a severe hyperglycemia results in beta-cell reduction (positive feedback). When a time-delay passes a bifurcation point, Hopf bifurcation occurs. Furthermore, we present an optimal control problem for external insulin infusions to minimize prolonged high blood sugar levels. Simulations demonstrate that delays in insulin response and glucose reduction can produce complex behaviors, such as biological oscillations, period-doubling, and chaos, which are linked to metabolic disorders like diabetes mellitus.