Optimal control and stability analysis of a glucose-insulin-FFA dynamic model with GLP-1 incretin effects

Obesity and its associated metabolic dysregulations, particularly Type 2 Diabetes Mellitus (T2DM), constitute a global health crisis. Understanding the intricate interplay of key metabolic components is crucial for effective management strategies. This study presents a novel mathematical model capturing the dynamic interactions among plasma glucose, insulin, and free fatty acids (FFAs), critically integrating the regulatory influence of Glucagon-Like Peptide-1 (GLP-1). Through qualitative analysis, we established the model’s physiological relevance and demonstrated the existence of a stable equilibrium point, confirmed by numerical simulations across various initial conditions. Sensitivity analysis revealed that FFA-related parameters (e.g., lipolysis rates and FFA-induced insulin impairment) and insulin secretion/clearance rates profoundly affect glucose homeostasis, underscoring the detrimental role of elevated FFAs in hyperglycemia. Furthermore, we applied optimal control theory, using Pontryagin’s Maximum Principle, to design GLP-1 receptor agonist intervention strategies. We evaluated two scenarios that balance the cost of intervention with the effectiveness of glucose regulation. Results show that GLP-1 agonism effectively lowers glucose and FFA levels, with greater glucose reduction achieved when control cost and glucose deviation are equally weighted. This research provides a comprehensive mathematical framework for analyzing complex glucose-insulin-FFA-GLP-1 dynamics. Our findings highlight the interconnectedness of insulin sensitivity, lipid metabolism, and incretin action in metabolic health and offer valuable insights for optimizing therapeutic interventions. The developed optimal control strategies suggest potential to improve glycemic control and to inform future clinical approaches to prevent and manage metabolic disorders.