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In this work, a nonlinear model is studied based on ordinary differential equations that describe the relationship between the mass of β cells and the secretion of epinephrine. It analyzes the impact of stress associated with the cause of increased blood pressure and glucose levels in the body. The mathematical analysis is based on the appliance of the nonlinear control theory to define the maximum load capacity for each state variable, establishing a bounded positive invariant domain through the Localization of Compact Invariants Sets (LCIS) method. The objective is to determine the effects of epinephrine secretion on the increase of blood glucose levels; therefore, this analysis's results define the necessary and sufficient conditions in which epinephrine raises insulin and glucose levels in the presence of β cells. The interest in studying this type of disease focuses on searching for a treatment or an analysis that guarantees complete control of glucose levels. This work's development and mathematical analysis strengthen current research on insulin-dependent diabetes mellitus around critical epinephrine factors that imply an increase in glucose in the body. [ABSTRACT FROM AUTHOR]