Narrow Results

A mathematical model for Intra-Venous Glucose Tolerance Test (IVGTT) with explicit glucose–insulin interaction is presented as a system of delay differential equation with discrete time delays and its important mathematical features are analyzed. This model includes the positivity and boundedness of the solution. An unique equilibrium point is found and its local stability is investigated. Using the Lyapunov functional approach, we show the global stability of the unique equilibrium point. The length of delay that preserves the stability is estimated. Sensitivity analysis is performed on a delay differential equation model for IVGTT that suggests the parameter value has a major impact on the model dynamics. Numerical calculations are performed to support and elaborate the analytical findings.

Calcium (${\mathrm{\text{Ca}}}^{2+}$) signaling is reported as a critical factor in the insulin secretion mechanism of pancreatic $\beta$-cells. Further, calcium signaling also has interactions with other similar signaling systems like adenosine triphosphate (ATP) to achieve the functions of $\beta$-cells. Disturbances in any of these two interactive signaling systems can cause disorders in the secretion mechanism of insulin, leading to the onset of type-2 diabetes. Therefore, this paper presents a one-dimensional spatio-temporal mathematical model designed to explore the dynamic interactions between ATP and ${\mathrm{\text{Ca}}}^{2+}$ within a pancreatic $\beta$-cell. The model under consideration comprises a set of nonlinear reaction–diffusion equations governing the behavior of ${\mathrm{\text{Ca}}}^{2+}$ and ATP. The formulation of the initial and boundary conditions takes into account the physical and physiological factors associated with the $\beta$-cell. Further, the model also incorporates adenosine diphosphate (ADP) production due to the hydrolysis of ATP and ${\mathrm{\text{Ca}}}^{2+}$-dependent insulin secretion of the $\beta$-cell. The numerical results are acquired through the utilization of the finite element method. The time derivative terms are resolved through the utilization of the Crank–Nicolson method. The various glycemic states caused by variational impacts of system parameters are demonstrated.