Nonlinear Analysis of the C-Peptide Variable Related to Type 1-Diabetes Mellitus

Abstract

Type-1 diabetes mellitus is a chronic disease that is constantly monitored worldwide by researchers who are strongly determined to establish mathematical and experimental strategies that lead to a breakthrough toward an immunological treatment or a mathematical model that would update the UVA/Padova algorithm. In this work, we aim at a nonlinear mathematical analysis related to a fifth-order ordinary differential equations model that describes the asymmetric relation between C-peptides, pancreatic cells, and the immunological response. The latter is based on both the Localization of Compact Invariant Set (LCIS) appliance and Lyapunov’s stability theory to discuss the viability of implementing a possible treatment that stabilizes a specific set of cell populations. Our main result is to establish conditions for the existence of a localizing compact invariant domain that contains all the dynamics of diabetes mellitus. These conditions become essential for the localizing domain and stabilize the cell populations within desired levels, i.e., a state where a patient with diabetes could consider a healthy stage. Moreover, these domains demonstrate the cell populations’ asymmetric behavior since both the dynamics and the localizing domain of each cell population are defined into the positive orthant. Furthermore, closed-loop analysis is discussed by proposing two regulatory inputs opening the possibility of nonlinear control. Additionally, numerical simulations show that all trajectories converge inside the positive domain once given an initial condition. Finally, there is a discussion about the biological implications derived from the analytical results.