Global Asymptotic Stability of a Non-linear Population Model of Diabetes Mellitus

Abstract

A preliminary mathematical model of diabetes has been proposed in [4], in which the evolution of the size of a population of diabetes mellitus patients and the number of patients with complications, has been modeled by second order system of nonlinear differential equations. The model, has already been analyzed for the linear local stability of the equilibria of the system. However, the global behavior of the flow of the nonlinear system has not been studied. The present article analyzes the global behavior of the trajectories of the population growth using Lyapunov stability analysis. Toward this, we construct a suitable Lyapunov function corresponding to an interior equilibrium point and show that it is asymptotically stable within the entire open first quadrant of the planar state space which is the region of interest. Further, transient or incremental stability in the phase plane has been studied via Lyapunov exponent analysis. The stability analysis has also been verified through numerical simulations, under various parameters. A physical interpretation of the parametric dependence of the flows of the nonlinear system is provided from the point of view of diabetic population dynamics.