Hopf bifurcation analysis for a diabetic population model with two delays and saturated treatment

Abstract

Diabetes is a global epidemic, rising at an alarming rate due to increased life expectancy, urbanization, and sedentary lifestyle. Complications of diabetes constitute a burden for the affected individuals and the whole community. If the current trend is uncontrolled, the community will have to allocate more budgets to treat diabetes and its complications. A delayed mathematical model of the diabetic population with a saturated treatment rate is developed to investigate the effect of limited medical resources. This model has a unique positive equilibrium point, and it is locally asymptotically stable under some conditions. By selecting time delay as the bifurcation parameter, the existence of Hopf bifurcation is discussed for three different cases. Afterward, the bifurcating oscillating solutions’ direction, stability, and period are derived by using the normal form theory and center manifold theorem. The correctness of the theoretical results is verified through numerical simulations. Furthermore, different scenarios of limited medical resources and the dynamical evolution of the diabetics with complications to the stage of recovering are illustrated. When an ample quantity of treatment is available in the community, the number of diabetics with complications decreased significantly.