On the dynamics of a diabetic population model with two delays and a general recovery rate of complications

Abstract

Once recognized as a disease suffered exclusively by older individuals, diabetes is now common among younger adults. It is highly associated with being overweight or obese, unhealthy diets, and low physical activities. In this study, a diabetic population model with a general treatment function is studied, including the slow progression of diabetes. The general treatment function is a dependence function of the people with diabetes with complications, including a saturating recovery rate as a particular case. It is shown that a unique positive equilibrium point exists and that this equilibrium point is locally and globally asymptotically stable in the absence of time delays. In the presence of time delays, threshold quantities are derived, which determine the occurrence of Hopf bifurcation. Using the time delay as the bifurcation parameter, we worked out an algorithm to establish the properties of Hopf bifurcation. Numerical simulations and some data are provided to substantiate the mathematical model and its theoretical results. Our findings underline the importance of diabetes education, lifestyle modification, and strict adherence to diabetes management to decrease the incidence rate of diabetes complications.