Global dynamics and sensitivity analysis of a diabetic population model with two-time delays

Diabetes is a chronic disease that can cause various long-term complications. This study revisits a four-state model of type-2 diabetic population with a saturating recovery rate of diabetes complications, and its qualitative properties are further analysed. The non-negativity and boundedness of the solution for delay and non-delay models are proved. However, the non-negativity of the solutions of the delay model can only be guaranteed if the model inputs satisfy certain conditions. The stability analysis of the non-delay model is performed, and the numerical simulation is conducted to illustrate and validate the findings. In the presence of two delay parameters, we discuss the characteristic equation of the delay model under the case of the first time delay equal to zero to obtain the stable region of the second time delay. The critical value corresponding to the delay parameter is derived. There are five conditions to characterize the stability properties of the (unique) equilibrium point (either locally asymptotically stable or unstable) and the occurrence of Hopf bifurcation. The delay values affect the stability of the equilibrium point. A locally asymptotically stable equilibrium point can become unstable under certain conditions, and a periodic orbit can arise from the equilibrium point as the model switches its stability. The sensitivity analysis shows that the overall diabetes cases can be reduced significantly by reducing the rate of developing diabetes, and the diabetics with complications will decrease if the parameter measuring the limited medical resources gets smaller.