A time-delay model of diabetic population: Dynamics analysis, sensitivity, and optimal control

Abstract

Diabetes, also known as diabetes mellitus, is a chronic degenerative disease with a variety of adverse complications. Due to its slow progression, a mathematical model of the diabetic population with time lag is developed. This novel study aims to analyze the stability of the diabetic equilibrium and also formulate an optimal control problem with lags. We show that under some values of parameters, a limit cycle arises through Hopf bifurcation. Sensitivity analysis, which used the direct differential method, reveals which parameters have a major impact on the model dynamics. An optimal control strategy is developed with two control variables for the description of the control strategy. A suitable objective function is formulated to minimize the total number of diabetics and the relative cost of implementing the controls. The resulting optimality system is solved numerically by using the Forward-Backward Sweep Method. Graphical results are examined with different parameter values and are compared with and without controls. The total number of diabetics continued to increase without controls, but the number of these individuals decreased after control. The optimal controls prevent the progression of diabetes complications by awareness, education, and medical care at the least cost.

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

“…Recently, in [18], we developed a three-state model of diabetes population considering D(t) as the number of diabetics who never experienced any complication at time t, D c (t) as the diabetics with complications, and D p (t) as the diabetics after the treatment of the complications. The dynamics are given by…”

“…where I is the incidence rate of diabetes assuming no complications, γ is the rate of the first incidence of complication, τ is the time delay in developing the first complication, κ is the treatment rate, σ is the recurring rate of complications, μ 1 is the rate of diabetes-related death among diabetics without complications, μ 2 is the rate of diabetes-related death among diabetics with complications, μ 3 is the rate of diabetes-related death among diabetics after the treatment of the complications, and μ is the rate of death due to other causes than diabetes. This system (5) exhibits a Hopf bifurcation phenomenon for some range of parameter values, see [18]. We used the model presented by Boutayeb et al [10] as the basis of model (5).…”

“…where the states C(t) and D(t) are the numbers of diabetics with and without complications, respectively, at time t. The parameter I is the incidence rate of diabetes, λ is the rate of developing complications, γ is the cure rate of complications, ν is the rate of becoming disabled, δ is the rate of mortality due to complications, and μ is the natural mortality rate. The reader may refer to [18] for the assumptions and development of model (5).…”

“…This treatment 3) indicates that when the treatment capacity is limited, the treatment will reach maximum if infected individuals are large enough. Motivated by the above works, we extend our previous model of the diabetic population with time delay in [18] by including the factor of limited medical resources. Time delay has a destabilizing effect that affects the stability of an equilibrium point [19].…”

“…Time delay has a destabilizing effect that affects the stability of an equilibrium point [19]. Since there is a time delay involved in [18], we further investigate the existence of Hopf bifurcation and its direction, stability, and period of the bifurcating periodic solutions. In the next section, we discuss the formulation of the model with the inclusion of the non-diabetics population.…”