Mathematical Analysis of the Effect of Control on the Dynamics of Diabetes Mellitus and Its Complications

Diabetes is a disorder in which the body becomes unable to control the amount of sugar in the blood. The pancreas (beta-cells) is not functioning normally, resulting in a partial or total lack of insulin which is the key to the mechanism converting sugar to energy. In this study, mathematical model for the dynamics of diabetes mellitus and its complications incorporating control is developed and analyzed. Positive lifestyle, which includes abstinence from alcohol, smoking and glutoning, and effective management of diabetes condition are incorporated as controls. The analytical solution of the model equations is obtained using Homotopy Perturbation Method. Numerical simulation of the model solution was done using Maple 18 Mathematical software. The parameters are varied and their effects on the model dynamics are presented graphically. The results showed that the two control measures can effectively be used to reduce the evolution of incidence of diabetes and occurrence of complications of diabetes thereby reducing the rate of morbidity and mortality due to diabetes complications.


Introduction
Diabetes is a metabolic disorder characterized by the inability of the body system to regulate the amount of glucose (special sugar in the body) in the body. This occurs when the insulin secretes by the organ (pancreas) in the beta cell is not sufficient or the body does not utilise the insulin produced effectively. The resulting effect is that the body system cannot function effectively and glucose level in the blood goes beyond normal [37], [36], [34], and [24]. The menace of diabetes and it effect on human, economy and social impact have become a serious problem to the world. International Diabetes with boundary condition of: (1.2) D denotes general differential operator, the boundary operator is E, the known analytical function is ( ) and ∏ denotes domain boundary of . Λ The operator D can be divided into two parts of G and H, the linear part is G, while H is a non-linear one. Equation (1.1) can be rewritten as follows: The stated HPM equation is as follows: The embedding parameter is In most cases, the series (1.5) converges. The convergent rate depends on the nonlinear operator ( ) w D [17].

Model formulation
The study considered the model proposed by [15] with some modifications and incorporate two control measures. The control measures are positive lifestyle and effective management of diabetes conditions. Based on their health status, the model population are classified into five classes. They are healthy class ( ),

t T
The assumption of the model was that the healthy individual will give birth to a healthy children that will be born into healthy compartment while parent who is diabetic or have history of diabetes will give birth to children with genetic factors that will be born into susceptible compartment. The proportion of children born into healthy compartment is denoted by θ while proportion of children that are born into susceptible compartment is denoted by The control parameters are . , 2 1 φ φ 1 φ is a measure of positive lifestyle in the susceptible class, such that .
indicate negative lifestyle and 1 1 = φ indicate positive lifestyle. 2 φ is a measure of effective management of diabetes condition in the compartment of diabetics without complications, such that .
indicate ineffective management of diabetes condition and 1 2 = φ indicate effective management of diabetes condition.

The model equations
The model equations are stated as follows in (2.1) to (2.5) The initial values conditions are ( )

Numerical Simulation and Result
Numerical simulation of the results obtained in Section 2.3 was carried out using mathematical software (MAPLE 18) and the graphical profiles of the system responses are presented below.     Figure 3.1 shows that, as control rate increases, diabetics with complications decreases faster. This shows that the more people adopt positive lifestyle such as good dieting and regular physical exercises, the less cases of diabetics with complications. It is observed that as the control rate increases, the diabetics with complications is almost zero. However, effort to increase control measure will go a long way to reduce complications arising from diabetes. Figure 3.2 shows that, as control rate increases, diabetics with complications decreases. This shows that high rate of effective management of diabetes condition results to less cases of diabetics with complications. Figure 3.3 shows that, as control rate increases, susceptible class increases. This can be attributed to high rate of positive lifestyle. It pointed to the fact that the more people adopt positive lifestyle such as good dieting and regular physical exercises, the less they are susceptible to diabetes while Figure 3.4 shows that as control rate increases, diabetics without complications class increases. This is attributed to high rate of effective management of diabetes condition which helps in controlling the transition from diabetics without complications state to diabetics with complications state. It shows that the more effective management of diabetes condition, the less cases of complications.

Conclusion
The study presented a modified deterministic model for controlling the incidence of diabetes and its complications in a population. Two control parameters were imposed on the model equations and the effect of the control measures on the dynamics of diabetes are analysed. The model equations were solved analytically using homotopy perturbation method and solutions were obtained. The result showed that if control parameters rate could be increased, the transition rate from susceptible class to diabetics without complications class and transition rate from diabetics without complications to diabetics with complications would be drastically reduced. This will significantly reduce the incidence of diabetes and occurrence of complications in a population and the number of deaths attributable to diabetes and its complications would be minimised.