Abstract
We propose an optimal control strategy by conducting awareness campaigns for diabetics about the severity of complications of diabetes and the negative impact of an unbalanced lifestyle and the surrounding environment, as well as treatment and psychological follow-up. Pontryagin’s maximum principle is used to characterize the optimal controls, and the optimality system is solved by an iterative method. Finally, some numerical simulations are performed to verify the theoretical analysis using MATLAB.
1. Introduction
Nowadays, diabetes is a chronic disease with a huge burden affecting individuals. According to the World Health Organisation (WHO) [1], diabetes is a disorder characterized by the presence of problems in the insulin hormone, which naturally results from the pancreas to help the body use glucose and fat and store some of them. According to the American Diabetes Association (ADA) [2], diabetes mellitus is a group of metabolic diseases characterized by hyperglycemia resulting from defects in insulin secretion, insulin action, or both. It is known that the proper level of glucose in the blood after fasting eight hours should be less than 108 mg/dl, while the borderline is 126 mg/dl. If a person’s blood glucose level is 126 mg/dL and above, in two or more tests, then that person is diagnosed with diabetes. Diabetes is divided into several different types; some more prevalent than others. The most common type of diabetes in the general population is type 2 diabetes, and type 1 diabetes is more common in children, and gestational diabetes is a form of diabetes that can occur during pregnancy. According to the latest statistics from the International Diabetes Federation (IDF) and as reported in the 9th edition of the Atlas Diabetes 2017 [3], diabetes is a constantly growing disease. There are more than 370 million people with diabetes worldwide (8.5% of the adult population) and about 463 million people in prediabetes (6.5% of the adult population), and more than 625 million are expected to be affected by to 2045.
Today, all countries of the world suffer from the high number of people with diabetes, which is increasing and expanding on the extreme level. When it is not treated well, all types of diabetes can lead to complications in many parts of the body, leading to an early death. When a diabetic knows how to control the level of glucose in the blood, this awareness plays a key role in reducing the serious complications of diabetes. According to IDF statistics, diabetes has serious and varied complications. For example, the risk of cardiovascular disease. Moreover, more than a third of diabetics have retinopathy which is the main cause of vision loss, in addition to the risk of kidney disease. In addition, the complications of diabetes are multiple and different depending on the degree of severity; there are complications that can be treated and others that have reached critical stages with which treatment is not beneficial. According to ADA , diabetes has economic and social burdens on the individual and society.
During the last decade, large mathematical models on diabetes have been developed to simulate, analyse, and understand the dynamics of a population of diabetics. In a related research work, Boutayeb and Chetouani [4] and Derouich et al. [5] introduced a mathematical model for the dynamics of the population of diabetes. And Kouidere et al. [6] proposed a discrete mathematical model highlighting the impact of living environment. Also, many researches have focused on this topic and other related topics ([7–12]).
As we said earlier, diabetes has many complications, and the nature of these complications are two types: treatable ones and those that have reached a critical stage which is impossible to cure completely.
According to IDF [3], diabetes is influenced by a complex interaction of behavioural, genetic, and socioeconomic factors; many of which are outside our individual control.
To achieve this objective, we considered a compartment model that describes the dynamics of a population of diabetics that is divided into six compartments, i.e., the healthy people the prediabetics through genetic factors the prediabetics through the negative effect of behavioral factors on diabetics patients and others and diabetics without complications are those who control blood sugar through diet and exercise before it is too late, and we divided diabetics with complications into two parts: diabetics with treatable complication and diabetics with serious complications.
We noticed that most of researchers about diabetes and its complications focused on continuous and discrete time models and described by differential equations. Recently, more and more attention has been paid to study the control optimal (see [13–21] and the references mentioned there).
In this paper, in Section 2, we represent a mathematical model that describes the dynamic of a population of diabetics. In Section 3, we presented an optimal control problem for the proposed model, where we gave some results concerning the existence and positivity of the optimal control and we characterized the optimal controls used Pontryagin’s maximum principle. Numerical simulations through MATLAB are given in Section 4. Finally, we conclude the paper in Section 5.
2. A Mathematical Model
We consider a mathematical model that describes the dynamics of a population of diabetics. We divide the population denoted by into six compartments.
2.1. Description of the Model
The graphical representation of the proposed model is shown in Figure 1.
The compartment are healthy people; the compartment is increased by (which is the recruitment rate of healthy people), and this compartment is decreased by (the number of prediabetic people through genetic factor) and also decreased by (the number of prediabetics through the negative effect of behavioral factors) and decreased by the amount (natural mortality).
The compartment are people who are likely to have diabetes through genetic factors. The compartment is increased by . This compartment is decreased by the amount (natural mortality) and also decreased by (the probability of developing diabetes) and by (the probability of developing diabetes at stage of complications).
Compartment are people who are likely to have diabetes through the negative effect of lifestyle or psychological problem factors and others (these are people at risk of developing diabetes, such as those who are obese, overweight, gestational diabetes, or due to family and work problems, in addition to the person without diabetes).The compartment increased by and decreased by (patients who become diabetics without complications because of the negative effect of lifestyle) and also decreased by (natural mortality).
Compartment is the number of diabetics without complications. The compartment increased by the amount and by the amount . This compartment is decreasing by (natural mortality) and (the rate of negative impact of ordinary people on diabetics without complications through improper nutrition, or practical or family social problems) and also decreased by (the probability of a diabetic person developing a complication) and also decreased by (the number of diabetics whose serious complications because of a sudden shock).
Compartment is the number of diabetics with treatable complications. The compartment increased by and by and also increased by and and decreased by (the rate negative impact of ordinary people on diabetics with treatable complications through improper nutrition, or practical or family social problems) and decreased by (the number of people developing of diabetics with complications can be treated to serious complications) and also by (natural mortality)
Compartment is the number of diabetics with serious complications, they are the diabetics who have serious complications such as retinopathy or renal failure. The compartment increased by and also by and by This compartment decreased by (mortality rate due to complications) and also by (natural mortality).
Hence, we present the diabetic model by the following system of differential equations: with , and
2.2. Positivity of Solutions
Theorem 1. If , and , the solutions , and of system (7) are positive for all .
Proof. It follows from the first equation of system (7) that
Both sides in the last inequality are multiplied with
We obtain then
Integrating this inequality from to gives then
Similarly, we prove that and
2.2.1. Boundedness of the Solutions
Theorem 2. The set is positively invariant under system (7) with initial conditions , and .
Proof. By adding the equations of system (7), we obtain
where represents the initial values of the total population.
Thus, . It implies that the region is a positively invariant set for system (7). So, we only need to consider the dynamics of the system on the set .
2.2.2. Existence of Solutions
Theorem 3. The system (7) that satisfies a given initial condition has a unique solution.
Proof. Let so the system (7) can be rewritten in the following form:
where
The second term on the right-hand side of (15) satisfies where then where
Thus, it follows that the function is uniformly Lipschitz continuous, and the restriction on , and , we see that a solution of the system (7) exists [22].
3. Formulation of the Model
Our objective in this proposed strategy of control is to minimize the number of people evolving from the stage of prediabetes to the stages of diabetes without complications and to diabetes with treatable complications. In this model, we include four controls , , , and for . The first control we note represented treatment that will work to treat diabetic patients with treatable complications. The second control represented the awareness program through media and education, by raising awareness and awareness of the seriousness of the negative impact of behavioral factors on diabetes. The third control represented treatment and education, by treating complications, with sensitivity to the negative effect of behavioral factors on diabetic patients with complications. The fourth control represented the awareness program, through raising awareness on nondiabetics against the negative impact of behavioral, economic, and social factors that lead to diabetes at time
So the controlled mathematical system is given by the following system of differential equations:
The problem is to minimize the objective functional where , and are the cost coefficients; they are selected to weigh the relative importance of , and at time , and is the final time.
In other words, we seek the optimal controls , , and such that where is the set of admissible controls defined by
4. The Optimal Control: Existence and Characterization
We first show the existence of solutions of the system (18),thereafter, we will prove the existence of optimal control.
4.1. Existence of an Optimal Control
Theorem 4. Consider the control problem with system (18). There exists an optimal control such that
Proof. The existence of the optimal control can be obtained using a result by Fleming and Rishel [23], checking the following steps: (i)It follows that the set of controls and corresponding state variables is not empty. we will use a simplified version of an existence result ([24], Theorem 7.1.1)(ii) is convex in (iii)The control space is convex and closed by definition(iv)All the right-hand sides of equations of system are continuous, bounded above by a sum of bounded control and state, and can be written as a linear function of , and with coefficients depending on the time and state(v)The integrand in the objective functional, , is clearly convex on (vi)It rests to show that there exist constants and such that satisfies
The state variables are being bounded; let and ; then, it follows that
Then, from Fleming and Rishel [23], we conclude that there exists an optimal control.
4.2. Characterization of the Optimal Control
In order to derive the necessary conditions for the optimal control, we apply Pontryagin’s maximum principle to the Hamiltonian at time defined by where is the right side of the difference equation of the state variable
Theorem 5. Given the optimal controls and the solutions , and of the corresponding state system (18), there exists adjoint variables , , , and satisfying
With the transversality conditions at time , , and
Furthermore, for , the optimal controls , and are given by
Proof. The Hamiltonian is defined as follows: where
,
For the adjoint equations and transversality conditions can be obtained by using Pontryagin’s maximum principle [13, 25] such that
For, the optimal controls , and can be solved from the optimality condition, that are
We have
,
By the bounds in of the controls, it is easy to obtain , and and are given in (13–16) the form of system and in the form of system (18).
5. Numerical Simulation
In this section, we present the results obtained by solving numerically the optimality system (18). In our control problem, we have initial conditions for the state variables and terminal conditions for the adjoints. That is, the optimality system is a two-point boundary value problem with separated boundary conditions at times step and . We solve the optimality system by an iterative method with forward solving of the state system followed by backward solving of the adjoint system. We start with an initial guess for the controls at the first iteration, and then before the next iteration, we update the controls by using the characterization. We continue until convergence of successive iterates is achieved. A code is written and compiled in MATLAB using the following data.
Different simulations can be carried out using various values of parameters. In the present numerical approach, we use the following parameters values taken from [6].
Since control and state functions are on different scales, the weight constant value is chosen as follows: , and and with the intial value of and 2000000 (Figure 2).
After the parameter values (Tables 1 and 2), we noted that diabetics without complications after 120 months decreased from to (Figure 2) This transformation is due to three main things: first, it is by the genetic factors. Second, due to the negative impact of behavioral factors on the patient (nutrition pattern and psychological and moral problems) and the third by sudden shock (family problem, work problem), we noted that diabetics with treatable complications are increasing. Indeed, we noted that the number of the transition becomes from to (Figure 2) and, as mentioned above, has disease progression for diabetics without complications, and also a sudden shift in the potential for people diagnosed with diabetes by means of genetics and with negative impact of behavioral factors.
We noted that diabetics with serious complications are increasing and that the number of the transition becomes from to (Figure 1) and, as mentioned above, has disease progression for diabetics without complications by sudden shock and with negative impact of behavioral factors, and by developing the disease of diabetics with treatable complications.
In this formulation, there are initial conditions for the state variables and terminal conditions for the adjoints.
That is, the optimality system is a two-point boundary value problem with separated boundary conditions at time steps and . We solve the optimality system by an iterative method with forward solving of the state system followed by backward solving of the adjoint system. We start with an initial guess for the controls at the first iteration, and then before the next iteration, we update the controls by using the characterization.
We continue until convergence of successive iterates is achieved.
The proposed control strategy in this work helps to achieve several objectives.
5.1. Strategy A
In this strategy, we applied two controls and in order to reduced the number of diabetics with treatable complications to diabetics without complications, through Figure 2, we noted that after applied different strategics, the number of diabetics with treatable complications decreased from to by the end of the strategy (Figure 3).
The reason of this increase was justified by the fact that the number of diabetics with treatable complications will become diabetics without complications. For improving the effectiveness of this strategy, we added the elements of follow-up and psychological support and education about the negative impact of behavioral factors which are represented in the proposed strategy by the optimal controls variables and (Figure 3 and Table 3), combining follow-up and psychological support with treatment and education results in an obvious decreased in the number of diabetics with treatable complications.
5.2. Strategy C: Control with Awareness Program,Treatment, and Psychological Support with Follow-Up
We combined three optimal controls and .
In this strategy, the three optimal controls and are activated at the same time, in order to reduced the number of diabetics with treatable complications to diabetics without complications (Figure 4).
In this strategy (Figure 4 and Table 4), we used three controls optimal , and . That is, we combined the previous two strategies to achieved better results that represented treatment, and psychological support with follow-up, and also awareness program through education and media for lower the negative impact of behavioral factors. In Figure 4 and Table 4, we observe that the number of diabetics with treatable complications is decreasing from to and also, the number of diabetics with serious complications is decreasing from to
5.3. Strategy D: Prevention and Protection from Diabetes
We we use only the optimal control
In this strategy, we focus the effort of the awareness campaign to reduce the negative impact of behavioral factors (Figure 5).
In this strategy, we used control (Figure 5 and Table 5); the objective of this control is to raise awareness campaigns for this target group on the risks of diabetes and its complications as cardiovascular disease, blindness, kidney failure, and lower limb amputation, with tracking healthy and balanced diet program. after Figure 5 and Table 5, we observed that the number of diabetics without complications is decreasing from to
Remark 6. We could also merge multiple assemblies as and and thus get a variety of results.
6. Conclusion
In this paper, we formulated a mathematical model of populations of diabetics, having six compartments: prediabetics through the genetics effects and others by behavioral factors, diabetics without complications, and diabetics with treatable and serious complications, in order to minimize the number of diabetics with treatable complications, and reduce the effect of behavioral factors. We also introduced four controls which, respectively, represent awareness program through education and media, treatment, and psychological support with follow-up. We applied the results of the control theory, and we managed to obtain the characterizations of the optimal controls. The numerical simulation of the obtained results showed the effectiveness of the proposed control strategies.
Data Availability
The disciplinary data used to support the Öndings of this study have been deposited in the Network Repository (http://www.networkrepository.com).
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank Dr. Bruno Carpentieri and all the members of the Editorial Board who were responsible for dealing with this paper, and the anonymous referees for their valuable comments and suggestions, improving the content of this paper.