Abstract
Nowadays, Diabetes is one of the most common diseases, which has a huge and growing socio-economic burden affecting individuals, families, and the whole society. In this paper, we propose an optimal control approach modeling the evolution from pre-diabetes to diabetes with and without complications and the effect of living environment. We show the existence of an optimal control and then use a numerical implicit finite-difference method to monitor the size of population in each compartment.
1. Introduction
Today, all countries of the world suffer from the high number of people with diabetes, which is increasing and expanding at the extreme level. According to the latest statistics from the International Diabetes Federation (IDF) and as reported in the eighth edition of the Atlas Diabetes 2017 [1], diabetes ranks fourth in diseases leading to death which means that it is a serious disease, and all social and age groups suffer from it. Also, IDF says that the number of people with diabetes is more than 425 million people, most of them are 65 years old, which increases the risk of the disease that does not exclude children and adolescents under 20 years. The last estimates of the number of people with diabetes in this category are that more than one million people have diabetes type I or type II. This is required to raise the degree of danger and preparedness to find solutions to it and reduce its expansion because the number of people with diabetes could increase to 629 million in the horizon in the year 2045. Despite these predictions, the number of people with the disease has begun to decline in some high-income countries. At the same time, despite this decline, there are about 352 million other people suffering from the disease for a variety of reasons such as impaired glucose tolerance, and these people are more at risk of diabetes, very large, and there are about 10 million adults and 8 million elderly people over the age of 65 years suffering from diabetes in recent years compared to 2015.
Diabetics are more likely to have a variety of other serious diseases; 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. Diabetics have an increased risk of a number of serious life-threatening health problems that have psychological, moral, and behavioral effects, leading to increased costs of medical care as well as serious physical deterioration, due to the increased risk of chronic disease with severe complications in various types such as cardiovascular disease, blindness, kidney failure, and amputation of the lower extremities, and so on. According to IFD statistics, diabetics have a two- to three-fold increased risk of cardiovascular disease, as well in more than a third of them, the incidence of retinopathy among all people with diabetes is the main cause of vision loss of adults at work and furthermore they accuse an increasing incidence of end-stage renal diseases (ESRD) up to 10 times in the disease. Besides, complications need to be treated and intensive care and expensive long-term care which have a material impact on individuals, families, and society as a whole have to be considered. The American Diabetes Association estimates that the annual cost of treating a person with diabetes is more than five times higher than the cost of a person who is not diabetic. Other studies suggest that the treatment of diabetic patients with complications is two to five times higher than the treatment of diabetes without complications, compounding burdens that have exceeded the limits of economic problems by incurring indirect and intangible costs [11]. Thus, to ease the social and economic burdens, several studies must be conducted for controlling the number of people developing from pre-diabetes to diabetes stages with and without complications.
During the last decade, a large number of mathematical models have been developed to simulate, analyse, and understand the dynamics of a population of diabetics; in a related research work, Boutayeb and Chetouani [2] and Derouich et al. [3] proposed a mathematical model for the dynamics of the population of diabetes patients with or without complications using a system of ordinary differential equations. Also, many researches have focused on this topic and other related topics [4, 5, 12–17].
Specialists overlooked the living environment as one of the main causes of the development of complications of diabetes, which plays a key role in the development of complications of diabetes in various types, according to studies carried out by the IDF. Therefore, in our proposed model we wanted to highlight the impact of the living environment on diabetic patients (availability of healthy food, diet, exercise, weight for instance) and its main role in the development of complications, which often ends either with loss of vision or amputation of the toes, feet, and lower legs or Paraplegia and so on. To achieve this objective, we consider a compartment model that describes the dynamic of a population of diabetics that is divided into four classes, i.e: the potential diabetic specialy pre-diabetics by through genetics diabetics without complications , diabetics with complications , and we add a compartment people who are likely to have diabetes through the effect of living environment or psychological problems.
We notice that most of researchers about diabetes and its complications focused on continuous time models, and described by differential equations. Recently, more and more attention has been paid to discrete time models (see [6, 7] and the references mentioned there). Reasons for adopting discrete models are as follows: First, statistical data are collected separately for moments (day, week, month, or year). Therefore, it is more direct, more accurate, and timely to describe the disease using discrete time models instead of continuous time models. Second, discrete time models can be used to avoid some mathematical complexities such as choosing the function space and regularity of the solution. Third, numerical simulations of continuous time models are obtained by abstraction.
In this paper, in Section 2, we propose a discrete Mathematical that describes the dynamic of a population of diabetics. In Section 3, we present an optimal control problem for the proposed model where we give some result concerning the existence of the optimal control and we characterize the optimal controls using the Pontryagin maximum principle in discrete time. Numerical simulations through MATLAB are given in Section 4. Finally, we conclude the paper in Section 5.
2. A Mathematical Model
We consider a discrete mathematical model that describes the dynamics of a population of diabetics; we highlight the impact of the living environment, such as unhealthy food and health habits on diabetics without complications. We divide the population denoted by into four compartments.
2.1. Description of the Model
The potential diabetic refers to people who are likely to have diabetes through genetics, is increasing by (denotes the incidence of pre-diabetes) and decreasing by the amount (natural mortality), (the rate of patients who develop diabetes without complications), and (the probability of developing diabetes at stage of complications).
Compartment is the number of diabetics without complications is increasing by amount and by the amount (patients who become diabetics without complications because of the effect of the living environment), and decreasing by (natural mortality) and (the rate of patients who become diabetic with complication because of the bad contact with the living environment) and decreasing by (the probability of a diabetic person developing a complication).
Compartment refers to people who are likely to have diabetes through the effect of living environment or psychological problems that describes the dynamics of a population of diabetics; we highlight the impact of the living environment, leading to an increase in the number of people diabetics to diabetics without complications and to diabetics with complications as people who live in the Middle East, Europe, or the USA, so in this compartment, it is increasing by (denote the incidence of effect environment) decreasing by (the rate of probability developing diabetes) and also decreased by (natural mortality)
Compartment is the number of diabetics with complications is increasing by and increasing and also increasing by (the rate of patients who become diabetics with complication because of the bad contact of the living environment) and decreasing by (mortality rate due to complications) and also decreasing by (natural mortality)
The following diagram proves the flow directions of diabetics among the compartments. These directions are going to be represented by directed arrows in figure of compartement.
2.2. Model Equations
Through the addition of the rates at which diabetics enter the compartment and also by subtracting the rates at which people vacate the compartment, we obtain an equation of difference for the rate at which the patients of each compartment change over discrete time. Hence, we present the diabetic model by the following system of difference equations:
The model is presented
And : final time with
3. Formulation of the Model
The strategy of control we adopt consists of an awareness program through to correct environment effect in diabetics people without complication. Our main goal is to minimize the number of people evolving from the stage of pre-diabetes to the stages of diabetes with and without complications. In this model, we include three controls , , and , that represent consecutively the awareness program through media and education, treatment, and psychological support with follow-up as measures at time . So the controlled mathematical system is given by the following system of difference equations:
4. The Optimal Control Problem
There are three controls , , and . The first control can be interpreted as the proportion to be subjected to treatment. So, we note that is the proportion of diabetics at stage of complications who moved to diabetics without complications at time step . The second control can be interpreted as the proportion to be adopted for the awareness program through media and education. So, we note that is the proportion of bad contact in living environment in diabetics without complications to complications, who will move to reduce the number of patients, diabetics with complications. The third control is the rate of probability developing diabetes, who will move to reduce the number of patients becoming diabetics without complications because of the effect of living environment at time step .
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 is the final time.
In other words, we seek the optimal controls ,,and such that
where is the set of admissible controls defined by
In order to derive the necessary condition for optimal control, pontryagin’s maximum principle, in discrete time, given in was used. This principle converts into a problem of minimizing a Hamiltonian, at time step defined by
where is the right side of the difference equation of the state variable at time step
5. The Optimal Control: Existence
We first show the existence of solutions of the system, after that we will prove the existence of optimal control [8, 9].
Theorem 1. Consider the control problem with the system. There are three optimal controls such that
Given the optimal controls and the solutions , and of the corresponding state system (6), there exists adjoint variables , , , and satisfying:
With the transversality conditions at time : , , , and
Furthermore, for , the optimal controls , , and are given by
Proof. Since the coefficients of the state equations are bounded and there are a finite number of time steps, , , , and are uniformly bounded for all in the control set thus is bounded for all . Since is bounded, is finite, and there exists a sequence such that and corresponding sequences of states , , , and . Since there is a finite number of uniformly bounded sequences, there exist and , , , and such that, on a subsequence, , , , , and . Finally, due to the finite dimensional structure of system (2) and the objective function , and is an optimal control with corresponding states , , , and . Therefore is achieved.
The Hamiltonian at time step is given by
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 in the form of system.
6. Numerical Simulation
Algorithm 1. In this section, we present the results obtained by solving numerically the optimality system.
This system consists of the state system, adjoint system, initial and final time conditions, and control characterization.
So, the optimality system is given by the following:
Step 1. , , , , , , , and given , , and
Step 2. For do:
end for
Step 3. For write:
, , , and
,, and
Different simulations can be carried out using various values of parameters. In the present numerical approach, we use the following parameter values taken from [2]:
, , , , , , , , and
Since control and state functions are on different scales, the weight constant value is chosen as follows: , , and (Figure 1)
After the parameter values (Table 1), we note that diabetics without complications after 120 months decreased from 10.2 × 106 to 2.31 × 106 (Figure 1) This transformation is due to two main things: first is the genetic factors; second, due to the negative impact of the living environment on the patient (nutrition pattern, psychological and moral problems). We note that the number of patients with diabetes complications is increasing. Indeed, we note the number of the transition becomes from 6.66 × 106 to 1.53 × 108 (Figure 1) and, as mentioned above, to come to disease progression for patients with diabetes without complications and also a sudden shift in the potential for people diagnosed with diabetes by means of genetics.
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 using characterization.
We continue until convergence of successive iterates is achieved.
The proposed control strategy in this work helps to achieve several objectives.
6.1. Objective: Prevention: To Protect Diabetic Patients without Complications from the Negative Impact of the Environment
Awareness programs are to lower the effect of bad contact with living environment, we propose an optimal strategy for this purpose. Hence, we activate the optimal control variable which represents the awareness program for diabetics without complications. Figure 2 compares the evolution of the number of diabetics with complications with and without control in which is the effect of the proposed awareness program through a diet program for diabetics and to keep them as far as possible about problems and family pressures and process.
6.2. Strategy B: Control with Treatment and Psychological
In this strategy, we apply in order to reduce the number of diabetics with complication to diabetics without complications; through Figure 3 note that after applying different Strategic, which is a medicine and psychologist support, the number of diabetics with complication dropped from 1.53 × 108 to 7.96 × 107 the end of the strategic.
The reason for this increase is justified by the fact that the number of diabetics with complications will becomes diabetics without complications. For improving the effectiveness of this strategy, we add the elements of follow-up and psychological support which are represented in the proposed strategy by the optimal control variable (Figure 3). Combining follow-up and psychological support with treatment results in an obvious decrease in the number of diabetics with complications.
6.3. Strategy C: Control with Awareness Program,Treatment, and Psychological Support with Follow-Up
In this strategy, we use two controls optimal and (Figure 4) that we combine the previous two strategies to achieve better results that is represent awareness program through education and media for lowering the effect of bad contact with living environment, and also treatment, and psychological support with follow up. In Figure 4, we observe that the number of diabetics with complications is decreasing from 1.53 × 108 to 7.29 × 107.
6.4. Strategy D: Prevention: Protection from Diabetes
6.4.1. With Three Controls , and
In this starategy we use three controls , , and (Figure 5), the objectif of the two controls , and as we say above, and the object if of the third control is the aim is to raise awareness campaigns for this target group on the risks of diabetes and its complications with giving recipes for health nutrition.
Remark 1. We can also merge multiple assemblies as , and thus get a variety of results.
7. Conclusion
In this paper, we introduced a discrete modeling of populations diabetics with and without complications and effect of living environment in order to minimize the number of diabetics with complications, and lower the effect of bad contact with living environment.
We also introduced three 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.