Optimal Control Strategy for a Discrete Time to the Dynamics of a Population of Diabetics with Highlighting the Impact of Living Environment

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Introduction
Today, all countries of the world su er 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 su er 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.e 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.
is is required to raise the degree of danger and preparedness to nd 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 su ering 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 su ering 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 e ects, 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.e American Diabetes Association estimates that the annual cost of treating a person with diabetes is more than ve 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 ve 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].us, 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 di erential equations.Also, many researches have focused on this topic and other related topics [4,5,[12][13][14][15][16][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.erefore, 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 o en 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 e ect 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 di erential 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).erefore, 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.ird, 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.

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.

Description of the Model.
e potential diabetic ( ) refers to people who are likely to have diabetes through genetics, is increasing by Λ 1 (denotes the incidence of pre-diabetes) and decreasing by the amount (natural mortality), 1 (the rate of patients who develop diabetes without complications), and 3 (the probability of developing diabetes at stage of complications).
Compartment ( ) is the number of diabetics without compli- cations = is increasing by amount 1 and by the amount (patients who become diabetics without complications because of the e ect 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 2 (the probability of a diabetic person developing a complication).
Compartment ( ) refers to people who are likely to have dia- betes through the e ect 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 Λ 2 (denote the incidence of e ect 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 3 and increasing 2 and also increasing by ( / ) (the rate of patients who become dia- betics 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) e following diagram proves the ow directions of diabetics among the compartments.ese directions are going to be represented by directed arrows in gure of compartement.

Model Equations.
rough 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 di erence for the rate at which the patients of each (1) ( compartment change over discrete time.Hence, we present the diabetic model by the following system of di erence equations: e model is presented And : nal time with 0 .0 .0 , 0 ≥ 0.

Formulation of the Model
e strategy of control we adopt consists of an awareness program through to correct environment e ect 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 , v , and w , 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 di erence equations:

The Optimal Control Problem
, and w = w 0 , w 1 , . . ., w −1 .e rst con- trol 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 .e second control can be interpreted as the proportion to be adopted for the awareness program through media and education.So, we note that v( / ) is the proportion of bad contact in living (5) environment in diabetics without complications to complications, who will move to reduce the number of patients, diabetics with complications.e third control w is the rate of probability developing diabetes, who will move to reduce the number of patients becoming diabetics without complications because of the e ect of living environment at time step .e problem is to minimize the objective functional where , , and are the cost coe cients.ey are selected to weigh the relative importance of , v , and w at time , is the nal time.
In other words, we seek the optimal controls * ,v * ,and w * such that where is the set of admissible controls de ned by In order to derive the necessary condition for optimal control, pontryagin's maximum principle, in discrete time, given in was used. is principle converts into a problem of minimizing a Hamiltonian, at time step de ned by where , +1 is the right side of the di erence equation of the ℎ state variable at time step + 1.

The Optimal Control: Existence
We rst show the existence of solutions of the system, a er that we will prove the existence of optimal control [8,9].
Theorem 1.Consider the control problem with the system.ere are three optimal controls * , v * , w * ∈ 3 such that Given the optimal controls * , v * , w * and the solutions * , * , * , and * of the corresponding state system (6), there exists adjoint variables 1, , 2, , 3, , and 4, satisfying: , , , , For, = 0, 1, . . ., − 1 the optimal controls , v , and w can be solved from the optimality condition, at are we have By the bounds in of the controls, it is easy to obtain * , v * , and w * in the form of system.

Numerical Simulation
Algorithm 1.In this section, we present the results obtained by solving numerically the optimality system.
is system consists of the state system, adjoint system, initial and nal time conditions, and control characterization.
e Hamiltonian at time step is given by ( 13) Since control and state functions are on di erent scales, the weight constant value is chosen as follows: = 100, = 100, and = 100 (Figure 1).
A er the parameter values (Table 1), we note that diabetics without complications a er 120 months decreased from 10.2 × 10 6 to 2.31 × 10 6 (Figure 1) is transformation is due to two main things: rst 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 × 10 6 to 1.53 × 10 8 (Figure 1) and, as mentioned above, to come to disease progression for patients with diabetes without complications and also a sudden shi 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.
at is, the optimality system is a two-point boundary value problem with separated boundary conditions at time steps = 0 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 rst iteration and then end for Step 3.For = 0; 1; . . .; write: * = , * = , * = , and * = * = ,v * = v , and w * = w Di erent simulations can be carried out using various values of parameters.In the present numerical approach, we use the following parameter values taken from [2]: (0) = 6660000, (0) = 10200000, (0) = 10000000, (0) = 5500000, = 100, 1 ( ) = 0, 2 ( ) = −1, 3 ( ) = 0, and complications.Figure 2 compares the evolution of the number of diabetics with complications with and without control v in which is the e ect 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.

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 a er applying di erent Strategic, which is a medicine and psychologist support, the number of diabetics with complication dropped from 1.53 × 10 8 to 7.96 × 10 7 the end of the strategic.
before the next iteration we update the controls using characterization.
We continue until convergence of successive iterates is achieved.
e proposed control strategy in this work helps to achieve several objectives.

Objective: Prevention: To Protect Diabetic Patients
without Complications from the Negative Impact of the Environment.Awareness programs are to lower the e ect of bad contact with living environment, we propose an optimal strategy for this purpose.Hence, we activate the optimal control variable v which represents the awareness program for diabetics without for lowering the e ect 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 × 10 8 to 7.29 × 10 7 .
6.4.Strategy D: Prevention: Protection from Diabetes 6.4.1.With ree Controls , v and w .In this starategy we use three controls , v , and w (Figure 5), the objectif of the two controls , and v as we say above, and the object if of the third control w 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 , w , v , w and thus get a variety of results.
e reason for this increase is justi ed by the fact that the number of diabetics with complications will becomes diabetics without complications.For improving the e ectiveness 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.

Strategy C: Control with Awareness Program,Treatment,
and Psychological Support with Follow-Up.In this strategy, we use two controls optimal and v (Figure 4), that we combine the previous two strategies to achieve better results that is represent awareness program through education and media  5: e evolution of the number of diabetics without complications with three controls , v , and w .

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.e numerical simulation of the obtained results showed the effectiveness of the proposed control strategies.

T 1 :
Parameter values used in numerical simulation.evolution of the number of diabetics with and without complications without controls.

CF 3 :
with control v(k)F2: e evolution of the number of diabetics with and without complications with control v .e evolution of the number of diabetics with and without complications with control .

9 D 16 CE
without controls D with two control u(k) and v(k) without controls C with two control u(k) and v(k) F 4: e evolution of the number of diabetics with and without complications with two controls and v .D with three controls u(k), v(k) and w(k) with control w(k) E without control