# A New Mathematical Modeling with Optimal Control Strategy for the Dynamics of Population of Diabetics and Its Complications with Effect of Behavioral Factors

Authors: Abdelfatah Kouidere; Abderrahim Labzai; Hanane Ferjouchia; Omar Balatif; Mostafa Rachik
Journal: Journal of Applied Mathematics
 (2020)

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## 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.


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## Body

## 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 ADA20, 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 peopleH, the prediabetics through genetic factors P, the prediabetics through the negative effect of behavioral factors on diabetics patients and others E, and D 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: CT diabetics with treatable complication and CS 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 Section2, we represent a HPEDCTCS 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 modelHPEDCTCS that describes the dynamics of a population of diabetics. We divide the population denoted by N into six compartments.


### 2.1. Description of the Model

The graphical representation of the proposed model is shown in Figure1.Figure 1
The dynamics of a population of diabetics.The compartmentH are healthy people; the compartment H is increased by I (which is the recruitment rate of healthy people), and this compartment H is decreased by θ1Ht (the number of prediabetic people through genetic factor) and also decreased by θ2Ht (the number of prediabetics through the negative effect of behavioral factors) and decreased by the amount μ (natural mortality).
(1)dHtdt=I−μ+θ1+θ2Ht.The compartmentP are people who are likely to have diabetes through genetic factors. The compartment P is increased by θ1Ht. This compartment P is decreased by the amount μ (natural mortality) and also decreased by β1Pt (the probability of developing diabetes) and by β3CTt (the probability of developing diabetes at stage of complications).
(2)dPtdt=θ1Ht−μ+β1+β3Pt.CompartmentE 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 E increased by θ2Ht and decreased by γEt (patients who become diabetics without complications because of the negative effect of lifestyle) and also decreased by μ (natural mortality).
(3)dEtdt=θ2Ht−μ+γEt.CompartmentD is the number of diabetics without complications. The compartment Dt increased by the amount β1Pt and by the amount γEt. This compartment D is decreasing by μ (natural mortality) and α1 (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 β2Dt (the probability of a diabetic person developing a complication) and also decreased by η2Dt(the number of diabetics whose serious complications because of a sudden shock).
(4)dDtdt=β1Pt+γEt−α1DtEtN−μ+β2+η2Dt.CompartmentCT is the number of diabetics with treatable complications. The compartment CT increased by β3Pt and byβ2Dt and also increased by α1 and and decreased by α2 (the rate negative impact of ordinary people on diabetics with treatable complications through improper nutrition, or practical or family social problems) and decreased by η1CTt (the number of people developing of diabetics with complications can be treated to serious complications) and also by μ (natural mortality)
(5)dCTtdt=β3Pt+β2Dt+α1DtEtN−α2CTtEtN−μ+η1CTt.CompartmentCS is the number of diabetics with serious complications, they are the diabetics who have serious complications such as retinopathy or renal failure. The compartment CS=CSt increased by η2Dt and also by η1CTt and by α2. This compartment CS decreased by δ (mortality rate due to complications) and also by μ (natural mortality).
(6)dCStdt=η2Dt+η1CTt+α2CTtEtN−μ+δCSt.Hence, we present the diabetic model by the following system of differential equations:(7)dHtdt=I−μ+θ1+θ2Ht,dPtdt=θ1Ht−μ+β1+β3Pt,dEtdt=θ2Ht−μ+γEt,dDtdt=β1Pt+γEt−α1DtEtN−μ+β2+η2Dt,dCTtdt=β3Pt+β2Dt+α1DtEtN−α2CTtEtN−μ+η1CTt,dCStdt=η2Dt+η1CTt+α2CTtEtN−μ+δCSt,with H0≥0,P0≥0,E0≥0,D0≥0,CT0≥0, and CS0≥0.


### 2.2. Positivity of Solutions

Theorem 1.
IfH0≥0,P0≥0,E0≥0,D0≥0,CT0≥0, and CS0>0, the solutions Ht,Pt,Et,Dt,CTt, and CSt of system (7) are positive for all t≥0.Proof.
It follows from the first equation of system (7) that
(8)dHtdt=I−μ+θ1+θ2Ht≥−μ+θ1+θ2HtdHtdt+μ+θ1+θ2Ht≥0Both sides in the last inequality are multiplied withexpμ+θ1+θ2t.We obtain(9)expμ+θ1+θ2t·dHtdt+μ+θ1+θ2expμ+θ1+θ2t·Ht≥0,then
(10)ddtexpμ+θ1+θ2t⋅Ht≥0.Integrating this inequality from0 to t gives
(11)∫0tddsexpμ+θ1+θ2s⋅Hsds≥0,then
(12)Ht≥H0expμ+θ1+θ2t⇒Ht>0.Similarly, we prove thatPt≥0,Et≥0,Dt≥0,CTt≥0, and CSt≥0.


#### 2.2.1. Boundedness of the Solutions

Theorem 2.
The setΩ=H,P,E,D,CT,CS∈ℝ5/0≤H+P+E+D+CT+CS≤I/μ is positively invariant under system (7) with initial conditions H0≥0,P0≥0,E0≥0,D0≥0,CT0≥0, and CS0>0.Proof.
By adding the equations of system (7), we obtain
(13)dNdt=I−μN−δCS≤I−μN⇒Nt≤Iμ+N0e−μt,whereN0 represents the initial values of the total population.Thus,limt→∞supNt=I/μ. It implies that the region Ω is a positively invariant set for system (7). So, we only need to consider the dynamics of the system on the set Ω.


#### 2.2.2. Existence of Solutions

Theorem 3.
The system (7) that satisfies a given initial condition H0,P0,E0,D0,CT0,andCS0 has a unique solution.Proof.
Let(14)X=StPtEtDtCTtCSt,φX=dHt/dtdPt/dtdEt/dtdDt/dtdCTt/dtdCSt/dt,so the system (7) can be rewritten in the following form:
(15)φX·=X=AX+BX,where(16)A=−μ+θ1+θ200000θ1−μ+β1+β30000θ20−μ+γ0000β1γ−μ+β2+η2000β30β2−μ+η10000η2η1−μ+δ,BX=I00−α1DtEtNα1DtEtN−α2CTtEtNα2CTtEtN.The second term on the right-hand side of (15) satisfies
(17)BX1−BX2≤MD1t−D2t+CT.1t−CT.2t,BX1−BX2≤M⋅X1−X2,where M=2I/μα1/N+α2/N;α1/N+α2/N,then
(18)φX1−φX2≤V⋅X1−X2,where V=maxM,A<∞.Thus, it follows that the functionφ is uniformly Lipschitz continuous, and the restriction on Ht≥0,Pt≥0,Et≥0,Dt≥0,CTt≥0, and CSt≥0, we see that a solution of the system (7) exists [22].


## 2.1. Description of the Model

The graphical representation of the proposed model is shown in Figure1.Figure 1
The dynamics of a population of diabetics.The compartmentH are healthy people; the compartment H is increased by I (which is the recruitment rate of healthy people), and this compartment H is decreased by θ1Ht (the number of prediabetic people through genetic factor) and also decreased by θ2Ht (the number of prediabetics through the negative effect of behavioral factors) and decreased by the amount μ (natural mortality).
(1)dHtdt=I−μ+θ1+θ2Ht.The compartmentP are people who are likely to have diabetes through genetic factors. The compartment P is increased by θ1Ht. This compartment P is decreased by the amount μ (natural mortality) and also decreased by β1Pt (the probability of developing diabetes) and by β3CTt (the probability of developing diabetes at stage of complications).
(2)dPtdt=θ1Ht−μ+β1+β3Pt.CompartmentE 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 E increased by θ2Ht and decreased by γEt (patients who become diabetics without complications because of the negative effect of lifestyle) and also decreased by μ (natural mortality).
(3)dEtdt=θ2Ht−μ+γEt.CompartmentD is the number of diabetics without complications. The compartment Dt increased by the amount β1Pt and by the amount γEt. This compartment D is decreasing by μ (natural mortality) and α1 (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 β2Dt (the probability of a diabetic person developing a complication) and also decreased by η2Dt(the number of diabetics whose serious complications because of a sudden shock).
(4)dDtdt=β1Pt+γEt−α1DtEtN−μ+β2+η2Dt.CompartmentCT is the number of diabetics with treatable complications. The compartment CT increased by β3Pt and byβ2Dt and also increased by α1 and and decreased by α2 (the rate negative impact of ordinary people on diabetics with treatable complications through improper nutrition, or practical or family social problems) and decreased by η1CTt (the number of people developing of diabetics with complications can be treated to serious complications) and also by μ (natural mortality)
(5)dCTtdt=β3Pt+β2Dt+α1DtEtN−α2CTtEtN−μ+η1CTt.CompartmentCS is the number of diabetics with serious complications, they are the diabetics who have serious complications such as retinopathy or renal failure. The compartment CS=CSt increased by η2Dt and also by η1CTt and by α2. This compartment CS decreased by δ (mortality rate due to complications) and also by μ (natural mortality).
(6)dCStdt=η2Dt+η1CTt+α2CTtEtN−μ+δCSt.Hence, we present the diabetic model by the following system of differential equations:(7)dHtdt=I−μ+θ1+θ2Ht,dPtdt=θ1Ht−μ+β1+β3Pt,dEtdt=θ2Ht−μ+γEt,dDtdt=β1Pt+γEt−α1DtEtN−μ+β2+η2Dt,dCTtdt=β3Pt+β2Dt+α1DtEtN−α2CTtEtN−μ+η1CTt,dCStdt=η2Dt+η1CTt+α2CTtEtN−μ+δCSt,with H0≥0,P0≥0,E0≥0,D0≥0,CT0≥0, and CS0≥0.


## 2.2. Positivity of Solutions

Theorem 1.
IfH0≥0,P0≥0,E0≥0,D0≥0,CT0≥0, and CS0>0, the solutions Ht,Pt,Et,Dt,CTt, and CSt of system (7) are positive for all t≥0.Proof.
It follows from the first equation of system (7) that
(8)dHtdt=I−μ+θ1+θ2Ht≥−μ+θ1+θ2HtdHtdt+μ+θ1+θ2Ht≥0Both sides in the last inequality are multiplied withexpμ+θ1+θ2t.We obtain(9)expμ+θ1+θ2t·dHtdt+μ+θ1+θ2expμ+θ1+θ2t·Ht≥0,then
(10)ddtexpμ+θ1+θ2t⋅Ht≥0.Integrating this inequality from0 to t gives
(11)∫0tddsexpμ+θ1+θ2s⋅Hsds≥0,then
(12)Ht≥H0expμ+θ1+θ2t⇒Ht>0.Similarly, we prove thatPt≥0,Et≥0,Dt≥0,CTt≥0, and CSt≥0.


### 2.2.1. Boundedness of the Solutions

Theorem 2.
The setΩ=H,P,E,D,CT,CS∈ℝ5/0≤H+P+E+D+CT+CS≤I/μ is positively invariant under system (7) with initial conditions H0≥0,P0≥0,E0≥0,D0≥0,CT0≥0, and CS0>0.Proof.
By adding the equations of system (7), we obtain
(13)dNdt=I−μN−δCS≤I−μN⇒Nt≤Iμ+N0e−μt,whereN0 represents the initial values of the total population.Thus,limt→∞supNt=I/μ. It implies that the region Ω is a positively invariant set for system (7). So, we only need to consider the dynamics of the system on the set Ω.


### 2.2.2. Existence of Solutions

Theorem 3.
The system (7) that satisfies a given initial condition H0,P0,E0,D0,CT0,andCS0 has a unique solution.Proof.
Let(14)X=StPtEtDtCTtCSt,φX=dHt/dtdPt/dtdEt/dtdDt/dtdCTt/dtdCSt/dt,so the system (7) can be rewritten in the following form:
(15)φX·=X=AX+BX,where(16)A=−μ+θ1+θ200000θ1−μ+β1+β30000θ20−μ+γ0000β1γ−μ+β2+η2000β30β2−μ+η10000η2η1−μ+δ,BX=I00−α1DtEtNα1DtEtN−α2CTtEtNα2CTtEtN.The second term on the right-hand side of (15) satisfies
(17)BX1−BX2≤MD1t−D2t+CT.1t−CT.2t,BX1−BX2≤M⋅X1−X2,where M=2I/μα1/N+α2/N;α1/N+α2/N,then
(18)φX1−φX2≤V⋅X1−X2,where V=maxM,A<∞.Thus, it follows that the functionφ is uniformly Lipschitz continuous, and the restriction on Ht≥0,Pt≥0,Et≥0,Dt≥0,CTt≥0, and CSt≥0, we see that a solution of the system (7) exists [22].


## 2.2.1. Boundedness of the Solutions

Theorem 2.
The setΩ=H,P,E,D,CT,CS∈ℝ5/0≤H+P+E+D+CT+CS≤I/μ is positively invariant under system (7) with initial conditions H0≥0,P0≥0,E0≥0,D0≥0,CT0≥0, and CS0>0.Proof.
By adding the equations of system (7), we obtain
(13)dNdt=I−μN−δCS≤I−μN⇒Nt≤Iμ+N0e−μt,whereN0 represents the initial values of the total population.Thus,limt→∞supNt=I/μ. It implies that the region Ω is a positively invariant set for system (7). So, we only need to consider the dynamics of the system on the set Ω.


## 2.2.2. Existence of Solutions

Theorem 3.
The system (7) that satisfies a given initial condition H0,P0,E0,D0,CT0,andCS0 has a unique solution.Proof.
Let(14)X=StPtEtDtCTtCSt,φX=dHt/dtdPt/dtdEt/dtdDt/dtdCTt/dtdCSt/dt,so the system (7) can be rewritten in the following form:
(15)φX·=X=AX+BX,where(16)A=−μ+θ1+θ200000θ1−μ+β1+β30000θ20−μ+γ0000β1γ−μ+β2+η2000β30β2−μ+η10000η2η1−μ+δ,BX=I00−α1DtEtNα1DtEtN−α2CTtEtNα2CTtEtN.The second term on the right-hand side of (15) satisfies
(17)BX1−BX2≤MD1t−D2t+CT.1t−CT.2t,BX1−BX2≤M⋅X1−X2,where M=2I/μα1/N+α2/N;α1/N+α2/N,then
(18)φX1−φX2≤V⋅X1−X2,where V=maxM,A<∞.Thus, it follows that the functionφ is uniformly Lipschitz continuous, and the restriction on Ht≥0,Pt≥0,Et≥0,Dt≥0,CTt≥0, and CSt≥0, 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 controlsu1t, u2t, u3t, and u4t for t∈0,Tf. The first control we note u1 represented treatment that will work to treat diabetic patients with treatable complications. The second control u2 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 u3 represented treatment and education, by treating complications, with sensitivity to the negative effect of behavioral factors on diabetic patients with complications. The fourth control u4 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 t.So the controlled mathematical system is given by the following system of differential equations:(19)dHtdt=I−μ+θ1+θ2Ht,dPtdt=θ1Ht−μ+β1+β3Pt,dEtdt=θ2Ht−μEt−γ1−u4tEt,dDtdt=β1Pt+γ1−u4tEt−α11−u2tDtEtN−μ+β2+η2Dt+u1tCTt,dCTtdt=β3Pt+β2Dt+α11−u2tDtEtN−α21−u3tCTtEtN−μ+η1+u1tCTt,dCStdt=η2Dt+η1CTt+α21−u3tCTtEtN−μ+δCSt.The problem is to minimize the objective functional(20)Ju1,u2,u3,u4=CTTf−DTf−ETf+∫0TfCTt−Dt−Et+A2u12t+B2u22t+F2u32t+G2u42tdt,where A>0, B>0,F>0, and G>0 are the cost coefficients; they are selected to weigh the relative importance of u1t, u2t,u3t, and u4t at time t, and Tf is the final time.In other words, we seek the optimal controlsu1∗t, u2∗t,u3∗t, and u4∗t such that
(21)Ju1∗,u2∗,u3∗,u4∗=minu1,u2,u3,u4∈UJu1,u2,u3,u4,where U is the set of admissible controls defined by
(22)U=u1,u2,u3,u4/0≤u1min≤u1t≤u1max≤1,0≤u2min≤u2t≤u2max≤1,0≤u3min≤u3t≤u3max≤1,0≤u4min≤u4t≤u4max≤1/t∈0,Tf


## 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 u1∗,u2∗,u3∗,u4∗∈U4such that
(23)Ju1∗,u2∗,u3∗,u4∗=minu1,u2,u3,u4∈UJu1,u2,u3,u4.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)
Ju1,u2,u3,u4 is convex in U(iii)
The control spaceU=u1,u2,u3,u4/u1,u2,u3,u4ismeasurable,0≤u1min≤u1t≤u1max≤1,0≤u2min≤u2t≤u2max≤1,and0≤u3min≤u3t≤u3max≤1,0≤u4min≤u4t≤u4max≤1/t∈0,Tf 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 ofu,v, and w with coefficients depending on the time and state(v)
The integrand in the objective functional,CTt−Dt−Et+A/2u12t+B/2u22t+F/2u32t+G/2u42t, is clearly convex on U(vi)
It rests to show that there exist constantsζ1,ζ2,ζ3,ζ4,ζ5>0 and ζ such that CTt−Dt−Et+A/2u12t+B/2u22t+F/2u32t+G/2u42t satisfies
(24)CTt−Dt−Et+A2u12t+B2u22t+F2u32t+G2u42t≥−ζ1+ζ2u1ζ+ζ3u2ζ+ζ4u3ζ+ζ5u4ζThe state variables are being bounded; letζ1=supt∈0,TfCTt−Dt−Et,ζ2=A,ζ3=B,ζ4=F,ζ5=G and ζ=2; then, it follows that
(25)CTt−Dt−Et+A2u12t+B2u22t+F2u32t+G2u42t≥−ζ1+ζ2u1ζ+ζ3u2ζ+ζ4u3ζ+ζ5u4ζ.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 HamiltonianH^ at time t defined by
(26)H^t=CTt−Dt−Et+A2u12t+B2u22t+F2u32t+G2u42t+∑i=16λitfiH,P,E,D,CT,CS,where fi is the right side of the difference equation of the ith state variable.Theorem 5.
Given the optimal controlsu1∗,u2∗,u3∗,u4∗ and the solutions H∗,P∗,E∗,D∗,CT∗ , and CS∗ of the corresponding state system (18), there exists adjoint variables λ1,λ2, λ3, λ4,λ5, and λ6 satisfying
(27)λ1′=λ1μ+θ1+θ2−λ2θ1−λ3θ2,λ2′=λ2μ+β1+β3−λ4β1−λ5β3,λ3′=1+λ3μ+γ1−u4t−λ4γ1−u4t−α11−u2tDtN−λ5α11−u2tDtN−α21−u3tCTtN−λ6α21−u3tCTtN,λ4′=1−λ4α11−u2tEtN+μ+β2+η2−λ5β2+α11−u2tEtN−λ6η2,λ5′=−1−λ4u1t+λ5−α21−u3tEtN−μ+η1+u1t−λ6η1+α21−u3tEtN,λ6′=λ6μ+δWith the transversality conditions at timeTf, λ1Tf=0,λ2Tf=0,λ3Tf=1,λ4Tf=1,λ5Tf=−1, and λ6Tf=0.Furthermore, fort∈0,Tf, the optimal controls u1∗,u2∗,u3∗, and u4∗ are given by
(28)u1∗=min1,max0,λ5−λ4ACTt,u2∗=min1,max0,α1×λ5−λ4B×DtEtN,u3∗=min1,max0,α2×λ6−λ5F×CTtEtN,u4∗=min1,max0,λ4−λ3G×γEt,Proof.
The Hamiltonian is defined as follows:(29)H^t=CTt−Dt−Et+A2u12t+B2u22t+F2u32t+G2u42t+∑i=16λitfiH,P,E,D,CT,CS,wheref1H,P,E,D,CT,CS=I−μ+θ1+θ2Htf2H,P,E,D,CT,CS=θ1Ht−μ+β1+β3Pt,f3H,P,E,D,CT,CS=θ2Ht−μEt+γ1−u4tEt,f4H,P,E,D,CT,CS=β1Pt+γ1−u4tEt−α11−u2tDtEt/N−μ+β2+η2Dt+u1tCTt,f5H,P,E,D,CT,CS=β3Pt+β2Dt+α11−u2tDtEt/N−α21−u3tCTtEt/N−μ+η1CTt−u1tCTt,f6H,P,E,D,CT,CS=η2Dt+η1CTt+α21−u3tCTtEt/N−μ+δCSt.Fort∈0,Tf, the adjoint equations and transversality conditions can be obtained by using Pontryagin’s maximum principle [13, 25] such that
(30)λ1′=−∂H^∂H=λ1μ+θ1+θ2−λ2θ1−λ3θ2,λ2′=−∂H^∂P=λ2μ+β1+β3−λ4β1−λ5β3,(31)λ3′=−∂H^∂E=1+λ3μ+γ1−u4t−λ4γ1−u4t−α11−u2tDtN−λ5α11−u2tDtN−α21−u3tCTtN−λ6α21−u3tCTtN,(32)λ4′=−∂H^∂D=1−λ4α11−u2tEtN+μ+β2+η2−λ5β2+α11−u2tEtN−λ6η2,λ5′=−∂H^∂CT=−1−λ4u1t+λ5−α21−u3tEtN−μ+η1+u1t−λ6η1+α21−u3tEtN,λ6′=−∂H^∂CS=λ6μ+δ.For,t∈0,Tf the optimal controls u1∗,u2∗,u3∗, and u4∗ can be solved from the optimality condition,
(33)∂H^∂u1=0,∂H∂u2=0,∂H^∂u3=0,∂H^∂u4=0,that are
(34)−∂H^∂u1=−Au1t+λ2−λ4CTt=0,−∂H^∂u2=−Bu2t+α1λ5−λ4DtEtN=0,−∂H^∂u3=−Fu3t+α2λ6−λ5CTtEtN=0,−∂H^∂u4=−Gu4t+λ4−λ3γEt=0.We have(35)u1t=λ2−λ4ACTt,u2t=α1×λ5−λ4/B×DtEt/N,
(36)u3t=α2×λ6−λ5F×CTtEtN,u4t=λ4−λ3G×γEt.By the bounds inU of the controls, it is easy to obtain u1∗,u2∗,u3∗, and u4∗ and are given in (13–16) the form of system and in the form of system (18).


## 4.1. Existence of an Optimal Control

Theorem 4.
Consider the control problem with system (18). There exists an optimal control u1∗,u2∗,u3∗,u4∗∈U4such that
(23)Ju1∗,u2∗,u3∗,u4∗=minu1,u2,u3,u4∈UJu1,u2,u3,u4.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)
Ju1,u2,u3,u4 is convex in U(iii)
The control spaceU=u1,u2,u3,u4/u1,u2,u3,u4ismeasurable,0≤u1min≤u1t≤u1max≤1,0≤u2min≤u2t≤u2max≤1,and0≤u3min≤u3t≤u3max≤1,0≤u4min≤u4t≤u4max≤1/t∈0,Tf 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 ofu,v, and w with coefficients depending on the time and state(v)
The integrand in the objective functional,CTt−Dt−Et+A/2u12t+B/2u22t+F/2u32t+G/2u42t, is clearly convex on U(vi)
It rests to show that there exist constantsζ1,ζ2,ζ3,ζ4,ζ5>0 and ζ such that CTt−Dt−Et+A/2u12t+B/2u22t+F/2u32t+G/2u42t satisfies
(24)CTt−Dt−Et+A2u12t+B2u22t+F2u32t+G2u42t≥−ζ1+ζ2u1ζ+ζ3u2ζ+ζ4u3ζ+ζ5u4ζThe state variables are being bounded; letζ1=supt∈0,TfCTt−Dt−Et,ζ2=A,ζ3=B,ζ4=F,ζ5=G and ζ=2; then, it follows that
(25)CTt−Dt−Et+A2u12t+B2u22t+F2u32t+G2u42t≥−ζ1+ζ2u1ζ+ζ3u2ζ+ζ4u3ζ+ζ5u4ζ.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 HamiltonianH^ at time t defined by
(26)H^t=CTt−Dt−Et+A2u12t+B2u22t+F2u32t+G2u42t+∑i=16λitfiH,P,E,D,CT,CS,where fi is the right side of the difference equation of the ith state variable.Theorem 5.
Given the optimal controlsu1∗,u2∗,u3∗,u4∗ and the solutions H∗,P∗,E∗,D∗,CT∗ , and CS∗ of the corresponding state system (18), there exists adjoint variables λ1,λ2, λ3, λ4,λ5, and λ6 satisfying
(27)λ1′=λ1μ+θ1+θ2−λ2θ1−λ3θ2,λ2′=λ2μ+β1+β3−λ4β1−λ5β3,λ3′=1+λ3μ+γ1−u4t−λ4γ1−u4t−α11−u2tDtN−λ5α11−u2tDtN−α21−u3tCTtN−λ6α21−u3tCTtN,λ4′=1−λ4α11−u2tEtN+μ+β2+η2−λ5β2+α11−u2tEtN−λ6η2,λ5′=−1−λ4u1t+λ5−α21−u3tEtN−μ+η1+u1t−λ6η1+α21−u3tEtN,λ6′=λ6μ+δWith the transversality conditions at timeTf, λ1Tf=0,λ2Tf=0,λ3Tf=1,λ4Tf=1,λ5Tf=−1, and λ6Tf=0.Furthermore, fort∈0,Tf, the optimal controls u1∗,u2∗,u3∗, and u4∗ are given by
(28)u1∗=min1,max0,λ5−λ4ACTt,u2∗=min1,max0,α1×λ5−λ4B×DtEtN,u3∗=min1,max0,α2×λ6−λ5F×CTtEtN,u4∗=min1,max0,λ4−λ3G×γEt,Proof.
The Hamiltonian is defined as follows:(29)H^t=CTt−Dt−Et+A2u12t+B2u22t+F2u32t+G2u42t+∑i=16λitfiH,P,E,D,CT,CS,wheref1H,P,E,D,CT,CS=I−μ+θ1+θ2Htf2H,P,E,D,CT,CS=θ1Ht−μ+β1+β3Pt,f3H,P,E,D,CT,CS=θ2Ht−μEt+γ1−u4tEt,f4H,P,E,D,CT,CS=β1Pt+γ1−u4tEt−α11−u2tDtEt/N−μ+β2+η2Dt+u1tCTt,f5H,P,E,D,CT,CS=β3Pt+β2Dt+α11−u2tDtEt/N−α21−u3tCTtEt/N−μ+η1CTt−u1tCTt,f6H,P,E,D,CT,CS=η2Dt+η1CTt+α21−u3tCTtEt/N−μ+δCSt.Fort∈0,Tf, the adjoint equations and transversality conditions can be obtained by using Pontryagin’s maximum principle [13, 25] such that
(30)λ1′=−∂H^∂H=λ1μ+θ1+θ2−λ2θ1−λ3θ2,λ2′=−∂H^∂P=λ2μ+β1+β3−λ4β1−λ5β3,(31)λ3′=−∂H^∂E=1+λ3μ+γ1−u4t−λ4γ1−u4t−α11−u2tDtN−λ5α11−u2tDtN−α21−u3tCTtN−λ6α21−u3tCTtN,(32)λ4′=−∂H^∂D=1−λ4α11−u2tEtN+μ+β2+η2−λ5β2+α11−u2tEtN−λ6η2,λ5′=−∂H^∂CT=−1−λ4u1t+λ5−α21−u3tEtN−μ+η1+u1t−λ6η1+α21−u3tEtN,λ6′=−∂H^∂CS=λ6μ+δ.For,t∈0,Tf the optimal controls u1∗,u2∗,u3∗, and u4∗ can be solved from the optimality condition,
(33)∂H^∂u1=0,∂H∂u2=0,∂H^∂u3=0,∂H^∂u4=0,that are
(34)−∂H^∂u1=−Au1t+λ2−λ4CTt=0,−∂H^∂u2=−Bu2t+α1λ5−λ4DtEtN=0,−∂H^∂u3=−Fu3t+α2λ6−λ5CTtEtN=0,−∂H^∂u4=−Gu4t+λ4−λ3γEt=0.We have(35)u1t=λ2−λ4ACTt,u2t=α1×λ5−λ4/B×DtEt/N,
(36)u3t=α2×λ6−λ5F×CTtEtN,u4t=λ4−λ3G×γEt.By the bounds inU of the controls, it is easy to obtain u1∗,u2∗,u3∗, and u4∗ 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 i=0 and i=Tf. 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:A=10000, B=10000,F=10000, and G=10000 and with the intial value of H0=14000000,P0=6660000,E0=13000000,D0=6200000,CT0=4500000, and CS0=2000000 (Figure 2).Figure 2
The evolution of the number of diabetics with and without complications without controls.After the parameter values (Tables1 and 2), we noted that diabetics without complications after 120 months decreased from 6.2×106 to 3.05×106 (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 4.5×106 to 1.49×107 (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.Table 1
Parameter values used in numerical simulation.
ParamaterValue in mth-1Descriptionμ0 : 02Natural mortalityδ0 : 001Mortality rate due to complicationsβ10 : 2The probability of developing diabetesβ20.08The probability of a diabetic person developing a complicationβ30.01The probability of developing diabetes at stage of complicationsa10 : 4The rate of negative impact on diabetics without complicationsY0.06Rate of patients become diabetic without complications through lifestyleI2000000Denote the incidence of healthy peoplea20 : 6The rate of negative impact on diabetics with treatable complicationsθ10 : 1Rate of prediabetic people through genetic factorθ10 : 2Rate of prediabetic people through lifestyle factorη10 : 6The probability of a diabetic person developing a serious complicationη20 : 3Rate of diabetics whose serious complications are because of a sudden shockTable 2
Evolution of number of diabetics without control after 120 days.
Population without control after 120 daysWithout controlDiabetics without complications3:05×106Diabetics with treatable complications1:49×107Diabetics with serious complications1×108We noted that diabetics with serious complications are increasing and that the number of the transition becomes from2×106 to 1×108 (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 stepsi=0 and i=T. 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 controlsu1t and u2t 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 1.49×107 to 1.26×107 by the end of the strategy (Figure 3).Figure 3
The evolution of the number of diabetics with and without complications without controls.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 variablesu1t and u2t (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.Table 3
Evolution of the number of diabetics with two controlsu1t and u2t after 120 days.
Population without control after 120 daysWithout controlWith two controlsu1t and u2tDiabetics without complications1:26×1061.78×107Diabetics with treatable complications1072.14×107


### 5.2. Strategy C: Control with Awareness Program,Treatment, and Psychological Support with Follow-Up

We combined three optimal controlsu1t,u2t, and u3t.In this strategy, the three optimal controlsu1t,u2t, and u3t are activated at the same time, in order to reduced the number of diabetics with treatable complications to diabetics without complications (Figure 4).Figure 4
The evolution of the number of diabetics with treatable and serious complications with three controlsu1t,u2t, and u3t.In this strategy (Figure4 and Table 4), we used three controls optimal u1t,u2t, and u3t. 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 1.5×107 to 1.28×107, and also, the number of diabetics with serious complications is decreasing from 108 to 7.32×107.Table 4
Evolution of number of diabetics with three controlsu1t,u2t, and u3t after 120 days.
Population without control after 120 daysWithout controlWith three controlsu1t,u2t, and u3tDiabetics with treatable complications1:5×1071.28×107Diabetics with serious complications1087:32×107


### 5.3. Strategy D: Prevention and ProtectionE from Diabetes

We we use only the optimal controlu4t.In this strategy, we focus the effort of the awareness campaign to reduce the negative impact of behavioral factors (Figure5).Figure 5
The evolution of the number of diabetics without complications with controlu4t.In this strategy, we used controlu4t (Figure 5 and Table 5); the objective of this control u4t 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 1.29×106 to 106.Remark 6.
We could also merge multiple assemblies asu1t,u2t,u3t, and u1t,u2t,u3t,u4t and thus get a variety of results.Table 5
Evolution of number of diabetics with controlu4t after 120 days.
Population without control after 120 daysWithout controlWith controlu4tDiabetics without complications1:29×106106Prediabetics due to effect of behavioral factors1.72×1071.86×107


## 5.1. Strategy A

In this strategy, we applied two controlsu1t and u2t 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 1.49×107 to 1.26×107 by the end of the strategy (Figure 3).Figure 3
The evolution of the number of diabetics with and without complications without controls.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 variablesu1t and u2t (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.Table 3
Evolution of the number of diabetics with two controlsu1t and u2t after 120 days.
Population without control after 120 daysWithout controlWith two controlsu1t and u2tDiabetics without complications1:26×1061.78×107Diabetics with treatable complications1072.14×107


## 5.2. Strategy C: Control with Awareness Program,Treatment, and Psychological Support with Follow-Up

We combined three optimal controlsu1t,u2t, and u3t.In this strategy, the three optimal controlsu1t,u2t, and u3t are activated at the same time, in order to reduced the number of diabetics with treatable complications to diabetics without complications (Figure 4).Figure 4
The evolution of the number of diabetics with treatable and serious complications with three controlsu1t,u2t, and u3t.In this strategy (Figure4 and Table 4), we used three controls optimal u1t,u2t, and u3t. 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 1.5×107 to 1.28×107, and also, the number of diabetics with serious complications is decreasing from 108 to 7.32×107.Table 4
Evolution of number of diabetics with three controlsu1t,u2t, and u3t after 120 days.
Population without control after 120 daysWithout controlWith three controlsu1t,u2t, and u3tDiabetics with treatable complications1:5×1071.28×107Diabetics with serious complications1087:32×107


## 5.3. Strategy D: Prevention and ProtectionE from Diabetes

We we use only the optimal controlu4t.In this strategy, we focus the effort of the awareness campaign to reduce the negative impact of behavioral factors (Figure5).Figure 5
The evolution of the number of diabetics without complications with controlu4t.In this strategy, we used controlu4t (Figure 5 and Table 5); the objective of this control u4t 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 1.29×106 to 106.Remark 6.
We could also merge multiple assemblies asu1t,u2t,u3t, and u1t,u2t,u3t,u4t and thus get a variety of results.Table 5
Evolution of number of diabetics with controlu4t after 120 days.
Population without control after 120 daysWithout controlWith controlu4tDiabetics without complications1:29×106106Prediabetics due to effect of behavioral factors1.72×1071.86×107


## 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.



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