Research ArticleA New Mathematical Modeling with Optimal ControlStrategy for the Dynamics of Population of Diabetics and ItsComplications with Effect of Behavioral Factors

Abdelfatah Kouidere ,1 Abderrahim Labzai ,1 Hanane Ferjouchia,1 Omar Balatif ,2

and Mostafa Rachik 1

1Laboratory of Analysis Modeling and Simulation, Department of Mathematics and Computer Science, Faculty of Sciences BenM’Sik,Hassan II University, Sidi Othman, Casablanca, Morocco2Laboratory of Dynamical Systems, Mathematical Engineering Team (INMA), Department of Mathematics, Faculty of SciencesEl Jadida, Chouaib Doukkali University, El Jadida, Morocco

Correspondence should be addressed to Abdelfatah Kouidere; kouidere89@gmail.com

Received 3 March 2020; Revised 23 April 2020; Accepted 16 May 2020; Published 4 June 2020

Academic Editor: Bruno Carpentieri

Copyright © 2020 Abdelfatah Kouidere et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

We propose an optimal control strategy by conducting awareness campaigns for diabetics about the severity of complicationsof diabetes and the negative impact of an unbalanced lifestyle and the surrounding environment, as well as treatment andpsychological follow-up. Pontryagin’s maximum principle is used to characterize the optimal controls, and the optimalitysystem is solved by an iterative method. Finally, some numerical simulations are performed to verify the theoretical analysisusing MATLAB.

1. Introduction

Nowadays, diabetes is a chronic disease with a huge burdenaffecting individuals. According to theWorld Health Organi-sation (WHO) [1], diabetes is a disorder characterized by thepresence of problems in the insulin hormone, which natu-rally results from the pancreas to help the body use glucoseand fat and store some of them. According to the AmericanDiabetes Association (ADA) [2], diabetes mellitus is a groupof metabolic diseases characterized by hyperglycemia result-ing from defects in insulin secretion, insulin action, or both.It is known that the proper level of glucose in the blood afterfasting eight hours should be less than 108mg/dl, while theborderline is 126mg/dl. If a person’s blood glucose level is126mg/dL and above, in two or more tests, then that personis diagnosed with diabetes. Diabetes is divided into severaldifferent types; some more prevalent than others. The mostcommon type of diabetes in the general population is type2 diabetes, and type 1 diabetes is more common in children,

and gestational diabetes is a form of diabetes that can occurduring pregnancy. According to the latest statistics fromthe International Diabetes Federation (IDF) and as reportedin the 9th edition of the Atlas Diabetes 2017 [3], diabetes isa constantly growing disease. There are more than 370 mil-lion people with diabetes worldwide (8.5% of the adult pop-ulation) and about 463 million people in prediabetes (6.5%of the adult population), and more than 625 million areexpected to be affected by to 2045.

Today, all countries of the world suffer from the highnumber of people with diabetes, which is increasing andexpanding on the extreme level. When it is not treated well,all types of diabetes can lead to complications in many partsof the body, leading to an early death. When a diabetic knowshow to control the level of glucose in the blood, this aware-ness plays a key role in reducing the serious complicationsof diabetes. According to IDF statistics, diabetes has seriousand varied complications. For example, the risk of cardiovas-cular disease. Moreover, more than a third of diabetics have

HindawiJournal of Applied MathematicsVolume 2020, Article ID 1943410, 12 pageshttps://doi.org/10.1155/2020/1943410

retinopathy which is the main cause of vision loss, in additionto the risk of kidney disease. In addition, the complications ofdiabetes are multiple and different depending on the degreeof severity; there are complications that can be treated andothers that have reached critical stages with which treatmentis not beneficial. According to ADA ½20�, diabetes has eco-nomic and social burdens on the individual and society.

During the last decade, large mathematical models on dia-betes have been developed to simulate, analyse, and under-stand the dynamics of a population of diabetics. In a relatedresearch work, Boutayeb and Chetouani [4] and Derouichet al. [5] introduced a mathematical model for the dynamicsof the population of diabetes. And Kouidere et al. [6] pro-posed a discrete mathematical model highlighting the impactof living environment. Also, many researches have focusedon this topic and other related topics ([7–12]).

As we said earlier, diabetes has many complications, andthe nature of these complications are two types: treatableones and those that have reached a critical stage which isimpossible to cure completely.

According to IDF [3], diabetes is influenced by a complexinteraction of behavioural, genetic, and socioeconomic fac-tors; many of which are outside our individual control.

To achieve this objective, we considered a compartmentmodel that describes the dynamics of a population of diabeticsthat is divided into six compartments, i.e., the healthy peopleðHÞ, the prediabetics through genetic factors ðPÞ, the predia-betics through the negative effect of behavioral factors on dia-betics patients and others ðEÞ, and ðDÞ diabetics withoutcomplications are those who control blood sugar through dietand exercise before it is too late, and we divided diabetics withcomplications into two parts: CT diabetics with treatablecomplication and CS diabetics with serious complications.

We noticed that most of researchers about diabetes andits complications focused on continuous and discrete timemodels and described by differential equations. Recently, moreand 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 HPEDCTCSmathematical model that describes the dynamic of a popula-

tion of diabetics. In Section 3, we presented an optimal con-trol problem for the proposed model, where we gave someresults concerning the existence and positivity of the opti-mal control and we characterized the optimal controls usedPontryagin’s maximum principle. Numerical simulationsthrough MATLAB are given in Section 4. Finally, we con-clude the paper in Section 5.

2. A Mathematical Model

We consider a mathematical modelHPEDCTCS that describesthe dynamics of a population of diabetics. We divide the pop-ulation denoted by N into six compartments.

2.1. Description of the Model. The graphical representation ofthe proposed model is shown in Figure 1.

The compartment ðHÞ are healthy people; the compart-ment H is increased by I (which is the recruitment rate ofhealthy people), and this compartment H is decreased by θ1HðtÞ (the number of prediabetic people through genetic fac-tor) and also decreased by θ2HðtÞ (the number of predia-betics through the negative effect of behavioral factors) anddecreased by the amount μ (natural mortality).

dH tð Þdt

= I − μ + θ1 + θ2ð ÞH tð Þ: ð1Þ

The compartment ðPÞ are people who are likely tohave diabetes through genetic factors. The compartmentP is increased by θ1HðtÞ. This compartment P is decreasedby the amount μ (natural mortality) and also decreasedby β1PðtÞ (the probability of developing diabetes) and byβ3CTðtÞ (the probability of developing diabetes at stage ofcomplications).

dP tð Þdt

= θ1H tð Þ − μ + β1 + β3ð ÞP tð Þ: ð2Þ

I

P D

E

H

𝛽3

𝛽1

𝜃1

𝛽2

𝜂2

𝜂1𝛿

𝜇

𝜇

𝜇

𝜇𝜇

𝜇

𝜃2

𝛾E

DEN

𝛼1

DTE

N𝛼2

CS

CT

Figure 1: The dynamics of a population of diabetics.

2 Journal of Applied Mathematics

Compartment ðEÞ are people who are likely to have dia-betes through the negative effect of lifestyle or psychologicalproblem factors and others (these are people at risk of devel-oping diabetes, such as those who are obese, overweight, ges-tational diabetes, or due to family and work problems, inaddition to the person without diabetes).The compartmentE increased by θ2HðtÞ and decreased by γEðtÞ (patientswho become diabetics without complications because of thenegative effect of lifestyle) and also decreased by μ (naturalmortality).

dE tð Þdt

= θ2H tð Þ − μ + γð ÞE tð Þ: ð3Þ

Compartment ðDÞ is the number of diabetics withoutcomplications. The compartment DðtÞ increased by theamount β1PðtÞ and by the amount γEðtÞ. This compartmentD is decreasing by μ (natural mortality) and α1 (the rate ofnegative impact of ordinary people on diabetics withoutcomplications through improper nutrition, or practical orfamily social problems) and also decreased by β2DðtÞ (theprobability of a diabetic person developing a complication)and also decreased by η2DðtÞ(the number of diabetics whoseserious complications because of a sudden shock).

dD tð Þdt

= β1P tð Þ + γE tð Þ − α1D tð ÞE tð Þ

N− μ + β2 + η2ð ÞD tð Þ:

ð4Þ

Compartment ðCTÞ is the number of diabetics withtreatable complications. The compartment CT increased byβ3PðtÞ and byβ2DðtÞ and also increased by α1 and anddecreased by α2 (the rate negative impact of ordinary peopleon diabetics with treatable complications through impropernutrition, or practical or family social problems) anddecreased by η1CTðtÞ (the number of people developing ofdiabetics with complications can be treated to serious com-plications) and also by μ (natural mortality)

dCT tð Þdt

= β3P tð Þ + β2D tð Þ + α1D tð ÞE tð Þ

N

− α2CT tð ÞE tð Þ

N− μ + η1ð ÞCT tð Þ:

ð5Þ

Compartment ðCSÞ is the number of diabetics with seri-ous complications, they are the diabetics who have seriouscomplications such as retinopathy or renal failure. The com-partment CS =CSðtÞ increased by η2DðtÞ and also by η1CTðtÞand by α2: This compartment CS decreased by δ (mortalityrate due to complications) and also by μ (natural mortality).

dCS tð Þdt

= η2D tð Þ + η1CT tð Þ + α2CT tð ÞE tð Þ

N− μ + δð ÞCS tð Þ:

ð6Þ

Hence, we present the diabetic model by the followingsystem of differential equations:

with Hð0Þ ≥ 0, Pð0Þ ≥ 0, Eð0Þ ≥ 0, Dð0Þ ≥ 0, CTð0Þ ≥ 0, andCSð0Þ ≥ 0:

2.2. Positivity of Solutions

Theorem 1. If Hð0Þ ≥ 0, Pð0Þ ≥ 0, Eð0Þ ≥ 0,Dð0Þ ≥ 0, CTð0Þ≥ 0, and CSð0Þ > 0, the solutions HðtÞ, PðtÞ, EðtÞ, DðtÞ,CTðtÞ, and CSðtÞ of system (7) are positive for all t ≥ 0.

Proof. It follows from the first equation of system (7) that

dH tð Þdt

= I − μ + θ1 + θ2ð ÞH tð Þ ≥ − μ + θ1 + θ2ð ÞH tð ÞdH tð Þdt

+ μ + θ1 + θ2ð ÞH tð Þ ≥ 0ð8Þ

dH tð Þdt

= I − μ + θ1 + θ2ð ÞH tð Þ,dP tð Þdt

= θ1H tð Þ − μ + β1 + β3ð ÞP tð Þ,dE tð Þdt

= θ2H tð Þ − μ + γð ÞE tð Þ,dD tð Þdt

= β1P tð Þ + γE tð Þ − α1D tð ÞE tð Þ

N− μ + β2 + η2ð ÞD tð Þ,

dCT tð Þdt

= β3P tð Þ + β2D tð Þ + α1D tð ÞE tð Þ

N− α2

CT tð ÞE tð ÞN

− μ + η1ð ÞCT tð Þ,dCS tð Þdt

= η2D tð Þ + η1CT tð Þ + α2CT tð ÞE tð Þ

N− μ + δð ÞCS tð Þ,

8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:

ð7Þ

3Journal of Applied Mathematics

Both sides in the last inequality are multiplied with expððμ + θ1 + θ2ÞtÞ:

We obtain

exp μ + θ1 + θ2ð Þtð Þ · dH tð Þdt

+ μ + θ1 + θ2ð Þ exp μ + θ1 + θ2ð Þtð Þ ·H tð Þ ≥ 0,ð9Þ

then

ddt

exp μ + θ1 + θ2ð Þtð Þ ⋅H tð Þð Þ ≥ 0: ð10Þ

Integrating this inequality from 0 to t gives

ðt0

dds

exp μ + θ1 + θ2ð Þsð Þ ⋅H sð Þð Þds ≥ 0, ð11Þ

then

H tð Þ ≥H 0ð Þ exp μ + θ1 + θ2ð Þtð Þ⇒H tð Þ > 0: ð12Þ

Similarly, we prove that PðtÞ ≥ 0, EðtÞ ≥ 0,DðtÞ ≥ 0,CTðtÞ ≥ 0, and CSðtÞ ≥ 0:

2.2.1. Boundedness of the Solutions

Theorem 2. The set Ω = fðH, P, E,D, CT , CSÞ ∈ℝ5/0 ≤H +P + E +D + CT + CS ≤ I/μg is positively invariant under sys-tem (7) with initial conditions Hð0Þ ≥ 0, Pð0Þ ≥ 0, Eð0Þ ≥ 0,Dð0Þ ≥ 0, CTð0Þ ≥ 0, and CSð0Þ > 0.

Proof. By adding the equations of system (7), we obtain

dNdt

= I − μN − δCS ≤ I − μN ⇒N tð Þ ≤ Iμ+N 0ð Þe−μt , ð13Þ

where Nð0Þ represents the initial values of the totalpopulation.

Thus, limt→∞

sup NðtÞ = ðI/μÞ. It implies that the region Ω

is a positively invariant set for system (7). So, we only needto 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 condi-tion ðHð0Þ, Pð0Þ, Eð0Þ,Dð0Þ, CTð0Þ, and CSð0ÞÞ has a uniquesolution.

Proof. Let

X =

S tð ÞP tð ÞE tð ÞD tð ÞCT tð ÞCS tð Þ

0BBBBBBBBBBB@

1CCCCCCCCCCCA

,

φ Xð Þ =

dH tð Þ/dtdP tð Þ/dtdE tð Þ/dtdD tð Þ/dtdCT tð Þ/dtdCS tð Þ/dt

0BBBBBBBBBBB@

1CCCCCCCCCCCA

,

ð14Þ

so the system (7) can be rewritten in the following form:

φ X·� �

= X = AX + B Xð Þ, ð15Þ

where

A =

− μ + θ1 + θ2ð Þ 0 0 0 0 0θ1 − μ + β1 + β3ð Þ 0 0 0 0θ2 0 −μ + γ 0 0 00 β1 γ − μ + β2 + η2ð Þ 0 00 β3 0 β2 − μ + η1ð Þ 00 0 0 η2 η1 − μ + δð Þ

0BBBBBBBBBBB@

1CCCCCCCCCCCA

,

B Xð Þ =

I

00

−α1D tð ÞE tð Þ

N

α1D tð ÞE tð Þ

N− α2

CT tð ÞE tð ÞN

α2CT tð ÞE tð Þ

N

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCA

:

ð16Þ

4 Journal of Applied Mathematics

The second term on the right-hand side of (15) satisfies

B X1ð Þ − B X2ð Þj j ≤M D1 tð Þ −D2 tð Þj j + CT:1 tð Þ −CT:2 tð Þj jð Þ,B X1ð Þ − B X2ð Þj j ≤M ⋅ X1 − X2k k,

ð17Þ

where M = 2ðI/μÞðjα1/Nj + jα2/Nj ; jα1/Nj + jα2/NjÞ,then

φ X1ð Þ − φ X2ð Þk k ≤V ⋅ X1 − X2k k, ð18Þ

where V =max ðM, kAkÞ <∞:Thus, it follows that the function φ is uniformly Lipschitz

continuous, and the restriction on HðtÞ ≥ 0, PðtÞ ≥ 0, EðtÞ ≥0,DðtÞ ≥ 0,CTðtÞ ≥ 0, and CSðtÞ ≥ 0, we see that a solutionof the system (7) exists [22].

3. Formulation of the Model

Our objective in this proposed strategy of control is to mini-mize the number of people evolving from the stage of predi-abetes to the stages of diabetes without complications and todiabetes with treatable complications. In this model, weinclude four controls u1ðtÞ, u2ðtÞ, u3ðtÞ, and u4ðtÞ for t ∈ ½0,T f �. The first control we note u1 represented treatment thatwill work to treat diabetic patients with treatable complica-tions. The second control u2 represented the awareness pro-gram throughmedia and education, by raising awareness andawareness of the seriousness of the negative impact of behav-ioral factors on diabetes. The third control u3 representedtreatment and education, by treating complications, withsensitivity to the negative effect of behavioral factors on dia-betic patients with complications. The fourth control u4 rep-resented the awareness program, through raising awarenesson 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 fol-lowing system of differential equations:

The problem is to minimize the objective functional

J u1, u2, u3, u4ð Þ = CT T f

� �−D Tf

� �− E T f

� �

+ðT f

0CT tð Þ −D tð Þ − E tð Þ + A

2 u21 tð Þ

+ B2 u

22 tð Þ + F

2 u23 tð Þ + G

2 u24 tð Þ

�dt,

ð20Þ

where A > 0, B > 0, F > 0, and G > 0 are the cost coefficients;they are selected to weigh the relative importance of u1ðtÞ,u2ðtÞ, u3ðtÞ, and u4ðtÞ at time t, and T f is the final time.

In other words, we seek the optimal controls u∗1 ðtÞ,u∗2 ðtÞ, u∗3 ðtÞ, and u∗4 ðtÞ such that

J u∗1 , u∗2 , u∗3 , u∗4ð Þ = minu1,u2,u3,u4∈U

J u1, u2, u3, u4ð Þ, ð21Þ

where U is the set of admissible controls defined by

U = u1, u2, u3, u4ð Þ/0 ≤ u1 min ≤ u1 tð Þ ≤ u1 maxf≤ 1, 0 ≤ u2 min ≤ u2 tð Þ ≤ u2 max ≤ 1, 0 ≤ u3 min≤ u3 tð Þ ≤ u3 max ≤ 1, 0 ≤ u4 min ≤ u4 tð Þ ≤ u4 max≤ 1/t ∈ 0, T f �g

�ð22Þ

4. The Optimal Control: Existenceand 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 ðu∗1 , u∗2 , u∗3 , u∗4 Þ ∈U4such that

J u∗1 , u∗2 , u∗3 , u∗4ð Þ = minu1 ,u2 ,u3 ,u4∈U

J u1, u2, u3, u4ð Þ: ð23Þ

dH tð Þdt

= I − μ + θ1 + θ2ð ÞH tð Þ,dP tð Þdt

= θ1H tð Þ − μ + β1 + β3ð ÞP tð Þ,dE tð Þdt

= θ2H tð Þ − μE tð Þ − γ 1 − u4 tð Þð ÞE tð Þ,dD tð Þdt

= β1P tð Þ + γ 1 − u4 tð Þð ÞE tð Þ − α1 1 − u2 tð Þð ÞD tð ÞE tð ÞN

− μ + β2 + η2ð ÞD tð Þ + u1 tð ÞCT tð Þ,dCT tð Þdt

= β3P tð Þ + β2D tð Þ + α1 1 − u2 tð Þð ÞD tð ÞE tð ÞN

− α2 1 − u3 tð Þð ÞCT tð ÞE tð ÞN

− μ + η1 + u1 tð Þð ÞCT tð Þ,dCS tð Þdt

= η2D tð Þ + η1CT tð Þ + α2 1 − u3 tð Þð ÞCT tð ÞE tð ÞN

− μ + δð ÞCS tð Þ:

8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:

ð19Þ

5Journal of Applied Mathematics

Proof. The existence of the optimal control can be obtainedusing a result by Fleming and Rishel [23], checking the fol-lowing steps:

(i) It follows that the set of controls and correspondingstate variables is not empty. we will use a simplifiedversion of an existence result ([24], Theorem 7.1.1)

(ii) Jðu1, u2, u3, u4Þ is convex in U

(iii) The control space U = fðu1, u2, u3, u4/u1, u2, u3, u4Þis measurable, 0 ≤ u1 min ≤ u1ðtÞ ≤ u1 max ≤ 1, 0 ≤u2 min ≤ u2ðtÞ ≤ u2 max ≤ 1, and 0 ≤ u3 min ≤ u3ðtÞ ≤u3 max ≤ 1, 0 ≤ u4 min ≤ u4ðtÞ ≤ u4 max ≤ 1/t ∈ ½0, T f �gis convex and closed by definition

(iv) All the right-hand sides of equations of system arecontinuous, bounded above by a sum of boundedcontrol and state, and can be written as a linear func-tion of u, v, andwwith coefficients depending on thetime and state

(v) The integrand in the objective functional, CTðtÞ −DðtÞ − EðtÞ + ðA/2Þu21ðtÞ + ðB/2Þu22ðtÞ + ðF/2Þu23ðtÞ+ ðG/2Þu24ðtÞ, is clearly convex on U

(vi) It rests to show that there exist constants ζ1, ζ2,ζ3, ζ4, ζ5 > 0 and ζ such that CTðtÞ −DðtÞ − EðtÞ +ðA/2Þu21ðtÞ + ðB/2Þu22ðtÞ + ðF/2Þu23ðtÞ + ðG/2Þu24ðtÞsatisfies

CT tð Þ −D tð Þ − E tð Þ + A2 u

21 tð Þ + B

2 u22 tð Þ + F

2 u23 tð Þ + G

2 u24 tð Þ

≥ −ζ1 + ζ2 u1j jζ + ζ3 u2j jζ + ζ4 u3j jζ + ζ5 u4j jζð24Þ

The state variables are being bounded; let ζ1 = supt∈½0,T f �

ðCTðtÞ

−DðtÞ − EðtÞÞ, ζ2 = A, ζ3 = B, ζ4 = F, ζ5 =G and ζ = 2; then,it follows that

CT tð Þ −D tð Þ − E tð Þ + A2 u

21 tð Þ + B

2 u22 tð Þ + F

2 u23 tð Þ + G

2 u24 tð Þ

≥ −ζ1 + ζ2 u1j jζ + ζ3 u2j jζ + ζ4 u3j jζ + ζ5 u4j jζ:ð25Þ

Then, from Fleming and Rishel [23], we conclude thatthere exists an optimal control.

4.2. Characterization of the Optimal Control. In order toderive the necessary conditions for the optimal control, weapply Pontryagin’s maximum principle to the HamiltonianH at time t defined by

H tð Þ = CT tð Þ −D tð Þ − E tð Þ + A2 u

21 tð Þ + B

2 u22 tð Þ + F

2 u23 tð Þ

+ G2 u

24 tð Þ + 〠

6

i=1λi tð Þf i H, P, E,D, CT , CSð Þ,

ð26Þ

where f i is the right side of the difference equation of the ith

state variable:

Theorem 5. Given the optimal controls ðu∗1 , u∗2 , u∗3 , u∗4 Þ andthe solutionsH∗, P∗, E∗,D∗, C∗

T , and C∗S of the corresponding

state system (18), there exists adjoint variables λ1, λ2, λ3, λ4,λ5, and λ6 satisfying

λ1′ = λ1 μ + θ1 + θ2ð Þ − λ2θ1 − λ3θ2,

λ2′ = λ2 μ + β1 + β3ð Þ − λ4β1 − λ5β3,

λ3′ = 1 + λ3 μ + γ 1 − u4 tð Þð Þ½ �− λ4 γ 1 − u4 tð Þð Þ − α1 1 − u2 tð Þð ÞD tð Þ

N

� �

− λ5 α1 1 − u2 tð Þð ÞD tð ÞN

− α2 1 − u3 tð Þð ÞCT tð ÞN

� �

− λ6 α2 1 − u3 tð Þð ÞCT tð ÞN

� �,

λ4′ = 1 − λ4

�α1 1 − u2 tð Þð ÞE tð Þ

N+ μ + β2 + η2ð Þ

�

− λ5 β2 + α1 1 − u2 tð Þð ÞE tð ÞN

� �− λ6η2,

λ5′ = −1 − λ4u1 tð Þ + λ5

�−α2 1 − u3 tð Þð Þ E tð Þ

N− μ + η1ð

+ u1 tð ÞÞ�− λ6 η1 + α2 1 − u3 tð Þð Þ E tð Þ

N

� �,

λ6′ = λ6 μ + δð Þ

ð27Þ

With the transversality conditions at time T f , λ1ðT f Þ= 0, λ2ðT f Þ = 0, λ3ðT f Þ = 1, λ4ðT f Þ = 1, λ5ðT f Þ = −1, and λ6ðT f Þ = 0:

Furthermore, for t ∈ ½0, T f �, the optimal controls u∗1 , u∗2 ,u∗3 , and u∗4 are given by

u∗1 = min 1, max 0, λ5 − λ4ð ÞA

CT tð Þ

,

u∗2 = min 1, max 0, α1 ×λ5 − λ4ð Þ

B× D tð ÞE tð Þ

N

,

u∗3 = min 1, max 0, α2 ×λ6 − λ5ð Þ

F× CT tð ÞE tð Þ

N

,

u∗4 = min 1, max 0, λ4 − λ3ð ÞG

× γE tð Þ

,

ð28Þ

6 Journal of Applied Mathematics

Proof. The Hamiltonian is defined as follows:

H tð Þ = CT tð Þ −D tð Þ − E tð Þ + A2 u

21 tð Þ + B

2 u22 tð Þ + F

2 u23 tð Þ

+ G2 u

24 tð Þ + 〠

6

i=1λi tð Þf i H, P, E,D, CT , CSð Þ,

ð29Þ

where

f1ðH, P, E,D, CT , CSÞ = I − ðμ + θ1 + θ2ÞHðtÞf2ðH, P, E,D, CT , CSÞ = θ1HðtÞ − ðμ + β1 + β3ÞPðtÞ,f3ðH, P, E, D, CT , CSÞ = θ2HðtÞ − μEðtÞ + γð1 − u4ðtÞÞ

EðtÞ,f4ðH, P, E,D, CT , CSÞ = β1PðtÞ + γð1 − u4ðtÞÞEðtÞ − α1ð1

− u2ðtÞÞðDðtÞEðtÞ/NÞ − ðμ + β2 + η2ÞDðtÞ + u1ðtÞCTðtÞ,f5ðH, P, E,D, CT , CSÞ = β3PðtÞ + β2DðtÞ + α1ð1 − u2ðtÞÞ

ðDðtÞEðtÞ/NÞ − α2ð1 − u3ðtÞÞðCTðtÞEðtÞ/NÞ − ðμ + η1ÞCTðtÞ− u1ðtÞCTðtÞ,

f6ðH, P, E,D, CT , CSÞ = η2DðtÞ + η1CTðtÞ + α2ð1 − u3ðtÞÞðCTðtÞEðtÞ/NÞ − ðμ + δÞCSðtÞ:

For t ∈ ½0, T f �, the adjoint equations and transversalityconditions can be obtained by using Pontryagin’s maximumprinciple [13, 25] such that

λ1′ = −∂H∂H

= λ1 μ + θ1 + θ2ð Þ − λ2θ1 − λ3θ2,

λ2′ = −∂H∂P

= λ2 μ + β1 + β3ð Þ − λ4β1 − λ5β3,ð30Þ

λ3′ = −∂H∂E

= 1 + λ3 μ + γ 1 − u4 tð Þð Þ½ �

− λ4 γ 1 − u4 tð Þð Þ − α1 1 − u2 tð Þð ÞD tð ÞN

� �

− λ5 α1 1 − u2 tð Þð ÞD tð ÞN

− α2 1 − u3 tð Þð ÞCT tð ÞN

� �

− λ6 α2 1 − u3 tð Þð ÞCT tð ÞN

� �,

ð31Þ

CT (t

)

0.4

0.6

0.8

1

1.2

1.4

1.6

CS (t

)

0

2

4

6

8

10

12

Months0 20 40 60 80 100 120

D (t

)

×106 ×107

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Diabetics without complications

Months0 20 40 60 80 100 120

Diabetics with treatable complications

Months0 20 40 60 80 100 120

Diabetics with several complications

×107

Figure 2: The evolution of the number of diabetics with and without complications without controls.

7Journal of Applied Mathematics

λ4′ = −∂H∂D

= 1 − λ4 α1 1 − u2 tð Þð Þ E tð ÞN

+ μ + β2 + η2ð Þ� �

− λ5 β2 + α1 1 − u2 tð Þð Þ E tð ÞN

� �− λ6η2,

λ5′ = −∂H∂CT

= −1 − λ4u1 tð Þ + λ5

�−α2 1 − u3 tð Þð Þ E tð Þ

N

− μ + η1 + u1 tð Þð Þ�− λ6 η1 + α2 1 − u3 tð Þð Þ E tð Þ

N

� �,

λ6′ = −∂H∂CS

= λ6 μ + δð Þ:

ð32ÞFor, t ∈ ½0, T f � the optimal controls u∗1 , u∗2 , u∗3 , and u∗4 can

be solved from the optimality condition,

∂H∂u1

= 0,

∂H∂u2

= 0,

∂H∂u3

= 0,

∂H∂u4

= 0,

ð33Þ

that are

−∂H∂u1

= −Au1 tð Þ + λ2 − λ4ð ÞCT tð Þ = 0,

−∂H∂u2

= −Bu2 tð Þ + α1 λ5 − λ4ð ÞD tð ÞE tð ÞN

= 0,

−∂H∂u3

= −Fu3 tð Þ + α2 λ6 − λ5ð ÞCT tð ÞE tð ÞN

= 0,

−∂H∂u4

= −Gu4 tð Þ + λ4 − λ3ð ÞγE tð Þ = 0:

ð34Þ

We have

u1 tð Þ = λ2 − λ4ð ÞA

CT tð Þ, ð35Þ

u2ðtÞ = α1 × ððλ5 − λ4Þ/BÞ × ðDðtÞEðtÞ/NÞ,

u3 tð Þ = α2 ×λ6 − λ5ð Þ

F× CT tð ÞE tð Þ

N,

u4 tð Þ = λ4 − λ3ð ÞG

× γE tð Þ:ð36Þ

By the bounds in U of the controls, it is easy to obtainu∗1 , u∗2 , u∗3 , and u∗4 and are given in (13–16) the form of sys-tem and in the form of system (18).

5. Numerical Simulation

In this section, we present the results obtained by solvingnumerically the optimality system (18). In our control prob-lem, we have initial conditions for the state variables and ter-minal conditions for the adjoints. That is, the optimality

Table 2: Evolution of number of diabetics without control after 120days.

Population without control after 120 days Without control

Diabetics without complications 3 : 05 × 106

Diabetics with treatable complications 1 : 49 × 107

Diabetics with serious complications 1 × 108

Table 1: Parameter values used in numerical simulation.

Paramater Value in mth-1 Description

μ 0 : 02 Natural mortality

δ 0 : 001 Mortality rate due to complications

β1 0 : 2 The probability of developing diabetes

β2 0.08 The probability of a diabetic person developing a complication

β3 0.01 The probability of developing diabetes at stage of complications

a1 0 : 4 The rate of negative impact on diabetics without complications

Y 0.06 Rate of patients become diabetic without complications through lifestyle

I 2000000 Denote the incidence of healthy people

a2 0 : 6 The rate of negative impact on diabetics with treatable complications

θ1 0 : 1 Rate of prediabetic people through genetic factor

θ1 0 : 2 Rate of prediabetic people through lifestyle factor

η1 0 : 6 The probability of a diabetic person developing a serious complication

η2 0 : 3 Rate of diabetics whose serious complications are because of a sudden shock

8 Journal of Applied Mathematics

system is a two-point boundary value problem with sepa-rated boundary conditions at times step i = 0 and i = T f .We solve the optimality system by an iterative method withforward solving of the state system followed by backwardsolving of the adjoint system. We start with an initial guessfor the controls at the first iteration, and then before the nextiteration, we update the controls by using the characteriza-tion. We continue until convergence of successive iterates isachieved. A code is written and compiled in MATLAB usingthe following data.

Different simulations can be carried out using variousvalues 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, andG = 10000 and with the intial valueof Hð0Þ = 14000000,Pð0Þ = 6660000, Eð0Þ = 13000000, Dð0Þ= 6200000,CTð0Þ = 4500000, andCSð0Þ =2000000 (Figure 2).

After the parameter values (Tables 1 and 2), we notedthat diabetics without complications after 120 monthsdecreased from 6:2 × 106 to 3:05 × 106 (Figure 2) This trans-formation is due to three main things: first, it is by thegenetic factors. Second, due to the negative impact of behav-ioral factors on the patient (nutrition pattern and psycholog-ical and moral problems) and the third by sudden shock(family problem, work problem), we noted that diabeticswith treatable complications are increasing. Indeed, we noted

that the number of the transition becomes from 4:5 × 106 to1:49 × 107 (Figure 2) and, as mentioned above, has diseaseprogression for diabetics without complications, and also asudden shift in the potential for people diagnosed with diabe-tes by means of genetics and with negative impact of behav-ioral factors.

We noted that diabetics with serious complications areincreasing and that the number of the transition becomesfrom 2 × 106 to 1 × 108 (Figure 1) and, as mentioned above,has disease progression for diabetics without complicationsby sudden shock and with negative impact of behavioral fac-tors, and by developing the disease of diabetics with treatablecomplications.

In this formulation, there are initial conditions for thestate variables and terminal conditions for the adjoints.

That is, the optimality system is a two-point boundaryvalue problem with separated boundary conditions at timesteps i = 0 and i = T . We solve the optimality system by aniterative method with forward solving of the state systemfollowed by backward solving of the adjoint system. We startwith an initial guess for the controls at the first iteration, andthen before the next iteration, we update the controls byusing the characterization.

We continue until convergence of successive iterates isachieved.

The proposed control strategy in this work helps toachieve several objectives.

5.1. Strategy A. In this strategy, we applied two controls u1ðtÞand u2ðtÞ in order to reduced the number of diabetics withtreatable complications to diabetics without complications,through Figure 2, we noted that after applied different strate-gics, the number of diabetics with treatable complicationsdecreased from 1:49 × 107 to 1:26 × 107 by the end of thestrategy (Figure 3).

The reason of this increase was justified by the fact thatthe number of diabetics with treatable complications willbecome diabetics without complications. For improving the

CT (t

)

0.4

0.6

0.8

1

1.2

1.4

1.6

D (t

)

0

2

4

6

8

10

12

14

16

18

Months0 20 40 60 80 100 120

D with control u1 and u2D without control

×106 ×107

Months0 20 40 60 80 100 120

CT without controlCT with two controls u1 and u2

Figure 3: The evolution of the number of diabetics with and without complications without controls.

Table 3: Evolution of the number of diabetics with two controlsu1ðtÞ and u2ðtÞ after 120 days.

Population without control after120 days

Withoutcontrol

With two controlsu1 tð Þ and u2 tð Þ

Diabetics without complications 1 : 26 × 106 1:78 × 107

Diabetics with treatablecomplications

107 2:14 × 107

9Journal of Applied Mathematics

effectiveness of this strategy, we added the elements offollow-up and psychological support and education aboutthe negative impact of behavioral factors which are repre-sented in the proposed strategy by the optimal controls vari-ables u1ðtÞ and u2ðtÞ (Figure 3 and Table 3), combiningfollow-up and psychological support with treatment andeducation results in an obvious decreased in the number ofdiabetics with treatable complications.

5.2. Strategy C: Control with Awareness Program,Treatment,and Psychological Support with Follow-Up. We combinedthree optimal controls u1ðtÞ, u2ðtÞ, and u3ðtÞ.

In this strategy, the three optimal controls u1ðtÞ, u2ðtÞ,and u3ðtÞ are activated at the same time, in order to reducedthe number of diabetics with treatable complications to dia-betics without complications (Figure 4).

In this strategy (Figure 4 and Table 4), we used three con-trols optimal u1ðtÞ, u2ðtÞ, and u3ðtÞ. That is, we combined theprevious two strategies to achieved better results that repre-sented treatment, and psychological support with follow-up, and also awareness program through education andmedia for lower the negative impact of behavioral factors.In Figure 4 and Table 4, we observe that the number of dia-betics with treatable complications is decreasing from 1:5 ×

107 to 1:28 × 107, and also, the number of diabetics with seri-ous complications is decreasing from 108 to 7:32 × 107:

5.3. Strategy D: Prevention and Protection E from Diabetes.We we use only the optimal control u4ðtÞ:

In this strategy, we focus the effort of the awareness cam-paign to reduce the negative impact of behavioral factors(Figure 5).

In this strategy, we used control u4ðtÞ (Figure 5 andTable 5); the objective of this control u4ðtÞ is to raise aware-ness campaigns for this target group on the risks of diabetesand its complications as cardiovascular disease, blindness,kidney failure, and lower limb amputation, with trackinghealthy and balanced diet program. after Figure 5 andTable 5, we observed that the number of diabetics withoutcomplications is decreasing from 1:29 × 106 to 106:

Remark 6. We could also merge multiple assemblies asðu1ðtÞ, u2ðtÞ, u3ðtÞÞ, and ðu1ðtÞ, u2ðtÞ, u3ðtÞ, u4ðtÞÞ and thusget a variety of results.

6. Conclusion

In this paper, we formulated a mathematical model of pop-ulations of diabetics, having six compartments: prediabeticsthrough the genetics effects and others by behavioral fac-tors, diabetics without complications, and diabetics withtreatable and serious complications, in order to minimizethe number of diabetics with treatable complications, andreduce the effect of behavioral factors. We also introducedfour controls which, respectively, represent awareness pro-gram through education and media, treatment, and psycho-logical support with follow-up. We applied the results of thecontrol theory, and we managed to obtain the characteriza-tions of the optimal controls. The numerical simulation ofthe obtained results showed the effectiveness of the pro-posed control strategies.

CT (t

)

0.4

0.6

0.8

1

1.2

1.4

1.6 ×107

Months0 20 40 60 80 100 120

CT without controlCT with three controls u1,u2 and u3

CS (t

)

0

2

4

6

8

10

12

Months0 20 40 60 80 100 120

×107

Cs without controlCs with control u3

Figure 4: The evolution of the number of diabetics with treatable and serious complications with three controls u1ðtÞ, u2ðtÞ, and u3ðtÞ:

Table 4: Evolution of number of diabetics with three controlsu1ðtÞ, u2ðtÞ, and u3ðtÞ after 120 days.

Population without controlafter 120 days

Withoutcontrol

With three controlsu1 tð Þ, u2 tð Þ, and u3 tð Þ

Diabetics with treatablecomplications

1 : 5 × 107 1:28 × 107

Diabetics with seriouscomplications

108 7 : 32 × 107

10 Journal of Applied Mathematics

Data Availability

The disciplinary data used to support the Öndings of thisstudy 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 Carpentieriand all the members of the Editorial Board who were respon-sible for dealing with this paper, and the anonymous refereesfor their valuable comments and suggestions, improving thecontent of this paper.

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11Journal of Applied Mathematics

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12 Journal of Applied Mathematics