Research ArticleA Discrete Mathematical Modeling of the Influence of AlcoholTreatment Centers on the Drinking Dynamics UsingOptimal Control

Bouchaib Khajji 1 Abderrahim Labzai 1 Abdelfatah Kouidere 1 Omar Balatif 2

and Mostafa Rachik1

1Laboratory of Analysis Modeling and Simulation Department of Mathematics and Computer ScienceFaculty of Sciences Ben MrsquoSik Hassan II University Sidi Othman Casablanca Morocco2Mathematical Engineering Team (INMA) Laboratory of Dynamical Systems Department of MathematicsFaculty of Sciences El Jadida Chouaib Doukkali University El Jadida Morocco

Correspondence should be addressed to Bouchaib Khajji khajjibouchaibgmailcom

Received 12 November 2019 Revised 5 January 2020 Accepted 23 January 2020 Published 19 February 2020

Academic Editor Fernando Simotildees

Copyright copy 2020 Bouchaib Khajji et al is 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

In this paper we propose a discrete mathematical model that describes the interaction between the classes of drinkers namelypotential drinkers (P) moderate drinkers (M) heavy drinkers (H) poor heavy drinkers (Tp) rich heavy drinkers (Tr) andquitters of drinking (Q) We also focus on the importance of treatment within addiction treatment centers aiming to find theoptimal strategies to minimize the number of drinkers and maximize the number of heavy drinkers who join addiction treatmentcenters We use three controls which represent awareness programs through media and education for the potential drinkersefforts to encourage the heavy drinkers to join addiction treatment centers and psychological support with follow-up for theindividuals who quit drinking We use Pontryaginrsquos maximum principle in discrete time to characterize these optimal controlse resulting optimality system is solved numerically by Matlab Consequently the obtained results confirm the performance ofthe optimization strategy

1 Introduction

Addiction is a social phenomenon and a psychologicalscourge that all societies suffer from It affects all socialclasses and all people with different educational levels andage groups ere are many types of addiction such asaddiction to smoking addiction to eating sugar addictionaddiction to video games addiction to social networkingsites mobile phone addiction addiction to sleep and ad-diction to drinking In this work we will investigate thephenomenon of addiction to drinking alcohol which is aproblem that has received the attention of several re-searchers and scholars in many fields including psychologysociology psychiatry and mathematics in an attempt tohighlight the reasons behind this phenomenon and identify

effective methods of prevention and treatment to thisproblem

According to the World Health Organization (WHO)report in 2016 alcohol addiction causes the death of morethan 3 million people annually which represents about 6percent of all deaths worldwide In spite of its fatal harmglobal consumption of alcohol is expected to increase in thenext 10 years [1] Currently about 23 billion people aredrinking alcohol including 237 million men and 46 millionwomen around the world who drink excessively or badlyaccording to the World Health Organization in its HealthReport in 2018 [1] In some places drinking alcohol is morecommon such as in Europe and the Americas which come atthe top of the list followed by Russia according to the sameorganization Alcohol-induced disorders exist in richer

HindawiJournal of Applied MathematicsVolume 2020 Article ID 9284698 13 pageshttpsdoiorg10115520209284698

countries more than the poor ones In some countries likeRussia the annual consumption of alcohol (measured inliters of pure ethanol) per capita is decreasing yearly as itmoved from 187 liters in 2005 to 117 liters in 2016 isldquodramatic decreaserdquo can be explained by the implementationof measures recommended byWHO such as the prohibitionof the sale of alcoholic beverages at service stations YetWHO in its report expects that the global alcohol con-sumption will rise over the next decade especially in theSoutheast Asia Western Pacific and the Americas and thisincrease will surely affect the number of deaths all over theworld [1]

In Morocco according to the WHO Global Status Re-port on Alcohol and Health 2018 Moroccans are followingthe global trend in alcohol consumption us we note thatthe consumption of alcohol has decreased from 18 liter perindividual in 2010 to 11 liter in 2016 for men and 02 liter in2010 to 01 liter in 2016 for women [2]

Addiction to drinking alcohol has many health andsocial consequences including chronic diseases gastroin-testinal disorders increased risk of pancreatitis and ulcerssevere damage to the liver damage to the brain and thenervous system muddled thinking and loss of memoryharmful hormonal performance increased risk of heartdisease and stroke family problems work problems socialmarginalization poverty road accidents and undesirablemorals [1]

Alcohol can be prevented through a number of measuressuch as regulating and limiting its sales increasing its costand offering inexpensive treatment e treatment can passthrough several steps due to medical issues that may ariseduring the period of quitting alcohol therefore carefulcontrol must be exercised on the drinkers to help them stopalcohol consumption Some medications can be given to theaddict while heshe is in a health facility or from time to timewhile the person lives within the community under closesupervision Psychological disorders or other addictions cancomplicate the treatment so after quitting alcohol supportand follow-up are used in group therapy or support groupsto help the person quit drinking again e most commontypes of support are given by psychiatrists and alcoholicswho gave up alcohol also by giving medicaments such asAcamprosate Disulfiram or Naltrexone

Some researchers in mathematics draw a comparisonbetween the spread of the drinking phenomenon and thespread of infectious diseases Accordingly several mathe-maticians did a lot of work in order to understand thedynamics of drinking and reduce its harm on the drinkerand society as well as minimizing the number of addicteddrinkers For example Huo and Wang [3] developed anonlinear mathematical model with the effect of awarenessprograms on the binge drinking where they show thatawareness programs are an effective measure in reducingalcohol problems Ma et al [4] modeled alcoholism as acontagious disease and used an optimal control to studytheir mathematical model with awareness programs andtime delay Wang et al [5] proposed and analyzed a non-linear alcoholism model and used optimal control for thepurpose of hindering interaction between susceptible

individuals and infected individuals Huo et al [6] proposeda new social epidemic model to depict alcoholism withmedia coverage which was proven to be an effective way inpushing people to quit drinking Huo and Song [7] dividedheavy drinkers in their study into two types those whoconfess drinking and those who do not and they proposed atwo-stage model for binge drinking problem taking intoconsideration the transition of drinkers from the class ofsusceptible individuals towards the class of admittingdrinkers Adu et al [8] used a nonlinear SHTRmathematicalmodel to study the dynamics of drinking epidemic theydivided their population into four classes nondrinkers (S)heavy drinkers (H) drinkers receiving treatment (T) andrecovered drinkers (R) ey discussed the existence andstability of drinking-free and endemic equilibrium Othermathematical models have also been widely used to studythis phenomenon (for example [5 9ndash12])

In addition most of these previous researches have fo-cused on continuous-time modeling In this research we willadopt the discrete-time modeling as the statistical data arecollected at discrete time (day week month and year) and thetreatment and vaccination of some patients are done indiscrete time So it is more direct more convenient andmoreaccurate to describe the phenomena by using the discrete-time modeling than the continuous-time modeling and theuse of discrete-time models may avoid some mathematicalcomplexities such as the choice of a function space andregularity of the solution Hence difference equations appearas a more natural way to describe the epidemic modelsMoreover numerical solutions of differential equations usediscretization and this encourages us to employ differenceequations directlye numerical exploration of discrete-timemodels is rather straightforward and therefore can be easilyimplemented by nonmathematicians

Besides these works we will study the dynamics of amathematical alcohol model PMHTrTpQ which containsthe following additions

(i) A discrete-time mathematical modeling(ii) A compartment Tp that represents the number of

the poor heavy drinkers who join public addictiontreatment centers which may not afford sophisti-cated equipment and high-quality treatment serv-icesmdashespecially in the developing countriesmdashandthose individuals who do not have the financialcapacity to go to private centers

(iii) A compartment Tr that represents the number ofthe rich heavy drinkers who join private addictiontreatment centers which have special facilities thatprovide advanced treatment and individuals whohave sufficient financial capacity to join thesecenters

(iv) e death rate induced by the heavy drinkers δ

e drinkers classes of this model are divided into sixcompartments potential drinkers (P) moderate drinkers(M) heavy drinkers (H) the rich heavy drinkers who joinprivate addiction treatment centers (Tr) the poor heavydrinkers who join public addiction treatment centers (Tp)

2 Journal of Applied Mathematics

and quitters of drinking (Q) roughout this research weseek to find the optimal strategies to minimize the number ofdrinkers and maximize the number of rich and poor heavydrinkers who join addiction treatment centers

In order to achieve this purpose we use optimal controlstrategies associated with three types of controls the firstrepresents awareness programs for potential drinkers thesecond is the effort to encourage the rich to go to privatetreatment centers and the poor heavy drinkers to go to publictreatment centers and the third represents follow-up and thepsychological support for temporary quitters of drinking

e paper is organized as follows In Section 3 wepresent our PMHTrTpQ discrete mathematical model thatdescribes the classes of drinkers In Sections 4 and 5 wepresent the optimal control problem for the proposed modelwhere we give some results concerning the existence of theoptimal controls and we characterize these optimal controlsusing Pontryaginrsquos maximum principle in discrete timeNumerical simulations are given in Section 6 Finally weconclude the paper in Section 6

2 Model Formulation

We propose a discrete model PMHTrTpQ to describe thedynamics of population and the transmission of drinkinge population is divided into six compartments denoted byP M H Tr Tp and Q

e graphical representation of the proposed model isshown in Figure 1

e mathematical representation of the model consistsof a system of nonlinear difference equations

Pk+1 b minus β1PkMk

Nk

+(1 minus μ)Pk

Mk+1 β1PkMk

Nk

+ θQk + 1 minus μ minus β2( 1113857Mk

Hk+1 β2Mk + 1 minus μ minus δ minus α1 minus α2 minus α3( 1113857Hk

Trk+1 α1Hk + 1 minus μ minus c1( 1113857Tr

k

Tp

k+1 α2Hk + 1 minus μ minus c2( 1113857Tp

k

Qk+1 c1Trk + c2T

p

k + α3Hk +(1 minus μ minus θ)Qk

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where P0 ge 0 M0 ge 0 H0 ge 0 Tr0 ge 0 T

p0 ge 0 and Q0 ge 0

e compartment P contains the potential drinkers whorepresent individuals whose age is over adolescence andadulthood and may become drinkers is compartment isincreased by the recruitment rate denoted by b and decreasedby an effective contact with the moderate drinkers at β1 rateand natural death μ It is assumed that potential drinkers canacquire drinking behavior and can becomemoderate drinkersthrough effective contact with moderate drinkers in somesocial occasions like weddings celebrating graduation cere-monies weekend parties and end of the year celebration Inother words it is assumed that the acquisition of a drinkingbehavior is analogous to acquiring disease infection

e compartment M is composed of the moderatedrinkers who drink alcohol in a controlled manner duringsome events and occasions or in a way that is unapparent totheir social environment It is increased by potential drinkerswho turn to be moderate drinkers at β1 rate and those whoquit addiction to alcohol temporarily at a rate θ iscompartment is decreased when moderate drinkers becomeheavy drinkers at a rate β2 and also by natural death at rate μ

e compartment H comprises the heavy drinkers iscompartment becomes larger as the number of heavydrinkers increases by the rate β2 and decreases when some ofthem give up drinking at a rate α3 and decreases also by therate α1 (α1 is a rate the heavy drinkers join private treatmentcenters) and decreases also by the rate α2 (α2 is a rate theheavy drinkers join public treatment centers) In additionthis compartment decreases by natural death μ and due todeaths caused by diseases resulted from excessive alcoholintake at a rate δ

e compartment Tp represents the number of heavydrinkers who join public treatment centers of alcohol ad-diction which may not provide advanced treatment and thatare marked by a shortage of equipment and low-qualityservicesmdashespecially in the developing countriesmdashand it alsocontains individuals who do not have the financial capacityto join the private centers is compartment is increased bythe rate α2 and decreased by the rate c2which representsindividuals who have been treated in the public treatmentcenters and also by natural death at rate μ

e compartment Tr contains the number of heavydrinkers who take advantage of their financial potentials to joinprivate treatment centers of alcohol addiction that are veryoften well-equipped and provide good-quality services iscompartment is increased by the rate α1 and decreased by therate c1 which represents individuals who have been treated inthe private treatment centers and also by natural death at rate μ

e compartment Q encompasses the individuals whoquit drinking temporarily and permanently It is increasedwith the recruitment of individuals who have been treated intreatment centers of alcohol addiction at rates c1 and c2 Italso increases at the rate α3 by those who quit alcoholwithout resorting to treatment centers and decreases at therate μ due to natural deaths and at rate θ (which representsdrinkers who quit drinking temporarily and revert back tomoderate drinking) We note that θQk is the proportion of

b

μ μ

μ

μ

μ

μ + δ

HMP Q

Tr

β1 β2MPMN

α1H

α2H

α3H

γ2 TpTp

γ1Tr

θQ

Figure 1 Schematic diagram of the six drinking classes in themodel

Journal of Applied Mathematics 3

the temporary quitters of drinking who moved to moderatedrinking at time step k

e total population size at time k is denoted by Nk withNk Pk + Mk + Hk + Tr

k + Tp

k + Qk and it is supposed to beconstant

3 The Optimal Control Problem

e objective of the proposed control strategy is to minimizethe number of heavy drinkers (H) and maximize thenumber of the rich heavy drinkers (Tr) and the poor heavydrinkers (Tp) who join private or public treatment centersof alcohol addiction and the number of the permanentquitters of drinking (Q) in the community during the timesteps k 0 to k T e cost spent in the awareness pro-grams is also to be minimized

In order to achieve these objectives we introduce threecontrol variables e first control u1 represents the effort ofthe awareness programs to protect the potential drinkers notto be drinkers e second control u2 measures the effortmade to make the heavy drinkers know private and publictreatment centers of alcohol addiction and to encourage themto join those centers We note that the control function ϵu2represents the fraction of the heavy drinkers who will betreated in private addiction treatment centers and the fraction(1 minus ϵ)u2 of those leaving the heavy drinkers class who willreceive treatment in public addiction treatment centers ethird control u3 represents the effort of the follow-up and thepsychological support for the temporary quitters of drinkingwith a purpose of protecting them to not go back to drinking

So the controlled mathematical system is given by thefollowing system of difference equations

Pk+1 b minus β1PkMk

Nk

+(1 minus μ)Pk minus u1kPk

Mk+1 β1PkMk

Nk

+ 1 minus u3k1113872 1113873θQk + 1 minus μ minus β2( 1113857Mk

Hk+1 β2Mk + 1 minus μ minus δ minus α1 minus α2 minus α3( 1113857Hk minus u2kHk

Trk+1 α1Hk + 1 minus μ minus c1( 1113857Tr

k + ϵu2kHk

Tp

k+1 α2Hk + 1 minus μ minus c2( 1113857Tp

k +(1 minus ϵ)u2kHk

Qk+1 c1Trk + α3Hk + c2T

p

k +(1 minus μ)Qk + u1kPk minus 1 minus u3k1113872 1113873θQk

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where P0 ge 0 M0 ge 0 H0 ge 0 Tr0 ge 0 T

p0 ge 0 and Q0 ge 0

e optimal control problem to minimize the objectivefunctional is given by

J u1 u2 u3( 1113857 HT minus TrT minus T

p

T minus QT

+ 1113944

Tminus 1

k0Hk minus T

rk minus T

p

k minus Qk1113872

+Aku2

1k

2+

Bku22k

2+

Cku23k

21113889

(3)

where the parameters Ak gt 0 Bk gt 0 and Ck gt 0 are selectedto weigh the relative importance of the cost of awarenessprograms encouragement programs and follow-up andpsychological support respectively

e aim is to find an optimal control ulowast1 ulowast2 and ulowast3 -such that

J ulowast1 ulowast2 ulowast3( 1113857 min

u1 u2 u3( )isinUad

J u1 u2 u3( 1113857 (4)

where Uad is the set of admissible controls defined by

Uad uj uj0 uj1 ujTminus 11113872 1113873

0le ujmin le ujk le ujmax le 1

⎧⎨

⎩

for j 1 2 3 k 0 1 T minus 11113897

(5)

e sufficient condition for the existence of an optimalcontrol (ulowast1 ulowast2 ulowast3 ) for problems (2) and (3) comes from thefollowing theorem

Theorem 1 5ere exists the optimal control (ulowast1 ulowast2 ulowast3 )

such that

J ulowast1 ulowast2 ulowast3( 1113857 min

u1 u2 u3( )isinUad

J u1 u2 u3( 1113857 (6)

subject to the control system (2) with initial conditions

Proof Since the coefficients of the state equations arebounded and there are a finite number of time stepsP (P0 P1 P2 PT) M (M0 M1 M2 MT) H

(H0 H1 H2 HT) Tr (Tr0 Tr

1 Tr2 Tr

T) Tp

(Tp0 T

p1 T

p2 T

p

T) and Q (Q0 Q1 Q2 QT) areuniformly bounded for all (u1 u2 u3) in the control set Uadthus J(u1 u2 u3) is bounded for all (u1 u2 u3) isin Uad SinceJ(u1 u2 u3) is bounded inf(u1 u2 u3)isinUad

J(u1 u2 u3) is finiteand there exists a sequence (u

j1 u

j2 u

j3) isin Uad such that

limj⟶+infin(uj1 u

j2 u

j3) inf(u1 u2u3)isinUad

J(u1 u2 u3) and cor-responding sequences of states Pj Mj Hj Trj Tpj and QjSince there are a finite number of uniformly bounded se-quences there exist (ulowast1 ulowast2 ulowast3 ) isin Uad and Plowast Mlowast Hlowast

Trlowast Tplowast and Qlowast isin IRT+1 such that on a subsequencelimj⟶+infin(u

j1 u

j2 u

j3) (ulowast1 ulowast2 ulowast3 ) limj⟶+infinPj Plowast

limj⟶+infinMj Mlowast limj⟶+infinHj Hlowast limj⟶+infinTrj

Trlowast limj⟶+infinTpj Tplowast and limj⟶+infinQj Qlowast Finallydue to the finite dimensional structure of system (2) and theobjective function J(u1 u2 u3) (ulowast1 ulowast2 ulowast3 ) is an optimalcontrol with corresponding states Plowast Mlowast Hlowast Trlowast Tplowast andQlowast erefore inf(u1 u2 u3)isinUad

J(u1 u2 u3) is achieved

4 Characterization of the Optimal Controls

We apply the discrete version of Pontryaginrsquos maximumprinciple [13ndash18] e key idea is introducing the adjointfunction to attach the system of difference equations to theobjective functional resulting in the formation of a functioncalled the Hamiltonian is principle converts the problem

4 Journal of Applied Mathematics

of finding the control to optimize the objective functionalsubject to the state difference equation with initial conditionto find the control to optimize Hamiltonian pointwise (withrespect to the control)

Now we have the Hamiltonian 1113954H at time step k definedby

1113954Hk Hk minus Trk minus T

p

k minus Qk +Aku2

1k

2+

Bku22k

2+

Cku23k

2

+ 11139446

i1λik+1fik+1

(7)

where fik+1 is the right-hand side of the system of differenceequation (2) of the ith state variable at time step k + 1

Theorem 2 Given an optimal control (ulowast1 ulowast2 ulowast3 ) isin Uadand solutions Plowastk M

lowastk Hlowastk T

rlowastk Tplowast

k and Qlowastk of correspondingstate system [13] there exist adjoint functions λ1 λ2 λ3 λ4 λ5and λ6 satisfying the following equations

λ1k z 1113954Hk

zPk

λ2k+1 minus λ1k+11113872 1113873β1Mk

Nk

+ λ6k+1 minus λ1k+11113872 1113873u1k + λ1k+1(1 minus μ)

λ2k z 1113954Hk

zMk

λ2k+1 minus λ1k+11113872 1113873β1Pk

Nk

+ β2λ3k+1 + λ2k+1 1 minus μ minus β2( 1113857

λ3k z 1113954Hk

zHk

1 + λ3k+1 1 minus μ minus δ minus α1 minus α2 minus α3 minus u2k1113872 1113873 + λ4k+1 α1 + ϵu2k1113872 1113873 + λ5k+1 α2 +(1 minus ϵ)u2kHk1113960 1113961 + α3λ6k+1

λ4k z 1113954Hk

zTrk

minus 1 + c1λ6k+1 + λ4k+1 1 minus μ minus c1( 1113857

λ5k z 1113954Hk

zTp

k

minus 1 + c2λ6k+1 + λ5k+1 1 minus μ minus c2( 1113857

λ6k z 1113954Hk

zQk

minus 1 + λ2k+1 1 minus u3k1113872 1113873θ + λ6k+1 1 minus μ minus θ + θu3k1113872 1113873

(8)

with the transversality conditions at time T

λ1(T) 0

λ2(T) 0

λ3(T) 1

λ4(T) minus 1

λ5(T) minus 1

λ6(T) minus 1

(9)

Furthermore for k 0 1 T minus 1 we obtain the opti-mal control (ulowast1k ulowast2k ulowast3k) as

ulowast1k min max

λ1k+1 minus λ6k+11113872 1113873Pk

Ak

u1min⎧⎨

⎩

⎫⎬

⎭ u1max⎧⎨

⎩

⎫⎬

⎭

ulowast2k min max

λ3k+1 minus λ5k+11113872 1113873Hk + λ5k+1 minus λ4k+11113872 1113873ϵHk

Bk

u2min⎧⎨

⎩

⎫⎬

⎭ u2max⎧⎨

⎩

⎫⎬

⎭

ulowast3k min max

λ2k+1 minus λ6k+11113872 1113873θQk

Ck

u3min⎧⎨

⎩

⎫⎬

⎭ u3max⎧⎨

⎩

⎫⎬

⎭

(10)

Journal of Applied Mathematics 5

Proof e Hamiltonian 1113954Hk at time step k is given by

1113954Hk Hk minus Trk minus T

p

k minus Qk +Aku2

1k

2+

Bku22k

2+

Cku23k

2

+ λ1k+1 b minus β1PkMk

Nk

+(1 minus μ)Pk minus u1kPk1113890 1113891

+ λ2k+1 β1PkMk

Nk

+ 1 minus u3k1113872 1113873θQk + 1 minus μ minus β2( 1113857Mk1113890 1113891

+ λ3k+1 β2Mk + 1 minus μ minus δ minus α1 minus α2 minus α3( 1113857Hk minus u2kHk1113960 1113961

+ λ4k+1 α1Hk + 1 minus μ minus c1( 1113857Trk + ϵu2kHk1113960 1113961

+ λ5k+1 α2Hk + 1 minus μ minus c2( 1113857Tp

k +(1 minus ϵ)u2kHk1113960 1113961

+ λ6k+1 c1Trk + α3Hk + c2T

p

k +(1 minus μ)Qk + u1kPk minus 1 minus u3k1113872 1113873θQk1113960 1113961

(11)

For k 0 1 T minus 1 the adjoint equations andtransversality conditions can be obtained by using Pon-tryaginrsquos maximum principle in discrete time given in[13ndash18] such that

λ1k z 1113954Hk

zPk

λ1(T)

λ2k z 1113954Hk

zMk

λ2(T)

λ3k z 1113954Hk

zHk

λ3(T)

λ4k z 1113954Hk

zTrk

λ4(T)

λ5k z 1113954Hk

zTp

k

λ5(T)

λ6k z 1113954Hk

zQk

λ6(T)

(12)

For k 0 1 T minus 1 the optimal controls ulowast1k ulowast2kand ulowast3k can be solved from the optimality condition

z 1113954Hk

zu1k

Aku1k minus λ1k+1Pk + λ6k+1Pk 0

z 1113954Hk

zu2k

Bku2k minus λ3k+1Hk + λ4k+1ϵHk + λ5k+1(1 minus ϵ)Hk 0

z 1113954Hk

zu3k

Cku3k minus λ2k+1θQk + λ6k+1θQk 0

(13)

So we have

u1k λ1k+1 minus λ6k+11113872 1113873Pk

Ak

u2k λ3k+1 minus λ5k+11113872 1113873Hk + λ5k+1 minus λ4k+11113872 1113873ϵHk

Bk

u3k λ2k+1 minus λ6k+11113872 1113873θQk

Ck

(14)

By the bounds in Uad of the controls it is easy to obtainulowast1k ulowast2k and ulowast3k in the form of (10)

5 Numerical Simulation

In this section we shall solve numerically the optimalcontrol problem for our PMHTrTpQ model Here weobtain the optimality system from the state and adjointequationse proposed optimal control strategy is obtainedby solving the optimal system which consists of six differ-ence equations and boundary conditions e optimalitysystem can be solved by using an iterative method Using aninitial guess for the control variables u1k u2k and u3k thestate variables P M H Tr Tp and Q are solved forwardand the adjoint variables λi for i 1 2 3 4 5 6 are solvedbackwards at time steps k 0 and k T If the new values ofthe state and adjoint variables differ from the previousvalues the new values are used to update u1k u2k and u3kand the process is repeated until the system converges

e numerical solution of model (1) is executed usingMatlab with the following parameter values and initial valuesof state variable in Table 1

We begin by presenting the solution evolution of ourmodel (1) without controls that are represented in Figure 2

6 Journal of Applied Mathematics

Figure 2(a) shows that the number of moderate drinkersincreases in the beginning but then decreases clearly isdecrease leads to an increase in the number of heavy drinkersfrom the beginning and approaches a given value of 46230as we see in Figure 2(b) Finally Figure 2(c) and Figures 2(d)and 2(e) show that the number of the rich and poor heavydrinkers joining private and public centers for alcohol ad-diction treatment and the quitters of drinking decreases andapproaches certain unacceptable values

In order to display the influence of alcohol treatmentcenters on the population drinking dynamics we presentsome figures about the effects of the parameters α1 α2 c1and c2 on system (1)

Figure 2(f ) shows that the number of heavy drinkersincreases when the number of the heavy drinkers who joinaddiction treatment centers decreases Also we observe inFigure 2(g) that the number of quitters of drinking increaseswhen the number of the heavy drinkers who join addictiontreatment centers increases

From these two figures the importance of addictiontreatment centers is clear in reducing the number of heavydrinkers and thus increasing the number of quitters ofdrinking erefore we need to make an effort to introducethese centers to people and encourage heavy drinkers to jointhem and this is shown through the following objectives

51 First Objective Protecting and Preventing PotentialDrinkers from Falling into Alcohol Addiction To realize thisobjective we apply only the control u1 ie the imple-mentation of awareness information and educationalprograms on potential drinkers and making them know therisks of this phenomenon and the resulting health and socialdamages Figure 3(a) shows that the number of moderatedrinkers decreases from 23116 (without control u1) to12459 (with control u1) at the end of the proposed programand the number of heavy drinkers decreases from 46229(without control u1) to 24902 (with control u1) at the end ofthe proposed control strategy (see Figure 3(b)) Also weobserve in Figure 3(c) that the number of people who quitdrinking increases and has reached the value 42978 (withcontrol u1) compared to the situation when there is nocontrol 568 at the end of the proposed strategy So ourobjective has been achieved

52 Second Objective Increasing the Number of HeavyDrinkers Who Join Treatment Centers of Alcohol AddictionTo achieve this objective we only use the control u2 ieencouraging heavy drinkers to know and join public andprivate treatment centers In Figure 4(a) it is observed thatthere is a significant decrease in the number of heavydrinkers with control compared to a situation when there isno control where the decrease reaches 4120 at the end ofthe proposed control strategy Figure 4(b) shows that thenumber of the poor heavy drinkers who join public treat-ment centers increased from 691 (Tp without control u2) to5970 (Tp with control u2) and that the number of rich heavydrinkers who join private treatment centers increased from706 (without control u2) to 23372 (with control u2) at the

end of the proposed control (see Figure 4(c)) and Figure 4(d)shows that the number of people who quit drinking withoutcontrol u2 decreases and approaches a value of 568 After 5days a slight increase appears with control u2 is increaseis not enough as the number of people who quit drinkingbecomes moderate drinkers To improve this result we canadd another control that represents follow-up for temporaryquitters of drinking

53 5ird Objective Encouraging Heavy Drinkers to JoinTreatment Centers of Alcohol Addiction and Following Up theQuitters of Drinking In order to accomplish this aim weuse the controls u2 and u3 ie encouraging heavydrinkers to know and join public and private treatmentcenters and following up the temporary quitters ofdrinking e number of the poor heavy drinkers who joinpublic treatment centers increases from 691 (withoutcontrols) to 3217 (with controls) at the end of the pro-posed strategy (see Figure 5(c)) Figure 5(d) demonstratesthat the number of the rich heavy drinkers who joinprivate treatment centers increases from 706 (withoutcontrols) to 12592 (with controls) at the end of theproposed strategy e number of the quitters of drinkingin Figure 5(e) increases significantly from 568 (withoutcontrols) to 43011 (with controls) rough the use ofthose controls the abovementioned objective has beenachieved

54 Fourth Objective Prevention Treatment and Follow-Upfor the Addicted Individuals To meet this objective we usethe controls u1 u2 and u3 ie awareness programs for thepotential drinkers encouraging heavy drinkers to know andjoin public and private treatment centers and follow-up forthe quitters of drinking Figure 6(a) shows that the numberof the moderate drinkers decreases starting from the earlydays by 3460 Also Figure 6(b) shows that the number ofthe heavy drinkers increases starting from the early days butthen decreases from 46229 (without controls) to 6581 (withcontrols) e number of the rich and poor heavy drinkerswho join private and public treatment centers increases by1145 and 298 respectively (see Figures 6(c) and 6(d))Figure 6(e) depicts clearly an increase in the number of thequitters of drinking from 568 (without controls) to 55063(with controls) As a result the objective set before has beenachieved

We remark that the numbers of the rich and poor heavydrinkers (Tr and Tp) decrease when we applied threecontrols (u1 u2 u3) compared to using only the control(u2) is decrease is due to the positive effect of the controlu1 on the compartment P and the control u3 on θQ (tem-porary quitters) which prevents temporary quitters of

Table 1 e parameters used for model (1)

P0 M0 H0 Tr0 T

p0 Q0 b N δ

500 300 100 60 30 10 65 1000 0002μ β1 β2 α1 α2 α3 c1 c2 θ0065 075 014 0001 0001 0001 0001 0002 002

Journal of Applied Mathematics 7

M without controls

200

250

300

350

400

450M

oder

ate d

rinke

rs (M

)

10 20 30 40 50 60 70 80 90 100Time (days)

(a)

H without controls

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

10 20 30 40 50 60 70 80 90 100Time (days)

(b)

Tp without controls

10 20 30 40 50 60 70 80 90 100Time (days)

5

10

15

20

25

30

The p

oor h

eavy

drin

kers

(Tp )

(c)

Tr without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

The r

ich

heav

y dr

inke

rs (T

r )

(d)

Q without controls

55556

657

758

859

9510

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

(e)

100 20 30 40 50 60 70 80 90 100Time (days)

50100150200250300350400450500

Hea

vy d

rinke

rs (H

)

H without treatment centers of alcohol addictionH with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002H with α1 = 03 α2 = 01 γ1 = 006 γ2 = 003H with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02H with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03

(f )

Figure 2 Continued

8 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

Q with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03Q with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02Q with α1 = 03 α2 = 01 γ1 = 005 γ2 = 02Q with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002Q without treatment centers of alcohol addiction

0

50

100

150

200

250

300

350

400

Qui

tters

of d

rinki

ng (Q

)

(g)

Figure 2 Representing the drinkers class without control

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

M with control u1M without controls

(a)

H with control u1H without controls

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

(b)

Q with control u1Q without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

(c)

Figure 3 Representing the drinkers class with and without control u1

Journal of Applied Mathematics 9

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500H

eavy

drin

kers

(H)

H with control u2H without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

e p

oor h

eavy

drin

kers

(Tp )

Tp with control u2Tp without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

e r

ich

heav

y dr

inke

rs (T

r )

Tr with control u2Tr without controls

(c)

55

6

65

7

75

8

85

9

95

10

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

Q with control u2Q without controls

(d)

Figure 4 Representing the drinkers class with and without control u2

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

The p

oor h

eavy

drin

kers

(Tp )

Tp with controls u2 and u3Tp without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

The r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u2 and u3Tr without controls

(b)

Figure 5 Continued

10 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

Q with controls u2 and u3Q without controls

(c)

Figure 5 Representing the drinkers class with controls u2 and u3 and without controls

50

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

10 20 30 40 50 60 70 80 90 100Time (days)

M with controls u1 u2 and u3M without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

50

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

H with controls u1 u2 and u3H without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

510152025303540455055

e p

oor h

eavy

drin

kers

(Tp )

Tp with controls u1 u2 and u3Tp without controls

(c)

10 20 30 40 50 60 70 80 90 100Time (days)

0

20

40

60

80

100

120

140

160

180

e r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u1 u2 and u3Tr without controls

(d)

Figure 6 Continued

Journal of Applied Mathematics 11

drinking from returning to drinking So the number ofindividuals that enter the compartment Tr and Tp de-creases erefore applying a combination of the threecontrols is more effective than one control

6 Conclusion

In this research paper we introduced a discrete modelingof drinking for the purpose of minimizing the number ofdrinkers and maximizing the number of the rich and poorheavy drinkers who join private and public treatmentcenters of alcohol addiction and subsequently the numberof quitters of drinking Unlike some other previousmodels we have taken into account the impact of privateand public addiction treatment centers on alcoholics eresults showed that those centers have substantial influ-ence on the dynamics of alcoholism and can greatlyimpact the spread of drinking us it is crucial to urgepeople to know and join private and public addictiontreatment centers to quit drinking We also presentedthree controls which respectively represent awarenessprograms encouragement and follow-up We applied theresults of the control theory and we managed to obtain thecharacterizations of the optimal controls e numericalsimulation of the obtained results showed the effective-ness of the proposed control strategies

Data Availability

e disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpnetworkrepositorycomprofilephp)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] WHOWorld Health Organization Report on Drinking WHOGeneva Switzerland 2016

[2] World Health Organization Alcohol World Health Organi-zation Geneva Switzerland 2018 httpswwwwhointnews-roomfact-sheetsdetailalcohol

[3] H F Huo and Q Wang ldquoModelling the influence ofawareness programs by media on the drinking dynamicsrdquoAbstract and Applied Analysis vol 2014 Article ID 9380808 pages 2014

[4] S-H Ma H-F Huo and X-Y Meng ldquoModelling alcoholismas a contagious disease a mathematical model with awarenessprograms and time delayrdquo Discrete Dynamics in Nature andSociety vol 2015 Article ID 260195 13 pages 2015

[5] X-Y Wang H-F Huo Q-K Kong andW-X Shi ldquoOptimalcontrol strategies in an alcoholism modelrdquo Abstract andApplied Analysis vol 2014 Article ID 954069 18 pages 2014

[6] H-F Huo S-R Huang X-Y Wang and H Xiang ldquoOptimalcontrol of a social epidemic model with media coveragerdquoJournal of Biological Dynamics vol 11 no 1 pp 226ndash2432017

[7] H F Huo and N N Song ldquoGlobal stability for a bingedrinking model with two stagesrdquo Discrete Dynamics in Natureand Society vol 2012 Article ID 829386 15 pages 2012

[8] I K Adu M Osman and C Yang ldquoMathematical model ofdrinking epidemicrdquo British Journal of Mathematics ComputerScience vol 22 no 5 pp 1ndash10 2017

[9] H-F Huo and Y-P Liu ldquoe analysis of the SIRS alcoholismmodels with relapse on weightedrdquo SpringerPlus vol 5 no 1p 722 2016

[10] K Mursquotamar ldquoOptimal control strategy for alcoholism modelwith two infected compartmentsrdquo IOSR Journal of Mathe-matics vol 14 no 3 pp 58ndash67 2018

[11] S Sharma and G P Samanta ldquoDrinking as an epidemic amathematical model with dynamic behaviourrdquo Journal ofApplied Mathematics amp Informatics vol 31 no 1-2 pp 1ndash252013

[12] S Sharma and G P Samanta ldquoAnalysis of a drinking epi-demic modelrdquo International Journal of Dynamics and Controlvol 3 no 3 pp 288ndash305 2015

10 20 30 40 50 60 70 80 90 100Time (days)

0

100

200

300

400

500

600

Qui

tters

of d

rinki

ng (Q

)

Q with controls u1 u2 and u3Q without controls

(e)

Figure 6 Representing the drinkers class with and without optimal controls u1 u2 and u3

12 Journal of Applied Mathematics

and quitters of drinking (Q) roughout this research weseek to find the optimal strategies to minimize the number ofdrinkers and maximize the number of rich and poor heavydrinkers who join addiction treatment centers

In order to achieve this purpose we use optimal controlstrategies associated with three types of controls the firstrepresents awareness programs for potential drinkers thesecond is the effort to encourage the rich to go to privatetreatment centers and the poor heavy drinkers to go to publictreatment centers and the third represents follow-up and thepsychological support for temporary quitters of drinking

e paper is organized as follows In Section 3 wepresent our PMHTrTpQ discrete mathematical model thatdescribes the classes of drinkers In Sections 4 and 5 wepresent the optimal control problem for the proposed modelwhere we give some results concerning the existence of theoptimal controls and we characterize these optimal controlsusing Pontryaginrsquos maximum principle in discrete timeNumerical simulations are given in Section 6 Finally weconclude the paper in Section 6

2 Model Formulation

We propose a discrete model PMHTrTpQ to describe thedynamics of population and the transmission of drinkinge population is divided into six compartments denoted byP M H Tr Tp and Q

e graphical representation of the proposed model isshown in Figure 1

e mathematical representation of the model consistsof a system of nonlinear difference equations

Pk+1 b minus β1PkMk

Nk

+(1 minus μ)Pk

Mk+1 β1PkMk

Nk

+ θQk + 1 minus μ minus β2( 1113857Mk

Hk+1 β2Mk + 1 minus μ minus δ minus α1 minus α2 minus α3( 1113857Hk

Trk+1 α1Hk + 1 minus μ minus c1( 1113857Tr

k

Tp

k+1 α2Hk + 1 minus μ minus c2( 1113857Tp

k

Qk+1 c1Trk + c2T

p

k + α3Hk +(1 minus μ minus θ)Qk

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where P0 ge 0 M0 ge 0 H0 ge 0 Tr0 ge 0 T

p0 ge 0 and Q0 ge 0

e compartment P contains the potential drinkers whorepresent individuals whose age is over adolescence andadulthood and may become drinkers is compartment isincreased by the recruitment rate denoted by b and decreasedby an effective contact with the moderate drinkers at β1 rateand natural death μ It is assumed that potential drinkers canacquire drinking behavior and can becomemoderate drinkersthrough effective contact with moderate drinkers in somesocial occasions like weddings celebrating graduation cere-monies weekend parties and end of the year celebration Inother words it is assumed that the acquisition of a drinkingbehavior is analogous to acquiring disease infection

e compartment M is composed of the moderatedrinkers who drink alcohol in a controlled manner duringsome events and occasions or in a way that is unapparent totheir social environment It is increased by potential drinkerswho turn to be moderate drinkers at β1 rate and those whoquit addiction to alcohol temporarily at a rate θ iscompartment is decreased when moderate drinkers becomeheavy drinkers at a rate β2 and also by natural death at rate μ

e compartment H comprises the heavy drinkers iscompartment becomes larger as the number of heavydrinkers increases by the rate β2 and decreases when some ofthem give up drinking at a rate α3 and decreases also by therate α1 (α1 is a rate the heavy drinkers join private treatmentcenters) and decreases also by the rate α2 (α2 is a rate theheavy drinkers join public treatment centers) In additionthis compartment decreases by natural death μ and due todeaths caused by diseases resulted from excessive alcoholintake at a rate δ

e compartment Tp represents the number of heavydrinkers who join public treatment centers of alcohol ad-diction which may not provide advanced treatment and thatare marked by a shortage of equipment and low-qualityservicesmdashespecially in the developing countriesmdashand it alsocontains individuals who do not have the financial capacityto join the private centers is compartment is increased bythe rate α2 and decreased by the rate c2which representsindividuals who have been treated in the public treatmentcenters and also by natural death at rate μ

e compartment Tr contains the number of heavydrinkers who take advantage of their financial potentials to joinprivate treatment centers of alcohol addiction that are veryoften well-equipped and provide good-quality services iscompartment is increased by the rate α1 and decreased by therate c1 which represents individuals who have been treated inthe private treatment centers and also by natural death at rate μ

e compartment Q encompasses the individuals whoquit drinking temporarily and permanently It is increasedwith the recruitment of individuals who have been treated intreatment centers of alcohol addiction at rates c1 and c2 Italso increases at the rate α3 by those who quit alcoholwithout resorting to treatment centers and decreases at therate μ due to natural deaths and at rate θ (which representsdrinkers who quit drinking temporarily and revert back tomoderate drinking) We note that θQk is the proportion of

b

μ μ

μ

μ

μ

μ + δ

HMP Q

Tr

β1 β2MPMN

α1H

α2H

α3H

γ2 TpTp

γ1Tr

θQ

Figure 1 Schematic diagram of the six drinking classes in themodel

Journal of Applied Mathematics 3

the temporary quitters of drinking who moved to moderatedrinking at time step k

e total population size at time k is denoted by Nk withNk Pk + Mk + Hk + Tr

k + Tp

k + Qk and it is supposed to beconstant

3 The Optimal Control Problem

e objective of the proposed control strategy is to minimizethe number of heavy drinkers (H) and maximize thenumber of the rich heavy drinkers (Tr) and the poor heavydrinkers (Tp) who join private or public treatment centersof alcohol addiction and the number of the permanentquitters of drinking (Q) in the community during the timesteps k 0 to k T e cost spent in the awareness pro-grams is also to be minimized

In order to achieve these objectives we introduce threecontrol variables e first control u1 represents the effort ofthe awareness programs to protect the potential drinkers notto be drinkers e second control u2 measures the effortmade to make the heavy drinkers know private and publictreatment centers of alcohol addiction and to encourage themto join those centers We note that the control function ϵu2represents the fraction of the heavy drinkers who will betreated in private addiction treatment centers and the fraction(1 minus ϵ)u2 of those leaving the heavy drinkers class who willreceive treatment in public addiction treatment centers ethird control u3 represents the effort of the follow-up and thepsychological support for the temporary quitters of drinkingwith a purpose of protecting them to not go back to drinking

So the controlled mathematical system is given by thefollowing system of difference equations

Pk+1 b minus β1PkMk

Nk

+(1 minus μ)Pk minus u1kPk

Mk+1 β1PkMk

Nk

+ 1 minus u3k1113872 1113873θQk + 1 minus μ minus β2( 1113857Mk

Hk+1 β2Mk + 1 minus μ minus δ minus α1 minus α2 minus α3( 1113857Hk minus u2kHk

Trk+1 α1Hk + 1 minus μ minus c1( 1113857Tr

k + ϵu2kHk

Tp

k+1 α2Hk + 1 minus μ minus c2( 1113857Tp

k +(1 minus ϵ)u2kHk

Qk+1 c1Trk + α3Hk + c2T

p

k +(1 minus μ)Qk + u1kPk minus 1 minus u3k1113872 1113873θQk

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where P0 ge 0 M0 ge 0 H0 ge 0 Tr0 ge 0 T

p0 ge 0 and Q0 ge 0

e optimal control problem to minimize the objectivefunctional is given by

J u1 u2 u3( 1113857 HT minus TrT minus T

p

T minus QT

+ 1113944

Tminus 1

k0Hk minus T

rk minus T

p

k minus Qk1113872

+Aku2

1k

2+

Bku22k

2+

Cku23k

21113889

(3)

where the parameters Ak gt 0 Bk gt 0 and Ck gt 0 are selectedto weigh the relative importance of the cost of awarenessprograms encouragement programs and follow-up andpsychological support respectively

e aim is to find an optimal control ulowast1 ulowast2 and ulowast3 -such that

J ulowast1 ulowast2 ulowast3( 1113857 min

u1 u2 u3( )isinUad

J u1 u2 u3( 1113857 (4)

where Uad is the set of admissible controls defined by

Uad uj uj0 uj1 ujTminus 11113872 1113873

0le ujmin le ujk le ujmax le 1

⎧⎨

⎩

for j 1 2 3 k 0 1 T minus 11113897

(5)

e sufficient condition for the existence of an optimalcontrol (ulowast1 ulowast2 ulowast3 ) for problems (2) and (3) comes from thefollowing theorem

Theorem 1 5ere exists the optimal control (ulowast1 ulowast2 ulowast3 )

such that

J ulowast1 ulowast2 ulowast3( 1113857 min

u1 u2 u3( )isinUad

J u1 u2 u3( 1113857 (6)

subject to the control system (2) with initial conditions

Proof Since the coefficients of the state equations arebounded and there are a finite number of time stepsP (P0 P1 P2 PT) M (M0 M1 M2 MT) H

(H0 H1 H2 HT) Tr (Tr0 Tr

1 Tr2 Tr

T) Tp

(Tp0 T

p1 T

p2 T

p

T) and Q (Q0 Q1 Q2 QT) areuniformly bounded for all (u1 u2 u3) in the control set Uadthus J(u1 u2 u3) is bounded for all (u1 u2 u3) isin Uad SinceJ(u1 u2 u3) is bounded inf(u1 u2 u3)isinUad

J(u1 u2 u3) is finiteand there exists a sequence (u

j1 u

j2 u

j3) isin Uad such that

limj⟶+infin(uj1 u

j2 u

j3) inf(u1 u2u3)isinUad

J(u1 u2 u3) and cor-responding sequences of states Pj Mj Hj Trj Tpj and QjSince there are a finite number of uniformly bounded se-quences there exist (ulowast1 ulowast2 ulowast3 ) isin Uad and Plowast Mlowast Hlowast

Trlowast Tplowast and Qlowast isin IRT+1 such that on a subsequencelimj⟶+infin(u

j1 u

j2 u

j3) (ulowast1 ulowast2 ulowast3 ) limj⟶+infinPj Plowast

limj⟶+infinMj Mlowast limj⟶+infinHj Hlowast limj⟶+infinTrj

Trlowast limj⟶+infinTpj Tplowast and limj⟶+infinQj Qlowast Finallydue to the finite dimensional structure of system (2) and theobjective function J(u1 u2 u3) (ulowast1 ulowast2 ulowast3 ) is an optimalcontrol with corresponding states Plowast Mlowast Hlowast Trlowast Tplowast andQlowast erefore inf(u1 u2 u3)isinUad

J(u1 u2 u3) is achieved

4 Characterization of the Optimal Controls

We apply the discrete version of Pontryaginrsquos maximumprinciple [13ndash18] e key idea is introducing the adjointfunction to attach the system of difference equations to theobjective functional resulting in the formation of a functioncalled the Hamiltonian is principle converts the problem

4 Journal of Applied Mathematics

of finding the control to optimize the objective functionalsubject to the state difference equation with initial conditionto find the control to optimize Hamiltonian pointwise (withrespect to the control)

Now we have the Hamiltonian 1113954H at time step k definedby

1113954Hk Hk minus Trk minus T

p

k minus Qk +Aku2

1k

2+

Bku22k

2+

Cku23k

2

+ 11139446

i1λik+1fik+1

(7)

where fik+1 is the right-hand side of the system of differenceequation (2) of the ith state variable at time step k + 1

Theorem 2 Given an optimal control (ulowast1 ulowast2 ulowast3 ) isin Uadand solutions Plowastk M

lowastk Hlowastk T

rlowastk Tplowast

k and Qlowastk of correspondingstate system [13] there exist adjoint functions λ1 λ2 λ3 λ4 λ5and λ6 satisfying the following equations

λ1k z 1113954Hk

zPk

λ2k+1 minus λ1k+11113872 1113873β1Mk

Nk

+ λ6k+1 minus λ1k+11113872 1113873u1k + λ1k+1(1 minus μ)

λ2k z 1113954Hk

zMk

λ2k+1 minus λ1k+11113872 1113873β1Pk

Nk

+ β2λ3k+1 + λ2k+1 1 minus μ minus β2( 1113857

λ3k z 1113954Hk

zHk

1 + λ3k+1 1 minus μ minus δ minus α1 minus α2 minus α3 minus u2k1113872 1113873 + λ4k+1 α1 + ϵu2k1113872 1113873 + λ5k+1 α2 +(1 minus ϵ)u2kHk1113960 1113961 + α3λ6k+1

λ4k z 1113954Hk

zTrk

minus 1 + c1λ6k+1 + λ4k+1 1 minus μ minus c1( 1113857

λ5k z 1113954Hk

zTp

k

minus 1 + c2λ6k+1 + λ5k+1 1 minus μ minus c2( 1113857

λ6k z 1113954Hk

zQk

minus 1 + λ2k+1 1 minus u3k1113872 1113873θ + λ6k+1 1 minus μ minus θ + θu3k1113872 1113873

(8)

with the transversality conditions at time T

λ1(T) 0

λ2(T) 0

λ3(T) 1

λ4(T) minus 1

λ5(T) minus 1

λ6(T) minus 1

(9)

Furthermore for k 0 1 T minus 1 we obtain the opti-mal control (ulowast1k ulowast2k ulowast3k) as

ulowast1k min max

λ1k+1 minus λ6k+11113872 1113873Pk

Ak

u1min⎧⎨

⎩

⎫⎬

⎭ u1max⎧⎨

⎩

⎫⎬

⎭

ulowast2k min max

λ3k+1 minus λ5k+11113872 1113873Hk + λ5k+1 minus λ4k+11113872 1113873ϵHk

Bk

u2min⎧⎨

⎩

⎫⎬

⎭ u2max⎧⎨

⎩

⎫⎬

⎭

ulowast3k min max

λ2k+1 minus λ6k+11113872 1113873θQk

Ck

u3min⎧⎨

⎩

⎫⎬

⎭ u3max⎧⎨

⎩

⎫⎬

⎭

(10)

Journal of Applied Mathematics 5

Proof e Hamiltonian 1113954Hk at time step k is given by

1113954Hk Hk minus Trk minus T

p

k minus Qk +Aku2

1k

2+

Bku22k

2+

Cku23k

2

+ λ1k+1 b minus β1PkMk

Nk

+(1 minus μ)Pk minus u1kPk1113890 1113891

+ λ2k+1 β1PkMk

Nk

+ 1 minus u3k1113872 1113873θQk + 1 minus μ minus β2( 1113857Mk1113890 1113891

+ λ3k+1 β2Mk + 1 minus μ minus δ minus α1 minus α2 minus α3( 1113857Hk minus u2kHk1113960 1113961

+ λ4k+1 α1Hk + 1 minus μ minus c1( 1113857Trk + ϵu2kHk1113960 1113961

+ λ5k+1 α2Hk + 1 minus μ minus c2( 1113857Tp

k +(1 minus ϵ)u2kHk1113960 1113961

+ λ6k+1 c1Trk + α3Hk + c2T

p

k +(1 minus μ)Qk + u1kPk minus 1 minus u3k1113872 1113873θQk1113960 1113961

(11)

For k 0 1 T minus 1 the adjoint equations andtransversality conditions can be obtained by using Pon-tryaginrsquos maximum principle in discrete time given in[13ndash18] such that

λ1k z 1113954Hk

zPk

λ1(T)

λ2k z 1113954Hk

zMk

λ2(T)

λ3k z 1113954Hk

zHk

λ3(T)

λ4k z 1113954Hk

zTrk

λ4(T)

λ5k z 1113954Hk

zTp

k

λ5(T)

λ6k z 1113954Hk

zQk

λ6(T)

(12)

For k 0 1 T minus 1 the optimal controls ulowast1k ulowast2kand ulowast3k can be solved from the optimality condition

z 1113954Hk

zu1k

Aku1k minus λ1k+1Pk + λ6k+1Pk 0

z 1113954Hk

zu2k

Bku2k minus λ3k+1Hk + λ4k+1ϵHk + λ5k+1(1 minus ϵ)Hk 0

z 1113954Hk

zu3k

Cku3k minus λ2k+1θQk + λ6k+1θQk 0

(13)

So we have

u1k λ1k+1 minus λ6k+11113872 1113873Pk

Ak

u2k λ3k+1 minus λ5k+11113872 1113873Hk + λ5k+1 minus λ4k+11113872 1113873ϵHk

Bk

u3k λ2k+1 minus λ6k+11113872 1113873θQk

Ck

(14)

By the bounds in Uad of the controls it is easy to obtainulowast1k ulowast2k and ulowast3k in the form of (10)

5 Numerical Simulation

In this section we shall solve numerically the optimalcontrol problem for our PMHTrTpQ model Here weobtain the optimality system from the state and adjointequationse proposed optimal control strategy is obtainedby solving the optimal system which consists of six differ-ence equations and boundary conditions e optimalitysystem can be solved by using an iterative method Using aninitial guess for the control variables u1k u2k and u3k thestate variables P M H Tr Tp and Q are solved forwardand the adjoint variables λi for i 1 2 3 4 5 6 are solvedbackwards at time steps k 0 and k T If the new values ofthe state and adjoint variables differ from the previousvalues the new values are used to update u1k u2k and u3kand the process is repeated until the system converges

e numerical solution of model (1) is executed usingMatlab with the following parameter values and initial valuesof state variable in Table 1

We begin by presenting the solution evolution of ourmodel (1) without controls that are represented in Figure 2

6 Journal of Applied Mathematics

Figure 2(a) shows that the number of moderate drinkersincreases in the beginning but then decreases clearly isdecrease leads to an increase in the number of heavy drinkersfrom the beginning and approaches a given value of 46230as we see in Figure 2(b) Finally Figure 2(c) and Figures 2(d)and 2(e) show that the number of the rich and poor heavydrinkers joining private and public centers for alcohol ad-diction treatment and the quitters of drinking decreases andapproaches certain unacceptable values

In order to display the influence of alcohol treatmentcenters on the population drinking dynamics we presentsome figures about the effects of the parameters α1 α2 c1and c2 on system (1)

Figure 2(f ) shows that the number of heavy drinkersincreases when the number of the heavy drinkers who joinaddiction treatment centers decreases Also we observe inFigure 2(g) that the number of quitters of drinking increaseswhen the number of the heavy drinkers who join addictiontreatment centers increases

From these two figures the importance of addictiontreatment centers is clear in reducing the number of heavydrinkers and thus increasing the number of quitters ofdrinking erefore we need to make an effort to introducethese centers to people and encourage heavy drinkers to jointhem and this is shown through the following objectives

51 First Objective Protecting and Preventing PotentialDrinkers from Falling into Alcohol Addiction To realize thisobjective we apply only the control u1 ie the imple-mentation of awareness information and educationalprograms on potential drinkers and making them know therisks of this phenomenon and the resulting health and socialdamages Figure 3(a) shows that the number of moderatedrinkers decreases from 23116 (without control u1) to12459 (with control u1) at the end of the proposed programand the number of heavy drinkers decreases from 46229(without control u1) to 24902 (with control u1) at the end ofthe proposed control strategy (see Figure 3(b)) Also weobserve in Figure 3(c) that the number of people who quitdrinking increases and has reached the value 42978 (withcontrol u1) compared to the situation when there is nocontrol 568 at the end of the proposed strategy So ourobjective has been achieved

52 Second Objective Increasing the Number of HeavyDrinkers Who Join Treatment Centers of Alcohol AddictionTo achieve this objective we only use the control u2 ieencouraging heavy drinkers to know and join public andprivate treatment centers In Figure 4(a) it is observed thatthere is a significant decrease in the number of heavydrinkers with control compared to a situation when there isno control where the decrease reaches 4120 at the end ofthe proposed control strategy Figure 4(b) shows that thenumber of the poor heavy drinkers who join public treat-ment centers increased from 691 (Tp without control u2) to5970 (Tp with control u2) and that the number of rich heavydrinkers who join private treatment centers increased from706 (without control u2) to 23372 (with control u2) at the

end of the proposed control (see Figure 4(c)) and Figure 4(d)shows that the number of people who quit drinking withoutcontrol u2 decreases and approaches a value of 568 After 5days a slight increase appears with control u2 is increaseis not enough as the number of people who quit drinkingbecomes moderate drinkers To improve this result we canadd another control that represents follow-up for temporaryquitters of drinking

53 5ird Objective Encouraging Heavy Drinkers to JoinTreatment Centers of Alcohol Addiction and Following Up theQuitters of Drinking In order to accomplish this aim weuse the controls u2 and u3 ie encouraging heavydrinkers to know and join public and private treatmentcenters and following up the temporary quitters ofdrinking e number of the poor heavy drinkers who joinpublic treatment centers increases from 691 (withoutcontrols) to 3217 (with controls) at the end of the pro-posed strategy (see Figure 5(c)) Figure 5(d) demonstratesthat the number of the rich heavy drinkers who joinprivate treatment centers increases from 706 (withoutcontrols) to 12592 (with controls) at the end of theproposed strategy e number of the quitters of drinkingin Figure 5(e) increases significantly from 568 (withoutcontrols) to 43011 (with controls) rough the use ofthose controls the abovementioned objective has beenachieved

54 Fourth Objective Prevention Treatment and Follow-Upfor the Addicted Individuals To meet this objective we usethe controls u1 u2 and u3 ie awareness programs for thepotential drinkers encouraging heavy drinkers to know andjoin public and private treatment centers and follow-up forthe quitters of drinking Figure 6(a) shows that the numberof the moderate drinkers decreases starting from the earlydays by 3460 Also Figure 6(b) shows that the number ofthe heavy drinkers increases starting from the early days butthen decreases from 46229 (without controls) to 6581 (withcontrols) e number of the rich and poor heavy drinkerswho join private and public treatment centers increases by1145 and 298 respectively (see Figures 6(c) and 6(d))Figure 6(e) depicts clearly an increase in the number of thequitters of drinking from 568 (without controls) to 55063(with controls) As a result the objective set before has beenachieved

We remark that the numbers of the rich and poor heavydrinkers (Tr and Tp) decrease when we applied threecontrols (u1 u2 u3) compared to using only the control(u2) is decrease is due to the positive effect of the controlu1 on the compartment P and the control u3 on θQ (tem-porary quitters) which prevents temporary quitters of

Table 1 e parameters used for model (1)

P0 M0 H0 Tr0 T

p0 Q0 b N δ

500 300 100 60 30 10 65 1000 0002μ β1 β2 α1 α2 α3 c1 c2 θ0065 075 014 0001 0001 0001 0001 0002 002

Journal of Applied Mathematics 7

M without controls

200

250

300

350

400

450M

oder

ate d

rinke

rs (M

)

10 20 30 40 50 60 70 80 90 100Time (days)

(a)

H without controls

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

10 20 30 40 50 60 70 80 90 100Time (days)

(b)

Tp without controls

10 20 30 40 50 60 70 80 90 100Time (days)

5

10

15

20

25

30

The p

oor h

eavy

drin

kers

(Tp )

(c)

Tr without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

The r

ich

heav

y dr

inke

rs (T

r )

(d)

Q without controls

55556

657

758

859

9510

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

(e)

100 20 30 40 50 60 70 80 90 100Time (days)

50100150200250300350400450500

Hea

vy d

rinke

rs (H

)

H without treatment centers of alcohol addictionH with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002H with α1 = 03 α2 = 01 γ1 = 006 γ2 = 003H with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02H with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03

(f )

Figure 2 Continued

8 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

Q with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03Q with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02Q with α1 = 03 α2 = 01 γ1 = 005 γ2 = 02Q with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002Q without treatment centers of alcohol addiction

0

50

100

150

200

250

300

350

400

Qui

tters

of d

rinki

ng (Q

)

(g)

Figure 2 Representing the drinkers class without control

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

M with control u1M without controls

(a)

H with control u1H without controls

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

(b)

Q with control u1Q without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

(c)

Figure 3 Representing the drinkers class with and without control u1

Journal of Applied Mathematics 9

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500H

eavy

drin

kers

(H)

H with control u2H without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

e p

oor h

eavy

drin

kers

(Tp )

Tp with control u2Tp without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

e r

ich

heav

y dr

inke

rs (T

r )

Tr with control u2Tr without controls

(c)

55

6

65

7

75

8

85

9

95

10

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

Q with control u2Q without controls

(d)

Figure 4 Representing the drinkers class with and without control u2

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

The p

oor h

eavy

drin

kers

(Tp )

Tp with controls u2 and u3Tp without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

The r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u2 and u3Tr without controls

(b)

Figure 5 Continued

10 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

Q with controls u2 and u3Q without controls

(c)

Figure 5 Representing the drinkers class with controls u2 and u3 and without controls

50

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

10 20 30 40 50 60 70 80 90 100Time (days)

M with controls u1 u2 and u3M without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

50

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

H with controls u1 u2 and u3H without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

510152025303540455055

e p

oor h

eavy

drin

kers

(Tp )

Tp with controls u1 u2 and u3Tp without controls

(c)

10 20 30 40 50 60 70 80 90 100Time (days)

0

20

40

60

80

100

120

140

160

180

e r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u1 u2 and u3Tr without controls

(d)

Figure 6 Continued

Journal of Applied Mathematics 11

drinking from returning to drinking So the number ofindividuals that enter the compartment Tr and Tp de-creases erefore applying a combination of the threecontrols is more effective than one control

6 Conclusion

In this research paper we introduced a discrete modelingof drinking for the purpose of minimizing the number ofdrinkers and maximizing the number of the rich and poorheavy drinkers who join private and public treatmentcenters of alcohol addiction and subsequently the numberof quitters of drinking Unlike some other previousmodels we have taken into account the impact of privateand public addiction treatment centers on alcoholics eresults showed that those centers have substantial influ-ence on the dynamics of alcoholism and can greatlyimpact the spread of drinking us it is crucial to urgepeople to know and join private and public addictiontreatment centers to quit drinking We also presentedthree controls which respectively represent awarenessprograms encouragement and follow-up We applied theresults of the control theory and we managed to obtain thecharacterizations of the optimal controls e numericalsimulation of the obtained results showed the effective-ness of the proposed control strategies

Data Availability

e disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpnetworkrepositorycomprofilephp)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] WHOWorld Health Organization Report on Drinking WHOGeneva Switzerland 2016

[2] World Health Organization Alcohol World Health Organi-zation Geneva Switzerland 2018 httpswwwwhointnews-roomfact-sheetsdetailalcohol

[3] H F Huo and Q Wang ldquoModelling the influence ofawareness programs by media on the drinking dynamicsrdquoAbstract and Applied Analysis vol 2014 Article ID 9380808 pages 2014

[4] S-H Ma H-F Huo and X-Y Meng ldquoModelling alcoholismas a contagious disease a mathematical model with awarenessprograms and time delayrdquo Discrete Dynamics in Nature andSociety vol 2015 Article ID 260195 13 pages 2015

[5] X-Y Wang H-F Huo Q-K Kong andW-X Shi ldquoOptimalcontrol strategies in an alcoholism modelrdquo Abstract andApplied Analysis vol 2014 Article ID 954069 18 pages 2014

[6] H-F Huo S-R Huang X-Y Wang and H Xiang ldquoOptimalcontrol of a social epidemic model with media coveragerdquoJournal of Biological Dynamics vol 11 no 1 pp 226ndash2432017

[7] H F Huo and N N Song ldquoGlobal stability for a bingedrinking model with two stagesrdquo Discrete Dynamics in Natureand Society vol 2012 Article ID 829386 15 pages 2012

[8] I K Adu M Osman and C Yang ldquoMathematical model ofdrinking epidemicrdquo British Journal of Mathematics ComputerScience vol 22 no 5 pp 1ndash10 2017

[9] H-F Huo and Y-P Liu ldquoe analysis of the SIRS alcoholismmodels with relapse on weightedrdquo SpringerPlus vol 5 no 1p 722 2016

[10] K Mursquotamar ldquoOptimal control strategy for alcoholism modelwith two infected compartmentsrdquo IOSR Journal of Mathe-matics vol 14 no 3 pp 58ndash67 2018

[11] S Sharma and G P Samanta ldquoDrinking as an epidemic amathematical model with dynamic behaviourrdquo Journal ofApplied Mathematics amp Informatics vol 31 no 1-2 pp 1ndash252013

[12] S Sharma and G P Samanta ldquoAnalysis of a drinking epi-demic modelrdquo International Journal of Dynamics and Controlvol 3 no 3 pp 288ndash305 2015

10 20 30 40 50 60 70 80 90 100Time (days)

0

100

200

300

400

500

600

Qui

tters

of d

rinki

ng (Q

)

Q with controls u1 u2 and u3Q without controls

(e)

Figure 6 Representing the drinkers class with and without optimal controls u1 u2 and u3

12 Journal of Applied Mathematics

the temporary quitters of drinking who moved to moderatedrinking at time step k

e total population size at time k is denoted by Nk withNk Pk + Mk + Hk + Tr

k + Tp

k + Qk and it is supposed to beconstant

3 The Optimal Control Problem

e objective of the proposed control strategy is to minimizethe number of heavy drinkers (H) and maximize thenumber of the rich heavy drinkers (Tr) and the poor heavydrinkers (Tp) who join private or public treatment centersof alcohol addiction and the number of the permanentquitters of drinking (Q) in the community during the timesteps k 0 to k T e cost spent in the awareness pro-grams is also to be minimized

In order to achieve these objectives we introduce threecontrol variables e first control u1 represents the effort ofthe awareness programs to protect the potential drinkers notto be drinkers e second control u2 measures the effortmade to make the heavy drinkers know private and publictreatment centers of alcohol addiction and to encourage themto join those centers We note that the control function ϵu2represents the fraction of the heavy drinkers who will betreated in private addiction treatment centers and the fraction(1 minus ϵ)u2 of those leaving the heavy drinkers class who willreceive treatment in public addiction treatment centers ethird control u3 represents the effort of the follow-up and thepsychological support for the temporary quitters of drinkingwith a purpose of protecting them to not go back to drinking

So the controlled mathematical system is given by thefollowing system of difference equations

Pk+1 b minus β1PkMk

Nk

+(1 minus μ)Pk minus u1kPk

Mk+1 β1PkMk

Nk

+ 1 minus u3k1113872 1113873θQk + 1 minus μ minus β2( 1113857Mk

Hk+1 β2Mk + 1 minus μ minus δ minus α1 minus α2 minus α3( 1113857Hk minus u2kHk

Trk+1 α1Hk + 1 minus μ minus c1( 1113857Tr

k + ϵu2kHk

Tp

k+1 α2Hk + 1 minus μ minus c2( 1113857Tp

k +(1 minus ϵ)u2kHk

Qk+1 c1Trk + α3Hk + c2T

p

k +(1 minus μ)Qk + u1kPk minus 1 minus u3k1113872 1113873θQk

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where P0 ge 0 M0 ge 0 H0 ge 0 Tr0 ge 0 T

p0 ge 0 and Q0 ge 0

e optimal control problem to minimize the objectivefunctional is given by

J u1 u2 u3( 1113857 HT minus TrT minus T

p

T minus QT

+ 1113944

Tminus 1

k0Hk minus T

rk minus T

p

k minus Qk1113872

+Aku2

1k

2+

Bku22k

2+

Cku23k

21113889

(3)

where the parameters Ak gt 0 Bk gt 0 and Ck gt 0 are selectedto weigh the relative importance of the cost of awarenessprograms encouragement programs and follow-up andpsychological support respectively

e aim is to find an optimal control ulowast1 ulowast2 and ulowast3 -such that

J ulowast1 ulowast2 ulowast3( 1113857 min

u1 u2 u3( )isinUad

J u1 u2 u3( 1113857 (4)

where Uad is the set of admissible controls defined by

Uad uj uj0 uj1 ujTminus 11113872 1113873

0le ujmin le ujk le ujmax le 1

⎧⎨

⎩

for j 1 2 3 k 0 1 T minus 11113897

(5)

e sufficient condition for the existence of an optimalcontrol (ulowast1 ulowast2 ulowast3 ) for problems (2) and (3) comes from thefollowing theorem

Theorem 1 5ere exists the optimal control (ulowast1 ulowast2 ulowast3 )

such that

J ulowast1 ulowast2 ulowast3( 1113857 min

u1 u2 u3( )isinUad

J u1 u2 u3( 1113857 (6)

subject to the control system (2) with initial conditions

Proof Since the coefficients of the state equations arebounded and there are a finite number of time stepsP (P0 P1 P2 PT) M (M0 M1 M2 MT) H

(H0 H1 H2 HT) Tr (Tr0 Tr

1 Tr2 Tr

T) Tp

(Tp0 T

p1 T

p2 T

p

T) and Q (Q0 Q1 Q2 QT) areuniformly bounded for all (u1 u2 u3) in the control set Uadthus J(u1 u2 u3) is bounded for all (u1 u2 u3) isin Uad SinceJ(u1 u2 u3) is bounded inf(u1 u2 u3)isinUad

J(u1 u2 u3) is finiteand there exists a sequence (u

j1 u

j2 u

j3) isin Uad such that

limj⟶+infin(uj1 u

j2 u

j3) inf(u1 u2u3)isinUad

J(u1 u2 u3) and cor-responding sequences of states Pj Mj Hj Trj Tpj and QjSince there are a finite number of uniformly bounded se-quences there exist (ulowast1 ulowast2 ulowast3 ) isin Uad and Plowast Mlowast Hlowast

Trlowast Tplowast and Qlowast isin IRT+1 such that on a subsequencelimj⟶+infin(u

j1 u

j2 u

j3) (ulowast1 ulowast2 ulowast3 ) limj⟶+infinPj Plowast

limj⟶+infinMj Mlowast limj⟶+infinHj Hlowast limj⟶+infinTrj

Trlowast limj⟶+infinTpj Tplowast and limj⟶+infinQj Qlowast Finallydue to the finite dimensional structure of system (2) and theobjective function J(u1 u2 u3) (ulowast1 ulowast2 ulowast3 ) is an optimalcontrol with corresponding states Plowast Mlowast Hlowast Trlowast Tplowast andQlowast erefore inf(u1 u2 u3)isinUad

J(u1 u2 u3) is achieved

4 Characterization of the Optimal Controls

We apply the discrete version of Pontryaginrsquos maximumprinciple [13ndash18] e key idea is introducing the adjointfunction to attach the system of difference equations to theobjective functional resulting in the formation of a functioncalled the Hamiltonian is principle converts the problem

4 Journal of Applied Mathematics

of finding the control to optimize the objective functionalsubject to the state difference equation with initial conditionto find the control to optimize Hamiltonian pointwise (withrespect to the control)

Now we have the Hamiltonian 1113954H at time step k definedby

1113954Hk Hk minus Trk minus T

p

k minus Qk +Aku2

1k

2+

Bku22k

2+

Cku23k

2

+ 11139446

i1λik+1fik+1

(7)

where fik+1 is the right-hand side of the system of differenceequation (2) of the ith state variable at time step k + 1

Theorem 2 Given an optimal control (ulowast1 ulowast2 ulowast3 ) isin Uadand solutions Plowastk M

lowastk Hlowastk T

rlowastk Tplowast

k and Qlowastk of correspondingstate system [13] there exist adjoint functions λ1 λ2 λ3 λ4 λ5and λ6 satisfying the following equations

λ1k z 1113954Hk

zPk

λ2k+1 minus λ1k+11113872 1113873β1Mk

Nk

+ λ6k+1 minus λ1k+11113872 1113873u1k + λ1k+1(1 minus μ)

λ2k z 1113954Hk

zMk

λ2k+1 minus λ1k+11113872 1113873β1Pk

Nk

+ β2λ3k+1 + λ2k+1 1 minus μ minus β2( 1113857

λ3k z 1113954Hk

zHk

1 + λ3k+1 1 minus μ minus δ minus α1 minus α2 minus α3 minus u2k1113872 1113873 + λ4k+1 α1 + ϵu2k1113872 1113873 + λ5k+1 α2 +(1 minus ϵ)u2kHk1113960 1113961 + α3λ6k+1

λ4k z 1113954Hk

zTrk

minus 1 + c1λ6k+1 + λ4k+1 1 minus μ minus c1( 1113857

λ5k z 1113954Hk

zTp

k

minus 1 + c2λ6k+1 + λ5k+1 1 minus μ minus c2( 1113857

λ6k z 1113954Hk

zQk

minus 1 + λ2k+1 1 minus u3k1113872 1113873θ + λ6k+1 1 minus μ minus θ + θu3k1113872 1113873

(8)

with the transversality conditions at time T

λ1(T) 0

λ2(T) 0

λ3(T) 1

λ4(T) minus 1

λ5(T) minus 1

λ6(T) minus 1

(9)

Furthermore for k 0 1 T minus 1 we obtain the opti-mal control (ulowast1k ulowast2k ulowast3k) as

ulowast1k min max

λ1k+1 minus λ6k+11113872 1113873Pk

Ak

u1min⎧⎨

⎩

⎫⎬

⎭ u1max⎧⎨

⎩

⎫⎬

⎭

ulowast2k min max

λ3k+1 minus λ5k+11113872 1113873Hk + λ5k+1 minus λ4k+11113872 1113873ϵHk

Bk

u2min⎧⎨

⎩

⎫⎬

⎭ u2max⎧⎨

⎩

⎫⎬

⎭

ulowast3k min max

λ2k+1 minus λ6k+11113872 1113873θQk

Ck

u3min⎧⎨

⎩

⎫⎬

⎭ u3max⎧⎨

⎩

⎫⎬

⎭

(10)

Journal of Applied Mathematics 5

Proof e Hamiltonian 1113954Hk at time step k is given by

1113954Hk Hk minus Trk minus T

p

k minus Qk +Aku2

1k

2+

Bku22k

2+

Cku23k

2

+ λ1k+1 b minus β1PkMk

Nk

+(1 minus μ)Pk minus u1kPk1113890 1113891

+ λ2k+1 β1PkMk

Nk

+ 1 minus u3k1113872 1113873θQk + 1 minus μ minus β2( 1113857Mk1113890 1113891

+ λ3k+1 β2Mk + 1 minus μ minus δ minus α1 minus α2 minus α3( 1113857Hk minus u2kHk1113960 1113961

+ λ4k+1 α1Hk + 1 minus μ minus c1( 1113857Trk + ϵu2kHk1113960 1113961

+ λ5k+1 α2Hk + 1 minus μ minus c2( 1113857Tp

k +(1 minus ϵ)u2kHk1113960 1113961

+ λ6k+1 c1Trk + α3Hk + c2T

p

k +(1 minus μ)Qk + u1kPk minus 1 minus u3k1113872 1113873θQk1113960 1113961

(11)

For k 0 1 T minus 1 the adjoint equations andtransversality conditions can be obtained by using Pon-tryaginrsquos maximum principle in discrete time given in[13ndash18] such that

λ1k z 1113954Hk

zPk

λ1(T)

λ2k z 1113954Hk

zMk

λ2(T)

λ3k z 1113954Hk

zHk

λ3(T)

λ4k z 1113954Hk

zTrk

λ4(T)

λ5k z 1113954Hk

zTp

k

λ5(T)

λ6k z 1113954Hk

zQk

λ6(T)

(12)

For k 0 1 T minus 1 the optimal controls ulowast1k ulowast2kand ulowast3k can be solved from the optimality condition

z 1113954Hk

zu1k

Aku1k minus λ1k+1Pk + λ6k+1Pk 0

z 1113954Hk

zu2k

Bku2k minus λ3k+1Hk + λ4k+1ϵHk + λ5k+1(1 minus ϵ)Hk 0

z 1113954Hk

zu3k

Cku3k minus λ2k+1θQk + λ6k+1θQk 0

(13)

So we have

u1k λ1k+1 minus λ6k+11113872 1113873Pk

Ak

u2k λ3k+1 minus λ5k+11113872 1113873Hk + λ5k+1 minus λ4k+11113872 1113873ϵHk

Bk

u3k λ2k+1 minus λ6k+11113872 1113873θQk

Ck

(14)

By the bounds in Uad of the controls it is easy to obtainulowast1k ulowast2k and ulowast3k in the form of (10)

5 Numerical Simulation

In this section we shall solve numerically the optimalcontrol problem for our PMHTrTpQ model Here weobtain the optimality system from the state and adjointequationse proposed optimal control strategy is obtainedby solving the optimal system which consists of six differ-ence equations and boundary conditions e optimalitysystem can be solved by using an iterative method Using aninitial guess for the control variables u1k u2k and u3k thestate variables P M H Tr Tp and Q are solved forwardand the adjoint variables λi for i 1 2 3 4 5 6 are solvedbackwards at time steps k 0 and k T If the new values ofthe state and adjoint variables differ from the previousvalues the new values are used to update u1k u2k and u3kand the process is repeated until the system converges

e numerical solution of model (1) is executed usingMatlab with the following parameter values and initial valuesof state variable in Table 1

We begin by presenting the solution evolution of ourmodel (1) without controls that are represented in Figure 2

6 Journal of Applied Mathematics

Figure 2(a) shows that the number of moderate drinkersincreases in the beginning but then decreases clearly isdecrease leads to an increase in the number of heavy drinkersfrom the beginning and approaches a given value of 46230as we see in Figure 2(b) Finally Figure 2(c) and Figures 2(d)and 2(e) show that the number of the rich and poor heavydrinkers joining private and public centers for alcohol ad-diction treatment and the quitters of drinking decreases andapproaches certain unacceptable values

In order to display the influence of alcohol treatmentcenters on the population drinking dynamics we presentsome figures about the effects of the parameters α1 α2 c1and c2 on system (1)

Figure 2(f ) shows that the number of heavy drinkersincreases when the number of the heavy drinkers who joinaddiction treatment centers decreases Also we observe inFigure 2(g) that the number of quitters of drinking increaseswhen the number of the heavy drinkers who join addictiontreatment centers increases

From these two figures the importance of addictiontreatment centers is clear in reducing the number of heavydrinkers and thus increasing the number of quitters ofdrinking erefore we need to make an effort to introducethese centers to people and encourage heavy drinkers to jointhem and this is shown through the following objectives

51 First Objective Protecting and Preventing PotentialDrinkers from Falling into Alcohol Addiction To realize thisobjective we apply only the control u1 ie the imple-mentation of awareness information and educationalprograms on potential drinkers and making them know therisks of this phenomenon and the resulting health and socialdamages Figure 3(a) shows that the number of moderatedrinkers decreases from 23116 (without control u1) to12459 (with control u1) at the end of the proposed programand the number of heavy drinkers decreases from 46229(without control u1) to 24902 (with control u1) at the end ofthe proposed control strategy (see Figure 3(b)) Also weobserve in Figure 3(c) that the number of people who quitdrinking increases and has reached the value 42978 (withcontrol u1) compared to the situation when there is nocontrol 568 at the end of the proposed strategy So ourobjective has been achieved

52 Second Objective Increasing the Number of HeavyDrinkers Who Join Treatment Centers of Alcohol AddictionTo achieve this objective we only use the control u2 ieencouraging heavy drinkers to know and join public andprivate treatment centers In Figure 4(a) it is observed thatthere is a significant decrease in the number of heavydrinkers with control compared to a situation when there isno control where the decrease reaches 4120 at the end ofthe proposed control strategy Figure 4(b) shows that thenumber of the poor heavy drinkers who join public treat-ment centers increased from 691 (Tp without control u2) to5970 (Tp with control u2) and that the number of rich heavydrinkers who join private treatment centers increased from706 (without control u2) to 23372 (with control u2) at the

end of the proposed control (see Figure 4(c)) and Figure 4(d)shows that the number of people who quit drinking withoutcontrol u2 decreases and approaches a value of 568 After 5days a slight increase appears with control u2 is increaseis not enough as the number of people who quit drinkingbecomes moderate drinkers To improve this result we canadd another control that represents follow-up for temporaryquitters of drinking

53 5ird Objective Encouraging Heavy Drinkers to JoinTreatment Centers of Alcohol Addiction and Following Up theQuitters of Drinking In order to accomplish this aim weuse the controls u2 and u3 ie encouraging heavydrinkers to know and join public and private treatmentcenters and following up the temporary quitters ofdrinking e number of the poor heavy drinkers who joinpublic treatment centers increases from 691 (withoutcontrols) to 3217 (with controls) at the end of the pro-posed strategy (see Figure 5(c)) Figure 5(d) demonstratesthat the number of the rich heavy drinkers who joinprivate treatment centers increases from 706 (withoutcontrols) to 12592 (with controls) at the end of theproposed strategy e number of the quitters of drinkingin Figure 5(e) increases significantly from 568 (withoutcontrols) to 43011 (with controls) rough the use ofthose controls the abovementioned objective has beenachieved

54 Fourth Objective Prevention Treatment and Follow-Upfor the Addicted Individuals To meet this objective we usethe controls u1 u2 and u3 ie awareness programs for thepotential drinkers encouraging heavy drinkers to know andjoin public and private treatment centers and follow-up forthe quitters of drinking Figure 6(a) shows that the numberof the moderate drinkers decreases starting from the earlydays by 3460 Also Figure 6(b) shows that the number ofthe heavy drinkers increases starting from the early days butthen decreases from 46229 (without controls) to 6581 (withcontrols) e number of the rich and poor heavy drinkerswho join private and public treatment centers increases by1145 and 298 respectively (see Figures 6(c) and 6(d))Figure 6(e) depicts clearly an increase in the number of thequitters of drinking from 568 (without controls) to 55063(with controls) As a result the objective set before has beenachieved

We remark that the numbers of the rich and poor heavydrinkers (Tr and Tp) decrease when we applied threecontrols (u1 u2 u3) compared to using only the control(u2) is decrease is due to the positive effect of the controlu1 on the compartment P and the control u3 on θQ (tem-porary quitters) which prevents temporary quitters of

Table 1 e parameters used for model (1)

P0 M0 H0 Tr0 T

p0 Q0 b N δ

500 300 100 60 30 10 65 1000 0002μ β1 β2 α1 α2 α3 c1 c2 θ0065 075 014 0001 0001 0001 0001 0002 002

Journal of Applied Mathematics 7

M without controls

200

250

300

350

400

450M

oder

ate d

rinke

rs (M

)

10 20 30 40 50 60 70 80 90 100Time (days)

(a)

H without controls

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

10 20 30 40 50 60 70 80 90 100Time (days)

(b)

Tp without controls

10 20 30 40 50 60 70 80 90 100Time (days)

5

10

15

20

25

30

The p

oor h

eavy

drin

kers

(Tp )

(c)

Tr without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

The r

ich

heav

y dr

inke

rs (T

r )

(d)

Q without controls

55556

657

758

859

9510

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

(e)

100 20 30 40 50 60 70 80 90 100Time (days)

50100150200250300350400450500

Hea

vy d

rinke

rs (H

)

H without treatment centers of alcohol addictionH with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002H with α1 = 03 α2 = 01 γ1 = 006 γ2 = 003H with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02H with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03

(f )

Figure 2 Continued

8 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

Q with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03Q with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02Q with α1 = 03 α2 = 01 γ1 = 005 γ2 = 02Q with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002Q without treatment centers of alcohol addiction

0

50

100

150

200

250

300

350

400

Qui

tters

of d

rinki

ng (Q

)

(g)

Figure 2 Representing the drinkers class without control

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

M with control u1M without controls

(a)

H with control u1H without controls

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

(b)

Q with control u1Q without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

(c)

Figure 3 Representing the drinkers class with and without control u1

Journal of Applied Mathematics 9

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500H

eavy

drin

kers

(H)

H with control u2H without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

e p

oor h

eavy

drin

kers

(Tp )

Tp with control u2Tp without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

e r

ich

heav

y dr

inke

rs (T

r )

Tr with control u2Tr without controls

(c)

55

6

65

7

75

8

85

9

95

10

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

Q with control u2Q without controls

(d)

Figure 4 Representing the drinkers class with and without control u2

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

The p

oor h

eavy

drin

kers

(Tp )

Tp with controls u2 and u3Tp without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

The r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u2 and u3Tr without controls

(b)

Figure 5 Continued

10 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

Q with controls u2 and u3Q without controls

(c)

Figure 5 Representing the drinkers class with controls u2 and u3 and without controls

50

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

10 20 30 40 50 60 70 80 90 100Time (days)

M with controls u1 u2 and u3M without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

50

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

H with controls u1 u2 and u3H without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

510152025303540455055

e p

oor h

eavy

drin

kers

(Tp )

Tp with controls u1 u2 and u3Tp without controls

(c)

10 20 30 40 50 60 70 80 90 100Time (days)

0

20

40

60

80

100

120

140

160

180

e r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u1 u2 and u3Tr without controls

(d)

Figure 6 Continued

Journal of Applied Mathematics 11

drinking from returning to drinking So the number ofindividuals that enter the compartment Tr and Tp de-creases erefore applying a combination of the threecontrols is more effective than one control

6 Conclusion

In this research paper we introduced a discrete modelingof drinking for the purpose of minimizing the number ofdrinkers and maximizing the number of the rich and poorheavy drinkers who join private and public treatmentcenters of alcohol addiction and subsequently the numberof quitters of drinking Unlike some other previousmodels we have taken into account the impact of privateand public addiction treatment centers on alcoholics eresults showed that those centers have substantial influ-ence on the dynamics of alcoholism and can greatlyimpact the spread of drinking us it is crucial to urgepeople to know and join private and public addictiontreatment centers to quit drinking We also presentedthree controls which respectively represent awarenessprograms encouragement and follow-up We applied theresults of the control theory and we managed to obtain thecharacterizations of the optimal controls e numericalsimulation of the obtained results showed the effective-ness of the proposed control strategies

Data Availability

e disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpnetworkrepositorycomprofilephp)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] WHOWorld Health Organization Report on Drinking WHOGeneva Switzerland 2016

[2] World Health Organization Alcohol World Health Organi-zation Geneva Switzerland 2018 httpswwwwhointnews-roomfact-sheetsdetailalcohol

[3] H F Huo and Q Wang ldquoModelling the influence ofawareness programs by media on the drinking dynamicsrdquoAbstract and Applied Analysis vol 2014 Article ID 9380808 pages 2014

[4] S-H Ma H-F Huo and X-Y Meng ldquoModelling alcoholismas a contagious disease a mathematical model with awarenessprograms and time delayrdquo Discrete Dynamics in Nature andSociety vol 2015 Article ID 260195 13 pages 2015

[5] X-Y Wang H-F Huo Q-K Kong andW-X Shi ldquoOptimalcontrol strategies in an alcoholism modelrdquo Abstract andApplied Analysis vol 2014 Article ID 954069 18 pages 2014

[6] H-F Huo S-R Huang X-Y Wang and H Xiang ldquoOptimalcontrol of a social epidemic model with media coveragerdquoJournal of Biological Dynamics vol 11 no 1 pp 226ndash2432017

[7] H F Huo and N N Song ldquoGlobal stability for a bingedrinking model with two stagesrdquo Discrete Dynamics in Natureand Society vol 2012 Article ID 829386 15 pages 2012

[8] I K Adu M Osman and C Yang ldquoMathematical model ofdrinking epidemicrdquo British Journal of Mathematics ComputerScience vol 22 no 5 pp 1ndash10 2017

[9] H-F Huo and Y-P Liu ldquoe analysis of the SIRS alcoholismmodels with relapse on weightedrdquo SpringerPlus vol 5 no 1p 722 2016

[10] K Mursquotamar ldquoOptimal control strategy for alcoholism modelwith two infected compartmentsrdquo IOSR Journal of Mathe-matics vol 14 no 3 pp 58ndash67 2018

[11] S Sharma and G P Samanta ldquoDrinking as an epidemic amathematical model with dynamic behaviourrdquo Journal ofApplied Mathematics amp Informatics vol 31 no 1-2 pp 1ndash252013

[12] S Sharma and G P Samanta ldquoAnalysis of a drinking epi-demic modelrdquo International Journal of Dynamics and Controlvol 3 no 3 pp 288ndash305 2015

10 20 30 40 50 60 70 80 90 100Time (days)

0

100

200

300

400

500

600

Qui

tters

of d

rinki

ng (Q

)

Q with controls u1 u2 and u3Q without controls

(e)

Figure 6 Representing the drinkers class with and without optimal controls u1 u2 and u3

12 Journal of Applied Mathematics

of finding the control to optimize the objective functionalsubject to the state difference equation with initial conditionto find the control to optimize Hamiltonian pointwise (withrespect to the control)

Now we have the Hamiltonian 1113954H at time step k definedby

1113954Hk Hk minus Trk minus T

p

k minus Qk +Aku2

1k

2+

Bku22k

2+

Cku23k

2

+ 11139446

i1λik+1fik+1

(7)

where fik+1 is the right-hand side of the system of differenceequation (2) of the ith state variable at time step k + 1

Theorem 2 Given an optimal control (ulowast1 ulowast2 ulowast3 ) isin Uadand solutions Plowastk M

lowastk Hlowastk T

rlowastk Tplowast

k and Qlowastk of correspondingstate system [13] there exist adjoint functions λ1 λ2 λ3 λ4 λ5and λ6 satisfying the following equations

λ1k z 1113954Hk

zPk

λ2k+1 minus λ1k+11113872 1113873β1Mk

Nk

+ λ6k+1 minus λ1k+11113872 1113873u1k + λ1k+1(1 minus μ)

λ2k z 1113954Hk

zMk

λ2k+1 minus λ1k+11113872 1113873β1Pk

Nk

+ β2λ3k+1 + λ2k+1 1 minus μ minus β2( 1113857

λ3k z 1113954Hk

zHk

1 + λ3k+1 1 minus μ minus δ minus α1 minus α2 minus α3 minus u2k1113872 1113873 + λ4k+1 α1 + ϵu2k1113872 1113873 + λ5k+1 α2 +(1 minus ϵ)u2kHk1113960 1113961 + α3λ6k+1

λ4k z 1113954Hk

zTrk

minus 1 + c1λ6k+1 + λ4k+1 1 minus μ minus c1( 1113857

λ5k z 1113954Hk

zTp

k

minus 1 + c2λ6k+1 + λ5k+1 1 minus μ minus c2( 1113857

λ6k z 1113954Hk

zQk

minus 1 + λ2k+1 1 minus u3k1113872 1113873θ + λ6k+1 1 minus μ minus θ + θu3k1113872 1113873

(8)

with the transversality conditions at time T

λ1(T) 0

λ2(T) 0

λ3(T) 1

λ4(T) minus 1

λ5(T) minus 1

λ6(T) minus 1

(9)

Furthermore for k 0 1 T minus 1 we obtain the opti-mal control (ulowast1k ulowast2k ulowast3k) as

ulowast1k min max

λ1k+1 minus λ6k+11113872 1113873Pk

Ak

u1min⎧⎨

⎩

⎫⎬

⎭ u1max⎧⎨

⎩

⎫⎬

⎭

ulowast2k min max

λ3k+1 minus λ5k+11113872 1113873Hk + λ5k+1 minus λ4k+11113872 1113873ϵHk

Bk

u2min⎧⎨

⎩

⎫⎬

⎭ u2max⎧⎨

⎩

⎫⎬

⎭

ulowast3k min max

λ2k+1 minus λ6k+11113872 1113873θQk

Ck

u3min⎧⎨

⎩

⎫⎬

⎭ u3max⎧⎨

⎩

⎫⎬

⎭

(10)

Journal of Applied Mathematics 5

Proof e Hamiltonian 1113954Hk at time step k is given by

1113954Hk Hk minus Trk minus T

p

k minus Qk +Aku2

1k

2+

Bku22k

2+

Cku23k

2

+ λ1k+1 b minus β1PkMk

Nk

+(1 minus μ)Pk minus u1kPk1113890 1113891

+ λ2k+1 β1PkMk

Nk

+ 1 minus u3k1113872 1113873θQk + 1 minus μ minus β2( 1113857Mk1113890 1113891

+ λ3k+1 β2Mk + 1 minus μ minus δ minus α1 minus α2 minus α3( 1113857Hk minus u2kHk1113960 1113961

+ λ4k+1 α1Hk + 1 minus μ minus c1( 1113857Trk + ϵu2kHk1113960 1113961

+ λ5k+1 α2Hk + 1 minus μ minus c2( 1113857Tp

k +(1 minus ϵ)u2kHk1113960 1113961

+ λ6k+1 c1Trk + α3Hk + c2T

p

k +(1 minus μ)Qk + u1kPk minus 1 minus u3k1113872 1113873θQk1113960 1113961

(11)

For k 0 1 T minus 1 the adjoint equations andtransversality conditions can be obtained by using Pon-tryaginrsquos maximum principle in discrete time given in[13ndash18] such that

λ1k z 1113954Hk

zPk

λ1(T)

λ2k z 1113954Hk

zMk

λ2(T)

λ3k z 1113954Hk

zHk

λ3(T)

λ4k z 1113954Hk

zTrk

λ4(T)

λ5k z 1113954Hk

zTp

k

λ5(T)

λ6k z 1113954Hk

zQk

λ6(T)

(12)

For k 0 1 T minus 1 the optimal controls ulowast1k ulowast2kand ulowast3k can be solved from the optimality condition

z 1113954Hk

zu1k

Aku1k minus λ1k+1Pk + λ6k+1Pk 0

z 1113954Hk

zu2k

Bku2k minus λ3k+1Hk + λ4k+1ϵHk + λ5k+1(1 minus ϵ)Hk 0

z 1113954Hk

zu3k

Cku3k minus λ2k+1θQk + λ6k+1θQk 0

(13)

So we have

u1k λ1k+1 minus λ6k+11113872 1113873Pk

Ak

u2k λ3k+1 minus λ5k+11113872 1113873Hk + λ5k+1 minus λ4k+11113872 1113873ϵHk

Bk

u3k λ2k+1 minus λ6k+11113872 1113873θQk

Ck

(14)

By the bounds in Uad of the controls it is easy to obtainulowast1k ulowast2k and ulowast3k in the form of (10)

5 Numerical Simulation

In this section we shall solve numerically the optimalcontrol problem for our PMHTrTpQ model Here weobtain the optimality system from the state and adjointequationse proposed optimal control strategy is obtainedby solving the optimal system which consists of six differ-ence equations and boundary conditions e optimalitysystem can be solved by using an iterative method Using aninitial guess for the control variables u1k u2k and u3k thestate variables P M H Tr Tp and Q are solved forwardand the adjoint variables λi for i 1 2 3 4 5 6 are solvedbackwards at time steps k 0 and k T If the new values ofthe state and adjoint variables differ from the previousvalues the new values are used to update u1k u2k and u3kand the process is repeated until the system converges

e numerical solution of model (1) is executed usingMatlab with the following parameter values and initial valuesof state variable in Table 1

We begin by presenting the solution evolution of ourmodel (1) without controls that are represented in Figure 2

6 Journal of Applied Mathematics

Figure 2(a) shows that the number of moderate drinkersincreases in the beginning but then decreases clearly isdecrease leads to an increase in the number of heavy drinkersfrom the beginning and approaches a given value of 46230as we see in Figure 2(b) Finally Figure 2(c) and Figures 2(d)and 2(e) show that the number of the rich and poor heavydrinkers joining private and public centers for alcohol ad-diction treatment and the quitters of drinking decreases andapproaches certain unacceptable values

In order to display the influence of alcohol treatmentcenters on the population drinking dynamics we presentsome figures about the effects of the parameters α1 α2 c1and c2 on system (1)

Figure 2(f ) shows that the number of heavy drinkersincreases when the number of the heavy drinkers who joinaddiction treatment centers decreases Also we observe inFigure 2(g) that the number of quitters of drinking increaseswhen the number of the heavy drinkers who join addictiontreatment centers increases

From these two figures the importance of addictiontreatment centers is clear in reducing the number of heavydrinkers and thus increasing the number of quitters ofdrinking erefore we need to make an effort to introducethese centers to people and encourage heavy drinkers to jointhem and this is shown through the following objectives

51 First Objective Protecting and Preventing PotentialDrinkers from Falling into Alcohol Addiction To realize thisobjective we apply only the control u1 ie the imple-mentation of awareness information and educationalprograms on potential drinkers and making them know therisks of this phenomenon and the resulting health and socialdamages Figure 3(a) shows that the number of moderatedrinkers decreases from 23116 (without control u1) to12459 (with control u1) at the end of the proposed programand the number of heavy drinkers decreases from 46229(without control u1) to 24902 (with control u1) at the end ofthe proposed control strategy (see Figure 3(b)) Also weobserve in Figure 3(c) that the number of people who quitdrinking increases and has reached the value 42978 (withcontrol u1) compared to the situation when there is nocontrol 568 at the end of the proposed strategy So ourobjective has been achieved

52 Second Objective Increasing the Number of HeavyDrinkers Who Join Treatment Centers of Alcohol AddictionTo achieve this objective we only use the control u2 ieencouraging heavy drinkers to know and join public andprivate treatment centers In Figure 4(a) it is observed thatthere is a significant decrease in the number of heavydrinkers with control compared to a situation when there isno control where the decrease reaches 4120 at the end ofthe proposed control strategy Figure 4(b) shows that thenumber of the poor heavy drinkers who join public treat-ment centers increased from 691 (Tp without control u2) to5970 (Tp with control u2) and that the number of rich heavydrinkers who join private treatment centers increased from706 (without control u2) to 23372 (with control u2) at the

end of the proposed control (see Figure 4(c)) and Figure 4(d)shows that the number of people who quit drinking withoutcontrol u2 decreases and approaches a value of 568 After 5days a slight increase appears with control u2 is increaseis not enough as the number of people who quit drinkingbecomes moderate drinkers To improve this result we canadd another control that represents follow-up for temporaryquitters of drinking

53 5ird Objective Encouraging Heavy Drinkers to JoinTreatment Centers of Alcohol Addiction and Following Up theQuitters of Drinking In order to accomplish this aim weuse the controls u2 and u3 ie encouraging heavydrinkers to know and join public and private treatmentcenters and following up the temporary quitters ofdrinking e number of the poor heavy drinkers who joinpublic treatment centers increases from 691 (withoutcontrols) to 3217 (with controls) at the end of the pro-posed strategy (see Figure 5(c)) Figure 5(d) demonstratesthat the number of the rich heavy drinkers who joinprivate treatment centers increases from 706 (withoutcontrols) to 12592 (with controls) at the end of theproposed strategy e number of the quitters of drinkingin Figure 5(e) increases significantly from 568 (withoutcontrols) to 43011 (with controls) rough the use ofthose controls the abovementioned objective has beenachieved

54 Fourth Objective Prevention Treatment and Follow-Upfor the Addicted Individuals To meet this objective we usethe controls u1 u2 and u3 ie awareness programs for thepotential drinkers encouraging heavy drinkers to know andjoin public and private treatment centers and follow-up forthe quitters of drinking Figure 6(a) shows that the numberof the moderate drinkers decreases starting from the earlydays by 3460 Also Figure 6(b) shows that the number ofthe heavy drinkers increases starting from the early days butthen decreases from 46229 (without controls) to 6581 (withcontrols) e number of the rich and poor heavy drinkerswho join private and public treatment centers increases by1145 and 298 respectively (see Figures 6(c) and 6(d))Figure 6(e) depicts clearly an increase in the number of thequitters of drinking from 568 (without controls) to 55063(with controls) As a result the objective set before has beenachieved

We remark that the numbers of the rich and poor heavydrinkers (Tr and Tp) decrease when we applied threecontrols (u1 u2 u3) compared to using only the control(u2) is decrease is due to the positive effect of the controlu1 on the compartment P and the control u3 on θQ (tem-porary quitters) which prevents temporary quitters of

Table 1 e parameters used for model (1)

P0 M0 H0 Tr0 T

p0 Q0 b N δ

500 300 100 60 30 10 65 1000 0002μ β1 β2 α1 α2 α3 c1 c2 θ0065 075 014 0001 0001 0001 0001 0002 002

Journal of Applied Mathematics 7

M without controls

200

250

300

350

400

450M

oder

ate d

rinke

rs (M

)

10 20 30 40 50 60 70 80 90 100Time (days)

(a)

H without controls

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

10 20 30 40 50 60 70 80 90 100Time (days)

(b)

Tp without controls

10 20 30 40 50 60 70 80 90 100Time (days)

5

10

15

20

25

30

The p

oor h

eavy

drin

kers

(Tp )

(c)

Tr without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

The r

ich

heav

y dr

inke

rs (T

r )

(d)

Q without controls

55556

657

758

859

9510

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

(e)

100 20 30 40 50 60 70 80 90 100Time (days)

50100150200250300350400450500

Hea

vy d

rinke

rs (H

)

H without treatment centers of alcohol addictionH with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002H with α1 = 03 α2 = 01 γ1 = 006 γ2 = 003H with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02H with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03

(f )

Figure 2 Continued

8 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

Q with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03Q with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02Q with α1 = 03 α2 = 01 γ1 = 005 γ2 = 02Q with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002Q without treatment centers of alcohol addiction

0

50

100

150

200

250

300

350

400

Qui

tters

of d

rinki

ng (Q

)

(g)

Figure 2 Representing the drinkers class without control

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

M with control u1M without controls

(a)

H with control u1H without controls

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

(b)

Q with control u1Q without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

(c)

Figure 3 Representing the drinkers class with and without control u1

Journal of Applied Mathematics 9

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500H

eavy

drin

kers

(H)

H with control u2H without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

e p

oor h

eavy

drin

kers

(Tp )

Tp with control u2Tp without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

e r

ich

heav

y dr

inke

rs (T

r )

Tr with control u2Tr without controls

(c)

55

6

65

7

75

8

85

9

95

10

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

Q with control u2Q without controls

(d)

Figure 4 Representing the drinkers class with and without control u2

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

The p

oor h

eavy

drin

kers

(Tp )

Tp with controls u2 and u3Tp without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

The r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u2 and u3Tr without controls

(b)

Figure 5 Continued

10 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

Q with controls u2 and u3Q without controls

(c)

Figure 5 Representing the drinkers class with controls u2 and u3 and without controls

50

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

10 20 30 40 50 60 70 80 90 100Time (days)

M with controls u1 u2 and u3M without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

50

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

H with controls u1 u2 and u3H without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

510152025303540455055

e p

oor h

eavy

drin

kers

(Tp )

Tp with controls u1 u2 and u3Tp without controls

(c)

10 20 30 40 50 60 70 80 90 100Time (days)

0

20

40

60

80

100

120

140

160

180

e r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u1 u2 and u3Tr without controls

(d)

Figure 6 Continued

Journal of Applied Mathematics 11

drinking from returning to drinking So the number ofindividuals that enter the compartment Tr and Tp de-creases erefore applying a combination of the threecontrols is more effective than one control

6 Conclusion

In this research paper we introduced a discrete modelingof drinking for the purpose of minimizing the number ofdrinkers and maximizing the number of the rich and poorheavy drinkers who join private and public treatmentcenters of alcohol addiction and subsequently the numberof quitters of drinking Unlike some other previousmodels we have taken into account the impact of privateand public addiction treatment centers on alcoholics eresults showed that those centers have substantial influ-ence on the dynamics of alcoholism and can greatlyimpact the spread of drinking us it is crucial to urgepeople to know and join private and public addictiontreatment centers to quit drinking We also presentedthree controls which respectively represent awarenessprograms encouragement and follow-up We applied theresults of the control theory and we managed to obtain thecharacterizations of the optimal controls e numericalsimulation of the obtained results showed the effective-ness of the proposed control strategies

Data Availability

e disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpnetworkrepositorycomprofilephp)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] WHOWorld Health Organization Report on Drinking WHOGeneva Switzerland 2016

[2] World Health Organization Alcohol World Health Organi-zation Geneva Switzerland 2018 httpswwwwhointnews-roomfact-sheetsdetailalcohol

[3] H F Huo and Q Wang ldquoModelling the influence ofawareness programs by media on the drinking dynamicsrdquoAbstract and Applied Analysis vol 2014 Article ID 9380808 pages 2014

[4] S-H Ma H-F Huo and X-Y Meng ldquoModelling alcoholismas a contagious disease a mathematical model with awarenessprograms and time delayrdquo Discrete Dynamics in Nature andSociety vol 2015 Article ID 260195 13 pages 2015

[5] X-Y Wang H-F Huo Q-K Kong andW-X Shi ldquoOptimalcontrol strategies in an alcoholism modelrdquo Abstract andApplied Analysis vol 2014 Article ID 954069 18 pages 2014

[6] H-F Huo S-R Huang X-Y Wang and H Xiang ldquoOptimalcontrol of a social epidemic model with media coveragerdquoJournal of Biological Dynamics vol 11 no 1 pp 226ndash2432017

[7] H F Huo and N N Song ldquoGlobal stability for a bingedrinking model with two stagesrdquo Discrete Dynamics in Natureand Society vol 2012 Article ID 829386 15 pages 2012

[8] I K Adu M Osman and C Yang ldquoMathematical model ofdrinking epidemicrdquo British Journal of Mathematics ComputerScience vol 22 no 5 pp 1ndash10 2017

[9] H-F Huo and Y-P Liu ldquoe analysis of the SIRS alcoholismmodels with relapse on weightedrdquo SpringerPlus vol 5 no 1p 722 2016

[10] K Mursquotamar ldquoOptimal control strategy for alcoholism modelwith two infected compartmentsrdquo IOSR Journal of Mathe-matics vol 14 no 3 pp 58ndash67 2018

[11] S Sharma and G P Samanta ldquoDrinking as an epidemic amathematical model with dynamic behaviourrdquo Journal ofApplied Mathematics amp Informatics vol 31 no 1-2 pp 1ndash252013

[12] S Sharma and G P Samanta ldquoAnalysis of a drinking epi-demic modelrdquo International Journal of Dynamics and Controlvol 3 no 3 pp 288ndash305 2015

10 20 30 40 50 60 70 80 90 100Time (days)

0

100

200

300

400

500

600

Qui

tters

of d

rinki

ng (Q

)

Q with controls u1 u2 and u3Q without controls

(e)

Figure 6 Representing the drinkers class with and without optimal controls u1 u2 and u3

12 Journal of Applied Mathematics

Proof e Hamiltonian 1113954Hk at time step k is given by

1113954Hk Hk minus Trk minus T

p

k minus Qk +Aku2

1k

2+

Bku22k

2+

Cku23k

2

+ λ1k+1 b minus β1PkMk

Nk

+(1 minus μ)Pk minus u1kPk1113890 1113891

+ λ2k+1 β1PkMk

Nk

+ 1 minus u3k1113872 1113873θQk + 1 minus μ minus β2( 1113857Mk1113890 1113891

+ λ3k+1 β2Mk + 1 minus μ minus δ minus α1 minus α2 minus α3( 1113857Hk minus u2kHk1113960 1113961

+ λ4k+1 α1Hk + 1 minus μ minus c1( 1113857Trk + ϵu2kHk1113960 1113961

+ λ5k+1 α2Hk + 1 minus μ minus c2( 1113857Tp

k +(1 minus ϵ)u2kHk1113960 1113961

+ λ6k+1 c1Trk + α3Hk + c2T

p

k +(1 minus μ)Qk + u1kPk minus 1 minus u3k1113872 1113873θQk1113960 1113961

(11)

For k 0 1 T minus 1 the adjoint equations andtransversality conditions can be obtained by using Pon-tryaginrsquos maximum principle in discrete time given in[13ndash18] such that

λ1k z 1113954Hk

zPk

λ1(T)

λ2k z 1113954Hk

zMk

λ2(T)

λ3k z 1113954Hk

zHk

λ3(T)

λ4k z 1113954Hk

zTrk

λ4(T)

λ5k z 1113954Hk

zTp

k

λ5(T)

λ6k z 1113954Hk

zQk

λ6(T)

(12)

For k 0 1 T minus 1 the optimal controls ulowast1k ulowast2kand ulowast3k can be solved from the optimality condition

z 1113954Hk

zu1k

Aku1k minus λ1k+1Pk + λ6k+1Pk 0

z 1113954Hk

zu2k

Bku2k minus λ3k+1Hk + λ4k+1ϵHk + λ5k+1(1 minus ϵ)Hk 0

z 1113954Hk

zu3k

Cku3k minus λ2k+1θQk + λ6k+1θQk 0

(13)

So we have

u1k λ1k+1 minus λ6k+11113872 1113873Pk

Ak

u2k λ3k+1 minus λ5k+11113872 1113873Hk + λ5k+1 minus λ4k+11113872 1113873ϵHk

Bk

u3k λ2k+1 minus λ6k+11113872 1113873θQk

Ck

(14)

By the bounds in Uad of the controls it is easy to obtainulowast1k ulowast2k and ulowast3k in the form of (10)

5 Numerical Simulation

In this section we shall solve numerically the optimalcontrol problem for our PMHTrTpQ model Here weobtain the optimality system from the state and adjointequationse proposed optimal control strategy is obtainedby solving the optimal system which consists of six differ-ence equations and boundary conditions e optimalitysystem can be solved by using an iterative method Using aninitial guess for the control variables u1k u2k and u3k thestate variables P M H Tr Tp and Q are solved forwardand the adjoint variables λi for i 1 2 3 4 5 6 are solvedbackwards at time steps k 0 and k T If the new values ofthe state and adjoint variables differ from the previousvalues the new values are used to update u1k u2k and u3kand the process is repeated until the system converges

e numerical solution of model (1) is executed usingMatlab with the following parameter values and initial valuesof state variable in Table 1

We begin by presenting the solution evolution of ourmodel (1) without controls that are represented in Figure 2

6 Journal of Applied Mathematics

Figure 2(a) shows that the number of moderate drinkersincreases in the beginning but then decreases clearly isdecrease leads to an increase in the number of heavy drinkersfrom the beginning and approaches a given value of 46230as we see in Figure 2(b) Finally Figure 2(c) and Figures 2(d)and 2(e) show that the number of the rich and poor heavydrinkers joining private and public centers for alcohol ad-diction treatment and the quitters of drinking decreases andapproaches certain unacceptable values

In order to display the influence of alcohol treatmentcenters on the population drinking dynamics we presentsome figures about the effects of the parameters α1 α2 c1and c2 on system (1)

Figure 2(f ) shows that the number of heavy drinkersincreases when the number of the heavy drinkers who joinaddiction treatment centers decreases Also we observe inFigure 2(g) that the number of quitters of drinking increaseswhen the number of the heavy drinkers who join addictiontreatment centers increases

From these two figures the importance of addictiontreatment centers is clear in reducing the number of heavydrinkers and thus increasing the number of quitters ofdrinking erefore we need to make an effort to introducethese centers to people and encourage heavy drinkers to jointhem and this is shown through the following objectives

51 First Objective Protecting and Preventing PotentialDrinkers from Falling into Alcohol Addiction To realize thisobjective we apply only the control u1 ie the imple-mentation of awareness information and educationalprograms on potential drinkers and making them know therisks of this phenomenon and the resulting health and socialdamages Figure 3(a) shows that the number of moderatedrinkers decreases from 23116 (without control u1) to12459 (with control u1) at the end of the proposed programand the number of heavy drinkers decreases from 46229(without control u1) to 24902 (with control u1) at the end ofthe proposed control strategy (see Figure 3(b)) Also weobserve in Figure 3(c) that the number of people who quitdrinking increases and has reached the value 42978 (withcontrol u1) compared to the situation when there is nocontrol 568 at the end of the proposed strategy So ourobjective has been achieved

52 Second Objective Increasing the Number of HeavyDrinkers Who Join Treatment Centers of Alcohol AddictionTo achieve this objective we only use the control u2 ieencouraging heavy drinkers to know and join public andprivate treatment centers In Figure 4(a) it is observed thatthere is a significant decrease in the number of heavydrinkers with control compared to a situation when there isno control where the decrease reaches 4120 at the end ofthe proposed control strategy Figure 4(b) shows that thenumber of the poor heavy drinkers who join public treat-ment centers increased from 691 (Tp without control u2) to5970 (Tp with control u2) and that the number of rich heavydrinkers who join private treatment centers increased from706 (without control u2) to 23372 (with control u2) at the

end of the proposed control (see Figure 4(c)) and Figure 4(d)shows that the number of people who quit drinking withoutcontrol u2 decreases and approaches a value of 568 After 5days a slight increase appears with control u2 is increaseis not enough as the number of people who quit drinkingbecomes moderate drinkers To improve this result we canadd another control that represents follow-up for temporaryquitters of drinking

53 5ird Objective Encouraging Heavy Drinkers to JoinTreatment Centers of Alcohol Addiction and Following Up theQuitters of Drinking In order to accomplish this aim weuse the controls u2 and u3 ie encouraging heavydrinkers to know and join public and private treatmentcenters and following up the temporary quitters ofdrinking e number of the poor heavy drinkers who joinpublic treatment centers increases from 691 (withoutcontrols) to 3217 (with controls) at the end of the pro-posed strategy (see Figure 5(c)) Figure 5(d) demonstratesthat the number of the rich heavy drinkers who joinprivate treatment centers increases from 706 (withoutcontrols) to 12592 (with controls) at the end of theproposed strategy e number of the quitters of drinkingin Figure 5(e) increases significantly from 568 (withoutcontrols) to 43011 (with controls) rough the use ofthose controls the abovementioned objective has beenachieved

54 Fourth Objective Prevention Treatment and Follow-Upfor the Addicted Individuals To meet this objective we usethe controls u1 u2 and u3 ie awareness programs for thepotential drinkers encouraging heavy drinkers to know andjoin public and private treatment centers and follow-up forthe quitters of drinking Figure 6(a) shows that the numberof the moderate drinkers decreases starting from the earlydays by 3460 Also Figure 6(b) shows that the number ofthe heavy drinkers increases starting from the early days butthen decreases from 46229 (without controls) to 6581 (withcontrols) e number of the rich and poor heavy drinkerswho join private and public treatment centers increases by1145 and 298 respectively (see Figures 6(c) and 6(d))Figure 6(e) depicts clearly an increase in the number of thequitters of drinking from 568 (without controls) to 55063(with controls) As a result the objective set before has beenachieved

We remark that the numbers of the rich and poor heavydrinkers (Tr and Tp) decrease when we applied threecontrols (u1 u2 u3) compared to using only the control(u2) is decrease is due to the positive effect of the controlu1 on the compartment P and the control u3 on θQ (tem-porary quitters) which prevents temporary quitters of

Table 1 e parameters used for model (1)

P0 M0 H0 Tr0 T

p0 Q0 b N δ

500 300 100 60 30 10 65 1000 0002μ β1 β2 α1 α2 α3 c1 c2 θ0065 075 014 0001 0001 0001 0001 0002 002

Journal of Applied Mathematics 7

M without controls

200

250

300

350

400

450M

oder

ate d

rinke

rs (M

)

10 20 30 40 50 60 70 80 90 100Time (days)

(a)

H without controls

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

10 20 30 40 50 60 70 80 90 100Time (days)

(b)

Tp without controls

10 20 30 40 50 60 70 80 90 100Time (days)

5

10

15

20

25

30

The p

oor h

eavy

drin

kers

(Tp )

(c)

Tr without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

The r

ich

heav

y dr

inke

rs (T

r )

(d)

Q without controls

55556

657

758

859

9510

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

(e)

100 20 30 40 50 60 70 80 90 100Time (days)

50100150200250300350400450500

Hea

vy d

rinke

rs (H

)

H without treatment centers of alcohol addictionH with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002H with α1 = 03 α2 = 01 γ1 = 006 γ2 = 003H with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02H with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03

(f )

Figure 2 Continued

8 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

Q with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03Q with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02Q with α1 = 03 α2 = 01 γ1 = 005 γ2 = 02Q with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002Q without treatment centers of alcohol addiction

0

50

100

150

200

250

300

350

400

Qui

tters

of d

rinki

ng (Q

)

(g)

Figure 2 Representing the drinkers class without control

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

M with control u1M without controls

(a)

H with control u1H without controls

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

(b)

Q with control u1Q without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

(c)

Figure 3 Representing the drinkers class with and without control u1

Journal of Applied Mathematics 9

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500H

eavy

drin

kers

(H)

H with control u2H without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

e p

oor h

eavy

drin

kers

(Tp )

Tp with control u2Tp without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

e r

ich

heav

y dr

inke

rs (T

r )

Tr with control u2Tr without controls

(c)

55

6

65

7

75

8

85

9

95

10

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

Q with control u2Q without controls

(d)

Figure 4 Representing the drinkers class with and without control u2

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

The p

oor h

eavy

drin

kers

(Tp )

Tp with controls u2 and u3Tp without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

The r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u2 and u3Tr without controls

(b)

Figure 5 Continued

10 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

Q with controls u2 and u3Q without controls

(c)

Figure 5 Representing the drinkers class with controls u2 and u3 and without controls

50

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

10 20 30 40 50 60 70 80 90 100Time (days)

M with controls u1 u2 and u3M without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

50

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

H with controls u1 u2 and u3H without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

510152025303540455055

e p

oor h

eavy

drin

kers

(Tp )

Tp with controls u1 u2 and u3Tp without controls

(c)

10 20 30 40 50 60 70 80 90 100Time (days)

0

20

40

60

80

100

120

140

160

180

e r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u1 u2 and u3Tr without controls

(d)

Figure 6 Continued

Journal of Applied Mathematics 11

drinking from returning to drinking So the number ofindividuals that enter the compartment Tr and Tp de-creases erefore applying a combination of the threecontrols is more effective than one control

6 Conclusion

In this research paper we introduced a discrete modelingof drinking for the purpose of minimizing the number ofdrinkers and maximizing the number of the rich and poorheavy drinkers who join private and public treatmentcenters of alcohol addiction and subsequently the numberof quitters of drinking Unlike some other previousmodels we have taken into account the impact of privateand public addiction treatment centers on alcoholics eresults showed that those centers have substantial influ-ence on the dynamics of alcoholism and can greatlyimpact the spread of drinking us it is crucial to urgepeople to know and join private and public addictiontreatment centers to quit drinking We also presentedthree controls which respectively represent awarenessprograms encouragement and follow-up We applied theresults of the control theory and we managed to obtain thecharacterizations of the optimal controls e numericalsimulation of the obtained results showed the effective-ness of the proposed control strategies

Data Availability

e disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpnetworkrepositorycomprofilephp)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] WHOWorld Health Organization Report on Drinking WHOGeneva Switzerland 2016

[2] World Health Organization Alcohol World Health Organi-zation Geneva Switzerland 2018 httpswwwwhointnews-roomfact-sheetsdetailalcohol

[3] H F Huo and Q Wang ldquoModelling the influence ofawareness programs by media on the drinking dynamicsrdquoAbstract and Applied Analysis vol 2014 Article ID 9380808 pages 2014

[4] S-H Ma H-F Huo and X-Y Meng ldquoModelling alcoholismas a contagious disease a mathematical model with awarenessprograms and time delayrdquo Discrete Dynamics in Nature andSociety vol 2015 Article ID 260195 13 pages 2015

[5] X-Y Wang H-F Huo Q-K Kong andW-X Shi ldquoOptimalcontrol strategies in an alcoholism modelrdquo Abstract andApplied Analysis vol 2014 Article ID 954069 18 pages 2014

[6] H-F Huo S-R Huang X-Y Wang and H Xiang ldquoOptimalcontrol of a social epidemic model with media coveragerdquoJournal of Biological Dynamics vol 11 no 1 pp 226ndash2432017

[7] H F Huo and N N Song ldquoGlobal stability for a bingedrinking model with two stagesrdquo Discrete Dynamics in Natureand Society vol 2012 Article ID 829386 15 pages 2012

[8] I K Adu M Osman and C Yang ldquoMathematical model ofdrinking epidemicrdquo British Journal of Mathematics ComputerScience vol 22 no 5 pp 1ndash10 2017

[9] H-F Huo and Y-P Liu ldquoe analysis of the SIRS alcoholismmodels with relapse on weightedrdquo SpringerPlus vol 5 no 1p 722 2016

[10] K Mursquotamar ldquoOptimal control strategy for alcoholism modelwith two infected compartmentsrdquo IOSR Journal of Mathe-matics vol 14 no 3 pp 58ndash67 2018

[11] S Sharma and G P Samanta ldquoDrinking as an epidemic amathematical model with dynamic behaviourrdquo Journal ofApplied Mathematics amp Informatics vol 31 no 1-2 pp 1ndash252013

[12] S Sharma and G P Samanta ldquoAnalysis of a drinking epi-demic modelrdquo International Journal of Dynamics and Controlvol 3 no 3 pp 288ndash305 2015

10 20 30 40 50 60 70 80 90 100Time (days)

0

100

200

300

400

500

600

Qui

tters

of d

rinki

ng (Q

)

Q with controls u1 u2 and u3Q without controls

(e)

Figure 6 Representing the drinkers class with and without optimal controls u1 u2 and u3

12 Journal of Applied Mathematics

Figure 2(a) shows that the number of moderate drinkersincreases in the beginning but then decreases clearly isdecrease leads to an increase in the number of heavy drinkersfrom the beginning and approaches a given value of 46230as we see in Figure 2(b) Finally Figure 2(c) and Figures 2(d)and 2(e) show that the number of the rich and poor heavydrinkers joining private and public centers for alcohol ad-diction treatment and the quitters of drinking decreases andapproaches certain unacceptable values

In order to display the influence of alcohol treatmentcenters on the population drinking dynamics we presentsome figures about the effects of the parameters α1 α2 c1and c2 on system (1)

Figure 2(f ) shows that the number of heavy drinkersincreases when the number of the heavy drinkers who joinaddiction treatment centers decreases Also we observe inFigure 2(g) that the number of quitters of drinking increaseswhen the number of the heavy drinkers who join addictiontreatment centers increases

From these two figures the importance of addictiontreatment centers is clear in reducing the number of heavydrinkers and thus increasing the number of quitters ofdrinking erefore we need to make an effort to introducethese centers to people and encourage heavy drinkers to jointhem and this is shown through the following objectives

51 First Objective Protecting and Preventing PotentialDrinkers from Falling into Alcohol Addiction To realize thisobjective we apply only the control u1 ie the imple-mentation of awareness information and educationalprograms on potential drinkers and making them know therisks of this phenomenon and the resulting health and socialdamages Figure 3(a) shows that the number of moderatedrinkers decreases from 23116 (without control u1) to12459 (with control u1) at the end of the proposed programand the number of heavy drinkers decreases from 46229(without control u1) to 24902 (with control u1) at the end ofthe proposed control strategy (see Figure 3(b)) Also weobserve in Figure 3(c) that the number of people who quitdrinking increases and has reached the value 42978 (withcontrol u1) compared to the situation when there is nocontrol 568 at the end of the proposed strategy So ourobjective has been achieved

52 Second Objective Increasing the Number of HeavyDrinkers Who Join Treatment Centers of Alcohol AddictionTo achieve this objective we only use the control u2 ieencouraging heavy drinkers to know and join public andprivate treatment centers In Figure 4(a) it is observed thatthere is a significant decrease in the number of heavydrinkers with control compared to a situation when there isno control where the decrease reaches 4120 at the end ofthe proposed control strategy Figure 4(b) shows that thenumber of the poor heavy drinkers who join public treat-ment centers increased from 691 (Tp without control u2) to5970 (Tp with control u2) and that the number of rich heavydrinkers who join private treatment centers increased from706 (without control u2) to 23372 (with control u2) at the

end of the proposed control (see Figure 4(c)) and Figure 4(d)shows that the number of people who quit drinking withoutcontrol u2 decreases and approaches a value of 568 After 5days a slight increase appears with control u2 is increaseis not enough as the number of people who quit drinkingbecomes moderate drinkers To improve this result we canadd another control that represents follow-up for temporaryquitters of drinking

53 5ird Objective Encouraging Heavy Drinkers to JoinTreatment Centers of Alcohol Addiction and Following Up theQuitters of Drinking In order to accomplish this aim weuse the controls u2 and u3 ie encouraging heavydrinkers to know and join public and private treatmentcenters and following up the temporary quitters ofdrinking e number of the poor heavy drinkers who joinpublic treatment centers increases from 691 (withoutcontrols) to 3217 (with controls) at the end of the pro-posed strategy (see Figure 5(c)) Figure 5(d) demonstratesthat the number of the rich heavy drinkers who joinprivate treatment centers increases from 706 (withoutcontrols) to 12592 (with controls) at the end of theproposed strategy e number of the quitters of drinkingin Figure 5(e) increases significantly from 568 (withoutcontrols) to 43011 (with controls) rough the use ofthose controls the abovementioned objective has beenachieved

54 Fourth Objective Prevention Treatment and Follow-Upfor the Addicted Individuals To meet this objective we usethe controls u1 u2 and u3 ie awareness programs for thepotential drinkers encouraging heavy drinkers to know andjoin public and private treatment centers and follow-up forthe quitters of drinking Figure 6(a) shows that the numberof the moderate drinkers decreases starting from the earlydays by 3460 Also Figure 6(b) shows that the number ofthe heavy drinkers increases starting from the early days butthen decreases from 46229 (without controls) to 6581 (withcontrols) e number of the rich and poor heavy drinkerswho join private and public treatment centers increases by1145 and 298 respectively (see Figures 6(c) and 6(d))Figure 6(e) depicts clearly an increase in the number of thequitters of drinking from 568 (without controls) to 55063(with controls) As a result the objective set before has beenachieved

We remark that the numbers of the rich and poor heavydrinkers (Tr and Tp) decrease when we applied threecontrols (u1 u2 u3) compared to using only the control(u2) is decrease is due to the positive effect of the controlu1 on the compartment P and the control u3 on θQ (tem-porary quitters) which prevents temporary quitters of

Table 1 e parameters used for model (1)

P0 M0 H0 Tr0 T

p0 Q0 b N δ

500 300 100 60 30 10 65 1000 0002μ β1 β2 α1 α2 α3 c1 c2 θ0065 075 014 0001 0001 0001 0001 0002 002

Journal of Applied Mathematics 7

M without controls

200

250

300

350

400

450M

oder

ate d

rinke

rs (M

)

10 20 30 40 50 60 70 80 90 100Time (days)

(a)

H without controls

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

10 20 30 40 50 60 70 80 90 100Time (days)

(b)

Tp without controls

10 20 30 40 50 60 70 80 90 100Time (days)

5

10

15

20

25

30

The p

oor h

eavy

drin

kers

(Tp )

(c)

Tr without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

The r

ich

heav

y dr

inke

rs (T

r )

(d)

Q without controls

55556

657

758

859

9510

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

(e)

100 20 30 40 50 60 70 80 90 100Time (days)

50100150200250300350400450500

Hea

vy d

rinke

rs (H

)

H without treatment centers of alcohol addictionH with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002H with α1 = 03 α2 = 01 γ1 = 006 γ2 = 003H with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02H with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03

(f )

Figure 2 Continued

8 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

Q with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03Q with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02Q with α1 = 03 α2 = 01 γ1 = 005 γ2 = 02Q with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002Q without treatment centers of alcohol addiction

0

50

100

150

200

250

300

350

400

Qui

tters

of d

rinki

ng (Q

)

(g)

Figure 2 Representing the drinkers class without control

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

M with control u1M without controls

(a)

H with control u1H without controls

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

(b)

Q with control u1Q without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

(c)

Figure 3 Representing the drinkers class with and without control u1

Journal of Applied Mathematics 9

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500H

eavy

drin

kers

(H)

H with control u2H without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

e p

oor h

eavy

drin

kers

(Tp )

Tp with control u2Tp without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

e r

ich

heav

y dr

inke

rs (T

r )

Tr with control u2Tr without controls

(c)

55

6

65

7

75

8

85

9

95

10

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

Q with control u2Q without controls

(d)

Figure 4 Representing the drinkers class with and without control u2

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

The p

oor h

eavy

drin

kers

(Tp )

Tp with controls u2 and u3Tp without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

The r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u2 and u3Tr without controls

(b)

Figure 5 Continued

10 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

Q with controls u2 and u3Q without controls

(c)

Figure 5 Representing the drinkers class with controls u2 and u3 and without controls

50

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

10 20 30 40 50 60 70 80 90 100Time (days)

M with controls u1 u2 and u3M without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

50

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

H with controls u1 u2 and u3H without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

510152025303540455055

e p

oor h

eavy

drin

kers

(Tp )

Tp with controls u1 u2 and u3Tp without controls

(c)

10 20 30 40 50 60 70 80 90 100Time (days)

0

20

40

60

80

100

120

140

160

180

e r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u1 u2 and u3Tr without controls

(d)

Figure 6 Continued

Journal of Applied Mathematics 11

drinking from returning to drinking So the number ofindividuals that enter the compartment Tr and Tp de-creases erefore applying a combination of the threecontrols is more effective than one control

6 Conclusion

In this research paper we introduced a discrete modelingof drinking for the purpose of minimizing the number ofdrinkers and maximizing the number of the rich and poorheavy drinkers who join private and public treatmentcenters of alcohol addiction and subsequently the numberof quitters of drinking Unlike some other previousmodels we have taken into account the impact of privateand public addiction treatment centers on alcoholics eresults showed that those centers have substantial influ-ence on the dynamics of alcoholism and can greatlyimpact the spread of drinking us it is crucial to urgepeople to know and join private and public addictiontreatment centers to quit drinking We also presentedthree controls which respectively represent awarenessprograms encouragement and follow-up We applied theresults of the control theory and we managed to obtain thecharacterizations of the optimal controls e numericalsimulation of the obtained results showed the effective-ness of the proposed control strategies

Data Availability

e disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpnetworkrepositorycomprofilephp)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] WHOWorld Health Organization Report on Drinking WHOGeneva Switzerland 2016

[2] World Health Organization Alcohol World Health Organi-zation Geneva Switzerland 2018 httpswwwwhointnews-roomfact-sheetsdetailalcohol

[3] H F Huo and Q Wang ldquoModelling the influence ofawareness programs by media on the drinking dynamicsrdquoAbstract and Applied Analysis vol 2014 Article ID 9380808 pages 2014

[4] S-H Ma H-F Huo and X-Y Meng ldquoModelling alcoholismas a contagious disease a mathematical model with awarenessprograms and time delayrdquo Discrete Dynamics in Nature andSociety vol 2015 Article ID 260195 13 pages 2015

[5] X-Y Wang H-F Huo Q-K Kong andW-X Shi ldquoOptimalcontrol strategies in an alcoholism modelrdquo Abstract andApplied Analysis vol 2014 Article ID 954069 18 pages 2014

[6] H-F Huo S-R Huang X-Y Wang and H Xiang ldquoOptimalcontrol of a social epidemic model with media coveragerdquoJournal of Biological Dynamics vol 11 no 1 pp 226ndash2432017

[7] H F Huo and N N Song ldquoGlobal stability for a bingedrinking model with two stagesrdquo Discrete Dynamics in Natureand Society vol 2012 Article ID 829386 15 pages 2012

[8] I K Adu M Osman and C Yang ldquoMathematical model ofdrinking epidemicrdquo British Journal of Mathematics ComputerScience vol 22 no 5 pp 1ndash10 2017

[9] H-F Huo and Y-P Liu ldquoe analysis of the SIRS alcoholismmodels with relapse on weightedrdquo SpringerPlus vol 5 no 1p 722 2016

[10] K Mursquotamar ldquoOptimal control strategy for alcoholism modelwith two infected compartmentsrdquo IOSR Journal of Mathe-matics vol 14 no 3 pp 58ndash67 2018

[11] S Sharma and G P Samanta ldquoDrinking as an epidemic amathematical model with dynamic behaviourrdquo Journal ofApplied Mathematics amp Informatics vol 31 no 1-2 pp 1ndash252013

[12] S Sharma and G P Samanta ldquoAnalysis of a drinking epi-demic modelrdquo International Journal of Dynamics and Controlvol 3 no 3 pp 288ndash305 2015

10 20 30 40 50 60 70 80 90 100Time (days)

0

100

200

300

400

500

600

Qui

tters

of d

rinki

ng (Q

)

Q with controls u1 u2 and u3Q without controls

(e)

Figure 6 Representing the drinkers class with and without optimal controls u1 u2 and u3

12 Journal of Applied Mathematics

M without controls

200

250

300

350

400

450M

oder

ate d

rinke

rs (M

)

10 20 30 40 50 60 70 80 90 100Time (days)

(a)

H without controls

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

10 20 30 40 50 60 70 80 90 100Time (days)

(b)

Tp without controls

10 20 30 40 50 60 70 80 90 100Time (days)

5

10

15

20

25

30

The p

oor h

eavy

drin

kers

(Tp )

(c)

Tr without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

The r

ich

heav

y dr

inke

rs (T

r )

(d)

Q without controls

55556

657

758

859

9510

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

(e)

100 20 30 40 50 60 70 80 90 100Time (days)

50100150200250300350400450500

Hea

vy d

rinke

rs (H

)

H without treatment centers of alcohol addictionH with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002H with α1 = 03 α2 = 01 γ1 = 006 γ2 = 003H with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02H with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03

(f )

Figure 2 Continued

8 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

Q with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03Q with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02Q with α1 = 03 α2 = 01 γ1 = 005 γ2 = 02Q with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002Q without treatment centers of alcohol addiction

0

50

100

150

200

250

300

350

400

Qui

tters

of d

rinki

ng (Q

)

(g)

Figure 2 Representing the drinkers class without control

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

M with control u1M without controls

(a)

H with control u1H without controls

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

(b)

Q with control u1Q without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

(c)

Figure 3 Representing the drinkers class with and without control u1

Journal of Applied Mathematics 9

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500H

eavy

drin

kers

(H)

H with control u2H without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

e p

oor h

eavy

drin

kers

(Tp )

Tp with control u2Tp without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

e r

ich

heav

y dr

inke

rs (T

r )

Tr with control u2Tr without controls

(c)

55

6

65

7

75

8

85

9

95

10

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

Q with control u2Q without controls

(d)

Figure 4 Representing the drinkers class with and without control u2

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

The p

oor h

eavy

drin

kers

(Tp )

Tp with controls u2 and u3Tp without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

The r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u2 and u3Tr without controls

(b)

Figure 5 Continued

10 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

Q with controls u2 and u3Q without controls

(c)

Figure 5 Representing the drinkers class with controls u2 and u3 and without controls

50

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

10 20 30 40 50 60 70 80 90 100Time (days)

M with controls u1 u2 and u3M without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

50

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

H with controls u1 u2 and u3H without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

510152025303540455055

e p

oor h

eavy

drin

kers

(Tp )

Tp with controls u1 u2 and u3Tp without controls

(c)

10 20 30 40 50 60 70 80 90 100Time (days)

0

20

40

60

80

100

120

140

160

180

e r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u1 u2 and u3Tr without controls

(d)

Figure 6 Continued

Journal of Applied Mathematics 11

drinking from returning to drinking So the number ofindividuals that enter the compartment Tr and Tp de-creases erefore applying a combination of the threecontrols is more effective than one control

6 Conclusion

In this research paper we introduced a discrete modelingof drinking for the purpose of minimizing the number ofdrinkers and maximizing the number of the rich and poorheavy drinkers who join private and public treatmentcenters of alcohol addiction and subsequently the numberof quitters of drinking Unlike some other previousmodels we have taken into account the impact of privateand public addiction treatment centers on alcoholics eresults showed that those centers have substantial influ-ence on the dynamics of alcoholism and can greatlyimpact the spread of drinking us it is crucial to urgepeople to know and join private and public addictiontreatment centers to quit drinking We also presentedthree controls which respectively represent awarenessprograms encouragement and follow-up We applied theresults of the control theory and we managed to obtain thecharacterizations of the optimal controls e numericalsimulation of the obtained results showed the effective-ness of the proposed control strategies

Data Availability

e disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpnetworkrepositorycomprofilephp)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] WHOWorld Health Organization Report on Drinking WHOGeneva Switzerland 2016

[2] World Health Organization Alcohol World Health Organi-zation Geneva Switzerland 2018 httpswwwwhointnews-roomfact-sheetsdetailalcohol

[3] H F Huo and Q Wang ldquoModelling the influence ofawareness programs by media on the drinking dynamicsrdquoAbstract and Applied Analysis vol 2014 Article ID 9380808 pages 2014

[4] S-H Ma H-F Huo and X-Y Meng ldquoModelling alcoholismas a contagious disease a mathematical model with awarenessprograms and time delayrdquo Discrete Dynamics in Nature andSociety vol 2015 Article ID 260195 13 pages 2015

[5] X-Y Wang H-F Huo Q-K Kong andW-X Shi ldquoOptimalcontrol strategies in an alcoholism modelrdquo Abstract andApplied Analysis vol 2014 Article ID 954069 18 pages 2014

[6] H-F Huo S-R Huang X-Y Wang and H Xiang ldquoOptimalcontrol of a social epidemic model with media coveragerdquoJournal of Biological Dynamics vol 11 no 1 pp 226ndash2432017

[7] H F Huo and N N Song ldquoGlobal stability for a bingedrinking model with two stagesrdquo Discrete Dynamics in Natureand Society vol 2012 Article ID 829386 15 pages 2012

[8] I K Adu M Osman and C Yang ldquoMathematical model ofdrinking epidemicrdquo British Journal of Mathematics ComputerScience vol 22 no 5 pp 1ndash10 2017

[9] H-F Huo and Y-P Liu ldquoe analysis of the SIRS alcoholismmodels with relapse on weightedrdquo SpringerPlus vol 5 no 1p 722 2016

[10] K Mursquotamar ldquoOptimal control strategy for alcoholism modelwith two infected compartmentsrdquo IOSR Journal of Mathe-matics vol 14 no 3 pp 58ndash67 2018

[11] S Sharma and G P Samanta ldquoDrinking as an epidemic amathematical model with dynamic behaviourrdquo Journal ofApplied Mathematics amp Informatics vol 31 no 1-2 pp 1ndash252013

[12] S Sharma and G P Samanta ldquoAnalysis of a drinking epi-demic modelrdquo International Journal of Dynamics and Controlvol 3 no 3 pp 288ndash305 2015

10 20 30 40 50 60 70 80 90 100Time (days)

0

100

200

300

400

500

600

Qui

tters

of d

rinki

ng (Q

)

Q with controls u1 u2 and u3Q without controls

(e)

Figure 6 Representing the drinkers class with and without optimal controls u1 u2 and u3

12 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

Q with α1 = 07 α2 = 04 γ1 = 06 γ2 = 03Q with α1 = 06 α2 = 03 γ1 = 05 γ2 = 02Q with α1 = 03 α2 = 01 γ1 = 005 γ2 = 02Q with α1 = 02 α2 = 005 γ1 = 004 γ2 = 002Q without treatment centers of alcohol addiction

0

50

100

150

200

250

300

350

400

Qui

tters

of d

rinki

ng (Q

)

(g)

Figure 2 Representing the drinkers class without control

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

M with control u1M without controls

(a)

H with control u1H without controls

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

(b)

Q with control u1Q without controls

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

(c)

Figure 3 Representing the drinkers class with and without control u1

Journal of Applied Mathematics 9

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500H

eavy

drin

kers

(H)

H with control u2H without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

e p

oor h

eavy

drin

kers

(Tp )

Tp with control u2Tp without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

e r

ich

heav

y dr

inke

rs (T

r )

Tr with control u2Tr without controls

(c)

55

6

65

7

75

8

85

9

95

10

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

Q with control u2Q without controls

(d)

Figure 4 Representing the drinkers class with and without control u2

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

The p

oor h

eavy

drin

kers

(Tp )

Tp with controls u2 and u3Tp without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

The r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u2 and u3Tr without controls

(b)

Figure 5 Continued

10 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

Q with controls u2 and u3Q without controls

(c)

Figure 5 Representing the drinkers class with controls u2 and u3 and without controls

50

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

10 20 30 40 50 60 70 80 90 100Time (days)

M with controls u1 u2 and u3M without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

50

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

H with controls u1 u2 and u3H without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

510152025303540455055

e p

oor h

eavy

drin

kers

(Tp )

Tp with controls u1 u2 and u3Tp without controls

(c)

10 20 30 40 50 60 70 80 90 100Time (days)

0

20

40

60

80

100

120

140

160

180

e r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u1 u2 and u3Tr without controls

(d)

Figure 6 Continued

Journal of Applied Mathematics 11

drinking from returning to drinking So the number ofindividuals that enter the compartment Tr and Tp de-creases erefore applying a combination of the threecontrols is more effective than one control

6 Conclusion

In this research paper we introduced a discrete modelingof drinking for the purpose of minimizing the number ofdrinkers and maximizing the number of the rich and poorheavy drinkers who join private and public treatmentcenters of alcohol addiction and subsequently the numberof quitters of drinking Unlike some other previousmodels we have taken into account the impact of privateand public addiction treatment centers on alcoholics eresults showed that those centers have substantial influ-ence on the dynamics of alcoholism and can greatlyimpact the spread of drinking us it is crucial to urgepeople to know and join private and public addictiontreatment centers to quit drinking We also presentedthree controls which respectively represent awarenessprograms encouragement and follow-up We applied theresults of the control theory and we managed to obtain thecharacterizations of the optimal controls e numericalsimulation of the obtained results showed the effective-ness of the proposed control strategies

Data Availability

e disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpnetworkrepositorycomprofilephp)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] WHOWorld Health Organization Report on Drinking WHOGeneva Switzerland 2016

[2] World Health Organization Alcohol World Health Organi-zation Geneva Switzerland 2018 httpswwwwhointnews-roomfact-sheetsdetailalcohol

[3] H F Huo and Q Wang ldquoModelling the influence ofawareness programs by media on the drinking dynamicsrdquoAbstract and Applied Analysis vol 2014 Article ID 9380808 pages 2014

[4] S-H Ma H-F Huo and X-Y Meng ldquoModelling alcoholismas a contagious disease a mathematical model with awarenessprograms and time delayrdquo Discrete Dynamics in Nature andSociety vol 2015 Article ID 260195 13 pages 2015

[5] X-Y Wang H-F Huo Q-K Kong andW-X Shi ldquoOptimalcontrol strategies in an alcoholism modelrdquo Abstract andApplied Analysis vol 2014 Article ID 954069 18 pages 2014

[6] H-F Huo S-R Huang X-Y Wang and H Xiang ldquoOptimalcontrol of a social epidemic model with media coveragerdquoJournal of Biological Dynamics vol 11 no 1 pp 226ndash2432017

[7] H F Huo and N N Song ldquoGlobal stability for a bingedrinking model with two stagesrdquo Discrete Dynamics in Natureand Society vol 2012 Article ID 829386 15 pages 2012

[8] I K Adu M Osman and C Yang ldquoMathematical model ofdrinking epidemicrdquo British Journal of Mathematics ComputerScience vol 22 no 5 pp 1ndash10 2017

[9] H-F Huo and Y-P Liu ldquoe analysis of the SIRS alcoholismmodels with relapse on weightedrdquo SpringerPlus vol 5 no 1p 722 2016

[10] K Mursquotamar ldquoOptimal control strategy for alcoholism modelwith two infected compartmentsrdquo IOSR Journal of Mathe-matics vol 14 no 3 pp 58ndash67 2018

[11] S Sharma and G P Samanta ldquoDrinking as an epidemic amathematical model with dynamic behaviourrdquo Journal ofApplied Mathematics amp Informatics vol 31 no 1-2 pp 1ndash252013

[12] S Sharma and G P Samanta ldquoAnalysis of a drinking epi-demic modelrdquo International Journal of Dynamics and Controlvol 3 no 3 pp 288ndash305 2015

10 20 30 40 50 60 70 80 90 100Time (days)

0

100

200

300

400

500

600

Qui

tters

of d

rinki

ng (Q

)

Q with controls u1 u2 and u3Q without controls

(e)

Figure 6 Representing the drinkers class with and without optimal controls u1 u2 and u3

12 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

100

150

200

250

300

350

400

450

500H

eavy

drin

kers

(H)

H with control u2H without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

e p

oor h

eavy

drin

kers

(Tp )

Tp with control u2Tp without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

e r

ich

heav

y dr

inke

rs (T

r )

Tr with control u2Tr without controls

(c)

55

6

65

7

75

8

85

9

95

10

Qui

tters

of d

rinki

ng (Q

)

10 20 30 40 50 60 70 80 90 100Time (days)

Q with control u2Q without controls

(d)

Figure 4 Representing the drinkers class with and without control u2

10 20 30 40 50 60 70 80 90 100Time (days)

0

10

20

30

40

50

60

70

The p

oor h

eavy

drin

kers

(Tp )

Tp with controls u2 and u3Tp without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

The r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u2 and u3Tr without controls

(b)

Figure 5 Continued

10 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

Q with controls u2 and u3Q without controls

(c)

Figure 5 Representing the drinkers class with controls u2 and u3 and without controls

50

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

10 20 30 40 50 60 70 80 90 100Time (days)

M with controls u1 u2 and u3M without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

50

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

H with controls u1 u2 and u3H without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

510152025303540455055

e p

oor h

eavy

drin

kers

(Tp )

Tp with controls u1 u2 and u3Tp without controls

(c)

10 20 30 40 50 60 70 80 90 100Time (days)

0

20

40

60

80

100

120

140

160

180

e r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u1 u2 and u3Tr without controls

(d)

Figure 6 Continued

Journal of Applied Mathematics 11

drinking from returning to drinking So the number ofindividuals that enter the compartment Tr and Tp de-creases erefore applying a combination of the threecontrols is more effective than one control

6 Conclusion

In this research paper we introduced a discrete modelingof drinking for the purpose of minimizing the number ofdrinkers and maximizing the number of the rich and poorheavy drinkers who join private and public treatmentcenters of alcohol addiction and subsequently the numberof quitters of drinking Unlike some other previousmodels we have taken into account the impact of privateand public addiction treatment centers on alcoholics eresults showed that those centers have substantial influ-ence on the dynamics of alcoholism and can greatlyimpact the spread of drinking us it is crucial to urgepeople to know and join private and public addictiontreatment centers to quit drinking We also presentedthree controls which respectively represent awarenessprograms encouragement and follow-up We applied theresults of the control theory and we managed to obtain thecharacterizations of the optimal controls e numericalsimulation of the obtained results showed the effective-ness of the proposed control strategies

Data Availability

e disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpnetworkrepositorycomprofilephp)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] WHOWorld Health Organization Report on Drinking WHOGeneva Switzerland 2016

[2] World Health Organization Alcohol World Health Organi-zation Geneva Switzerland 2018 httpswwwwhointnews-roomfact-sheetsdetailalcohol

[3] H F Huo and Q Wang ldquoModelling the influence ofawareness programs by media on the drinking dynamicsrdquoAbstract and Applied Analysis vol 2014 Article ID 9380808 pages 2014

[4] S-H Ma H-F Huo and X-Y Meng ldquoModelling alcoholismas a contagious disease a mathematical model with awarenessprograms and time delayrdquo Discrete Dynamics in Nature andSociety vol 2015 Article ID 260195 13 pages 2015

[5] X-Y Wang H-F Huo Q-K Kong andW-X Shi ldquoOptimalcontrol strategies in an alcoholism modelrdquo Abstract andApplied Analysis vol 2014 Article ID 954069 18 pages 2014

[6] H-F Huo S-R Huang X-Y Wang and H Xiang ldquoOptimalcontrol of a social epidemic model with media coveragerdquoJournal of Biological Dynamics vol 11 no 1 pp 226ndash2432017

[7] H F Huo and N N Song ldquoGlobal stability for a bingedrinking model with two stagesrdquo Discrete Dynamics in Natureand Society vol 2012 Article ID 829386 15 pages 2012

[8] I K Adu M Osman and C Yang ldquoMathematical model ofdrinking epidemicrdquo British Journal of Mathematics ComputerScience vol 22 no 5 pp 1ndash10 2017

[9] H-F Huo and Y-P Liu ldquoe analysis of the SIRS alcoholismmodels with relapse on weightedrdquo SpringerPlus vol 5 no 1p 722 2016

[10] K Mursquotamar ldquoOptimal control strategy for alcoholism modelwith two infected compartmentsrdquo IOSR Journal of Mathe-matics vol 14 no 3 pp 58ndash67 2018

[11] S Sharma and G P Samanta ldquoDrinking as an epidemic amathematical model with dynamic behaviourrdquo Journal ofApplied Mathematics amp Informatics vol 31 no 1-2 pp 1ndash252013

[12] S Sharma and G P Samanta ldquoAnalysis of a drinking epi-demic modelrdquo International Journal of Dynamics and Controlvol 3 no 3 pp 288ndash305 2015

10 20 30 40 50 60 70 80 90 100Time (days)

0

100

200

300

400

500

600

Qui

tters

of d

rinki

ng (Q

)

Q with controls u1 u2 and u3Q without controls

(e)

Figure 6 Representing the drinkers class with and without optimal controls u1 u2 and u3

12 Journal of Applied Mathematics

10 20 30 40 50 60 70 80 90 100Time (days)

0

50

100

150

200

250

300

350

400

450

Qui

tters

of d

rinki

ng (Q

)

Q with controls u2 and u3Q without controls

(c)

Figure 5 Representing the drinkers class with controls u2 and u3 and without controls

50

100

150

200

250

300

350

400

450

Mod

erat

e drin

kers

(M)

10 20 30 40 50 60 70 80 90 100Time (days)

M with controls u1 u2 and u3M without controls

(a)

10 20 30 40 50 60 70 80 90 100Time (days)

50

100

150

200

250

300

350

400

450

500

Hea

vy d

rinke

rs (H

)

H with controls u1 u2 and u3H without controls

(b)

10 20 30 40 50 60 70 80 90 100Time (days)

510152025303540455055

e p

oor h

eavy

drin

kers

(Tp )

Tp with controls u1 u2 and u3Tp without controls

(c)

10 20 30 40 50 60 70 80 90 100Time (days)

0

20

40

60

80

100

120

140

160

180

e r

ich

heav

y dr

inke

rs (T

r )

Tr with controls u1 u2 and u3Tr without controls

(d)

Figure 6 Continued

Journal of Applied Mathematics 11

drinking from returning to drinking So the number ofindividuals that enter the compartment Tr and Tp de-creases erefore applying a combination of the threecontrols is more effective than one control

6 Conclusion

In this research paper we introduced a discrete modelingof drinking for the purpose of minimizing the number ofdrinkers and maximizing the number of the rich and poorheavy drinkers who join private and public treatmentcenters of alcohol addiction and subsequently the numberof quitters of drinking Unlike some other previousmodels we have taken into account the impact of privateand public addiction treatment centers on alcoholics eresults showed that those centers have substantial influ-ence on the dynamics of alcoholism and can greatlyimpact the spread of drinking us it is crucial to urgepeople to know and join private and public addictiontreatment centers to quit drinking We also presentedthree controls which respectively represent awarenessprograms encouragement and follow-up We applied theresults of the control theory and we managed to obtain thecharacterizations of the optimal controls e numericalsimulation of the obtained results showed the effective-ness of the proposed control strategies

Data Availability

e disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpnetworkrepositorycomprofilephp)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] WHOWorld Health Organization Report on Drinking WHOGeneva Switzerland 2016

[2] World Health Organization Alcohol World Health Organi-zation Geneva Switzerland 2018 httpswwwwhointnews-roomfact-sheetsdetailalcohol

[3] H F Huo and Q Wang ldquoModelling the influence ofawareness programs by media on the drinking dynamicsrdquoAbstract and Applied Analysis vol 2014 Article ID 9380808 pages 2014

[4] S-H Ma H-F Huo and X-Y Meng ldquoModelling alcoholismas a contagious disease a mathematical model with awarenessprograms and time delayrdquo Discrete Dynamics in Nature andSociety vol 2015 Article ID 260195 13 pages 2015

[5] X-Y Wang H-F Huo Q-K Kong andW-X Shi ldquoOptimalcontrol strategies in an alcoholism modelrdquo Abstract andApplied Analysis vol 2014 Article ID 954069 18 pages 2014

[6] H-F Huo S-R Huang X-Y Wang and H Xiang ldquoOptimalcontrol of a social epidemic model with media coveragerdquoJournal of Biological Dynamics vol 11 no 1 pp 226ndash2432017

[7] H F Huo and N N Song ldquoGlobal stability for a bingedrinking model with two stagesrdquo Discrete Dynamics in Natureand Society vol 2012 Article ID 829386 15 pages 2012

[8] I K Adu M Osman and C Yang ldquoMathematical model ofdrinking epidemicrdquo British Journal of Mathematics ComputerScience vol 22 no 5 pp 1ndash10 2017

[9] H-F Huo and Y-P Liu ldquoe analysis of the SIRS alcoholismmodels with relapse on weightedrdquo SpringerPlus vol 5 no 1p 722 2016

[10] K Mursquotamar ldquoOptimal control strategy for alcoholism modelwith two infected compartmentsrdquo IOSR Journal of Mathe-matics vol 14 no 3 pp 58ndash67 2018

[11] S Sharma and G P Samanta ldquoDrinking as an epidemic amathematical model with dynamic behaviourrdquo Journal ofApplied Mathematics amp Informatics vol 31 no 1-2 pp 1ndash252013

[12] S Sharma and G P Samanta ldquoAnalysis of a drinking epi-demic modelrdquo International Journal of Dynamics and Controlvol 3 no 3 pp 288ndash305 2015

10 20 30 40 50 60 70 80 90 100Time (days)

0

100

200

300

400

500

600

Qui

tters

of d

rinki

ng (Q

)

Q with controls u1 u2 and u3Q without controls

(e)

Figure 6 Representing the drinkers class with and without optimal controls u1 u2 and u3

12 Journal of Applied Mathematics

drinking from returning to drinking So the number ofindividuals that enter the compartment Tr and Tp de-creases erefore applying a combination of the threecontrols is more effective than one control

6 Conclusion

In this research paper we introduced a discrete modelingof drinking for the purpose of minimizing the number ofdrinkers and maximizing the number of the rich and poorheavy drinkers who join private and public treatmentcenters of alcohol addiction and subsequently the numberof quitters of drinking Unlike some other previousmodels we have taken into account the impact of privateand public addiction treatment centers on alcoholics eresults showed that those centers have substantial influ-ence on the dynamics of alcoholism and can greatlyimpact the spread of drinking us it is crucial to urgepeople to know and join private and public addictiontreatment centers to quit drinking We also presentedthree controls which respectively represent awarenessprograms encouragement and follow-up We applied theresults of the control theory and we managed to obtain thecharacterizations of the optimal controls e numericalsimulation of the obtained results showed the effective-ness of the proposed control strategies

Data Availability

e disciplinary data used to support the findings of thisstudy have been deposited in the Network Repository(httpnetworkrepositorycomprofilephp)

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] WHOWorld Health Organization Report on Drinking WHOGeneva Switzerland 2016

[2] World Health Organization Alcohol World Health Organi-zation Geneva Switzerland 2018 httpswwwwhointnews-roomfact-sheetsdetailalcohol

[3] H F Huo and Q Wang ldquoModelling the influence ofawareness programs by media on the drinking dynamicsrdquoAbstract and Applied Analysis vol 2014 Article ID 9380808 pages 2014

[4] S-H Ma H-F Huo and X-Y Meng ldquoModelling alcoholismas a contagious disease a mathematical model with awarenessprograms and time delayrdquo Discrete Dynamics in Nature andSociety vol 2015 Article ID 260195 13 pages 2015

[5] X-Y Wang H-F Huo Q-K Kong andW-X Shi ldquoOptimalcontrol strategies in an alcoholism modelrdquo Abstract andApplied Analysis vol 2014 Article ID 954069 18 pages 2014

[6] H-F Huo S-R Huang X-Y Wang and H Xiang ldquoOptimalcontrol of a social epidemic model with media coveragerdquoJournal of Biological Dynamics vol 11 no 1 pp 226ndash2432017

[7] H F Huo and N N Song ldquoGlobal stability for a bingedrinking model with two stagesrdquo Discrete Dynamics in Natureand Society vol 2012 Article ID 829386 15 pages 2012

[8] I K Adu M Osman and C Yang ldquoMathematical model ofdrinking epidemicrdquo British Journal of Mathematics ComputerScience vol 22 no 5 pp 1ndash10 2017

[9] H-F Huo and Y-P Liu ldquoe analysis of the SIRS alcoholismmodels with relapse on weightedrdquo SpringerPlus vol 5 no 1p 722 2016

[10] K Mursquotamar ldquoOptimal control strategy for alcoholism modelwith two infected compartmentsrdquo IOSR Journal of Mathe-matics vol 14 no 3 pp 58ndash67 2018

[11] S Sharma and G P Samanta ldquoDrinking as an epidemic amathematical model with dynamic behaviourrdquo Journal ofApplied Mathematics amp Informatics vol 31 no 1-2 pp 1ndash252013

[12] S Sharma and G P Samanta ldquoAnalysis of a drinking epi-demic modelrdquo International Journal of Dynamics and Controlvol 3 no 3 pp 288ndash305 2015

10 20 30 40 50 60 70 80 90 100Time (days)

0

100

200

300

400

500

600

Qui

tters

of d

rinki

ng (Q

)

Q with controls u1 u2 and u3Q without controls

(e)

Figure 6 Representing the drinkers class with and without optimal controls u1 u2 and u3

12 Journal of Applied Mathematics

[13] W Ding R Hendon B Cathey E Lancaster and R GermickldquoDiscrete time optimal control applied to pest controlproblemsrdquo Involve A Journal of Mathematics vol 7 no 4pp 479ndash489 2014

[14] V Guibout and A M Bloch ldquoA discrete maximum principlefor solving optimal control problemsrdquo in Proceedings of the2004 43rd IEEE Conference on Decision and Control (CDC)vol 2 pp 1806ndash1811 Nassau Bahamas December 2004

[15] C L Hwang and L T Fan ldquoA discrete version of pontryaginrsquosmaximum principlerdquo Operations Research vol 15 no 1pp 139ndash146 1967

[16] L S Pontryagin V G Boltyanskii R V Gamkrelidze andE F Mishchenko 5e Mathematical 5eory of OptimalProcesses John Wiley amp Sons London UK 1962

[17] S Lenhart and J T Workman Optimal Control Applied toBiological Models Chapman amp HallCRC Boca Raton FLUSA 2007

[18] D C Zhang and B Shi ldquoOscillation and global asymptoticstability in a discrete epidemic modelrdquo Journal of Mathe-matical Analysis and Applications vol 278 no 1 pp 194ndash2022003

Journal of Applied Mathematics 13