Research ArticleMathematical Modeling and Optimal Control Strategy for aDiscrete Time Drug Consumption Model

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

and Mostafa Rachik 1

1Laboratory of Analysis Modeling and Simulation Department of Mathematics and Computer ScienceFaculty of Sciences Ben MrsquoSik Sidi Othman Hassan II University Casablanca Morocco2Laboratory of Dynamical Systems Department of Mathematics Faculty of Sciences El Jadida Chouaib Doukkali UniversityEl Jadida Morocco

Correspondence should be addressed to Abderrahim Labzai labzaiabdo1977gmailcom

Received 17 December 2019 Accepted 22 June 2020 Published 6 August 2020

Academic Editor Ricardo Lopez-Ruiz

Copyright copy 2020 Abderrahim Labzai 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

(e aim of this paper is to study and investigate the optimal control strategy of a discrete mathematical model of drug con-sumption(e population that we are going to study is divided into six compartments potential drug users light drug users heavydrug users heavy drug users-dealers and providers temporary quitters of drug consumption and permanent quitters of drugconsumption Our objective is to find the best strategy to reduce the number of light drug users heavy drug users heavy drugusers-dealers and providers and temporary quitters of drug consumption We use four control strategies which are awarenessprograms through media and education preventing contact through security campaigns treatment and psychological supportalong with follow-up Pontryaginrsquos maximum principle in discrete time is used to characterize the optimal controls(e numericalsimulation is carried out using MATLAB Consequently the obtained results confirm the effectiveness of theoptimization strategy

1 Introduction

Drug addiction is a serious problem that affects a largepopulation of the world causing some people diseasespsychological disorders and in many cases it leads to deathDespite the efforts made to decrease the consumption ofdrugs there is an ongoing increase of the actual number ofpeople who use drugs by 20million people from 2015 to 2016according to the World Drug Report in 2018 [1] It is es-timated that 271 million people worldwide which representsapproximately 56 percent of the global population aged15ndash64 years used drugs at least once during 2017 About 35million of drugs users suffer from drug use disorders Asregards to the number of deaths the Global Burden ofDisease Study 2017 estimated that globally in 2017 therewere 585000 deaths and 42 million years of ldquohealthyrdquo lifelost as a result of the use of drugs [2 3] Drugs consumption

can cause some treated and untreated diseases such ashepatitis C heart disorders high blood pressure increase inthe proportion of toxins in the body impaired sexual ac-tivity chronic gastric infections dyspepsia and infections ofthe pancreas gland (ere are also some psychological andmental damages induced by drug use for example negli-gence of appearance excessive nervousness tension andirritation In addition the drug users are cut off from thefamily atmosphere and even from the whole society andtheir relationship with their families and friends collapsesTraffic accidents prevail in roads in many cases due to drugsconsumption

In terms of the financial effects of the spread of drug usedrug money can have negative consequences on economy inthe sense that it makes countries poorer especially whendrug-related revenues make up a large part of the totaleconomy of any society or country (is type of economy is

HindawiDiscrete Dynamics in Nature and SocietyVolume 2020 Article ID 5671493 10 pageshttpsdoiorg10115520205671493

called ldquoillicit economyrdquo as it weakens the rule of law andfacilitates corruption which in turn strengthens the illicitdrug sector [1]

In Morocco drugs remain a source of worry for thepeople and government as it remains illegal and forbidden inthe country Some epidemiological studies previously car-ried out in Morocco state that drug use is increasing amongyoung people and women So in order to come up with well-designed surveys on drug use in schools Morocco adoptedthe Mediterranean School Survey Project on Alcohol andOther Drugs (MedSPAD) [4] which is supported by thePompidou Group of the Council of Europe MedSPADsurveys were conducted in Morocco in two big cities (Rabatand Sale) following a pilot survey in 2003 before the firstnationwide study was undertaken in 2009 and the second in2013 MedSPAD aims were to determine the prevalence ofsubstance use among 15ndash 17-year-old young people inMorocco to determine the age of onset of drug use and tolearn about teenagersrsquo knowledge perspectives and be-haviors regarding drugs (e project worked on identifyingsome of the predictive factors of drug use to help in for-mulating strong policies for facilitating mental health sup-port and drug prevention in Moroccan schools (e surveyscarried out by MedSPAD found initial drug use starts from ayoung age which is alarming and indicates that preventiveand counseling programs need to be executed for very youngstudents (elementary school) prior to onset (erefore theeffective school prevention policies and community inter-vention programs (prevention treatment and rehabilita-tion) must be developed to find an end to this problem [4]

Mathematical modeling of drug consumption has beenstudied by many researchers [5ndash11] In fact most of thoseresearchers were interested in the continuous time modelsdescribed by the differential equations Recently more at-tention has been given to discrete time models (see[8 12ndash16] and the references cited therein) (e rationalebehind using discrete modeling can be summarized asfollows Firstly the statistical data are collected at discretemoments (day week month or year) So it is more directmore accurate and timely to describe the disease usingdiscrete time models compared to continuous time modelsSecondly the use of discrete time models can avoid somemathematical complexities such as choosing a functionspace and regularity of the solution (irdly the numericalsimulations of continuous time models are obtained by theway of discretization [8 12]

Based on the aforementioned reasons we will develop adiscrete time model in order to study the dynamics of thepopulation that uses drug and we will introduce two classesof drug users distributed into a compartment of heavy drugconsumers who do not influence other individuals and thecompartment of heavy drug users-dealers and providers ofdrug who do influence other individuals Also we add toour model an element which was not taken into consider-ation in most previous research studies namely a group ofheavy drug users who transform to heavy drug users-dealersand providers of drug

In addition in order to find the best strategy to reducethe number of light drug users heavy drug users and

temporary quitters of drugs we will use four controlstrategies awareness programs through media and educa-tion preventing contact through security campaignstreatment and psychological support along with follow-up

(e paper is organized as follows In Section 2 wepresent our discrete mathematical model that describes thedynamics of the population that uses drugs In Section 3 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 4 Finally theconclusion is given in Section 5

2 A Mathematical Model

In this section we present a discrete PkLkH+kH

minuskQ

tkQ

p

k

mathematical model of drug consumers (e populationunder investigation is divided into six compartments in-dividuals who are not yet drug users but interact with drugusers Pk light or occasional drug users Lk heavy drug userswho do not influence other individuals H+

k heavy drugusers-dealers and providers who do influence other indi-viduals Hminus

k individuals who temporarily quit drug con-sumption Qt

k and individuals who permanently quit drugconsumption Q

p

k respectively

21 Description of the Model

(i) e compartment P represents the potential drugusers whose age is over adolescence and adulthoodand who do not use drugs but may use drugssubsequently due to interaction with drug users It isassumed that potential drug users can acquire drugconsumption behavior and can become light drugconsumers through effective contact with occa-sional drug users in some social occasions such asweddings graduation ceremonies and week-endparties In other words it is assumed that the ac-quisition of a drug consumption behavior is anal-ogous to acquiring disease infection (iscompartment is increased by the recruitment ofindividuals at rate Λ and it is decreased by the ratesα1(PkLkN) α2(PkHminus

k N) Some people of thiscompartment leave at a constant death rate of μ dueto the total natural death rate μPk

(ii) e compartment L contains light drug users whocan control their consumption during some eventsand occasions or they use drugs in a way that isunapparent to their social environment (is cate-gory of light drug users does not face any problemsor negative consequences their friends or family donot complain about their drug intake Light drugusers neither think about drugs very often nor dothey feel a need to using drugs When using drugsthey are able to handle their drug consumptionwithout experiencing a loss of control Drugs do notdominate their thoughts and they do not need to setlimits when they use drugs (ey are not prone to

2 Discrete Dynamics in Nature and Society

extreme mood swings fighting or being violentand their number is increased when they start usingit with a rate α1(PkLkN) In this compartmentsome other individuals will leave at the ratesβ2(LkHminus

k N) β1Lk and μLk Here β1 and β2 are therates of light drug consumers who transform intoheavy drug consumers and drug dealers and pro-viders consecutively (is compartment is increasedby θQt

k (at the rate θ) due to temporary quitters whorevert back to using the drug

(iii) e compartment H+ encompasses heavy drugconsumers who do not influence other individualsand who are suffering from addiction to drugconsumption When an individual becomes a heavydrug user they face a great difficulty to control or setlimits to their consumption (eir job their familysocial circle and health are all endangered Despitethese negative consequences the heavy drug con-sumers are unable to quit using drugs (e heavydrug consumers may begin to disclaim that theyhave a problem this disclaim can make it even moredifficult for the person to get help (is compart-ment is increased by the rate β1(LkH+

k N) and isdecreased by the rate cH+

k and the rate μH+k due to

natural death(iv) e compartment Hminus includes heavy drug users

dealers and providers of drugs who do influenceother individuals (ey work to make big profitsthrough illegal ways ie a drug dealer can be de-fined as a person who sells drugs of any type orquantity in an illegal manner A dealer can be anoccasional seller of drugs by selling small quantitiesto cover the costs of their own drug use or they canbe highly organized groups and businessmen withinhigh-organized operations that run like a seriousbusiness (ey are increased by the rateα2(PkHminus

k N) β2(LkHminusk N) and decreased by the

rate cHminusk and the rate μHminus

k due to natural death(v) e compartment Qt is composed of the individ-

uals who temporarily quit drugs and are increased atthe rate c(1 minus σ1)H+

k λσ2Hminusk where (1 minus σ1) is the

fraction of drug users who temporarily quit usingdrug (at a rate c) σ1 is the remaining fraction ofheavy drug users who permanently and temporarilyquit using drug (at the rate c and λ) (is com-partment is decreased by the natural death μQt

k andthe rate θQt

k represents the persons who return to belight drug users

(vi) e compartment Qp includes individuals whopermanently quit drugs (is compartment is in-creased at the rate cσ1H+

k and λ(1 minus σ2)Hminusk where

(1 minus σ2) is the fraction of drug users who perma-nently quit using drugs (at a rate c) σ2 is theremaining fraction of heavy drug users who per-manently and temporarily quit using drugs (at a ratec) Some people of this compartment will die at therate μQ

p

k

(e variables Pk LkH+k Hminus

k Qtk andQ

p

k are the numbersof the individuals in the six classes at time k respectively(eunit k can correspond to periods phases or years It dependson the frequency of the survey studies as needed

(e following diagram will demonstrate the flow di-rections of individuals among the compartments (esedirections are going to be represented by directed arrows(see Figure 1)

(e total population size at time k is denoted by Nk withNk Pk + Lk + H+

k + Hminusk + Qt

k + Qp

k

22 Model Equations By the addition of the rates at whichindividuals enter the compartment and also by subtractingthe rates at which people leave the compartment we obtain adifference equation for the rate at which the individuals ofeach compartment change over discrete time Hence wepresent the drug users infection model by the followingsystem of difference equations

Pk+1 Λ +(1 minus μ)Pk minus α1PkLk

Nminus α2

PkHminusk

N

Lk+1 1 minus μ minus β1( 1113857Lk + θQtk + α1

PkLk

Nminus β2

LkHminusk

N

H+k+1 1 minus μ minus β3 minus c( 1113857H+

k + β1Lk + β2LkHminus

k

N+ α2

PkHminusk

N

Hminusk+1 (1 minus μ minus λ)Hminus

k + β3H+k

Qtk+1 (1 minus μ minus θ)Qt

k + c 1 minus σ1( 1113857H+k + λσ2Hminus

k

Qp

k+1 (1 minus μ)Qp

k + cσ1H+k + λ 1 minus σ2( 1113857Hminus

k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where P0 ge 0 L0 ge 0 H+0 ge 0 Hminus

0 ge 0 Qt0 ge 0 and Q

p0 ge 0 are

the given initial states

3 The Optimal Control Problem

(e strategies of control that we adopt consist of anawareness program through media and education contactprevention through security campaigns treatment andpsychological support along with follow-up Our main goalin adopting these strategies is to minimize the number oflight drug consumers heavy drug consumers heavy drugusers-dealers and the temporary quitters of drugs during thetime steps k 0 to T minus 1 and also minimize the cost spent inapplying these strategies In this model we include the fourcontrols u1k u2k u3k and u4k that represent consecutivelyawareness programs through media and education contactprevention through security campaigns treatment andpsychological support along with follow-up as measures attime k So the controlled mathematical system is given bythe following system of difference equations

Discrete Dynamics in Nature and Society 3

Pk+1 Λ +(1 minus μ)Pk minus α1 1 minus u1k1113872 1113873PkLk

Nminus α2 1 minus u2k1113872 1113873

PkHminusk

N

Lk+1 1 minus μ minus β1( 1113857Lk + θQtk + α1 1 minus u1k1113872 1113873

PkLk

Nminus β2 1 minus u2k1113872 1113873

LkHminusk

N

H+k+1 1 minus μ minus β3 minus c( 1113857H+

k + β1LkH+

k

N+ β2 1 minus u2k1113872 1113873

LkHminusk

N+ α2 1 minus vk( 1113857

PkHminusk

Nminus u3kH

+k

Hminusk+1 (1 minus μ minus λ)Hminus

k + β3H+k

Qtk+1 (1 minus μ minus θ)Qt

k + c 1 minus σ1( 1113857H+k + λσ2Hminus

k minus u4kQtk

Qp

k+1 (1 minus μ)Qp

k + cσ1H+k + λ 1 minus σ2( 1113857Hminus

k + u3kH+k + u4kQt

k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where P0 ge 0 L0 ge 0 H+0 ge 0 Hminus

0 ge 0 Qt0 ge 0 and Q

p0 ge 0 are

the given initial states(ere are four controls u1k (u10 u11 u1Tminus1)

u2k (u20 u21 u2Tminus1) u3k (u30 u31 u3Tminus1)and u4k (u40 u41 u4Tminus1) (e first control can beinterpreted as the proportion to be adopted to awarenessprograms through media and education So we note that(1 minus u1k)(PkLkN) is the proportion of the potential drugusers who are protected from contacting light drug con-sumers at time step k (e second control can be interpretedas the proportion of contact prevention through securitycampaigns We observe that (1 minus u2k)(LkHminus

k N) is the pro-portion of the individuals who will be prevented to move fromthe class of light drug consumers towards the class of heavydrug users dealers and providers at time step k (e thirdcontrol can be interpreted as the proportion of individuals to besubjected to treatment So we note that u3kH+

k is the pro-portion of the individuals who will move from the class ofheavy drug consumers towards the class of the individuals whopermanently quit using drug at time step k (e fourth controlcan be interpreted as the proportion of individuals who will getpsychological support alongwith follow-up So we observe thatu4kQt

k is the proportion of the individuals who temporarilyquit using drugs and who will transform into the individualswho permanently quit drug at time step k

(e challenge that we face here is how to minimize theobjective functional

J u1k u2k u3k u4k1113872 1113873 ATLT + BTH+T + CTH

minusT + DTQ

tT

+ 1113944Tminus1

k0AkLk + BkH

+k + CkH

minusk + DkQ

tk1113872 1113873

+ 1113944

Tminus1

k0

Ek

2u21k +

Fk

2u22k +

Gk

2u23k +

Mk

2u24k

(3)

where the parameters Ak gt 0 Bk gt 0 Ck gt 0 Dk gt 0 Ek gt 0Fk gt 0 and Mk gt 0 are the cost coefficients and they areselected to weigh the relative importance of Lk H+

k Hminusk Qt

ku1k u2k u3k and u4k at time k T is the final time

In other words we seek the optimal controls u1k u2ku3k and u4k such that

J ulowast1k ulowast2k ulowast3k ulowast4k1113872 1113873 min

u1 u2 u3 u4( )isinU4a d

J u1k u2k u3k u4k1113872 1113873

(4)

where Ua d is the set of admissible controls defined by

microPk microLk

β1Lk

β3H+

β2 (LkHkndashN)

α2 (PkHkndashN)

microHndash microQp

microQt

θQkt

microH+

γ (1 ndash σ1)Hk+

λ (1 ndash σ2)Hkndash

λσ2Hkndash

γσ1Hk+

α1(PkLkN)P L

H+

Hndash

Qt

Qp

Λ

Figure 1 (e flow between the five compartments PLH+Hminus QtQp

4 Discrete Dynamics in Nature and Society

Uad uik ui0 ui1 uiTminus11113872 11138731113966

for i 1 2 3 4 ai le uik le bi k 0 1 2 T minus 11113967

(5)

(e sufficient condition for the existence of the optimalcontrols (u1 u2 u3 u4) for problems (2) and (3) comes fromthe following theorem

Theorem 1 1ere exists the optimal controls(ulowast1k ulowast2k ulowast3k ulowast4k) such that

J ulowast1k ulowast2k ulowast3k ulowast4k1113872 1113873 min

u1 u2u3 u4( )isinU4ad

J u1 u2 u3 u4( 1113857

(6)

subject to the control system (2) with initial conditions

Proof Since the coefficients of the state equations arebounded and there is a finite number of time steps P

(P0 P1 PT) L (L0 L1 LT) H+ (H+0 H+

1

H+T) Hminus (Hminus

0 Hminus1 Hminus

T)Qt (Qt0 Qt

1 QtT) and

Qp (Qp0 Q

p1 Q

p

T) are uniformly bounded for all(u1 u2 u3 u4) in the control set Uad thus J(u1 u2 u3 u4) isbounded for all (u1 u2 u3 u4) isin U4

ad Since J(u1 u2 u3 u4)

is bounded inf(u1 u2u3 u4)isinU4ad

J(u1 u2 u3 u4) is finite and

there exists a sequence (uj1 u

j2 u

j3 u

j4) isin U4a d such that

limj⟶+infin

J(uj1 u

j2 u

j3 u

j4) inf

(u1 u2u3 u4)isinU4a d

J(u1 u2 u3 u4) and

corresponding sequences of states Pj Lj H+j Hminus j Qtj andQpj Since there is a finite number of uniformly boundedsequences there exist (ulowast1 ulowast2 ulowast3 ulowast4 ) isin U4

ad and Plowast Llowast H+lowast

Hminuslowast Qtlowast and Qplowast isin IRT+1 such that on a subsequence(u

j1 u

j2 u

j3 u

j4)⟶ (ulowast1 ulowast2 ulowast3 ulowast4 ) Pj⟶ Plowast Lj⟶ Llowast

H+j⟶ H+jlowast Hminusj⟶ Hminusjlowast Qtj⟶ Qtlowast and Qpj⟶Qplowast Finally due to the finite dimensional structure ofsystem (2) and the objective function J(u1 u2 u3 u4)(ulowast1 ulowast2 ulowast3 ulowast4 ) is an optimal control with correspondingstates Plowast Llowast H+jlowast Hminus jlowast and Qplowast (ereforeinf(u1u2 u3 u4)isinU4

adJ(u1 u2 u3 u4) is achieved

We apply the discrete version of Pontryaginrsquos Maxi-mum Principle [10 14 16ndash19] (e key idea is introducingthe adjoint function to attach the system of differenceequations to the objective functional resulting in theformation of a function called the Hamiltonian (is

principle converts the problem of finding the control tooptimize the objective functional subject to the statedifference equation with initial condition to find thecontrol to optimize Hamiltonian pointwise (with respectto the control)

We have the Hamiltonian Hk at time step k defined by

Hk AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k

+Gk

2u23k +

Mk

2u24k + 1113944

6

i1ζ ik+1fik+1

(7)

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

Theorem 2 Given the optimal controls (ulowast1k ulowast2k

ulowast3k ulowast4k) isin U4a d and the solutions Plowast Llowast H+jlowast Hminus jlowast Qtlowast

and Qplowast of the corresponding state system (2) thereexist adjoint functions ζ1k ζ2k ζ3k ζ4k ζ5k and ζ6k

satisfying

ζ1k ζ1k+1(1 minus μ) + α1 1 minus u1k1113872 1113873Lk

Nζ2k+1 minus ζ1k+11113872 1113873

ζ2k Ak + α1 ζ2k+1 minus ζ1k+11113872 1113873Pk

N+ β1 ζ3k+1 minus ζ2k+11113872 1113873

+ α2 1 minus u2k1113872 1113873Hminus

k

Nζ3k+1 minus ζ1k+11113872 1113873

ζ3k Bk + ζ3k+1 1 minus μ minus β3 minus c( 1113857 + ζ4k+1β3 + ζ5k+1c 1 minus σ1( 1113857 + ζ6k+1cσ1

+ β2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ2k+11113872 1113873Hminus

k

N+ ζ2k+1(1 minus μ)

ζ4k Ck + α2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ1k+11113872 1113873Pk

N+ β2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ2k+11113872 1113873

Lk

N

ζ5k Dk + ζ2k+1θ + ζ5k+1 1 minus μ minus θ minus u4k1113872 1113873 + ζ6k+1u4k minus ζ3k+1u3k

+ ζ4k+1(1 minus μ minus λ) + ζ5k+1λσ2 + ζ6k+1 λ 1 minus σ2( 1113857 + u3k1113872 1113873

ζ6k ζ6k+1(1 minus μ)

(8)

With the transversality conditions at time T ζ1T ζ6T

0 ζ2T AT ζ3T BT ζ4T CT and ζ5T DTFurthermore for k 0 1 2 T minus 1 the optimal con-

trols ulowast1k ulowast2k ulowast3k and ulowast4k are given by

Discrete Dynamics in Nature and Society 5

ulowast1k min b max a

1Ek

ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N1113876 11138771113888 11138891113890 1113891

ulowast2k min d max c

1NFk

α2PkHminusk ζ3k+1 minus ζ1k+11113872 1113873 + β2LkH

minusk( 1113857 ζ3k+1 minus ζ2k+11113872 11138731113960 11139611113888 11138891113890 1113891

ulowast3k min f max e

1Gk

ζ3k+1 minus ζ6k+11113872 1113873Hminusk1113960 11139611113888 11138891113890 1113891

ulowast4k min h max g

1Mk

ζ5k+1 minus ζ6k+11113872 1113873Qtk1113960 11139611113888 11138891113890 1113891

(9)

Proof (e Hamiltonian at time step k is given by

Hk AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k +

Gk

2u23k +

Mk

2u24k

+ ζ1k+1f1k+1 + ζ2k+1f2k+1 + ζ3k+1f3k+1 + ζ4k+1f4k+1

+ ζ5k+1f5k+1 + ζ6k+1f6k+1AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k

+Gk

2u23k +

Mk

2u24k + ζ1k+1 Λ +(1 minus μ)Pk minus α1 1 minus u1k1113872 1113873

PkLk

Nminus α2 1 minus u2k1113872 1113873

PkHminusk

N1113876 1113877

+ ζ2k+1 1 minus μ minus β1( 1113857Lk + θQtk + α1 1 minus uk( 1113857

PkLk

Nminus β2 1 minus u2k1113872 1113873

LkHminusk

N1113876 1113877

+ ζ3k+1 1 minus μ minus β3 minus c( 1113857H+k + β1Lk + β2 1 minus u2k1113872 1113873

LkHminusk

N1113876 1113877

+ ζ3k+1 α2 1 minus u2k1113872 1113873PkHminus

k

Nminus u3kH

minusk1113876 1113877 + ζ4k+1 (1 minus μ minus λ)H

minusk + β3H

+k1113858 1113859

+ ζ5k+1 (1 minus μ minus θ)Qtk + c 1 minus σ1( 1113857H

+k + λσ2H

minusk minus u4kQ

tk1113960 1113961

+ ζ6k+1 (1 minus μ)Qp

k + λ 1 minus σ2( 1113857Hminusk + cσ1H

+k + u3kH

minusk + u4kQ

tk1113960 1113961

(10)

For k 0 1 T minus 1 the optimal controls u1ku2k u3k and u4k can be solved from the optimalitycondition

zHk

zu1k

0

zHk

zu2k

0

zHk

zu3k

0

zHk

zu4k

0

(11)

which are

zHk

zu1k

Eku1k minus ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N 0

zHk

zu2k

Fku2k minus ζ3k+1 minus ζ1k+11113872 1113873α2PkHminus

k

N

minus ζ3k+1 minus ζ2k+11113872 1113873β2LkHminus

k

N 0

zHk

zu3k

Gku3k minus ζ3k+1Hminusk + ζ6k+1H

minusk 0

zHk

zu4k

Mku4k minus ζ5k+1 minus ζ6k+11113872 1113873Qtk 0

(12)

So we have

6 Discrete Dynamics in Nature and Society

u1k 1

Ek

ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N1113876 1113877

u2k 1

NFk

ζ3k+1 minus ζ1k+11113872 1113873α2PkHminusk + ζ3k+1 minus ζ2k+11113872 1113873β2LkH

minusk1113960 1113961

u3k 1

Gk

ζ3k+1 minus ζ6k+11113872 1113873Hminusk1113960 1113961

u4k 1

Mk

ζ5k+1 minus ζ6k+11113872 1113873Qtk1113960 1113961

(13)

By the bounds in Uad of the controls it is easy to obtainulowast1k ulowast2k ulowast3k and ulowast4k in the form of (9)

4 Simulation

In this section we present the results obtained by solvingnumerically the optimality system (is system consists ofthe state system adjoint system initial and final timeconditions and the control characterization

In this formulation there were initial conditions for thestate variables and terminal conditions for the adjoints (atis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k T We solve the optimality system by an it-erative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

41 Discussion In this section we study and analyse nu-merically the effects of the optimal control strategies such asawareness programs through media and education contactprevention through security campaigns treatment andpsychological support along with follow-up for the drugconsumers (e numerical solution of model (2) is executedusing Matlab with the following parameter values and initialvalues of state variable in Table 1

(e proposed control strategies in this work help toachieve several objectives

411 Objective A Protecting and Preventing Potential DrugUsers and the Light Drug Users from Falling into Drug Useand Addiction Due the importance of the awareness pro-grams throughmedia and education in restricting the spreadof drug use we propose an optimal strategy for this purposeHence we activate the optimal control variable u1 whichrepresents awareness programs for the light drug usersFigure 2(a) compares the evolution of the light drug userswith and without control u1 in which the effect of theproposed awareness programs throughmedia and education

is proven to be positive in decreasing the number of lightdrug users and preventing potential drug users from con-tacting light drug users (Figure 2(b))

412 Objective B Decreasing the Number of Heavy DrugUsers and Dealers by Preventing Contact through SecurityCampaigns When the number of drug users is so high itis obligatory to resort to some strategies such as pre-venting contact through security campaigns in order toreduce the number of heavy drug users-dealers(Figure 3(c)) and to protect light drug users from con-tacting heavy drug users-dealers through security cam-paigns which also has a positive effect on reducing thenumber of the heavy drug users (Figure 3(b)) (ereforewe propose an optimal strategy by using the optimalcontrol u2 in the beginning In spite of using the optimalcontrol u2 we observe that the number of light drugconsumers increases due to protecting them from con-tacting the heavy drug users-dealers (e reason of thisincrease is justified by the fact that light drug consumersrevert back to using drugs occasionally (Figure 3(a)) Alsothe proposed strategy has an additional effect in de-creasing clearly the number of heavy drug users and heavydrug users-dealers and providers

413 Objective C Treatment in the Addiction CentersGiven the importance and effectiveness of this strategy weuse the control strategy to encourage heavy drug users toknow about treatment centers and join them to decrease thespread of drug consumers

We propose treatment within addiction centers repre-sented by the strategy of optimal control u3 FromFigures 4(a) and 4(b) the decrease of the number of heavydrug users is clearly achieved which in turn had a positiveeffect on reducing the number of heavy drug users-dealersand providers

414 Objective D Control with Psychological Support alongwith Follow-Up Taking into consideration the importanceand the effectiveness of this strategy on the individuals whotemporarily quit drug we propose an optimal strategy byusing the optimal control u4 in the beginning which rep-resents follow-up and psychological support to prevent thetemporary quitters from reverting back to using the drugsoccasionally (θne 0) (Figure 5(b)) (e proposed strategy hasan additional effect in decreasing clearly the number oftemporary quitters of drug consumption Figure 5(a) showsthat the number of light drug users is decreased markedly(θ 0)

Note several optimal controls can be combined toachieve other objectives and implement other strategiesdepending on the phenomenon and the particularity of eachsociety

Discrete Dynamics in Nature and Society 7

3000

3500

4000

4500

5000

5500

6000

6500

The l

ight

dru

g co

nsum

ers

5 10 15 20 25 30Time

L without controlL with control u2

(a)

1000

1200

1400

1600

1800

2000

2200

e h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u2

(b)

1300

1400

1500

1600

1700

1800

1900

2000

The h

eavy

dru

g co

nsum

ers

5 10 15 20 25 30Time

H+ without controlH+ with control u2

(c)

Figure 3 (a)(e evolution of Lwith and without controls (b)(e evolution of theH- with and without controls (c)(e evolution of theH+

with and without controls

Table 1 (e description of parameters used for the definition of discrete time system (1) We used just arbitrary academic data

P0 L0 H+0 Hminus

0 Qt0 Q

p0 Λ α1 α2 μ β1 β2 β3 c λ σ1 σ2 θ

5103 3103 15103 1103 2103 1103 5102 045 05 004 0025 02 01 005 005 005 07 005

2000

2500

3000

3500

4000

4500

5000

5500

The o

ccas

iona

l dru

g us

ers (

L)

5 10 15 20 25 30Time

L without controlL with control u1

(a)

1500

2000

2500

3000

3500

4000

4500

5000

5 10 15 20 25 30Time

e p

oten

tial d

rug

user

s (P)

P with control u1

P without control

(b)

Figure 2 (a) (e evolution of the L with and without controls (b) (e evolution of the P with and without controls

8 Discrete Dynamics in Nature and Society

5 Conclusion

In this paper we introduced a discrete modeling of drugusers in order to minimize the number of light drug usersheavy drug users heavy drug users-dealers and temporaryquitters of drug consumption We also introduced fourcontrols which respectively represent awareness programsthrough education and media contact prevention throughsecurity campaigns treatment and psychological supportalong with follow-up We applied the results of the controltheory and wemanaged to obtain the characterisations of theoptimal controls (e numerical simulation of the obtainedresults showed the effectiveness of the proposed controlstrategies

Data Availability

No data were used to support this study

Disclosure

(is article was presented at the International Conferenceon Research in Applied Mathematics and Computer Sci-ence (ICRAMC) 2020 which took place on July 15ndash182020

Conflicts of Interest

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

References

[1] UNODC ldquoWorld drug reportmdashunited nations office on drugsand crimerdquo 2019 httpswwwunodcorgwdr2019prelaunchWDR19_Booklet_1_EXSUMpdf

400600800

100012001400160018002000

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ ne 0

(a)

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

400600800

1000120014001600180020002200

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ = 0

(b)

Figure 5 (a) (e evolution of the Qt with and without controls with θne 0 (b) (e evolution of the Qt with and without controls with θ 0

800

1000

1200

1400

1600

1800

2000

5 10 15 20 25 30Time

The h

eavy

dru

g us

ers

H+ without controlH+ with control u3

(a)

1000

1200

1400

1600

1800

2000

2200

The h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u3

(b)

Figure 4 (a) (e evolution of the H+ with and without controls (b) (e evolution of the Hminus with and without controls

Discrete Dynamics in Nature and Society 9

extreme mood swings fighting or being violentand their number is increased when they start usingit with a rate α1(PkLkN) In this compartmentsome other individuals will leave at the ratesβ2(LkHminus

k N) β1Lk and μLk Here β1 and β2 are therates of light drug consumers who transform intoheavy drug consumers and drug dealers and pro-viders consecutively (is compartment is increasedby θQt

k (at the rate θ) due to temporary quitters whorevert back to using the drug

(iii) e compartment H+ encompasses heavy drugconsumers who do not influence other individualsand who are suffering from addiction to drugconsumption When an individual becomes a heavydrug user they face a great difficulty to control or setlimits to their consumption (eir job their familysocial circle and health are all endangered Despitethese negative consequences the heavy drug con-sumers are unable to quit using drugs (e heavydrug consumers may begin to disclaim that theyhave a problem this disclaim can make it even moredifficult for the person to get help (is compart-ment is increased by the rate β1(LkH+

k N) and isdecreased by the rate cH+

k and the rate μH+k due to

natural death(iv) e compartment Hminus includes heavy drug users

dealers and providers of drugs who do influenceother individuals (ey work to make big profitsthrough illegal ways ie a drug dealer can be de-fined as a person who sells drugs of any type orquantity in an illegal manner A dealer can be anoccasional seller of drugs by selling small quantitiesto cover the costs of their own drug use or they canbe highly organized groups and businessmen withinhigh-organized operations that run like a seriousbusiness (ey are increased by the rateα2(PkHminus

k N) β2(LkHminusk N) and decreased by the

rate cHminusk and the rate μHminus

k due to natural death(v) e compartment Qt is composed of the individ-

uals who temporarily quit drugs and are increased atthe rate c(1 minus σ1)H+

k λσ2Hminusk where (1 minus σ1) is the

fraction of drug users who temporarily quit usingdrug (at a rate c) σ1 is the remaining fraction ofheavy drug users who permanently and temporarilyquit using drug (at the rate c and λ) (is com-partment is decreased by the natural death μQt

k andthe rate θQt

k represents the persons who return to belight drug users

(vi) e compartment Qp includes individuals whopermanently quit drugs (is compartment is in-creased at the rate cσ1H+

k and λ(1 minus σ2)Hminusk where

(1 minus σ2) is the fraction of drug users who perma-nently quit using drugs (at a rate c) σ2 is theremaining fraction of heavy drug users who per-manently and temporarily quit using drugs (at a ratec) Some people of this compartment will die at therate μQ

p

k

(e variables Pk LkH+k Hminus

k Qtk andQ

p

k are the numbersof the individuals in the six classes at time k respectively(eunit k can correspond to periods phases or years It dependson the frequency of the survey studies as needed

(e following diagram will demonstrate the flow di-rections of individuals among the compartments (esedirections are going to be represented by directed arrows(see Figure 1)

(e total population size at time k is denoted by Nk withNk Pk + Lk + H+

k + Hminusk + Qt

k + Qp

k

22 Model Equations By the addition of the rates at whichindividuals enter the compartment and also by subtractingthe rates at which people leave the compartment we obtain adifference equation for the rate at which the individuals ofeach compartment change over discrete time Hence wepresent the drug users infection model by the followingsystem of difference equations

Pk+1 Λ +(1 minus μ)Pk minus α1PkLk

Nminus α2

PkHminusk

N

Lk+1 1 minus μ minus β1( 1113857Lk + θQtk + α1

PkLk

Nminus β2

LkHminusk

N

H+k+1 1 minus μ minus β3 minus c( 1113857H+

k + β1Lk + β2LkHminus

k

N+ α2

PkHminusk

N

Hminusk+1 (1 minus μ minus λ)Hminus

k + β3H+k

Qtk+1 (1 minus μ minus θ)Qt

k + c 1 minus σ1( 1113857H+k + λσ2Hminus

k

Qp

k+1 (1 minus μ)Qp

k + cσ1H+k + λ 1 minus σ2( 1113857Hminus

k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where P0 ge 0 L0 ge 0 H+0 ge 0 Hminus

0 ge 0 Qt0 ge 0 and Q

p0 ge 0 are

the given initial states

3 The Optimal Control Problem

(e strategies of control that we adopt consist of anawareness program through media and education contactprevention through security campaigns treatment andpsychological support along with follow-up Our main goalin adopting these strategies is to minimize the number oflight drug consumers heavy drug consumers heavy drugusers-dealers and the temporary quitters of drugs during thetime steps k 0 to T minus 1 and also minimize the cost spent inapplying these strategies In this model we include the fourcontrols u1k u2k u3k and u4k that represent consecutivelyawareness programs through media and education contactprevention through security campaigns treatment andpsychological support along with follow-up as measures attime k So the controlled mathematical system is given bythe following system of difference equations

Discrete Dynamics in Nature and Society 3

Pk+1 Λ +(1 minus μ)Pk minus α1 1 minus u1k1113872 1113873PkLk

Nminus α2 1 minus u2k1113872 1113873

PkHminusk

N

Lk+1 1 minus μ minus β1( 1113857Lk + θQtk + α1 1 minus u1k1113872 1113873

PkLk

Nminus β2 1 minus u2k1113872 1113873

LkHminusk

N

H+k+1 1 minus μ minus β3 minus c( 1113857H+

k + β1LkH+

k

N+ β2 1 minus u2k1113872 1113873

LkHminusk

N+ α2 1 minus vk( 1113857

PkHminusk

Nminus u3kH

+k

Hminusk+1 (1 minus μ minus λ)Hminus

k + β3H+k

Qtk+1 (1 minus μ minus θ)Qt

k + c 1 minus σ1( 1113857H+k + λσ2Hminus

k minus u4kQtk

Qp

k+1 (1 minus μ)Qp

k + cσ1H+k + λ 1 minus σ2( 1113857Hminus

k + u3kH+k + u4kQt

k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where P0 ge 0 L0 ge 0 H+0 ge 0 Hminus

0 ge 0 Qt0 ge 0 and Q

p0 ge 0 are

the given initial states(ere are four controls u1k (u10 u11 u1Tminus1)

u2k (u20 u21 u2Tminus1) u3k (u30 u31 u3Tminus1)and u4k (u40 u41 u4Tminus1) (e first control can beinterpreted as the proportion to be adopted to awarenessprograms through media and education So we note that(1 minus u1k)(PkLkN) is the proportion of the potential drugusers who are protected from contacting light drug con-sumers at time step k (e second control can be interpretedas the proportion of contact prevention through securitycampaigns We observe that (1 minus u2k)(LkHminus

k N) is the pro-portion of the individuals who will be prevented to move fromthe class of light drug consumers towards the class of heavydrug users dealers and providers at time step k (e thirdcontrol can be interpreted as the proportion of individuals to besubjected to treatment So we note that u3kH+

k is the pro-portion of the individuals who will move from the class ofheavy drug consumers towards the class of the individuals whopermanently quit using drug at time step k (e fourth controlcan be interpreted as the proportion of individuals who will getpsychological support alongwith follow-up So we observe thatu4kQt

k is the proportion of the individuals who temporarilyquit using drugs and who will transform into the individualswho permanently quit drug at time step k

(e challenge that we face here is how to minimize theobjective functional

J u1k u2k u3k u4k1113872 1113873 ATLT + BTH+T + CTH

minusT + DTQ

tT

+ 1113944Tminus1

k0AkLk + BkH

+k + CkH

minusk + DkQ

tk1113872 1113873

+ 1113944

Tminus1

k0

Ek

2u21k +

Fk

2u22k +

Gk

2u23k +

Mk

2u24k

(3)

where the parameters Ak gt 0 Bk gt 0 Ck gt 0 Dk gt 0 Ek gt 0Fk gt 0 and Mk gt 0 are the cost coefficients and they areselected to weigh the relative importance of Lk H+

k Hminusk Qt

ku1k u2k u3k and u4k at time k T is the final time

In other words we seek the optimal controls u1k u2ku3k and u4k such that

J ulowast1k ulowast2k ulowast3k ulowast4k1113872 1113873 min

u1 u2 u3 u4( )isinU4a d

J u1k u2k u3k u4k1113872 1113873

(4)

where Ua d is the set of admissible controls defined by

microPk microLk

β1Lk

β3H+

β2 (LkHkndashN)

α2 (PkHkndashN)

microHndash microQp

microQt

θQkt

microH+

γ (1 ndash σ1)Hk+

λ (1 ndash σ2)Hkndash

λσ2Hkndash

γσ1Hk+

α1(PkLkN)P L

H+

Hndash

Qt

Qp

Λ

Figure 1 (e flow between the five compartments PLH+Hminus QtQp

4 Discrete Dynamics in Nature and Society

Uad uik ui0 ui1 uiTminus11113872 11138731113966

for i 1 2 3 4 ai le uik le bi k 0 1 2 T minus 11113967

(5)

(e sufficient condition for the existence of the optimalcontrols (u1 u2 u3 u4) for problems (2) and (3) comes fromthe following theorem

Theorem 1 1ere exists the optimal controls(ulowast1k ulowast2k ulowast3k ulowast4k) such that

J ulowast1k ulowast2k ulowast3k ulowast4k1113872 1113873 min

u1 u2u3 u4( )isinU4ad

J u1 u2 u3 u4( 1113857

(6)

subject to the control system (2) with initial conditions

Proof Since the coefficients of the state equations arebounded and there is a finite number of time steps P

(P0 P1 PT) L (L0 L1 LT) H+ (H+0 H+

1

H+T) Hminus (Hminus

0 Hminus1 Hminus

T)Qt (Qt0 Qt

1 QtT) and

Qp (Qp0 Q

p1 Q

p

T) are uniformly bounded for all(u1 u2 u3 u4) in the control set Uad thus J(u1 u2 u3 u4) isbounded for all (u1 u2 u3 u4) isin U4

ad Since J(u1 u2 u3 u4)

is bounded inf(u1 u2u3 u4)isinU4ad

J(u1 u2 u3 u4) is finite and

there exists a sequence (uj1 u

j2 u

j3 u

j4) isin U4a d such that

limj⟶+infin

J(uj1 u

j2 u

j3 u

j4) inf

(u1 u2u3 u4)isinU4a d

J(u1 u2 u3 u4) and

corresponding sequences of states Pj Lj H+j Hminus j Qtj andQpj Since there is a finite number of uniformly boundedsequences there exist (ulowast1 ulowast2 ulowast3 ulowast4 ) isin U4

ad and Plowast Llowast H+lowast

Hminuslowast Qtlowast and Qplowast isin IRT+1 such that on a subsequence(u

j1 u

j2 u

j3 u

j4)⟶ (ulowast1 ulowast2 ulowast3 ulowast4 ) Pj⟶ Plowast Lj⟶ Llowast

H+j⟶ H+jlowast Hminusj⟶ Hminusjlowast Qtj⟶ Qtlowast and Qpj⟶Qplowast Finally due to the finite dimensional structure ofsystem (2) and the objective function J(u1 u2 u3 u4)(ulowast1 ulowast2 ulowast3 ulowast4 ) is an optimal control with correspondingstates Plowast Llowast H+jlowast Hminus jlowast and Qplowast (ereforeinf(u1u2 u3 u4)isinU4

adJ(u1 u2 u3 u4) is achieved

We apply the discrete version of Pontryaginrsquos Maxi-mum Principle [10 14 16ndash19] (e key idea is introducingthe adjoint function to attach the system of differenceequations to the objective functional resulting in theformation of a function called the Hamiltonian (is

principle converts the problem of finding the control tooptimize the objective functional subject to the statedifference equation with initial condition to find thecontrol to optimize Hamiltonian pointwise (with respectto the control)

We have the Hamiltonian Hk at time step k defined by

Hk AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k

+Gk

2u23k +

Mk

2u24k + 1113944

6

i1ζ ik+1fik+1

(7)

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

Theorem 2 Given the optimal controls (ulowast1k ulowast2k

ulowast3k ulowast4k) isin U4a d and the solutions Plowast Llowast H+jlowast Hminus jlowast Qtlowast

and Qplowast of the corresponding state system (2) thereexist adjoint functions ζ1k ζ2k ζ3k ζ4k ζ5k and ζ6k

satisfying

ζ1k ζ1k+1(1 minus μ) + α1 1 minus u1k1113872 1113873Lk

Nζ2k+1 minus ζ1k+11113872 1113873

ζ2k Ak + α1 ζ2k+1 minus ζ1k+11113872 1113873Pk

N+ β1 ζ3k+1 minus ζ2k+11113872 1113873

+ α2 1 minus u2k1113872 1113873Hminus

k

Nζ3k+1 minus ζ1k+11113872 1113873

ζ3k Bk + ζ3k+1 1 minus μ minus β3 minus c( 1113857 + ζ4k+1β3 + ζ5k+1c 1 minus σ1( 1113857 + ζ6k+1cσ1

+ β2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ2k+11113872 1113873Hminus

k

N+ ζ2k+1(1 minus μ)

ζ4k Ck + α2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ1k+11113872 1113873Pk

N+ β2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ2k+11113872 1113873

Lk

N

ζ5k Dk + ζ2k+1θ + ζ5k+1 1 minus μ minus θ minus u4k1113872 1113873 + ζ6k+1u4k minus ζ3k+1u3k

+ ζ4k+1(1 minus μ minus λ) + ζ5k+1λσ2 + ζ6k+1 λ 1 minus σ2( 1113857 + u3k1113872 1113873

ζ6k ζ6k+1(1 minus μ)

(8)

With the transversality conditions at time T ζ1T ζ6T

0 ζ2T AT ζ3T BT ζ4T CT and ζ5T DTFurthermore for k 0 1 2 T minus 1 the optimal con-

trols ulowast1k ulowast2k ulowast3k and ulowast4k are given by

Discrete Dynamics in Nature and Society 5

ulowast1k min b max a

1Ek

ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N1113876 11138771113888 11138891113890 1113891

ulowast2k min d max c

1NFk

α2PkHminusk ζ3k+1 minus ζ1k+11113872 1113873 + β2LkH

minusk( 1113857 ζ3k+1 minus ζ2k+11113872 11138731113960 11139611113888 11138891113890 1113891

ulowast3k min f max e

1Gk

ζ3k+1 minus ζ6k+11113872 1113873Hminusk1113960 11139611113888 11138891113890 1113891

ulowast4k min h max g

1Mk

ζ5k+1 minus ζ6k+11113872 1113873Qtk1113960 11139611113888 11138891113890 1113891

(9)

Proof (e Hamiltonian at time step k is given by

Hk AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k +

Gk

2u23k +

Mk

2u24k

+ ζ1k+1f1k+1 + ζ2k+1f2k+1 + ζ3k+1f3k+1 + ζ4k+1f4k+1

+ ζ5k+1f5k+1 + ζ6k+1f6k+1AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k

+Gk

2u23k +

Mk

2u24k + ζ1k+1 Λ +(1 minus μ)Pk minus α1 1 minus u1k1113872 1113873

PkLk

Nminus α2 1 minus u2k1113872 1113873

PkHminusk

N1113876 1113877

+ ζ2k+1 1 minus μ minus β1( 1113857Lk + θQtk + α1 1 minus uk( 1113857

PkLk

Nminus β2 1 minus u2k1113872 1113873

LkHminusk

N1113876 1113877

+ ζ3k+1 1 minus μ minus β3 minus c( 1113857H+k + β1Lk + β2 1 minus u2k1113872 1113873

LkHminusk

N1113876 1113877

+ ζ3k+1 α2 1 minus u2k1113872 1113873PkHminus

k

Nminus u3kH

minusk1113876 1113877 + ζ4k+1 (1 minus μ minus λ)H

minusk + β3H

+k1113858 1113859

+ ζ5k+1 (1 minus μ minus θ)Qtk + c 1 minus σ1( 1113857H

+k + λσ2H

minusk minus u4kQ

tk1113960 1113961

+ ζ6k+1 (1 minus μ)Qp

k + λ 1 minus σ2( 1113857Hminusk + cσ1H

+k + u3kH

minusk + u4kQ

tk1113960 1113961

(10)

For k 0 1 T minus 1 the optimal controls u1ku2k u3k and u4k can be solved from the optimalitycondition

zHk

zu1k

0

zHk

zu2k

0

zHk

zu3k

0

zHk

zu4k

0

(11)

which are

zHk

zu1k

Eku1k minus ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N 0

zHk

zu2k

Fku2k minus ζ3k+1 minus ζ1k+11113872 1113873α2PkHminus

k

N

minus ζ3k+1 minus ζ2k+11113872 1113873β2LkHminus

k

N 0

zHk

zu3k

Gku3k minus ζ3k+1Hminusk + ζ6k+1H

minusk 0

zHk

zu4k

Mku4k minus ζ5k+1 minus ζ6k+11113872 1113873Qtk 0

(12)

So we have

6 Discrete Dynamics in Nature and Society

u1k 1

Ek

ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N1113876 1113877

u2k 1

NFk

ζ3k+1 minus ζ1k+11113872 1113873α2PkHminusk + ζ3k+1 minus ζ2k+11113872 1113873β2LkH

minusk1113960 1113961

u3k 1

Gk

ζ3k+1 minus ζ6k+11113872 1113873Hminusk1113960 1113961

u4k 1

Mk

ζ5k+1 minus ζ6k+11113872 1113873Qtk1113960 1113961

(13)

By the bounds in Uad of the controls it is easy to obtainulowast1k ulowast2k ulowast3k and ulowast4k in the form of (9)

4 Simulation

In this section we present the results obtained by solvingnumerically the optimality system (is system consists ofthe state system adjoint system initial and final timeconditions and the control characterization

In this formulation there were initial conditions for thestate variables and terminal conditions for the adjoints (atis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k T We solve the optimality system by an it-erative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

41 Discussion In this section we study and analyse nu-merically the effects of the optimal control strategies such asawareness programs through media and education contactprevention through security campaigns treatment andpsychological support along with follow-up for the drugconsumers (e numerical solution of model (2) is executedusing Matlab with the following parameter values and initialvalues of state variable in Table 1

(e proposed control strategies in this work help toachieve several objectives

411 Objective A Protecting and Preventing Potential DrugUsers and the Light Drug Users from Falling into Drug Useand Addiction Due the importance of the awareness pro-grams throughmedia and education in restricting the spreadof drug use we propose an optimal strategy for this purposeHence we activate the optimal control variable u1 whichrepresents awareness programs for the light drug usersFigure 2(a) compares the evolution of the light drug userswith and without control u1 in which the effect of theproposed awareness programs throughmedia and education

is proven to be positive in decreasing the number of lightdrug users and preventing potential drug users from con-tacting light drug users (Figure 2(b))

412 Objective B Decreasing the Number of Heavy DrugUsers and Dealers by Preventing Contact through SecurityCampaigns When the number of drug users is so high itis obligatory to resort to some strategies such as pre-venting contact through security campaigns in order toreduce the number of heavy drug users-dealers(Figure 3(c)) and to protect light drug users from con-tacting heavy drug users-dealers through security cam-paigns which also has a positive effect on reducing thenumber of the heavy drug users (Figure 3(b)) (ereforewe propose an optimal strategy by using the optimalcontrol u2 in the beginning In spite of using the optimalcontrol u2 we observe that the number of light drugconsumers increases due to protecting them from con-tacting the heavy drug users-dealers (e reason of thisincrease is justified by the fact that light drug consumersrevert back to using drugs occasionally (Figure 3(a)) Alsothe proposed strategy has an additional effect in de-creasing clearly the number of heavy drug users and heavydrug users-dealers and providers

413 Objective C Treatment in the Addiction CentersGiven the importance and effectiveness of this strategy weuse the control strategy to encourage heavy drug users toknow about treatment centers and join them to decrease thespread of drug consumers

We propose treatment within addiction centers repre-sented by the strategy of optimal control u3 FromFigures 4(a) and 4(b) the decrease of the number of heavydrug users is clearly achieved which in turn had a positiveeffect on reducing the number of heavy drug users-dealersand providers

414 Objective D Control with Psychological Support alongwith Follow-Up Taking into consideration the importanceand the effectiveness of this strategy on the individuals whotemporarily quit drug we propose an optimal strategy byusing the optimal control u4 in the beginning which rep-resents follow-up and psychological support to prevent thetemporary quitters from reverting back to using the drugsoccasionally (θne 0) (Figure 5(b)) (e proposed strategy hasan additional effect in decreasing clearly the number oftemporary quitters of drug consumption Figure 5(a) showsthat the number of light drug users is decreased markedly(θ 0)

Note several optimal controls can be combined toachieve other objectives and implement other strategiesdepending on the phenomenon and the particularity of eachsociety

Discrete Dynamics in Nature and Society 7

3000

3500

4000

4500

5000

5500

6000

6500

The l

ight

dru

g co

nsum

ers

5 10 15 20 25 30Time

L without controlL with control u2

(a)

1000

1200

1400

1600

1800

2000

2200

e h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u2

(b)

1300

1400

1500

1600

1700

1800

1900

2000

The h

eavy

dru

g co

nsum

ers

5 10 15 20 25 30Time

H+ without controlH+ with control u2

(c)

Figure 3 (a)(e evolution of Lwith and without controls (b)(e evolution of theH- with and without controls (c)(e evolution of theH+

with and without controls

Table 1 (e description of parameters used for the definition of discrete time system (1) We used just arbitrary academic data

P0 L0 H+0 Hminus

0 Qt0 Q

p0 Λ α1 α2 μ β1 β2 β3 c λ σ1 σ2 θ

5103 3103 15103 1103 2103 1103 5102 045 05 004 0025 02 01 005 005 005 07 005

2000

2500

3000

3500

4000

4500

5000

5500

The o

ccas

iona

l dru

g us

ers (

L)

5 10 15 20 25 30Time

L without controlL with control u1

(a)

1500

2000

2500

3000

3500

4000

4500

5000

5 10 15 20 25 30Time

e p

oten

tial d

rug

user

s (P)

P with control u1

P without control

(b)

Figure 2 (a) (e evolution of the L with and without controls (b) (e evolution of the P with and without controls

8 Discrete Dynamics in Nature and Society

5 Conclusion

In this paper we introduced a discrete modeling of drugusers in order to minimize the number of light drug usersheavy drug users heavy drug users-dealers and temporaryquitters of drug consumption We also introduced fourcontrols which respectively represent awareness programsthrough education and media contact prevention throughsecurity campaigns treatment and psychological supportalong with follow-up We applied the results of the controltheory and wemanaged to obtain the characterisations of theoptimal controls (e numerical simulation of the obtainedresults showed the effectiveness of the proposed controlstrategies

Data Availability

No data were used to support this study

Disclosure

(is article was presented at the International Conferenceon Research in Applied Mathematics and Computer Sci-ence (ICRAMC) 2020 which took place on July 15ndash182020

Conflicts of Interest

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

References

[1] UNODC ldquoWorld drug reportmdashunited nations office on drugsand crimerdquo 2019 httpswwwunodcorgwdr2019prelaunchWDR19_Booklet_1_EXSUMpdf

400600800

100012001400160018002000

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ ne 0

(a)

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

400600800

1000120014001600180020002200

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ = 0

(b)

Figure 5 (a) (e evolution of the Qt with and without controls with θne 0 (b) (e evolution of the Qt with and without controls with θ 0

800

1000

1200

1400

1600

1800

2000

5 10 15 20 25 30Time

The h

eavy

dru

g us

ers

H+ without controlH+ with control u3

(a)

1000

1200

1400

1600

1800

2000

2200

The h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u3

(b)

Figure 4 (a) (e evolution of the H+ with and without controls (b) (e evolution of the Hminus with and without controls

Discrete Dynamics in Nature and Society 9

Pk+1 Λ +(1 minus μ)Pk minus α1 1 minus u1k1113872 1113873PkLk

Nminus α2 1 minus u2k1113872 1113873

PkHminusk

N

Lk+1 1 minus μ minus β1( 1113857Lk + θQtk + α1 1 minus u1k1113872 1113873

PkLk

Nminus β2 1 minus u2k1113872 1113873

LkHminusk

N

H+k+1 1 minus μ minus β3 minus c( 1113857H+

k + β1LkH+

k

N+ β2 1 minus u2k1113872 1113873

LkHminusk

N+ α2 1 minus vk( 1113857

PkHminusk

Nminus u3kH

+k

Hminusk+1 (1 minus μ minus λ)Hminus

k + β3H+k

Qtk+1 (1 minus μ minus θ)Qt

k + c 1 minus σ1( 1113857H+k + λσ2Hminus

k minus u4kQtk

Qp

k+1 (1 minus μ)Qp

k + cσ1H+k + λ 1 minus σ2( 1113857Hminus

k + u3kH+k + u4kQt

k

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where P0 ge 0 L0 ge 0 H+0 ge 0 Hminus

0 ge 0 Qt0 ge 0 and Q

p0 ge 0 are

the given initial states(ere are four controls u1k (u10 u11 u1Tminus1)

u2k (u20 u21 u2Tminus1) u3k (u30 u31 u3Tminus1)and u4k (u40 u41 u4Tminus1) (e first control can beinterpreted as the proportion to be adopted to awarenessprograms through media and education So we note that(1 minus u1k)(PkLkN) is the proportion of the potential drugusers who are protected from contacting light drug con-sumers at time step k (e second control can be interpretedas the proportion of contact prevention through securitycampaigns We observe that (1 minus u2k)(LkHminus

k N) is the pro-portion of the individuals who will be prevented to move fromthe class of light drug consumers towards the class of heavydrug users dealers and providers at time step k (e thirdcontrol can be interpreted as the proportion of individuals to besubjected to treatment So we note that u3kH+

k is the pro-portion of the individuals who will move from the class ofheavy drug consumers towards the class of the individuals whopermanently quit using drug at time step k (e fourth controlcan be interpreted as the proportion of individuals who will getpsychological support alongwith follow-up So we observe thatu4kQt

k is the proportion of the individuals who temporarilyquit using drugs and who will transform into the individualswho permanently quit drug at time step k

(e challenge that we face here is how to minimize theobjective functional

J u1k u2k u3k u4k1113872 1113873 ATLT + BTH+T + CTH

minusT + DTQ

tT

+ 1113944Tminus1

k0AkLk + BkH

+k + CkH

minusk + DkQ

tk1113872 1113873

+ 1113944

Tminus1

k0

Ek

2u21k +

Fk

2u22k +

Gk

2u23k +

Mk

2u24k

(3)

where the parameters Ak gt 0 Bk gt 0 Ck gt 0 Dk gt 0 Ek gt 0Fk gt 0 and Mk gt 0 are the cost coefficients and they areselected to weigh the relative importance of Lk H+

k Hminusk Qt

ku1k u2k u3k and u4k at time k T is the final time

In other words we seek the optimal controls u1k u2ku3k and u4k such that

J ulowast1k ulowast2k ulowast3k ulowast4k1113872 1113873 min

u1 u2 u3 u4( )isinU4a d

J u1k u2k u3k u4k1113872 1113873

(4)

where Ua d is the set of admissible controls defined by

microPk microLk

β1Lk

β3H+

β2 (LkHkndashN)

α2 (PkHkndashN)

microHndash microQp

microQt

θQkt

microH+

γ (1 ndash σ1)Hk+

λ (1 ndash σ2)Hkndash

λσ2Hkndash

γσ1Hk+

α1(PkLkN)P L

H+

Hndash

Qt

Qp

Λ

Figure 1 (e flow between the five compartments PLH+Hminus QtQp

4 Discrete Dynamics in Nature and Society

Uad uik ui0 ui1 uiTminus11113872 11138731113966

for i 1 2 3 4 ai le uik le bi k 0 1 2 T minus 11113967

(5)

(e sufficient condition for the existence of the optimalcontrols (u1 u2 u3 u4) for problems (2) and (3) comes fromthe following theorem

Theorem 1 1ere exists the optimal controls(ulowast1k ulowast2k ulowast3k ulowast4k) such that

J ulowast1k ulowast2k ulowast3k ulowast4k1113872 1113873 min

u1 u2u3 u4( )isinU4ad

J u1 u2 u3 u4( 1113857

(6)

subject to the control system (2) with initial conditions

Proof Since the coefficients of the state equations arebounded and there is a finite number of time steps P

(P0 P1 PT) L (L0 L1 LT) H+ (H+0 H+

1

H+T) Hminus (Hminus

0 Hminus1 Hminus

T)Qt (Qt0 Qt

1 QtT) and

Qp (Qp0 Q

p1 Q

p

T) are uniformly bounded for all(u1 u2 u3 u4) in the control set Uad thus J(u1 u2 u3 u4) isbounded for all (u1 u2 u3 u4) isin U4

ad Since J(u1 u2 u3 u4)

is bounded inf(u1 u2u3 u4)isinU4ad

J(u1 u2 u3 u4) is finite and

there exists a sequence (uj1 u

j2 u

j3 u

j4) isin U4a d such that

limj⟶+infin

J(uj1 u

j2 u

j3 u

j4) inf

(u1 u2u3 u4)isinU4a d

J(u1 u2 u3 u4) and

corresponding sequences of states Pj Lj H+j Hminus j Qtj andQpj Since there is a finite number of uniformly boundedsequences there exist (ulowast1 ulowast2 ulowast3 ulowast4 ) isin U4

ad and Plowast Llowast H+lowast

Hminuslowast Qtlowast and Qplowast isin IRT+1 such that on a subsequence(u

j1 u

j2 u

j3 u

j4)⟶ (ulowast1 ulowast2 ulowast3 ulowast4 ) Pj⟶ Plowast Lj⟶ Llowast

H+j⟶ H+jlowast Hminusj⟶ Hminusjlowast Qtj⟶ Qtlowast and Qpj⟶Qplowast Finally due to the finite dimensional structure ofsystem (2) and the objective function J(u1 u2 u3 u4)(ulowast1 ulowast2 ulowast3 ulowast4 ) is an optimal control with correspondingstates Plowast Llowast H+jlowast Hminus jlowast and Qplowast (ereforeinf(u1u2 u3 u4)isinU4

adJ(u1 u2 u3 u4) is achieved

We apply the discrete version of Pontryaginrsquos Maxi-mum Principle [10 14 16ndash19] (e key idea is introducingthe adjoint function to attach the system of differenceequations to the objective functional resulting in theformation of a function called the Hamiltonian (is

principle converts the problem of finding the control tooptimize the objective functional subject to the statedifference equation with initial condition to find thecontrol to optimize Hamiltonian pointwise (with respectto the control)

We have the Hamiltonian Hk at time step k defined by

Hk AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k

+Gk

2u23k +

Mk

2u24k + 1113944

6

i1ζ ik+1fik+1

(7)

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

Theorem 2 Given the optimal controls (ulowast1k ulowast2k

ulowast3k ulowast4k) isin U4a d and the solutions Plowast Llowast H+jlowast Hminus jlowast Qtlowast

and Qplowast of the corresponding state system (2) thereexist adjoint functions ζ1k ζ2k ζ3k ζ4k ζ5k and ζ6k

satisfying

ζ1k ζ1k+1(1 minus μ) + α1 1 minus u1k1113872 1113873Lk

Nζ2k+1 minus ζ1k+11113872 1113873

ζ2k Ak + α1 ζ2k+1 minus ζ1k+11113872 1113873Pk

N+ β1 ζ3k+1 minus ζ2k+11113872 1113873

+ α2 1 minus u2k1113872 1113873Hminus

k

Nζ3k+1 minus ζ1k+11113872 1113873

ζ3k Bk + ζ3k+1 1 minus μ minus β3 minus c( 1113857 + ζ4k+1β3 + ζ5k+1c 1 minus σ1( 1113857 + ζ6k+1cσ1

+ β2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ2k+11113872 1113873Hminus

k

N+ ζ2k+1(1 minus μ)

ζ4k Ck + α2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ1k+11113872 1113873Pk

N+ β2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ2k+11113872 1113873

Lk

N

ζ5k Dk + ζ2k+1θ + ζ5k+1 1 minus μ minus θ minus u4k1113872 1113873 + ζ6k+1u4k minus ζ3k+1u3k

+ ζ4k+1(1 minus μ minus λ) + ζ5k+1λσ2 + ζ6k+1 λ 1 minus σ2( 1113857 + u3k1113872 1113873

ζ6k ζ6k+1(1 minus μ)

(8)

With the transversality conditions at time T ζ1T ζ6T

0 ζ2T AT ζ3T BT ζ4T CT and ζ5T DTFurthermore for k 0 1 2 T minus 1 the optimal con-

trols ulowast1k ulowast2k ulowast3k and ulowast4k are given by

Discrete Dynamics in Nature and Society 5

ulowast1k min b max a

1Ek

ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N1113876 11138771113888 11138891113890 1113891

ulowast2k min d max c

1NFk

α2PkHminusk ζ3k+1 minus ζ1k+11113872 1113873 + β2LkH

minusk( 1113857 ζ3k+1 minus ζ2k+11113872 11138731113960 11139611113888 11138891113890 1113891

ulowast3k min f max e

1Gk

ζ3k+1 minus ζ6k+11113872 1113873Hminusk1113960 11139611113888 11138891113890 1113891

ulowast4k min h max g

1Mk

ζ5k+1 minus ζ6k+11113872 1113873Qtk1113960 11139611113888 11138891113890 1113891

(9)

Proof (e Hamiltonian at time step k is given by

Hk AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k +

Gk

2u23k +

Mk

2u24k

+ ζ1k+1f1k+1 + ζ2k+1f2k+1 + ζ3k+1f3k+1 + ζ4k+1f4k+1

+ ζ5k+1f5k+1 + ζ6k+1f6k+1AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k

+Gk

2u23k +

Mk

2u24k + ζ1k+1 Λ +(1 minus μ)Pk minus α1 1 minus u1k1113872 1113873

PkLk

Nminus α2 1 minus u2k1113872 1113873

PkHminusk

N1113876 1113877

+ ζ2k+1 1 minus μ minus β1( 1113857Lk + θQtk + α1 1 minus uk( 1113857

PkLk

Nminus β2 1 minus u2k1113872 1113873

LkHminusk

N1113876 1113877

+ ζ3k+1 1 minus μ minus β3 minus c( 1113857H+k + β1Lk + β2 1 minus u2k1113872 1113873

LkHminusk

N1113876 1113877

+ ζ3k+1 α2 1 minus u2k1113872 1113873PkHminus

k

Nminus u3kH

minusk1113876 1113877 + ζ4k+1 (1 minus μ minus λ)H

minusk + β3H

+k1113858 1113859

+ ζ5k+1 (1 minus μ minus θ)Qtk + c 1 minus σ1( 1113857H

+k + λσ2H

minusk minus u4kQ

tk1113960 1113961

+ ζ6k+1 (1 minus μ)Qp

k + λ 1 minus σ2( 1113857Hminusk + cσ1H

+k + u3kH

minusk + u4kQ

tk1113960 1113961

(10)

For k 0 1 T minus 1 the optimal controls u1ku2k u3k and u4k can be solved from the optimalitycondition

zHk

zu1k

0

zHk

zu2k

0

zHk

zu3k

0

zHk

zu4k

0

(11)

which are

zHk

zu1k

Eku1k minus ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N 0

zHk

zu2k

Fku2k minus ζ3k+1 minus ζ1k+11113872 1113873α2PkHminus

k

N

minus ζ3k+1 minus ζ2k+11113872 1113873β2LkHminus

k

N 0

zHk

zu3k

Gku3k minus ζ3k+1Hminusk + ζ6k+1H

minusk 0

zHk

zu4k

Mku4k minus ζ5k+1 minus ζ6k+11113872 1113873Qtk 0

(12)

So we have

6 Discrete Dynamics in Nature and Society

u1k 1

Ek

ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N1113876 1113877

u2k 1

NFk

ζ3k+1 minus ζ1k+11113872 1113873α2PkHminusk + ζ3k+1 minus ζ2k+11113872 1113873β2LkH

minusk1113960 1113961

u3k 1

Gk

ζ3k+1 minus ζ6k+11113872 1113873Hminusk1113960 1113961

u4k 1

Mk

ζ5k+1 minus ζ6k+11113872 1113873Qtk1113960 1113961

(13)

By the bounds in Uad of the controls it is easy to obtainulowast1k ulowast2k ulowast3k and ulowast4k in the form of (9)

4 Simulation

In this section we present the results obtained by solvingnumerically the optimality system (is system consists ofthe state system adjoint system initial and final timeconditions and the control characterization

In this formulation there were initial conditions for thestate variables and terminal conditions for the adjoints (atis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k T We solve the optimality system by an it-erative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

41 Discussion In this section we study and analyse nu-merically the effects of the optimal control strategies such asawareness programs through media and education contactprevention through security campaigns treatment andpsychological support along with follow-up for the drugconsumers (e numerical solution of model (2) is executedusing Matlab with the following parameter values and initialvalues of state variable in Table 1

(e proposed control strategies in this work help toachieve several objectives

411 Objective A Protecting and Preventing Potential DrugUsers and the Light Drug Users from Falling into Drug Useand Addiction Due the importance of the awareness pro-grams throughmedia and education in restricting the spreadof drug use we propose an optimal strategy for this purposeHence we activate the optimal control variable u1 whichrepresents awareness programs for the light drug usersFigure 2(a) compares the evolution of the light drug userswith and without control u1 in which the effect of theproposed awareness programs throughmedia and education

is proven to be positive in decreasing the number of lightdrug users and preventing potential drug users from con-tacting light drug users (Figure 2(b))

412 Objective B Decreasing the Number of Heavy DrugUsers and Dealers by Preventing Contact through SecurityCampaigns When the number of drug users is so high itis obligatory to resort to some strategies such as pre-venting contact through security campaigns in order toreduce the number of heavy drug users-dealers(Figure 3(c)) and to protect light drug users from con-tacting heavy drug users-dealers through security cam-paigns which also has a positive effect on reducing thenumber of the heavy drug users (Figure 3(b)) (ereforewe propose an optimal strategy by using the optimalcontrol u2 in the beginning In spite of using the optimalcontrol u2 we observe that the number of light drugconsumers increases due to protecting them from con-tacting the heavy drug users-dealers (e reason of thisincrease is justified by the fact that light drug consumersrevert back to using drugs occasionally (Figure 3(a)) Alsothe proposed strategy has an additional effect in de-creasing clearly the number of heavy drug users and heavydrug users-dealers and providers

413 Objective C Treatment in the Addiction CentersGiven the importance and effectiveness of this strategy weuse the control strategy to encourage heavy drug users toknow about treatment centers and join them to decrease thespread of drug consumers

We propose treatment within addiction centers repre-sented by the strategy of optimal control u3 FromFigures 4(a) and 4(b) the decrease of the number of heavydrug users is clearly achieved which in turn had a positiveeffect on reducing the number of heavy drug users-dealersand providers

414 Objective D Control with Psychological Support alongwith Follow-Up Taking into consideration the importanceand the effectiveness of this strategy on the individuals whotemporarily quit drug we propose an optimal strategy byusing the optimal control u4 in the beginning which rep-resents follow-up and psychological support to prevent thetemporary quitters from reverting back to using the drugsoccasionally (θne 0) (Figure 5(b)) (e proposed strategy hasan additional effect in decreasing clearly the number oftemporary quitters of drug consumption Figure 5(a) showsthat the number of light drug users is decreased markedly(θ 0)

Note several optimal controls can be combined toachieve other objectives and implement other strategiesdepending on the phenomenon and the particularity of eachsociety

Discrete Dynamics in Nature and Society 7

3000

3500

4000

4500

5000

5500

6000

6500

The l

ight

dru

g co

nsum

ers

5 10 15 20 25 30Time

L without controlL with control u2

(a)

1000

1200

1400

1600

1800

2000

2200

e h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u2

(b)

1300

1400

1500

1600

1700

1800

1900

2000

The h

eavy

dru

g co

nsum

ers

5 10 15 20 25 30Time

H+ without controlH+ with control u2

(c)

Figure 3 (a)(e evolution of Lwith and without controls (b)(e evolution of theH- with and without controls (c)(e evolution of theH+

with and without controls

Table 1 (e description of parameters used for the definition of discrete time system (1) We used just arbitrary academic data

P0 L0 H+0 Hminus

0 Qt0 Q

p0 Λ α1 α2 μ β1 β2 β3 c λ σ1 σ2 θ

5103 3103 15103 1103 2103 1103 5102 045 05 004 0025 02 01 005 005 005 07 005

2000

2500

3000

3500

4000

4500

5000

5500

The o

ccas

iona

l dru

g us

ers (

L)

5 10 15 20 25 30Time

L without controlL with control u1

(a)

1500

2000

2500

3000

3500

4000

4500

5000

5 10 15 20 25 30Time

e p

oten

tial d

rug

user

s (P)

P with control u1

P without control

(b)

Figure 2 (a) (e evolution of the L with and without controls (b) (e evolution of the P with and without controls

8 Discrete Dynamics in Nature and Society

5 Conclusion

In this paper we introduced a discrete modeling of drugusers in order to minimize the number of light drug usersheavy drug users heavy drug users-dealers and temporaryquitters of drug consumption We also introduced fourcontrols which respectively represent awareness programsthrough education and media contact prevention throughsecurity campaigns treatment and psychological supportalong with follow-up We applied the results of the controltheory and wemanaged to obtain the characterisations of theoptimal controls (e numerical simulation of the obtainedresults showed the effectiveness of the proposed controlstrategies

Data Availability

No data were used to support this study

Disclosure

(is article was presented at the International Conferenceon Research in Applied Mathematics and Computer Sci-ence (ICRAMC) 2020 which took place on July 15ndash182020

Conflicts of Interest

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

References

[1] UNODC ldquoWorld drug reportmdashunited nations office on drugsand crimerdquo 2019 httpswwwunodcorgwdr2019prelaunchWDR19_Booklet_1_EXSUMpdf

400600800

100012001400160018002000

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ ne 0

(a)

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

400600800

1000120014001600180020002200

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ = 0

(b)

Figure 5 (a) (e evolution of the Qt with and without controls with θne 0 (b) (e evolution of the Qt with and without controls with θ 0

800

1000

1200

1400

1600

1800

2000

5 10 15 20 25 30Time

The h

eavy

dru

g us

ers

H+ without controlH+ with control u3

(a)

1000

1200

1400

1600

1800

2000

2200

The h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u3

(b)

Figure 4 (a) (e evolution of the H+ with and without controls (b) (e evolution of the Hminus with and without controls

Discrete Dynamics in Nature and Society 9

Uad uik ui0 ui1 uiTminus11113872 11138731113966

for i 1 2 3 4 ai le uik le bi k 0 1 2 T minus 11113967

(5)

(e sufficient condition for the existence of the optimalcontrols (u1 u2 u3 u4) for problems (2) and (3) comes fromthe following theorem

Theorem 1 1ere exists the optimal controls(ulowast1k ulowast2k ulowast3k ulowast4k) such that

J ulowast1k ulowast2k ulowast3k ulowast4k1113872 1113873 min

u1 u2u3 u4( )isinU4ad

J u1 u2 u3 u4( 1113857

(6)

subject to the control system (2) with initial conditions

Proof Since the coefficients of the state equations arebounded and there is a finite number of time steps P

(P0 P1 PT) L (L0 L1 LT) H+ (H+0 H+

1

H+T) Hminus (Hminus

0 Hminus1 Hminus

T)Qt (Qt0 Qt

1 QtT) and

Qp (Qp0 Q

p1 Q

p

T) are uniformly bounded for all(u1 u2 u3 u4) in the control set Uad thus J(u1 u2 u3 u4) isbounded for all (u1 u2 u3 u4) isin U4

ad Since J(u1 u2 u3 u4)

is bounded inf(u1 u2u3 u4)isinU4ad

J(u1 u2 u3 u4) is finite and

there exists a sequence (uj1 u

j2 u

j3 u

j4) isin U4a d such that

limj⟶+infin

J(uj1 u

j2 u

j3 u

j4) inf

(u1 u2u3 u4)isinU4a d

J(u1 u2 u3 u4) and

corresponding sequences of states Pj Lj H+j Hminus j Qtj andQpj Since there is a finite number of uniformly boundedsequences there exist (ulowast1 ulowast2 ulowast3 ulowast4 ) isin U4

ad and Plowast Llowast H+lowast

Hminuslowast Qtlowast and Qplowast isin IRT+1 such that on a subsequence(u

j1 u

j2 u

j3 u

j4)⟶ (ulowast1 ulowast2 ulowast3 ulowast4 ) Pj⟶ Plowast Lj⟶ Llowast

H+j⟶ H+jlowast Hminusj⟶ Hminusjlowast Qtj⟶ Qtlowast and Qpj⟶Qplowast Finally due to the finite dimensional structure ofsystem (2) and the objective function J(u1 u2 u3 u4)(ulowast1 ulowast2 ulowast3 ulowast4 ) is an optimal control with correspondingstates Plowast Llowast H+jlowast Hminus jlowast and Qplowast (ereforeinf(u1u2 u3 u4)isinU4

adJ(u1 u2 u3 u4) is achieved

We apply the discrete version of Pontryaginrsquos Maxi-mum Principle [10 14 16ndash19] (e key idea is introducingthe adjoint function to attach the system of differenceequations to the objective functional resulting in theformation of a function called the Hamiltonian (is

principle converts the problem of finding the control tooptimize the objective functional subject to the statedifference equation with initial condition to find thecontrol to optimize Hamiltonian pointwise (with respectto the control)

We have the Hamiltonian Hk at time step k defined by

Hk AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k

+Gk

2u23k +

Mk

2u24k + 1113944

6

i1ζ ik+1fik+1

(7)

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

Theorem 2 Given the optimal controls (ulowast1k ulowast2k

ulowast3k ulowast4k) isin U4a d and the solutions Plowast Llowast H+jlowast Hminus jlowast Qtlowast

and Qplowast of the corresponding state system (2) thereexist adjoint functions ζ1k ζ2k ζ3k ζ4k ζ5k and ζ6k

satisfying

ζ1k ζ1k+1(1 minus μ) + α1 1 minus u1k1113872 1113873Lk

Nζ2k+1 minus ζ1k+11113872 1113873

ζ2k Ak + α1 ζ2k+1 minus ζ1k+11113872 1113873Pk

N+ β1 ζ3k+1 minus ζ2k+11113872 1113873

+ α2 1 minus u2k1113872 1113873Hminus

k

Nζ3k+1 minus ζ1k+11113872 1113873

ζ3k Bk + ζ3k+1 1 minus μ minus β3 minus c( 1113857 + ζ4k+1β3 + ζ5k+1c 1 minus σ1( 1113857 + ζ6k+1cσ1

+ β2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ2k+11113872 1113873Hminus

k

N+ ζ2k+1(1 minus μ)

ζ4k Ck + α2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ1k+11113872 1113873Pk

N+ β2 1 minus u2k1113872 1113873 ζ3k+1 minus ζ2k+11113872 1113873

Lk

N

ζ5k Dk + ζ2k+1θ + ζ5k+1 1 minus μ minus θ minus u4k1113872 1113873 + ζ6k+1u4k minus ζ3k+1u3k

+ ζ4k+1(1 minus μ minus λ) + ζ5k+1λσ2 + ζ6k+1 λ 1 minus σ2( 1113857 + u3k1113872 1113873

ζ6k ζ6k+1(1 minus μ)

(8)

With the transversality conditions at time T ζ1T ζ6T

0 ζ2T AT ζ3T BT ζ4T CT and ζ5T DTFurthermore for k 0 1 2 T minus 1 the optimal con-

trols ulowast1k ulowast2k ulowast3k and ulowast4k are given by

Discrete Dynamics in Nature and Society 5

ulowast1k min b max a

1Ek

ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N1113876 11138771113888 11138891113890 1113891

ulowast2k min d max c

1NFk

α2PkHminusk ζ3k+1 minus ζ1k+11113872 1113873 + β2LkH

minusk( 1113857 ζ3k+1 minus ζ2k+11113872 11138731113960 11139611113888 11138891113890 1113891

ulowast3k min f max e

1Gk

ζ3k+1 minus ζ6k+11113872 1113873Hminusk1113960 11139611113888 11138891113890 1113891

ulowast4k min h max g

1Mk

ζ5k+1 minus ζ6k+11113872 1113873Qtk1113960 11139611113888 11138891113890 1113891

(9)

Proof (e Hamiltonian at time step k is given by

Hk AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k +

Gk

2u23k +

Mk

2u24k

+ ζ1k+1f1k+1 + ζ2k+1f2k+1 + ζ3k+1f3k+1 + ζ4k+1f4k+1

+ ζ5k+1f5k+1 + ζ6k+1f6k+1AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k

+Gk

2u23k +

Mk

2u24k + ζ1k+1 Λ +(1 minus μ)Pk minus α1 1 minus u1k1113872 1113873

PkLk

Nminus α2 1 minus u2k1113872 1113873

PkHminusk

N1113876 1113877

+ ζ2k+1 1 minus μ minus β1( 1113857Lk + θQtk + α1 1 minus uk( 1113857

PkLk

Nminus β2 1 minus u2k1113872 1113873

LkHminusk

N1113876 1113877

+ ζ3k+1 1 minus μ minus β3 minus c( 1113857H+k + β1Lk + β2 1 minus u2k1113872 1113873

LkHminusk

N1113876 1113877

+ ζ3k+1 α2 1 minus u2k1113872 1113873PkHminus

k

Nminus u3kH

minusk1113876 1113877 + ζ4k+1 (1 minus μ minus λ)H

minusk + β3H

+k1113858 1113859

+ ζ5k+1 (1 minus μ minus θ)Qtk + c 1 minus σ1( 1113857H

+k + λσ2H

minusk minus u4kQ

tk1113960 1113961

+ ζ6k+1 (1 minus μ)Qp

k + λ 1 minus σ2( 1113857Hminusk + cσ1H

+k + u3kH

minusk + u4kQ

tk1113960 1113961

(10)

For k 0 1 T minus 1 the optimal controls u1ku2k u3k and u4k can be solved from the optimalitycondition

zHk

zu1k

0

zHk

zu2k

0

zHk

zu3k

0

zHk

zu4k

0

(11)

which are

zHk

zu1k

Eku1k minus ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N 0

zHk

zu2k

Fku2k minus ζ3k+1 minus ζ1k+11113872 1113873α2PkHminus

k

N

minus ζ3k+1 minus ζ2k+11113872 1113873β2LkHminus

k

N 0

zHk

zu3k

Gku3k minus ζ3k+1Hminusk + ζ6k+1H

minusk 0

zHk

zu4k

Mku4k minus ζ5k+1 minus ζ6k+11113872 1113873Qtk 0

(12)

So we have

6 Discrete Dynamics in Nature and Society

u1k 1

Ek

ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N1113876 1113877

u2k 1

NFk

ζ3k+1 minus ζ1k+11113872 1113873α2PkHminusk + ζ3k+1 minus ζ2k+11113872 1113873β2LkH

minusk1113960 1113961

u3k 1

Gk

ζ3k+1 minus ζ6k+11113872 1113873Hminusk1113960 1113961

u4k 1

Mk

ζ5k+1 minus ζ6k+11113872 1113873Qtk1113960 1113961

(13)

By the bounds in Uad of the controls it is easy to obtainulowast1k ulowast2k ulowast3k and ulowast4k in the form of (9)

4 Simulation

In this section we present the results obtained by solvingnumerically the optimality system (is system consists ofthe state system adjoint system initial and final timeconditions and the control characterization

In this formulation there were initial conditions for thestate variables and terminal conditions for the adjoints (atis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k T We solve the optimality system by an it-erative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

41 Discussion In this section we study and analyse nu-merically the effects of the optimal control strategies such asawareness programs through media and education contactprevention through security campaigns treatment andpsychological support along with follow-up for the drugconsumers (e numerical solution of model (2) is executedusing Matlab with the following parameter values and initialvalues of state variable in Table 1

(e proposed control strategies in this work help toachieve several objectives

411 Objective A Protecting and Preventing Potential DrugUsers and the Light Drug Users from Falling into Drug Useand Addiction Due the importance of the awareness pro-grams throughmedia and education in restricting the spreadof drug use we propose an optimal strategy for this purposeHence we activate the optimal control variable u1 whichrepresents awareness programs for the light drug usersFigure 2(a) compares the evolution of the light drug userswith and without control u1 in which the effect of theproposed awareness programs throughmedia and education

is proven to be positive in decreasing the number of lightdrug users and preventing potential drug users from con-tacting light drug users (Figure 2(b))

412 Objective B Decreasing the Number of Heavy DrugUsers and Dealers by Preventing Contact through SecurityCampaigns When the number of drug users is so high itis obligatory to resort to some strategies such as pre-venting contact through security campaigns in order toreduce the number of heavy drug users-dealers(Figure 3(c)) and to protect light drug users from con-tacting heavy drug users-dealers through security cam-paigns which also has a positive effect on reducing thenumber of the heavy drug users (Figure 3(b)) (ereforewe propose an optimal strategy by using the optimalcontrol u2 in the beginning In spite of using the optimalcontrol u2 we observe that the number of light drugconsumers increases due to protecting them from con-tacting the heavy drug users-dealers (e reason of thisincrease is justified by the fact that light drug consumersrevert back to using drugs occasionally (Figure 3(a)) Alsothe proposed strategy has an additional effect in de-creasing clearly the number of heavy drug users and heavydrug users-dealers and providers

413 Objective C Treatment in the Addiction CentersGiven the importance and effectiveness of this strategy weuse the control strategy to encourage heavy drug users toknow about treatment centers and join them to decrease thespread of drug consumers

We propose treatment within addiction centers repre-sented by the strategy of optimal control u3 FromFigures 4(a) and 4(b) the decrease of the number of heavydrug users is clearly achieved which in turn had a positiveeffect on reducing the number of heavy drug users-dealersand providers

414 Objective D Control with Psychological Support alongwith Follow-Up Taking into consideration the importanceand the effectiveness of this strategy on the individuals whotemporarily quit drug we propose an optimal strategy byusing the optimal control u4 in the beginning which rep-resents follow-up and psychological support to prevent thetemporary quitters from reverting back to using the drugsoccasionally (θne 0) (Figure 5(b)) (e proposed strategy hasan additional effect in decreasing clearly the number oftemporary quitters of drug consumption Figure 5(a) showsthat the number of light drug users is decreased markedly(θ 0)

Note several optimal controls can be combined toachieve other objectives and implement other strategiesdepending on the phenomenon and the particularity of eachsociety

Discrete Dynamics in Nature and Society 7

3000

3500

4000

4500

5000

5500

6000

6500

The l

ight

dru

g co

nsum

ers

5 10 15 20 25 30Time

L without controlL with control u2

(a)

1000

1200

1400

1600

1800

2000

2200

e h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u2

(b)

1300

1400

1500

1600

1700

1800

1900

2000

The h

eavy

dru

g co

nsum

ers

5 10 15 20 25 30Time

H+ without controlH+ with control u2

(c)

Figure 3 (a)(e evolution of Lwith and without controls (b)(e evolution of theH- with and without controls (c)(e evolution of theH+

with and without controls

Table 1 (e description of parameters used for the definition of discrete time system (1) We used just arbitrary academic data

P0 L0 H+0 Hminus

0 Qt0 Q

p0 Λ α1 α2 μ β1 β2 β3 c λ σ1 σ2 θ

5103 3103 15103 1103 2103 1103 5102 045 05 004 0025 02 01 005 005 005 07 005

2000

2500

3000

3500

4000

4500

5000

5500

The o

ccas

iona

l dru

g us

ers (

L)

5 10 15 20 25 30Time

L without controlL with control u1

(a)

1500

2000

2500

3000

3500

4000

4500

5000

5 10 15 20 25 30Time

e p

oten

tial d

rug

user

s (P)

P with control u1

P without control

(b)

Figure 2 (a) (e evolution of the L with and without controls (b) (e evolution of the P with and without controls

8 Discrete Dynamics in Nature and Society

5 Conclusion

In this paper we introduced a discrete modeling of drugusers in order to minimize the number of light drug usersheavy drug users heavy drug users-dealers and temporaryquitters of drug consumption We also introduced fourcontrols which respectively represent awareness programsthrough education and media contact prevention throughsecurity campaigns treatment and psychological supportalong with follow-up We applied the results of the controltheory and wemanaged to obtain the characterisations of theoptimal controls (e numerical simulation of the obtainedresults showed the effectiveness of the proposed controlstrategies

Data Availability

No data were used to support this study

Disclosure

(is article was presented at the International Conferenceon Research in Applied Mathematics and Computer Sci-ence (ICRAMC) 2020 which took place on July 15ndash182020

Conflicts of Interest

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

References

[1] UNODC ldquoWorld drug reportmdashunited nations office on drugsand crimerdquo 2019 httpswwwunodcorgwdr2019prelaunchWDR19_Booklet_1_EXSUMpdf

400600800

100012001400160018002000

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ ne 0

(a)

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

400600800

1000120014001600180020002200

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ = 0

(b)

Figure 5 (a) (e evolution of the Qt with and without controls with θne 0 (b) (e evolution of the Qt with and without controls with θ 0

800

1000

1200

1400

1600

1800

2000

5 10 15 20 25 30Time

The h

eavy

dru

g us

ers

H+ without controlH+ with control u3

(a)

1000

1200

1400

1600

1800

2000

2200

The h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u3

(b)

Figure 4 (a) (e evolution of the H+ with and without controls (b) (e evolution of the Hminus with and without controls

Discrete Dynamics in Nature and Society 9

ulowast1k min b max a

1Ek

ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N1113876 11138771113888 11138891113890 1113891

ulowast2k min d max c

1NFk

α2PkHminusk ζ3k+1 minus ζ1k+11113872 1113873 + β2LkH

minusk( 1113857 ζ3k+1 minus ζ2k+11113872 11138731113960 11139611113888 11138891113890 1113891

ulowast3k min f max e

1Gk

ζ3k+1 minus ζ6k+11113872 1113873Hminusk1113960 11139611113888 11138891113890 1113891

ulowast4k min h max g

1Mk

ζ5k+1 minus ζ6k+11113872 1113873Qtk1113960 11139611113888 11138891113890 1113891

(9)

Proof (e Hamiltonian at time step k is given by

Hk AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k +

Gk

2u23k +

Mk

2u24k

+ ζ1k+1f1k+1 + ζ2k+1f2k+1 + ζ3k+1f3k+1 + ζ4k+1f4k+1

+ ζ5k+1f5k+1 + ζ6k+1f6k+1AkLk + BkH+k + CkH

minusk + DkQ

tk +

Ek

2u21k +

Fk

2u22k

+Gk

2u23k +

Mk

2u24k + ζ1k+1 Λ +(1 minus μ)Pk minus α1 1 minus u1k1113872 1113873

PkLk

Nminus α2 1 minus u2k1113872 1113873

PkHminusk

N1113876 1113877

+ ζ2k+1 1 minus μ minus β1( 1113857Lk + θQtk + α1 1 minus uk( 1113857

PkLk

Nminus β2 1 minus u2k1113872 1113873

LkHminusk

N1113876 1113877

+ ζ3k+1 1 minus μ minus β3 minus c( 1113857H+k + β1Lk + β2 1 minus u2k1113872 1113873

LkHminusk

N1113876 1113877

+ ζ3k+1 α2 1 minus u2k1113872 1113873PkHminus

k

Nminus u3kH

minusk1113876 1113877 + ζ4k+1 (1 minus μ minus λ)H

minusk + β3H

+k1113858 1113859

+ ζ5k+1 (1 minus μ minus θ)Qtk + c 1 minus σ1( 1113857H

+k + λσ2H

minusk minus u4kQ

tk1113960 1113961

+ ζ6k+1 (1 minus μ)Qp

k + λ 1 minus σ2( 1113857Hminusk + cσ1H

+k + u3kH

minusk + u4kQ

tk1113960 1113961

(10)

For k 0 1 T minus 1 the optimal controls u1ku2k u3k and u4k can be solved from the optimalitycondition

zHk

zu1k

0

zHk

zu2k

0

zHk

zu3k

0

zHk

zu4k

0

(11)

which are

zHk

zu1k

Eku1k minus ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N 0

zHk

zu2k

Fku2k minus ζ3k+1 minus ζ1k+11113872 1113873α2PkHminus

k

N

minus ζ3k+1 minus ζ2k+11113872 1113873β2LkHminus

k

N 0

zHk

zu3k

Gku3k minus ζ3k+1Hminusk + ζ6k+1H

minusk 0

zHk

zu4k

Mku4k minus ζ5k+1 minus ζ6k+11113872 1113873Qtk 0

(12)

So we have

6 Discrete Dynamics in Nature and Society

u1k 1

Ek

ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N1113876 1113877

u2k 1

NFk

ζ3k+1 minus ζ1k+11113872 1113873α2PkHminusk + ζ3k+1 minus ζ2k+11113872 1113873β2LkH

minusk1113960 1113961

u3k 1

Gk

ζ3k+1 minus ζ6k+11113872 1113873Hminusk1113960 1113961

u4k 1

Mk

ζ5k+1 minus ζ6k+11113872 1113873Qtk1113960 1113961

(13)

By the bounds in Uad of the controls it is easy to obtainulowast1k ulowast2k ulowast3k and ulowast4k in the form of (9)

4 Simulation

In this section we present the results obtained by solvingnumerically the optimality system (is system consists ofthe state system adjoint system initial and final timeconditions and the control characterization

In this formulation there were initial conditions for thestate variables and terminal conditions for the adjoints (atis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k T We solve the optimality system by an it-erative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

41 Discussion In this section we study and analyse nu-merically the effects of the optimal control strategies such asawareness programs through media and education contactprevention through security campaigns treatment andpsychological support along with follow-up for the drugconsumers (e numerical solution of model (2) is executedusing Matlab with the following parameter values and initialvalues of state variable in Table 1

(e proposed control strategies in this work help toachieve several objectives

411 Objective A Protecting and Preventing Potential DrugUsers and the Light Drug Users from Falling into Drug Useand Addiction Due the importance of the awareness pro-grams throughmedia and education in restricting the spreadof drug use we propose an optimal strategy for this purposeHence we activate the optimal control variable u1 whichrepresents awareness programs for the light drug usersFigure 2(a) compares the evolution of the light drug userswith and without control u1 in which the effect of theproposed awareness programs throughmedia and education

is proven to be positive in decreasing the number of lightdrug users and preventing potential drug users from con-tacting light drug users (Figure 2(b))

412 Objective B Decreasing the Number of Heavy DrugUsers and Dealers by Preventing Contact through SecurityCampaigns When the number of drug users is so high itis obligatory to resort to some strategies such as pre-venting contact through security campaigns in order toreduce the number of heavy drug users-dealers(Figure 3(c)) and to protect light drug users from con-tacting heavy drug users-dealers through security cam-paigns which also has a positive effect on reducing thenumber of the heavy drug users (Figure 3(b)) (ereforewe propose an optimal strategy by using the optimalcontrol u2 in the beginning In spite of using the optimalcontrol u2 we observe that the number of light drugconsumers increases due to protecting them from con-tacting the heavy drug users-dealers (e reason of thisincrease is justified by the fact that light drug consumersrevert back to using drugs occasionally (Figure 3(a)) Alsothe proposed strategy has an additional effect in de-creasing clearly the number of heavy drug users and heavydrug users-dealers and providers

413 Objective C Treatment in the Addiction CentersGiven the importance and effectiveness of this strategy weuse the control strategy to encourage heavy drug users toknow about treatment centers and join them to decrease thespread of drug consumers

We propose treatment within addiction centers repre-sented by the strategy of optimal control u3 FromFigures 4(a) and 4(b) the decrease of the number of heavydrug users is clearly achieved which in turn had a positiveeffect on reducing the number of heavy drug users-dealersand providers

414 Objective D Control with Psychological Support alongwith Follow-Up Taking into consideration the importanceand the effectiveness of this strategy on the individuals whotemporarily quit drug we propose an optimal strategy byusing the optimal control u4 in the beginning which rep-resents follow-up and psychological support to prevent thetemporary quitters from reverting back to using the drugsoccasionally (θne 0) (Figure 5(b)) (e proposed strategy hasan additional effect in decreasing clearly the number oftemporary quitters of drug consumption Figure 5(a) showsthat the number of light drug users is decreased markedly(θ 0)

Note several optimal controls can be combined toachieve other objectives and implement other strategiesdepending on the phenomenon and the particularity of eachsociety

Discrete Dynamics in Nature and Society 7

3000

3500

4000

4500

5000

5500

6000

6500

The l

ight

dru

g co

nsum

ers

5 10 15 20 25 30Time

L without controlL with control u2

(a)

1000

1200

1400

1600

1800

2000

2200

e h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u2

(b)

1300

1400

1500

1600

1700

1800

1900

2000

The h

eavy

dru

g co

nsum

ers

5 10 15 20 25 30Time

H+ without controlH+ with control u2

(c)

Figure 3 (a)(e evolution of Lwith and without controls (b)(e evolution of theH- with and without controls (c)(e evolution of theH+

with and without controls

Table 1 (e description of parameters used for the definition of discrete time system (1) We used just arbitrary academic data

P0 L0 H+0 Hminus

0 Qt0 Q

p0 Λ α1 α2 μ β1 β2 β3 c λ σ1 σ2 θ

5103 3103 15103 1103 2103 1103 5102 045 05 004 0025 02 01 005 005 005 07 005

2000

2500

3000

3500

4000

4500

5000

5500

The o

ccas

iona

l dru

g us

ers (

L)

5 10 15 20 25 30Time

L without controlL with control u1

(a)

1500

2000

2500

3000

3500

4000

4500

5000

5 10 15 20 25 30Time

e p

oten

tial d

rug

user

s (P)

P with control u1

P without control

(b)

Figure 2 (a) (e evolution of the L with and without controls (b) (e evolution of the P with and without controls

8 Discrete Dynamics in Nature and Society

5 Conclusion

In this paper we introduced a discrete modeling of drugusers in order to minimize the number of light drug usersheavy drug users heavy drug users-dealers and temporaryquitters of drug consumption We also introduced fourcontrols which respectively represent awareness programsthrough education and media contact prevention throughsecurity campaigns treatment and psychological supportalong with follow-up We applied the results of the controltheory and wemanaged to obtain the characterisations of theoptimal controls (e numerical simulation of the obtainedresults showed the effectiveness of the proposed controlstrategies

Data Availability

No data were used to support this study

Disclosure

(is article was presented at the International Conferenceon Research in Applied Mathematics and Computer Sci-ence (ICRAMC) 2020 which took place on July 15ndash182020

Conflicts of Interest

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

References

[1] UNODC ldquoWorld drug reportmdashunited nations office on drugsand crimerdquo 2019 httpswwwunodcorgwdr2019prelaunchWDR19_Booklet_1_EXSUMpdf

400600800

100012001400160018002000

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ ne 0

(a)

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

400600800

1000120014001600180020002200

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ = 0

(b)

Figure 5 (a) (e evolution of the Qt with and without controls with θne 0 (b) (e evolution of the Qt with and without controls with θ 0

800

1000

1200

1400

1600

1800

2000

5 10 15 20 25 30Time

The h

eavy

dru

g us

ers

H+ without controlH+ with control u3

(a)

1000

1200

1400

1600

1800

2000

2200

The h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u3

(b)

Figure 4 (a) (e evolution of the H+ with and without controls (b) (e evolution of the Hminus with and without controls

Discrete Dynamics in Nature and Society 9

u1k 1

Ek

ζ2k+1 minus ζ1k+11113872 1113873α1PkLk

N1113876 1113877

u2k 1

NFk

ζ3k+1 minus ζ1k+11113872 1113873α2PkHminusk + ζ3k+1 minus ζ2k+11113872 1113873β2LkH

minusk1113960 1113961

u3k 1

Gk

ζ3k+1 minus ζ6k+11113872 1113873Hminusk1113960 1113961

u4k 1

Mk

ζ5k+1 minus ζ6k+11113872 1113873Qtk1113960 1113961

(13)

By the bounds in Uad of the controls it is easy to obtainulowast1k ulowast2k ulowast3k and ulowast4k in the form of (9)

4 Simulation

In this section we present the results obtained by solvingnumerically the optimality system (is system consists ofthe state system adjoint system initial and final timeconditions and the control characterization

In this formulation there were initial conditions for thestate variables and terminal conditions for the adjoints (atis the optimality system is a two-point boundary valueproblem with separated boundary conditions at time stepsk 0 and k T We solve the optimality system by an it-erative method with forward solving of the state systemfollowed by backward solving of the adjoint systemWe startwith an initial guess for the controls at the first iteration andthen before the next iteration we update the controls byusing the characterization We continue until convergenceof successive iterates is achieved

41 Discussion In this section we study and analyse nu-merically the effects of the optimal control strategies such asawareness programs through media and education contactprevention through security campaigns treatment andpsychological support along with follow-up for the drugconsumers (e numerical solution of model (2) is executedusing Matlab with the following parameter values and initialvalues of state variable in Table 1

(e proposed control strategies in this work help toachieve several objectives

411 Objective A Protecting and Preventing Potential DrugUsers and the Light Drug Users from Falling into Drug Useand Addiction Due the importance of the awareness pro-grams throughmedia and education in restricting the spreadof drug use we propose an optimal strategy for this purposeHence we activate the optimal control variable u1 whichrepresents awareness programs for the light drug usersFigure 2(a) compares the evolution of the light drug userswith and without control u1 in which the effect of theproposed awareness programs throughmedia and education

is proven to be positive in decreasing the number of lightdrug users and preventing potential drug users from con-tacting light drug users (Figure 2(b))

412 Objective B Decreasing the Number of Heavy DrugUsers and Dealers by Preventing Contact through SecurityCampaigns When the number of drug users is so high itis obligatory to resort to some strategies such as pre-venting contact through security campaigns in order toreduce the number of heavy drug users-dealers(Figure 3(c)) and to protect light drug users from con-tacting heavy drug users-dealers through security cam-paigns which also has a positive effect on reducing thenumber of the heavy drug users (Figure 3(b)) (ereforewe propose an optimal strategy by using the optimalcontrol u2 in the beginning In spite of using the optimalcontrol u2 we observe that the number of light drugconsumers increases due to protecting them from con-tacting the heavy drug users-dealers (e reason of thisincrease is justified by the fact that light drug consumersrevert back to using drugs occasionally (Figure 3(a)) Alsothe proposed strategy has an additional effect in de-creasing clearly the number of heavy drug users and heavydrug users-dealers and providers

413 Objective C Treatment in the Addiction CentersGiven the importance and effectiveness of this strategy weuse the control strategy to encourage heavy drug users toknow about treatment centers and join them to decrease thespread of drug consumers

We propose treatment within addiction centers repre-sented by the strategy of optimal control u3 FromFigures 4(a) and 4(b) the decrease of the number of heavydrug users is clearly achieved which in turn had a positiveeffect on reducing the number of heavy drug users-dealersand providers

414 Objective D Control with Psychological Support alongwith Follow-Up Taking into consideration the importanceand the effectiveness of this strategy on the individuals whotemporarily quit drug we propose an optimal strategy byusing the optimal control u4 in the beginning which rep-resents follow-up and psychological support to prevent thetemporary quitters from reverting back to using the drugsoccasionally (θne 0) (Figure 5(b)) (e proposed strategy hasan additional effect in decreasing clearly the number oftemporary quitters of drug consumption Figure 5(a) showsthat the number of light drug users is decreased markedly(θ 0)

Note several optimal controls can be combined toachieve other objectives and implement other strategiesdepending on the phenomenon and the particularity of eachsociety

Discrete Dynamics in Nature and Society 7

3000

3500

4000

4500

5000

5500

6000

6500

The l

ight

dru

g co

nsum

ers

5 10 15 20 25 30Time

L without controlL with control u2

(a)

1000

1200

1400

1600

1800

2000

2200

e h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u2

(b)

1300

1400

1500

1600

1700

1800

1900

2000

The h

eavy

dru

g co

nsum

ers

5 10 15 20 25 30Time

H+ without controlH+ with control u2

(c)

Figure 3 (a)(e evolution of Lwith and without controls (b)(e evolution of theH- with and without controls (c)(e evolution of theH+

with and without controls

Table 1 (e description of parameters used for the definition of discrete time system (1) We used just arbitrary academic data

P0 L0 H+0 Hminus

0 Qt0 Q

p0 Λ α1 α2 μ β1 β2 β3 c λ σ1 σ2 θ

5103 3103 15103 1103 2103 1103 5102 045 05 004 0025 02 01 005 005 005 07 005

2000

2500

3000

3500

4000

4500

5000

5500

The o

ccas

iona

l dru

g us

ers (

L)

5 10 15 20 25 30Time

L without controlL with control u1

(a)

1500

2000

2500

3000

3500

4000

4500

5000

5 10 15 20 25 30Time

e p

oten

tial d

rug

user

s (P)

P with control u1

P without control

(b)

Figure 2 (a) (e evolution of the L with and without controls (b) (e evolution of the P with and without controls

8 Discrete Dynamics in Nature and Society

5 Conclusion

In this paper we introduced a discrete modeling of drugusers in order to minimize the number of light drug usersheavy drug users heavy drug users-dealers and temporaryquitters of drug consumption We also introduced fourcontrols which respectively represent awareness programsthrough education and media contact prevention throughsecurity campaigns treatment and psychological supportalong with follow-up We applied the results of the controltheory and wemanaged to obtain the characterisations of theoptimal controls (e numerical simulation of the obtainedresults showed the effectiveness of the proposed controlstrategies

Data Availability

No data were used to support this study

Disclosure

(is article was presented at the International Conferenceon Research in Applied Mathematics and Computer Sci-ence (ICRAMC) 2020 which took place on July 15ndash182020

Conflicts of Interest

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

References

[1] UNODC ldquoWorld drug reportmdashunited nations office on drugsand crimerdquo 2019 httpswwwunodcorgwdr2019prelaunchWDR19_Booklet_1_EXSUMpdf

400600800

100012001400160018002000

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ ne 0

(a)

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

400600800

1000120014001600180020002200

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ = 0

(b)

Figure 5 (a) (e evolution of the Qt with and without controls with θne 0 (b) (e evolution of the Qt with and without controls with θ 0

800

1000

1200

1400

1600

1800

2000

5 10 15 20 25 30Time

The h

eavy

dru

g us

ers

H+ without controlH+ with control u3

(a)

1000

1200

1400

1600

1800

2000

2200

The h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u3

(b)

Figure 4 (a) (e evolution of the H+ with and without controls (b) (e evolution of the Hminus with and without controls

Discrete Dynamics in Nature and Society 9

3000

3500

4000

4500

5000

5500

6000

6500

The l

ight

dru

g co

nsum

ers

5 10 15 20 25 30Time

L without controlL with control u2

(a)

1000

1200

1400

1600

1800

2000

2200

e h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u2

(b)

1300

1400

1500

1600

1700

1800

1900

2000

The h

eavy

dru

g co

nsum

ers

5 10 15 20 25 30Time

H+ without controlH+ with control u2

(c)

Figure 3 (a)(e evolution of Lwith and without controls (b)(e evolution of theH- with and without controls (c)(e evolution of theH+

with and without controls

Table 1 (e description of parameters used for the definition of discrete time system (1) We used just arbitrary academic data

P0 L0 H+0 Hminus

0 Qt0 Q

p0 Λ α1 α2 μ β1 β2 β3 c λ σ1 σ2 θ

5103 3103 15103 1103 2103 1103 5102 045 05 004 0025 02 01 005 005 005 07 005

2000

2500

3000

3500

4000

4500

5000

5500

The o

ccas

iona

l dru

g us

ers (

L)

5 10 15 20 25 30Time

L without controlL with control u1

(a)

1500

2000

2500

3000

3500

4000

4500

5000

5 10 15 20 25 30Time

e p

oten

tial d

rug

user

s (P)

P with control u1

P without control

(b)

Figure 2 (a) (e evolution of the L with and without controls (b) (e evolution of the P with and without controls

8 Discrete Dynamics in Nature and Society

5 Conclusion

In this paper we introduced a discrete modeling of drugusers in order to minimize the number of light drug usersheavy drug users heavy drug users-dealers and temporaryquitters of drug consumption We also introduced fourcontrols which respectively represent awareness programsthrough education and media contact prevention throughsecurity campaigns treatment and psychological supportalong with follow-up We applied the results of the controltheory and wemanaged to obtain the characterisations of theoptimal controls (e numerical simulation of the obtainedresults showed the effectiveness of the proposed controlstrategies

Data Availability

No data were used to support this study

Disclosure

(is article was presented at the International Conferenceon Research in Applied Mathematics and Computer Sci-ence (ICRAMC) 2020 which took place on July 15ndash182020

Conflicts of Interest

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

References

[1] UNODC ldquoWorld drug reportmdashunited nations office on drugsand crimerdquo 2019 httpswwwunodcorgwdr2019prelaunchWDR19_Booklet_1_EXSUMpdf

400600800

100012001400160018002000

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ ne 0

(a)

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

400600800

1000120014001600180020002200

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ = 0

(b)

Figure 5 (a) (e evolution of the Qt with and without controls with θne 0 (b) (e evolution of the Qt with and without controls with θ 0

800

1000

1200

1400

1600

1800

2000

5 10 15 20 25 30Time

The h

eavy

dru

g us

ers

H+ without controlH+ with control u3

(a)

1000

1200

1400

1600

1800

2000

2200

The h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u3

(b)

Figure 4 (a) (e evolution of the H+ with and without controls (b) (e evolution of the Hminus with and without controls

Discrete Dynamics in Nature and Society 9

5 Conclusion

In this paper we introduced a discrete modeling of drugusers in order to minimize the number of light drug usersheavy drug users heavy drug users-dealers and temporaryquitters of drug consumption We also introduced fourcontrols which respectively represent awareness programsthrough education and media contact prevention throughsecurity campaigns treatment and psychological supportalong with follow-up We applied the results of the controltheory and wemanaged to obtain the characterisations of theoptimal controls (e numerical simulation of the obtainedresults showed the effectiveness of the proposed controlstrategies

Data Availability

No data were used to support this study

Disclosure

(is article was presented at the International Conferenceon Research in Applied Mathematics and Computer Sci-ence (ICRAMC) 2020 which took place on July 15ndash182020

Conflicts of Interest

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

References

[1] UNODC ldquoWorld drug reportmdashunited nations office on drugsand crimerdquo 2019 httpswwwunodcorgwdr2019prelaunchWDR19_Booklet_1_EXSUMpdf

400600800

100012001400160018002000

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ ne 0

(a)

The i

ndiv

idua

ls w

ho te

mpo

raril

y q

uit d

rug

user

s

400600800

1000120014001600180020002200

5 10 15 20 25 30Time

Qt with control u4Qt without controlThe rate θ = 0

(b)

Figure 5 (a) (e evolution of the Qt with and without controls with θne 0 (b) (e evolution of the Qt with and without controls with θ 0

800

1000

1200

1400

1600

1800

2000

5 10 15 20 25 30Time

The h

eavy

dru

g us

ers

H+ without controlH+ with control u3

(a)

1000

1200

1400

1600

1800

2000

2200

The h

eavy

dru

g us

ers-

deal

ers

and

prov

ider

s

5 10 15 20 25 30Time

Hndash without controlHndash with control u3

(b)

Figure 4 (a) (e evolution of the H+ with and without controls (b) (e evolution of the Hminus with and without controls

Discrete Dynamics in Nature and Society 9

[2] WHO ldquoHIV 2016ndash2021mdashworld health organizationrdquo 2016httpsappswhointirisbitstreamhandle10665246178WHO-HIV-201605-engpdf

[3] EMCDDA ldquo(eEuropeanmonitoring centre for drugs and drugaddictionrdquo httpwwwemcddaeuropaeudatastats2019gps

[4] F El Omari and T Jallal ldquo(e mediterranean school surveyproject on alcohol and other drugs in Moroccordquo Addicta 1eTurkish Journal on Addictions vol 2 pp 30ndash39 2015

[5] K Bucher ldquoBernadette mathematically modeling the spreadof methamphetamine userdquo University of Alabama LibrariesTuscaloosa Alabama 2014

[6] F Guerrero F-J Santonja and R-J Villanueva ldquoAnalysingthe Spanish smoke-free legislation of 2006 a new method toquantify its impact using a dynamic modelrdquo InternationalJournal of Drug Policy vol 22 no 4 pp 247ndash251 2011

[7] Z Hu Z Teng and H Jiang ldquoStability analysis in a class ofdiscrete SIRS epidemic modelsrdquo Nonlinear Analysis RealWorld Applications vol 13 no 5 pp 2017ndash2033 2012

[8] A Labzai O Balatif andM Rachik ldquoOptimal control strategyfor a discrete time smoking model with specific saturatedincidence raterdquo Discrete Dynamics in Nature and Societyvol 2018 Article ID 5949303 10 pages 2018

[9] A Lahrouz L Omari D Kiouach and A Belmaati ldquoDe-terministic and stochastic stability of a mathematical model ofsmokingrdquo Statistics amp Probability Letters vol 81 no 8pp 1276ndash1284 2011

[10] 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

[11] J Boscoh H Njagarah and F Nyabadza ldquoModelling the roleof drug barons on the prevalence of drug epidemicsrdquoMathematical Biosciences and Engineering vol 10 no 3pp 843ndash860 2013

[12] O Balatif A Labzai andM Rachik ldquoA discrete mathematicalmodeling and optimal control of the electoral behavior withregard to a political partyrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 9649014 14 pages 2018

[13] V Guibout and A M Bloch ldquoA discrete maximum principlefor solving optimal control problemsrdquo in Proceedings of the43rd IEEE Conference on Decision and Control vol 2pp 1806ndash1811 Nassau Bahamas December 2004

[14] D Wandi R Hendon B Cathey E Lancaster andR Germick ldquoDiscrete time optimal control applied to pestcontrol problemsrdquo Involve a Journal of Mathematics vol 7no 4 pp 479ndash489 2014

[15] S Mushayabasa and G Tapedzesa ldquoModeling illicit drug usedynamics and its optimal control analysisrdquo Computationaland Mathematical Methods in Medicine vol 2015 Article ID383154 11 pages 2015

[16] A Zeb G Zaman and S Momani ldquoSquare-root dynamics ofa giving up smoking modelrdquo Applied Mathematical Model-ling vol 37 no 7 pp 5326ndash5334 2013

[17] L S Pontryagin V G Boltyanskii R V Gamkrelidze andE F Mishchenko 1e Mathematical 1eory of OptimalProcesses Wiley New York NY USA 1962

[18] M D Rafal and W F Stevens ldquoDiscrete dynamic optimi-zation applied to on-line optimal controlrdquo AlChE Journalvol 14 no 1 pp 85ndash91 1968

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

10 Discrete Dynamics in Nature and Society