research article optimal control strategies in an...

Research ArticleOptimal Control Strategies in an Alcoholism Model

Xun-Yang Wang12 Hai-Feng Huo12 Qing-Kai Kong3 and Wei-Xuan Shi2

1 College of Electrical and Information Engineering Lanzhou University of Technology Lanzhou Gansu 730050 China2Department of Applied Mathematics Lanzhou University of Technology Lanzhou Gansu 730050 China3 School of Electrical and Information Engineering Anhui Industrial University Maanshan Anhui 243032 China

Correspondence should be addressed to Hai-Feng Huo huohf1970gmailcom

Received 10 December 2013 Accepted 9 January 2014 Published 2 March 2014

Academic Editor Weiming Wang

Copyright copy 2014 Xun-Yang Wang et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a deterministic SATQ-type mathematical model (including susceptible alcoholism treating and quittingcompartments) for the spread of alcoholismwith two control strategies to gain insights into this increasingly concerned about healthand social phenomenon Some properties of the solutions to themodel including positivity existence and stability are analyzedTheoptimal control strategies are derived by proposing an objective functional and using Pontryaginrsquos Maximum Principle Numericalsimulations are also conducted in the analytic results

1 Introduction

Alcoholism also known as alcohol dependence is a diseasethat includes the desire for alcohol and continuing to drink itdespite its negative effect on individualrsquos health relationshipsand social status [1] Similar to all other drug addictionsalcoholism can be regarded as a treatable disease The WorldHealth Organization estimates that about 140 million peoplethroughout the world suffer from alcohol dependence withrelated problems such as being sick losing a job among ahost of other things [2] Particularly young peoplersquos alco-holism problem is a major concern to public health USsurveys indicate that approximately 90 of college studentshave consumed alcohol at least once [3] and more than40 of college students have engaged in binge drinking [45] Unfortunately the biological mechanisms underpinningalcoholism are not known however risk factors includesocial environment stress mental health genetic sensitivityage ethnic group and sex [6 7] Long-term alcohol abusewill produce negative changes in the brain such as toleranceand physical dependenceThe subtle changesmake it difficultfor the alcoholics to stop drinking and result in alcoholwithdrawal symptoms upon discontinuation of alcohol con-sumption Alcohol damages almost all parts of the body andcontribute to a number of human diseases including butnot limited to liver cirrhosis pancreatitis heart disease and

sexual dysfunction and can eventually be deadly [8] Damageto the central and peripheral nervous systems can take placefrom sustained alcohol consumption [9ndash13]

Although alcoholism is becoming more and more dan-gerous and serious as well as a widespread social phe-nomenon only much less work has been done in the mathe-maticalmodelling of alcoholism as a growing health problemincluding a few studies which offered some mathematicalapproaches to understand the growing burden of alcoholism[10 14ndash19] In [10] a SIR-type model was proposed theauthors used standard contact rate between susceptiblesand alcoholism getting alcoholism reproductive numberand discussing the existence and stability of two equilibriaIn [14] a framework where drinking was modeled as asocially contagious process in low- and high-risk connectedenvironments was introduced they found that high levelsof social interaction between light and moderate drinkers inlow-risk environments can diminish the importance of thedistribution of relative drinking times on the prevalence ofheavy drinking In [15] neurophysiological examinations of100 long-term alcohol dependent patients who were havingneuropsychiatric treatment showed symptoms of polytopicdamage of the peripheral and central nervous system Theresults showed that for recognition of the damage an exten-sive diagnostic programmemust be used In [16] the authorsconsidered a kind of binge drinking model with two equal

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 954069 18 pageshttpdxdoiorg1011552014954069

2 Abstract and Applied Analysis

infectivity drunk states mathematical analyses establishedthat the global dynamics of themodel were determined by thebasic reproduction number In [17] the authors modified themodel from [16] that is they considered different infectivityof two drunk states and a SEIR-type model of alcoholismwas thus presented in which two alcohol related states wereinvolved namely no alcohol dependent consumers 119863(119905)and alcohol dependent consumers 119860(119905) In [18] the authorsformulated a deterministic model for evaluating the impactof heavy alcohol drinking on the reemerging gonorrheaepidemic and both analytical and numerical results wereprovided to ascertain whether heavy alcohol drinking hadan impact on the transmission dynamics of gonorrhea Theapproach of the literature [18] was very meaningful sinceit provided a new direction of thinking when the cross-infection between alcoholism and other pathological diseasesoccurs In recent monograph [19] the authors also proposeda SIR-type model to investigate alcohol abuse phenomenonand generated some useful insights for example the basicreproductive number was not always the key to controllingdrinking within the population For other papers that studythe model of giving up smoking or quitting drinking pleasesee [20 21] and references cited therein

As living standard and health awareness get improvedmore and more people who fall into binge drinking stateare actively seeking the quitting alcoholism measures andtreatment methods [1 11 22] In [22] treatment strategy wasintroduced into a simple SIR-type alcoholics quitting modelin which the authors used the bilinear incidence to depict theldquoinfectionrdquo between the occasional drinkers 119878 and problemdrinkers 119863 Motivated by some aforementioned documents[10 19 22] in this paper we will formulate a more reasonablealcoholics quitting model The fact that our model is morereasonable is embodied from the following three aspects

(1) Taking into account that alcoholism is a widespreadsocial phenomenon so the standard incidence issuperior to bilinear incidence when we portray therelationship between the alcoholism and the suscep-tibles during the course of infection While in [22]the authors adopted bilinear incidence we will adoptstandard incidence in this paper

(2) Since alcohol is harmful to health moreover aswe all know alcoholism is treatable if we can takeapproximate measure in time for example artificialisolation from alcoholisms medications persuasionand education programing on alcoholism So it isnecessary to take effective measures to avoid alcoholor to treat after alcoholism Documents [10 19] havenot considered these aspects

(3) Since there is effective prevention and treatment indescribing the phenomenon of alcoholism there aresome people who will never drink due to successfulprevention or some people who no longer drink aftersuccessful treatment Therefore when we formulatethe model in this paper itrsquos reasonable to introducea new compartment119876 the people in which will neverdrink for ever Obviously themodels of [10 19 22] arenot involving the quitting compartment 119876

Based on the above considerations we will premeditatetwo treating methods namely prevention of susceptiblesfrom alcoholism and treatment on alcoholism as controlvariables hence we will derive a SATQ-type model Wenote the fact that many authors are interested in solvingoptimal control problems such as cost minimization andoptimal control of various disease especially with biologicalbackground and various mathematical models [22ndash24] Inthis paper we will propose an objective functional whichconsiders not only alcohol quitting effects but also the costof controlling alcohol Then we consider a range of issuesrelated to the optimal control with themethod of PontryaginrsquosMaximum Principle including optimal control existenceuniqueness and characterization

The organization of this paper is as follows In thenext section the alcoholism model with prevention for thesusceptibles and treatment for alcoholism is formulated InSection 3 the basic reproduction number and the existence ofequilibria are investigatedThe stability of the disease free andendemic equilibria is proved in Section 4 Optimal controlstrategies by the classic method of PMP (Pontryaginrsquos Maxi-mum Principle) are discussed in Section 5 In Section 6 wegive some numerical simulations We give some discussionsand conclusions in the last section

2 The Model Formulation and SomeFundamental Properties

In this section we introduce a mathematical model withprevention and treatment for the alcoholism and then studysome important properties such as the boundness and posi-tivity of its solutions

21 Model Formulation and Parameter Explanation Thetotal population is partitioned into four compartments thesusceptible compartment 119878 which refers to the personswho never drink or drink moderately without affecting thephysical health the alcoholism compartment 119860 which refersto the persons who binge drink and affect the physical healthseriously the treatment compartment 119879 which refers to thepersons who have been receiving treatments by taking pills orothermedical interventions after alcoholism and the quittingcompartment 119876 which refers to the persons who recoverfrom alcoholism after treatment and stay off alcohol hereafterIn this paper we focus on a closed environment such as acommunity a university or a village So the total number ofpopulation to be considered is a constant we denote it as119873The population flow among those compartments is shown inthe following diagram (Figure 1)

The schematic diagram leads to the following system ofordinary differential equations

1198781015840= 120583119873 minus (1 minus 119906

1)120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

Abstract and Applied Analysis 3

A

S Q

T

120583N

120583S

120585T

120583T

120583A

120583Q

u2A

120575T

(1 minus u1)120573SA

N

Figure 1 Transfer diagram for the dynamics of alcoholism model

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(1)

Here 120583119873 is the birth number of the population 120583 is thenatural death rate of the population 119906

1is the fraction of

the susceptible individuals who successfully avoid to stayoff the alcoholism 119906

2is the fraction of the alcoholics who

take part in treatments here 0 le 119906119894le 1 119894 = 1 2 and

they will be considered as two control variables in Section 5120573 is the transmission coefficient of the ldquoinfectionrdquo for thesusceptible individuals from the alcoholic individuals 120585 is therate coefficient of the person who fail to be treated and returnto the alcoholism compartmentmostly due to their ownweakwill 120575 is the rate coefficient of the person who have receivedeffective treatment and recovered from alcoholism forever

22 Boundedness of Solutions to System and Positively Invari-ant Region It is important to show positivity and bounded-ness for the system (1) as they represent populations Firstlywe present the positivity of the solutions System (1) can beput into the matrix form

1198831015840= 119866 (119883) (2)

where119883 = (119878 119860 119879 119876)119879isin 1198774 and 119866(119883) is given by

119866 (119883) = (

1198661(119883)

1198662(119883)

1198663(119883)

1198664(119883)

)

=(

120583119873 minus (1 minus 1199061)120573119878119860

119873minus 120583119878

(1 minus 1199061)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1199062119860 minus (120583 + 120585 + 120575) 119879

120575119879 minus 120583119876

)

(3)

It is easy to check that

119866119894(119883)

1003816100381610038161003816119883119894(119905)=0119883119905isin119862+ge 0 119894 = 1 2 3 4 (4)

Due to Lemma 2 in [25] any solution of (1) is 119883(119905) isin 1198774+for

all 119905 ge 0We denote 119873(119905) = 119878(119905) + 119860(119905) + 119879(119905) + 119876(119905) summing

equations in (1) yields

119889119873 (119905)

119889119905= 0 (5)

so 119873(119905) = 119878(119905) + 119860(119905) + 119879(119905) + 119876(119905) = constant (denoted as119873) and the set

Ω = (119878 119860 119879 119876) isin 1198774

+ 119878 + 119860 + 119879 + 119876 le 119873 (6)

is a positively invariant region for (1) Therefore we willconsider the global stability of (1) on the setΩ

3 The Basic Reproduction Number andExistence of Alcoholism Equilibria

31 The Basic Reproduction Number 1198770 In epidemiology

the basic reproduction number (sometimes called basicreproductive rate or basic reproductive ratio) of an infectionis the number of infectious cases that one infectious casegenerates on average over the course of its infectious periodWhile in this context it means the number of persons thatan alcoholic will ldquoinfectrdquo during his ldquoinfectiousrdquo period in thepure susceptible environment so that the infected personswillenter the alcoholism compartment It is easy to see that themodel has an alcohol free equilibrium 119864

0= (1198780 0 0 0) =

(119873 0 0 0) In the following the basic reproduction numberof system (1) will be obtained by the next generation matrixmethod formulated in [26]

Let 119909 = (119860 119879 119876 119878)119879 then system (1) can be written as

119889119909

119889119905= F (119909) minusV (119909) (7)

where

F (119909) = (

(1 minus 1199061)120573119878119860

1198730

0

0

)

V (119909) = (

(1199062+ 120583)119860 minus 120585119879

(120583 + 120585 + 120575) 119879 minus 1199062119860

120583119876 minus 120575119879

(1 minus 1199061)120573119878119860

119873+ 120583119878 minus 120583119873

)

(8)

The Jacobian matrices of F(119909) and V(119909) at the alcohol freeequilibrium 119864

0are respectively

119863F (1198640) = (

1198653times3

0

0 0)

119863V (1198640) = (

1198813times3

0

(1 minus 1199061) 120573 0 0 120583

)

(9)

where

119865 = (

(1 minus 1199061) 120573 0 0

0 0 0

0 0 0

) 119881 = (

1199062+ 120583 minus120585 0

minus1199062

120583 + 120585 + 120575 0

0 minus120575 120583

)

(10)

The basic reproduction number denoted by 1198770 is thus given

by

1198770= 120588 (119865119881

minus1) =

120573 (1 minus 1199061) (120583 + 120585 + 120575)

1199062(120583 + 120575) + 120583 (120583 + 120585 + 120575)

(11)


It is easy to see that both of the control parameters con-tributed to reducing the alcoholism From this point thecontrol measures are meaningful

32 Existence of Alcoholism Equilibrium The endemic equi-librium 119864

lowast(119878lowast 119860lowast 119879lowast 119876lowast) of system (1) is determined by

equations

120583119873 minus (1 minus 1199061)120573119878119860

119873minus 120583119878 = 0

(1 minus 1199061)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860 = 0

1199062119860 minus (120583 + 120585 + 120575) 119879 = 0

120575119879 minus 120583119876 = 0

(12)

The third equation in (12) leads to

119860 =120583 + 120585 + 120575

1199062

119879 (13)

From the last equation in (12) we have

119876 =120575119879

120583 (14)

From the first equation of (12) and together with (13) we canget

119878 =120583119873

120583 + (((1 minus 1199061) 120573) 119873)119860

=12058311987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879 (120583 + 120585 + 120575)

(15)

Substituting (13)ndash(15) into the second equation of (12) gives

120583119873 minus120583211987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879 (120583 + 120585 + 120575)

+ 120585119879 minus (120583 + 1199062)120583 + 120585 + 120575

1199062

119879 = 0

(16)

By simplifying (16) we can get

119879 [1199062120585 (1 minus 119906

1) 120573 (120583 + 120585 + 120575)

minus (1 minus 1199061) 120573(120583 + 120585 + 120575)

2

(120583 + 1199062)] 119879 + 120590 = 0

(17)

where120590 = 119906

2120583119873 (1 minus 119906

1) 120573 (120583 + 120585 + 120575)

+ (1199062)2

120583119873120585 minus 1205831198731199062(120583 + 119906

2) (120583 + 120585 + 120575)

(18)

Hence we get two explicit solutions to (17) one is 1198790= 0

which is corresponding to the alcohol free equilibria and theother is119879lowast= (120590) ((1 minus 119906

1) 120573(120583 + 120585 + 120575)

2

(120583 + 1199062)

minus1199062120585 (1 minus 119906

1) 120573 (120583 + 120585 + 120575) )

minus1

=120590

(1 minus 1199061) 120573 (120583 + 120585 + 120575) [120583120585 + (120575 + 120583) (120583 + 119906

2)]

(19)

which should be corresponding to the alcoholism equilibriaon condition that 119879lowast gt 0 otherwise the alcoholismequilibria are nonexistent It is enough to show the positivityof 120590 tomake sure the existence of alcoholism equilibria on thecondition 119877

0ge 1 By some simple calculations we simplify

the expression of 120590 to be

120590 = 1205831198731199062(120583 + 120585 + 120575) (1 minus 119906

1) 120573

minus [(120583 + 1199062) (120575 + 120583) + 120583120585]

(20)

Since 1198770gt 1 is equivalent to

120573 (1 minus 1199061) (120583 + 120585 + 120575) gt 119906

2(120583 + 120575) + 120583 (120583 + 120585 + 120575) (21)

the right side of this inequality is exactly equal to (120583+1199062)(120575+

120583) + 120583120585 Hence we have proved the existence of 119879lowast gt 0so are the alcoholism equilibria We summarize this result inTheorem 1

Theorem 1 For system (1) there is always an alcohol freeequilibrium 119864

0= (119873 0 0 0) When 119877

0gt 1 besides alcohol

free equilibrium 1198640 system (1) also has a unique alcoholism

equilibrium 119864lowast(119878lowast 119860lowast 119879lowast 119876lowast) where

119878lowast=

12058311987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879lowast (120583 + 120585 + 120575)

119860lowast=120583 + 120585 + 120575

1199062

119879lowast

119876lowast=120575119879lowast

120583

119879lowast=

120590

(1 minus 1199061) 120573 (120583 + 120585 + 120575) [120583120585 + (120575 + 120583) (120583 + 119906

2)]

(22)

4 Stability Analysis of Equilibria

For the convenience of subsequent proof we denote a vector119883 = (119860 119879 119876 119878)

119879 and

119891 (119883) =(

(1 minus 1199061)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1199062119860 minus (120583 + 120585 + 120575) 119879

120575119879 minus 120583119876

120583119873 minus (1 minus 1199061)120573119878119860

119873minus 120583119878

) (23)

So the Jacobian matrix of 119891(119909) about vector 119883 is as thefollowing

119869 =

120597119891 (119883)

120597119883

=(

(1 minus 1199061) 120573119878

119873

minus (120583 + 1199062) 120585 0

(1 minus 1199061) 120573119860

119873

1199062 minus (120583 + 120585 + 120575) 0 0

0 120575 minus120583 0

minus

(1 minus 1199061) 120573119878

119873

0 0 minus120583 minus

(1 minus 1199061) 120573119860

119873

)

(24)

Theorem 2 For system (1) the alcohol free equilibrium 1198640is

locally asymptotically stable if 1198770lt 1


Proof Since

119869 (1198640)

= (

(1 minus 1199061) 120573 minus (120583 + 119906

2) 120585 0 0

1199062

minus (120583 + 120585 + 120575) 0 0

0 120575 minus120583 0

minus (1 minus 1199061) 120573 0 0 minus120583

)

(25)

we can easily get that two of the eigenvalues are 1205821= 1205822=

minus120583 lt 0 while 1205823 1205824satisfy

1205822+ [2120583 + 120585 + 120575 + 119906

2minus (1 minus 119906

1) 120573] 120582

+ (120583 + 120585 + 120575) (120583 + 1199062minus (1 minus 119906

1) 120573) minus 119906

2120585 = 0

(26)

Thus

1205823+ 1205824= (1 minus 119906

1) 120573 minus (120583 + 120585 + 120575) minus (120583 + 119906

2) (27)

Since 1198770lt 1 is equivalent to

120573 (1 minus 1199061) (120583 + 120585 + 120575) lt 119906

2(120583 + 120575) + 120583 (120583 + 120585 + 120575)

lt (120583 + 1199062) (120583 + 120585 + 120575)

(28)

so

120573 (1 minus 1199061) lt 120583 + 119906

2 (29)

and then

1205823+ 1205824lt minus (120583 + 120585 + 120575) lt 0 (30)

while

12058231205824= (120583 + 120585 + 120575) (120583 + 119906


1) 120573) minus 119906

2120585 (31)

Similarly from 1198770lt 1 we can derive the inequality

minus120573 (1 minus 1199061) (120583 + 120585 + 120575) gt minus119906

2(120583 + 120575) minus 120583 (120583 + 120585 + 120575) (32)

so12058231205824gt (120583 + 119906

2) (120583 + 120585 + 120575)

minus 1199062120585 minus 1199062(120583 + 120575) minus 120583 (120583 + 120585 + 120575)

(33)

It reduces to

12058231205824gt 0 (34)

Hence Re 1205823lt 0 Re 120582

4lt 0 The proof is complete

Next we will turn to investigate the global stability of 1198640


globally asymptotically stable if 1198770lt 1

Proof Consider the subsystem of (1) as follows

1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(35)

Equation (35) can be rewritten as

(

) = (119865 minus 119881)(

119860

119879

119876

) minus (1 minus119878

119873)

times (

120573 (1 minus 1199061) 0 0

0 0 0

0 0 0

)(

119860

119879

119876

)

(36)

Since 119878 le 119873 and 0 le 1199061le 1 then for all 119905 gt 0 we can get

(

) le (119865 minus 119881)(

119860

119879

119876

) (37)

According to Lemma 1 in [26] all the eigenvalues of matrix119865 minus 119881 have negative real parts so the solutions of this sub-system are stable whenever 119877

0lt 1 So (119860(119905) 119879(119905) 119876(119905)) rarr

(0 0 0) as 119905 rarr infin By the comparison theorem [27] andbased on the fact that the total population is constant 119873 itfollows that (119860(119905) 119879(119905) 119876(119905)) rarr (0 0 0) and 119878(119905) rarr 119873

as 119905 rarr infin So the alcohol free equilibrium 1198640is globally

asymptotically stable the proof is complete

Theorem 4 For system (1) the alcoholism equilibrium119864lowast(119878lowast 119860lowast 119879lowast 119876lowast) is globally asymptotically stable if 119877

0gt 1

Proof Since the total population in model (1) is a constantnumber119873 in order to prove the global stability of system (1)it is sufficed to prove the corresponding stability of subsystem(38)

1198781015840= 120583119873 minus (1 minus 119906

1)120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

(38)

We make normalization transform and still use the samesymbols 119878 119860 119879 to denote the variables then (38) can betransformed into

1198781015840= 120583 minus (1 minus 119906

1) 120573119878119860 minus 120583119878

1199041198601015840= (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

(39)

From (39) we can easily know that the equilibria (119878lowast 119860lowast 119879lowast)satisfy the following three equalities to be used later

120583 = (1 minus 1199061) 120573119878lowast119860lowastminus 120583119878lowast

(1 minus 1199061) 120573119878lowast119860lowast+ 120585119879lowast= (1199062+ 120583)119860

lowast

1199062119860lowast= (120583 + 120585 + 120575) 119879

lowast

(40)


Let 119881 = 1199091(119878 minus 119878

lowastminus 119878lowast ln(119878119878lowast)) + 119909

2(119860 minus 119860

lowastminus

119860lowast ln(119860119860lowast)) + 119909

3(119879 minus 119879

lowastminus 119879lowast ln(119879119879lowast)) then

119881101584010038161003816100381610038161003816(39)

= 1199091[120583 minus (1 minus 119906

1) 120573119878119860 minus 120583119878

minus119878lowast

119878120583 + (1 minus 119906

1) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860

lowastminus119860lowast

119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091[ (1 minus 119906

1) 120573119878lowast119860lowast+ 120583119878lowastminus (1 minus 119906

1) 120573119878119860 minus 120583119878

minus119878lowast

119878((1 minus 119906

1) 120573119878lowast119860lowast+ 120583119878lowast)

+ (1 minus 1199061) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860


119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091120583119878lowast(2 minus

119878

119878lowastminus119878lowast

119878)

+ [1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast]

minus [1199091

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878+ 1199092(1 minus 119906

1) 120573119878119860

lowast

+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

+ 119878119860 [minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573]

+ 119860 [minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093]

+ 119879 [1205851199092minus (120583 + 120585 + 120575) 119909

3]

(41)

To eliminate the cross-term 119878119860 and two single-variable terms119860 and 119879 we let

minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573 = 0

minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093= 0

1205851199092minus (120583 + 120585 + 120575) 119909

3= 0

(42)

By solving them we can get

1199091= 1 119909

2= 1

1199093=

120585

120583 + 120585 + 120575=1199062+ 120583 minus (1 minus 119906

1) 120573119878lowast

1199062

(43)

Next we let

1198811015840

1= 1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast

1198811015840

2= minus[119909

1

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ 1199092(1 minus 119906

1) 120573119878119860

lowast+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

(44)

and then

1198811015840= 1199091120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2 (45)

Due to

(1199062+ 120583)119860

lowast= (1 minus 119906

1) 120573119878lowast119860lowast+ 120585119879lowast

1199093(120583 + 120585 + 120575) 119879

lowast= 120585119879lowast1199092= 120585119879lowast

(46)


so

1198811015840

1= 2 (1 minus 119906

1) 120573119878lowast119860lowast+ 2120585119879

lowast

1198811015840

2= minus[

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ (1 minus 1199061) 120573119878119860

lowast+119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

le minus2[(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878sdot (1 minus 119906

1) 120573119878119860

lowast]

12

minus 2 [119860lowast120585119879

119860sdot 1199093

119879lowast

1198791199062119860]

= minus2 (1 minus 1199061) 120573119878lowast119860lowastminus 2(1199093119860lowast1205851199062119879lowast)12

= minus2 (1 minus 1199061) 120573119878lowast119860lowastminus 2[120585119879

lowast(11990931199062119860lowast)]12

= minus2 (1 minus 1199061) 120573119878lowast119860lowast

minus 2120585119879lowast[(1199062+ 120583)119860

lowastminus (1 minus 119906

1) 120573119878lowast119860lowast]12

= minus2 (1 minus 1199061) 120573119878lowast119860lowastminus 2(120585119879

lowast120585119879lowast)12

= minus2 (1 minus 1199061) 120573119878lowast119860lowastminus 2120585119879

lowast

(47)

Hence

1198811015840= 120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2

le 120583119878lowast(2 minus

119878


119878) + 2 (1 minus 119906

1) 120573119878lowast119860lowast

+ 2120585119879lowastminus 2 (1 minus 119906

1) 120573119878lowast119860lowastminus 2120585119879

lowast

= 120583119878lowast(2 minus

119878


119878) le 0

(48)

1198811015840= 0 if and only if 119878 = 119878

lowast 119860 = 119860lowast 119879 = 119879

lowast Accordingto LaSallersquos invariance principle [28] we can derive theconclusion that the alcoholism equilibria 119864lowast(119878lowast 119860lowast 119879lowast 119876lowast)are globally asymptotically stable the proof is complete

5 Optimal Control Problem

51 The Existence of Optimal Control In order to investigatean effective campaign to control alcoholism in a communitywhich pursue the goals of the minimized alcoholisms andmore recovered individuals we will reconsider the system (1)and use two control variables to reduce the numbers of alco-holics The difference is that we will change the parameters1199061 1199062into control variable 119906

1(119905) 1199062(119905) Their aforementioned

definitions allow us to do so 1199061(119905) is used to limit the

proportion of the susceptible individual to contact withalcoholism usually by propaganda and education so that thesusceptible individual can stay off alcoholism consciously andbe free of ldquoinfectionrdquo we can understand the effect of 119906

1(119905)

is to prevent the the susceptible from contacting with thealcoholism The control variable 119906

2(119905) is used to control the

alcoholism to take appropriate treatment measures such astaking pills or seeking other medical help However just as acoin has two sides there will be a lot of costs generated duringthe control process So it is advisable to balance between thecosts and the alcohol effects In view of this our optimalcontrol problem tominimize the objective functional is givenby

119869 (1199061 1199062) = int

119905119891

0

[119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)] 119889119905 (49)

which subjects to system

1198781015840= 120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860

1198791015840= 1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(50)

with initial conditions

119878 (0) = 1198780 119860 (0) = 119860

0

119879 (0) = 1198790 119876 (0) = 119876

0

(51)

Here 119906119894(119905) isin 119880 ≜ (119906

1 1199062) | 119906119894(119905) is measurable and 0 le

119906119894(119905) le 1 for all 119905 isin [0 119905

119891] 119905119891is the end time to be

controlled 119880 is an admissible control set 119888119894 and 119894 = 1 2 are

weight factors (positive constants) that adjust the intensity oftwo different control measures

Next we will investigate the existence of the optimalcontrol of the above-mentioned problem

Theorem 5 There exists an optimal control pair 119906lowast

=

(119906lowast

1 119906lowast

2) isin 119880 such that

119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) 119906

1(119905) 1199062(119905) isin 119880 (52)

subjects to the control system (1) with initial conditions (50)

Proof Toprove the existence of an optimal control accordingto the classic literature [29] we have to show the following

(1) The control and state variables are nonnegative values(2) The control set 119880 is convex and closed(3) The right side of the state system is bounded by linear

function in the state and control variables(4) The integrand of the objective functional is concave

on 119880(5) There exist constants 119889

1 1198892gt 0 and 120572 gt 1 such

that the integrand 119871(119905 1199061 1199062) ≜ 119860(119905) + (119888

12)1199062

1(119905) +

(11988822)1199062

2(119905) of the objective functional satisfies

119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(53)


statements (1) (2) and (3) are obvious satisfied we only needto test and verify the latter two ones Since the four statevariables have been all proved to be up bounded by 119873 wewill get the following equalities

1198781015840le 120583119873 119860

1015840le (1 minus 119906

1(119905)) 120573119878 + 120585119879

1198791015840le 1199062(119905) 119860 119876

1015840le 120575119879

(54)

so the fourth condition is set up As for the last condition

119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(55)

is also true when we choose 1198891= min119888

12 11988822 and for all

1198892isin 119877+ 120572 = 2 The proof is complete

We next come to the core of this section

52 The Characterization of the Optimal Control With theexistence of the optimal control pairs established we nowpresent the optimality system and use a result from [30]we can easily know the existence of the solutions to theoptimality system (71) which will be gotten later Firstly wecome to discuss the theorem that relates to the character-ization of the optimal control The optimality system canbe used to compute candidates for optimal control pairsTo do this we begin by defining an augmented Hamilto-nian 119867 with penalty terms for the control constraints asfollows

119867 = 119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)

+ 1205821[120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878]

+ 1205822[(1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860]

+ 1205823[1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879] + 120582

4(120575119879 minus 120583119876)

minus 119908111199061(119905) minus 119908

12(1 minus 119906

1(119905))

minus 119908211199062(119905) minus 119908

22(1 minus 119906

2(119905))

(56)

where 119908119894119895(119905) ge 0 are the penalty multipliers satisfying

11990811(119905) 1199061(119905) = 119908

12(119905) (1 minus 119906

1(119905))

= 0 at optimal control 119906lowast1

11990821(119905) 1199062(119905) = 119908

22(119905) (1 minus 119906

2(119905))


(57)

Theorem6 Given optimal control pairs (119906lowast1 119906lowast

2) and solutions

119878(119905) 119860(119905) 119879(119905) 119876(119905) of the corresponding state system (50)there exist adjoint variables 120582

119894 119894 = 1 2 3 4 satisfying

1205821015840

1= 1205821(1 minus 119906

1(119905))

120573119860

119873+ 1205831205821minus 1205822(1 minus 119906

1(119905))

120573119860

119873

1205821015840

2= minus1 + 120582

1(1 minus 119906

1(119905))

120573119878

119873minus 1205822(1 minus 119906

1(119905))

120573119878

119873

+ 1205822(120583 + 119906

2(119905)) minus 120582

31199062(119905)

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

(58)

with the terminal conditions

120582119894(119905119891) = 0 119894 = 1 2 3 4 (59)

Furthermore (119906lowast1 119906lowast

2) are represented by

119906lowast

1= min(1max(0

120573119878119860 (1205822minus 1205821)

1198881119873

))

119906lowast

2= min(1max(0

119860 (1205822minus 1205823)

1198882

))

(60)

Proof According to Pontryagin Maximum Principle [29ndash31] we first differentiate the Hamiltonian operator 119867 withrespect to states Then the adjoint system can be written as

1205821015840

1= minus

120597119867

120597119878 120582

1015840

2= minus

120597119867

120597119860

1205821015840

3= minus

120597119867

120597119879 120582

1015840

4= minus

120597119867

120597119876

(61)

The terminal condition (56) of adjoint equations is given by120582119894(119905119891) = 0 119894 = 1 2 3 4

To obtain the necessary conditions of optimality (59) wealso differentiate theHamiltonian operator119867 with respect to119880 = (119906

1 1199062) and set them equal to zero then

120597119867

1205971199061

= 11988811199061(119905) + 120582

1

120573119878119860

119873minus 1205822

120573119878119860

119873minus 11990811+ 11990812= 0

120597119867

1205971199062

= 11988821199062(119905) minus 120582

2119860 + 120582

3119860 minus 119908

21+ 11990822= 0

(62)

By solving the optimal control we obtain

119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811minus 11990812] (63)

To determine an explicit expression for the optimalcontrol without119908

11and119908

12 a standard optimality technique

is utilized [29] We consider the following three cases

(i) On the set 119905 | 0 lt 119906lowast

1(119905) lt 1 we have 119908

11(119905) =


11990812(119905) = 0 Hence the optimal control is

119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873] (64)

(ii) On the set 119905 | 119906lowast1(119905) = 1 we have 119908

11(119905) = 0 Hence

1 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873minus 11990812] (65)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873ge 1 since 119908

12(119905) ge 0 (66)

(iii) On the set 119905 | 119906lowast1(119905) = 0 we have 119908

12(119905) = 0 Hence

0 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811] (67)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873le 0 since 119908

11(119905) ge 0 (68)

Combining these results the optimal control 119906lowast1(119905) is

characterized as

119906lowast

1= min1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (69)

Using the similar arguments we can also obtain the otheroptimal control function

119906lowast

2= min1max0

(1205822minus 1205823) 119860

1198882

(70)

The proof is complete

We point out that the optimality system consistsof the state system (50) with the initial conditions119878(0) 119860(0) 119879(0) 119876(0) the adjoint (or costate) system(58) with the terminal conditions (59) and the optimalitycondition (60) Any optimal control pairs must satisfythis optimality system For the convenience of subsequent

numerical simulation in Section 6 we give the optimalitysystem as follows

1198781015840= 120583119873 minus (1 minusmin1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119878119860

119873minus 120583119878

1198601015840= (1minusmin1max0 1

1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

120573119878119860

119873

+ 120585119879 minus (min1max0(1205822minus 1205823) 119860

1198882

(119905) + 120583)119860

1198791015840= min1max0

(1205822minus 1205823) 119860

1198882

(119905) 119860

minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

1205821015840

1= 1205821(1 minusmin1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119860

119873+ 1205831205821

minus 1205822(1minusmin1max0 1

1198881

[(1205822minus1205821)120573119878119860

119873](119905))

times120573119860

119873

1205821015840

2= minus1

+ 1205821(1minusmin1max0 1

1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

times120573119878

119873minus 1205823sdotmin1max0

(1205822minus 1205823) 119860

1198882

(119905)

minus 1205822(1 minusmin1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873])

times120573119878

119873+ 1205822(120583+min1max0

(1205822minus 1205823) 119860

1198882

(119905))

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

119878 (0) = 1198780 119860 (0) = 119860

0 119879 (0) = 119879

0

119876 (0) = 1198760 120582

119894(119905119891) = 0 119894 = 1 2 3 4

(71)

53 The Uniqueness of Optimal Control Due to the a prioriboundedness of the state adjoint functions and the resultingLipschitz structure of the ODEs we can obtain the unique-ness of the optimal control


Lemma 7 (see [23]) The function 119906lowast(119904) = min (119887max (119904 119886))is Lipschitz continuous in 119904 where 119886 lt 119887 are some fixed positiveconstants

Theorem 8 For all 119905 isin [0 119905119891] the solution to the optimality

system (71) is unique

Proof Suppose (119878 119860 119879 119876 1205821 1205822 1205823 1205824) and

(119878 119860 119879 119876 1205821 1205822 1205823 1205824) are two different solutions of

our optimality system (71) Let

119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

(72)

where 120582 gt 0 is to be chosenAccordingly we have

119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

(73)

Now we substitute 119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902

1205821= 119890minus120582119905119903 1205822= 119890minus120582119905119904 1205823= 119890minus120582119905119908 1205824= 119890minus120582119905V and 119878 =

119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902 1205821= 119890minus120582119905119903 1205822=

119890minus120582119905119904 1205823= 119890minus120582119905119908 1205824= 119890minus120582119905V into the first ODE of (71)

respectively then we can obtain

+ 120582119898 = 120583119873119890minus120582119905

minus (1 minusmin1max0120573119898119899 (119904 minus 119903) 119890

120582119905

1198881119873

)

times120573119898119899119890

120582119905

119873minus 120583119898

(74)

for119898 and119898 respectively Similarly we can derive

119899 + 120582119899

= (1 minusmin1max1120573119898119899 (119904 minus 119903) 119890

120582119905

1198881119873

)120573119898119899119890

120582119905

119873

+ 120585119901 minus (min1max0 119899 (119904 minus 119908)1198882

+ 120583) 119899

(75)

for 119899 and 119899 respectively

+ 120582119901 = min1max0 119899 (119904 minus 119908)1198882

119899

minus (120583 + 120585 + 120575) 119901

(76)


119902 + 120582119902 = 120575119901 minus 120583119902 (77)


119903 minus 120582119903 = 119903(1 minusmin1max0120573119898119899 (119904 minus 119903) 119890

120582119905

1198881119873

)120573119899119890120582119905

119873

+ 120583119903minus119904(1minusmin1max0120573119898119899 (119904 minus 119903) 119890

120582119905

1198881119873

)

times120573119899119890120582119905

119873

(78)


119904 minus 120582119904 = minus119890120582119905

+ 119903(1 minusmin1max0120573119898119899 (119904 minus 119903) 119890

120582119905

1198881119873

)120573119898119890120582119905

119873

minus 119904(1 minusmin1max0120573119898119899 (119904 minus 119903) 119890

120582119905

1198881119873

)

times120573119898119890120582119905

119873+ 119904(min1max0 119899 (119904 minus 119908)

1198882

+ 120583)

minus 119908(min1max0 119899 (119904 minus 119908)1198882

)

(79)


minus 120582119908 = minus119904120585 + (120583 + 120585 + 120575)119908 minus 120575V (80)


V minus 120582V = 120583V (81)

for V and V respectively


By Lemma 7 we can obtain

1003816100381610038161003816119906lowast

1(119905) minus 119906

lowast

1(119905)1003816100381610038161003816 le

120573119890120582119905

1198881119873|119898119899 (119904 minus 119903) minus 119898119899 (119904 minus 119903)|

1003816100381610038161003816119906lowast

2(119905) minus 119906

lowast

2(119905)1003816100381610038161003816 le

1

1198882

|119899 (119904 minus 119908) minus 119899 (119904 minus 119908)|

(82)

The equations for 119898 119899 119901 119902 119903 119904 119908 V and the equationsfor 119898 119899 119901 119902 119903 119904 119908 V are subtracted respectively then wemultiply each equation by appropriate difference of functionsand integrate from 0 to 119905

119891 Next we add all eight integral

equations and some inequality techniques to obtain unique-ness The following calculation is similar for the sake ofsimplicity we only take119898 and119898 for an example

minus + (120583 + 120582) (119898 minus 119898)

=120573119890120582119905

119873[minus(1 minusmin1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

)119898119899

+(1 minusmin1max0120573119898119899 (119904 minus 119903) 119890

120582119905

1198881119873

)119898119899]

(83)

Multiplying both sides of (83) by (119898minus119898) and integratingfrom 0 to 119905

119891gives

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873

times [minus (1 minus 119906lowast

1)119898119899 + (1 minus 119906

lowast

1)119898119899] 119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873[(119906lowast

1minus 1) (119898119899 minus 119898119899 + 119898119899 minus 119898119899)

+ 119898119899 (119906lowast

1minus 119906lowast

1)] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [119906lowast

1minus 1

100381610038161003816100381610038161003816100381610038161003816

(119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

100381610038161003816100381610038161003816100381610038161003816

times 119898119899 (119904 minus 119903) minus 119898119899 (119904 minus 119903) ] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 (119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

|119898119899 (119904 minus 119904) + (119898119899 minus 119898119899) 119904

minus (119898119899 (119903 minus 119903) + (119898119899 minus 119898119899) 119903)| ] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 |119898| |119898 minus 119898| |119899 minus 119899|

+ |119898 minus 119898|2

|119899| + |119861| |119904| |119899| |119898 minus 119898|2

+ |119861| |119898119899| |119898 minus 119898| |119903 minus 119903|

+ |119861| |119898119899| |119898 minus 119898| |119904 minus 119904|

+ |119861| |119904| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119899| |119898 minus 119898|2] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[

1003816100381610038161003816119906lowastminus 1

1003816100381610038161003816

2|119898| ((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119904 minus 119904)

2)

+|119861| |119904| |119898|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119904| |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119903 minus 119903)

2)

+|119861| |119898| |119903|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119903| |119899| (119898 minus 119898)2]119889119905

(84)

In the above derivation we use many scaling techniquesfor inequality or absolute inequality Particularly what shouldbe noted is that to get the first inequality of above derivationwe use the estimation of |119906lowast

1(119905) minus 119906

lowast

1(119905)| which has been given

before besides for the sake of convenience we note 119861 =

119898119899(120573119890120582119905119891119873) Furthermore we notice that the coefficients of

all the eight terms in the last formula (119898 minus 119898)2+ (119899 minus 119899)

2(119898 minus119898)

2 (119898 minus119898)2+ (119904 minus 119904)

2 (119898 minus119898)2+ (119899 minus 119899)

2 (119898 minus119898)2

(119898minus119898)2+(119903minus119903)

2 (119898minus119898)2+(119899minus119899)2 (119898minus119898)2 namely (|119906lowast1minus

1|2)|119898| |119899| (|119861||119898119899|)2 (|119861||119904||119898|)2 |119861||119904|119899 (|119861||119898119899|)2(|119861||119898||119903|)2 |119861||119903||119899| are nonnegative and bounded So thereexists a positive constant 119888

2such that

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905


le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)

Combining eight of these inequalities gives

1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)

Thus from the above inequality we can conclude that

(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)

where 119861 depends on the coefficients and the bounds dependon119898 119899 119901 119902 119903 119904 119908 V If we choose 120582 such that 120583+120582 gt 119861 then119898 = 119898 119899 = 119899 119901 = 119901 119902 = 119902 119903 = 119903 119904 = 119904 119908 = 119908 and V = VHence the solution to the optimality system is unique Theproof is complete

6 Numerical Simulation

61 The Simulation of State System (1) without ControlParameters For the sake of simplicity but without loss ofgenerality we will perform the numerical simulation of statesystem (1) with parameters 119906

1= 0 119906

2= 0 Before

illustrating the analytic properties of the alcoholism model(1) we will target the populations in the environment ofa community or a university for example the school ofmaterial science and engineering in our university that isLan zhou University of Technology (LUT for short) owing tothe accurate and available information we can obtain Refer-ring to the information provided by the admissions officeof LUT this school will enroll almost 1200 undergraduatesand almost 300 various postgraduates at the beginning of

fall semester at the same time there will be almost 1500various students graduated and left this school so the scaleof students in school remained almost 6000 we can take thetotal population 119873 = 6000 In this simulation we will takeSeptember as the initial time and units in one week periodin one year According to the investigations of the studentunion implemented in September every year we can takeinitial values as 119878(0) = 4500 119860(0) = 1000 119879(0) = 300and 119876(0) = 200 It seems that the alcoholism is a little bitmore but it is rather natural because many freshmen feelconfused when they are faced with the new environment anda new lifestyle many of them have no better choice but gathertogether to drink in small groups to mediate the anxiety andget to know each other over time some of them developthe habit of drinking To a certain extent for example thefrequent drinking badly affects their study we can classifythem into the alcoholism compartment Other initial valuesseem more reasonable so we need no more explanation Aswe know alcoholism death is seldom happen within oneyear so we omit mortality from alcoholism then how tounderstand the recruitment rate as well as natural death rate120583 We can treat the freshmen admission as the recruitmentpopulation and graduation students as the natural ldquodeathrdquoparts So we can take 120583 = 15006000 = 025 which is exactlyconsistent with the value in [16] As for the infection rate 120573and recovery rate 120575 we will let them be variables since thedrinking behaviors are related to many factors such as theseason and the pressure

According to the data we get from the student unionwe choose 120585 = 04 To summarize we list the values ofthe parameters in Table 1 Using the values of parameters inTable 1 we can plot Figures 2 and 3 which are on condition1198770lt 1 and 119877

0ge 1 respectively From Figure 2 we easily

know when 1198770lt 1 holds the solution of system (1) tends to

the alcohol free equilibrium1198640and verifies the global stability

of 1198640 While seen from Figure 3 we also know that if 119877

0ge

1 holds the solution of system (1) tends to the alcoholismequilibrium 119864

lowast and verifies the global stability of 119864lowast

62 The Sensitivity of 1198770about Two Control Parameters

Although from the expression of the model reproductionnumber 119877

0 we can easily find out the fact that the two

control variables that is 1199061and 119906

2 attribute to reducing the

severity of alcoholism we will still depict the graph between1198770and the two control variables to see more intuitiveness see

Figure 4 It seems from the figure that 1198770is a monotonically

decreasing function about two control parameters so it isadvisable to take two approaches simultaneously to controlthe alcoholics

63 The Simulation of Optimality System (71) In this subsec-tion we will investigate numerically the optimal solution tooptimality system (71) by numerical method from [32] theoptimality system is solved with a fourth-order Runge-Kuttascheme Beginning with a guess for the control variables thestate system is solved forward in time and then those values ofstate are used to solve the adjoint equations backward in timeThe controls are updated at the end of each iteration using


0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)

Figure 2 When 1198770lt 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable (120573 = 02 120575 = 03 and 1198770= 071698)

S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

Figure 3 When 1198770

ge 1 the alcoholism equilibrium 119864lowast

=

(4466 476 32 26) corresponding to the given parameters is globallyasymptotically stable (120573 = 03 120575 = 02 and 119877

0= 108511)

the values of optimal controls obtained lastly The iterationscontinue until convergence takes place

In the simulations we choose the available variable valuesas Table 1 shows besides 120573 = 03 120575 = 02 The initial valueof model (1) is assumed to be 119878(0) = 4500 119860(0) = 1000119879(0) = 300 and 119876(0) = 200 as before

The ideal weights in objective functional are very difficultto obtain in reality it needs much work on data mining andfittingHence the acquisition of appropriate practical weights

Table 1 The parameters description of model (1)

Parameter Description Values120583 Natural birth rate or death rate 025

120573Transmission coefficient betweenalcoholism and susceptibles Variables

119873 The total populations to be considered 6000

120575The rate of populations quitting fromalcoholism permanently after treatment Variables

120585The rate of populations failed intreatment and returned to be alcoholic 04

0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0

Figure 4 The relationship between 1198770and two control variables 119906

1

and 1199062

is still a difficult problem and remains for further investiga-tionsThe cost associated with119860(119905) and 119906

1(119905)mainly includes

the cost of dangerous behaviours during the alcoholism timeand educating the public while the cost associated with1199062(119905)mainly comes from health professional and the medical

resource includingmedicines andnursing care In viewof thisand taking the expressions of 119906

1 1199062into account after many

numerical simulations we finally give weighting coefficientsas 1198881= 102 1198882= 104 It should be pointed out that the

weights here are of only theoretical interest to reveal thecontrol strategies proposed in this paper Another point tonote is that themaximumcontrol is very difficult to achieve inreality so we will omit the situation of the maximum controlduring the series of simulations

Next we will make some necessary instructions andexplanations to the above simulation graphs Figures 5 67 and 8 depict the number of four compartments underdifferent control levels when we choose the weight coeffi-cients in objective function to be 119888

1= 102 1198882= 104 From

the four simulation graphs we can observe the followingsimple facts in reducing the total number of alcoholismsand increasing the number of susceptibles the effectivenessof various control measures is as follows optimal control isevidently better than middle control and middle control isbetter than single control 119906

2 single control 119906

2is better than

single control 1199061 while single control 119906

1is much better than

no controlFigures 9 and 10 depict the optimal control law of 119906

1 1199062

respectively In the beginning of the simulation the control


0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)

Constant control measure u1 = 0 u2 = 0

Single control measure u1 = 05 u2 = 0


Middle control measure u1 = u2 = 05

Optimal control

Figure 5Number of the susceptibles whenwe choose theweights inobjective function are 119888

1= 102 1198882= 104 and under different optimal

control strategies that is (1) without control 1199061= 1199062= 0 (2) single

control 1199061= 05 119906

2= 0 (3) single control 119906

1= 0 119906

2= 05 (4)

middle control 1199061= 1199062= 05 (5) optimal control 119906

1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)

Figure 6 Number of the alcoholics when we choose the weights inobjective function are 119888



control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control

Figure 7 Number of the persons in treatment when we choose theweights in objective function are 119888

1= 10

2 1198882= 10

4 and underdifferent optimal control strategies that is (1) without control 119906

1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906

1= 1199062= 05 (5) optimal control

1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)

Figure 8 Number of people in quitting compartment when wechoose the weights in objective function are 119888

1= 102 1198882= 104 and

under different optimal control strategies that is (1) without control1199061= 1199062= 0 (2) single control 119906

1= 05 119906

2= 0 (3) single control

1199061= 0 119906

2= 05 (4) middle control 119906

1= 1199062= 05 (5) optimal

control 1199061= 119906lowast

1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

Optimal control of u1

Figure 9 Figures of the optimal control 1199061when we choose the

weight in objective function are 1198881= 102 1198882= 104

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)




effort of 1199061should be decreased from 065 to 05 within the

first month and over the next week it should be increased tothe maximum control until 50 weeks then rapidly decreasedto 0 at the end of the simulation As for the control 119906

2 it

should start from around 05 due to the initial alcoholics thenincrease to 055 within one week since the rapid infection andnext decrease to almost 0 since the effectiveness of treatmentin three weeks but with the infection going on the controleffort of 119906

2should gradually increase to the maximum and

maintain this level until the tenth week for the purpose ofconsolidation therapy and preventing rebound hereafter itshould be gradually decreased to the level of almost 043 untilthe fifty weeks then quickly decreased to 0 in the end

0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control

Figure 11 Number of the susceptibles when we choose the weightsin objective function are 119888

1= 104 1198882= 102 and under different

optimal control strategies that is (1) without control 1199061= 1199062= 0

(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

In order to investigate the influence of different weightcoefficients in the objective functional on the effect ofcontrolling at the same time for a better comparison we willchange the weight coefficients in objective function into 119888

2=

104 1198881= 102 and we will list the corresponding numerical

simulation results as Figures 11ndash16 showWhen we change the weight coefficients in objective

function into 1198881= 104 1198882= 102 we find that the results

of simulations derived from the graphs are very similar tothe ones before We speculate that the most likely reasons ofthis result are due to three respects one is that the weightcoefficients are not too sensitive in the numerical simulationand another possible reason is that both of the two controlsare important in some sense equivalently importantThe lastbut not themost unlikely reason is that we have not found themost appropriate weight coefficients in the simulation whichis very difficult to find as previously mentioned

7 Conclusions

In this paper we formulate an alcoholics quitting modeland firstly investigate the variation discipline of variouspopulations from the perspective of global stability thenwe propose an objective functional to examine two differentcontrolmeasures (ie prevention and treatment) on the effectof alcohol The basic reproduction number of the model wasderived and the global stability of the two equilibria is givenFrom the expression of the basic reproductionnumber119877

0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control

Figure 13 Number of the persons in treatment when we choosethe weights in objective function are 119888

1= 104 1198882= 102 and under

different optimal control strategies that is (1) without control 1199061=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




related numerical simulation we can easily see that the twocontrol strategies are effective in the alcoholics process

Using Pontryaginrsquos Maximum Principle we firstly deter-mine the necessary conditions for existence of optimalcontrol pairsThe uniqueness of the solution to the optimalitysystem (71) is derived by the classical method of contradic-tion Numerical simulations of the model suggest that thetwo different groups of weights in the objective function have


0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)




much similar effects on the transmission of the alcoholismfrom this point the two control measures are almost equallyimportant in controlling the alcoholism although they willprobably have great influences on the cost of the objectivefunction From the simulation figures it seems that the effectof optimal control which is measured by the reduction inthe number of alcoholics and the increase in the number ofsusceptibles is much better than other control strategies asnoted earlier in the simulation section According to the real-time curve of two optimal controls we point out the specificimplementation methods of optimal control which can beachieved in practice

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the NSF ofGansu Province of China (2013GS09485) the NNSF ofChina (10961018) the NSF of Gansu Province of China(1107RJZA088) the NSF for Distinguished Young Scholarsof Gansu Province of China (1111RJDA003) the Special Fundfor the Basic Requirements in the Research of University ofGansu Province of China and the Development Programfor Hong Liu Distinguished Young Scholars in LanzhouUniversity of Technology

References

[1] R Room T Babor and J Rehm ldquoAlcohol and public healthrdquoThe Lancet vol 365 no 9458 pp 519ndash530 2005

[2] J B Saunders O G Aasland A Amundsen and M GrantldquoAlcohol consumption and related problems among primary

health care patients WHO collaborative project on early detec-tion of personswith harmful alcohol consumption IrdquoAddictionvol 88 no 3 pp 349ndash362 1993

[3] L D Johnston P M OMalley and J G Bachman NationalSurvey Results on Drug Use From the Monitoring the FutureStudy 1975ndash1992 National Institute on Drug Abuse RockvilleMd USA 1993

[4] H Wechsler J E Lee M Kuo and H Lee ldquoCollege bingedrinking in the 1990s a continuing problem Results of theHarvard School of Public Health 1999 College Alcohol StudyrdquoJournal of American College Health vol 48 no 5 pp 199ndash2102000

[5] P M OrsquoMalley and L D Johnston ldquoEpidemiology of alcoholand other drug use among American college studentsrdquo Journalof Studies on Alcohol vol 63 no 14 pp 23ndash39 2002

[6] K Agarwal-Kozlowski and D P Agarwal ldquoGenetic predisposi-tion to alcoholismrdquo Therapeutische Umschau vol 57 no 4 pp179ndash184 2000

[7] C Y Chen C L Storr and J C Anthony ldquoEarly-onset drug useand risk for drug dependence problemsrdquo Addictive Behaviorsvol 34 no 3 pp 319ndash322 2009

[8] M Glavas and J Weinberg ldquoStress alcohol consumption andthe hypothalamic-pituitary adrenal axisrdquo in Nutrients Stressand Medical Disorders pp 165ndash183 Humana Press New YorkNY USA 2006

[9] L N Blum N H Nielsen J A Riggs and L B BresolinldquoAlcoholism and alcohol abuse among women report of theCouncil on Scientific Affairsrdquo Journal of Womenrsquos Health vol7 no 7 pp 861ndash871 1998

[10] B Benedict ldquoModeling alcoholism as a contagious disease howldquoinfectedrdquo drinking buddies spread problem drinkingrdquo SIAMNews vol 40 no 3 2007

[11] H Walter K Gutierrez K Ramskogler I Hertling A Dvorakand O M Lesch ldquoGender-specific differences in alcoholismImplications for treatmentrdquo Archives of Womenrsquos Mental Healthvol 6 no 4 pp 253ndash258 2003

[12] D Muller R D Koch H von Specht w Volker and E MMunch ldquoNeurophisiologic findings in chronic alcohol abuserdquoPsychiatrie Neurologie und Medizinische Psychologie vol 37no 3 pp 129ndash132 1985

[13] G Testino ldquoAlcoholic diseases in hepato-gastroenterology apoint of viewrdquo Hepato-Gastroenterology vol 55 no 82-83 pp371ndash377 2008

[14] A Mubayi P E Greenwood C Castillo-Chavez P J Grue-newald and D M Gorman ldquoThe impact of relative residencetimes on the distribution of heavy drinkers in highly distinctenvironmentsrdquo Socio-Economic Planning Sciences vol 44 no 1pp 45ndash56 2010

[15] D Muller R D Koch H von Specht W Volker and E MMunch ldquoNeurophysiologic findings in chronic alcohol abuserdquoPsychiatrie Neurologie Und Medizinische Psychologie vol 37no 3 pp 129ndash132 1985 (Leipz)

[16] G Mulone and B Straughan ldquoModeling binge drinkingrdquoInternational Journal of Biomathematics vol 5 no 1 Article ID1250005 14 pages 2012

[17] H F Huo and N N Song ldquoGlobal stability for a binge drinkingmodel with two stagesrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 829386 15 pages 2012

[18] S S Mushayabasa and C P Bhunu ldquoModelling the effects ofheavy alcohol consumption on the transmission dynamics ofgonorrheardquo Nonlinear Dynamics vol 66 no 4 pp 695ndash7062011


[19] F S Sancheznchez XWang C Castillo-Chvez DM Gormanand P J Gruenewald ldquoDrinking as an epidemic a simplemathematical model with recovery and relapserdquo in TherapistsGuide to Evidence-Based Relapse Prevention K Witkiewitz andG Marlatt Eds Academic Press New York NY USA 2007

[20] H-F Huo and C-C Zhu ldquoInfluence of relapse in a givingup smoking modelrdquo Abstract and Applied Analysis vol 2013Article ID 525461 12 pages 2013

[21] H F Huo and Q Wang ldquoThe effects of awareness programs by17 media on the drinking dynamicsrdquo Submitted

[22] S Lee E Jung and C Castillo-Chavez ldquoOptimal controlintervention strategies in low- and high-risk problem drinkingpopulationsrdquo Socio-Economic Planning Sciences vol 44 no 4pp 258ndash265 2010

[23] K R Fister S Lenhart and J S McNally ldquoOptimizingchemotherapy in an HIV modelrdquo Electronic Journal of Differ-ential Equations vol 1998 no 32 pp 1ndash12 1998

[24] S Lenhart and J T Workman Optimal Control Applied toBiological Models Chapman amp HallCRC Mathematical andComputational Biology Series Chapman amp HallCRC BocaRaton Fla USA 2007

[25] X Yang L Chen and J Chen ldquoPermanence and positiveperiodic solution for the single-species nonautonomous delaydiffusive modelsrdquo Computers amp Mathematics with ApplicationsAn International Journal vol 32 no 4 pp 109ndash116 1996

[26] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[27] V Lakshmikantham S Leela and A A Martynyuk StabilityAnalysis of Nonlinear Systems vol 125 Marcel Dekker NewYork NY USA 1989

[28] J P LaSalle The Stability of Dynamical Systems vol 25 ofRegional Conference Series in Applied Mathematics Society forIndustrial and Applied Mathematics 1976

[29] W H Fleming and R W Rishel Deterministic and StochasticOptimal Control vol 1 ofApplications of Mathematics SpringerBerlin Germany 1975

[30] D L Lukes Differential Equations Classical to ControlledMathematics in Science and Engineering Academic Press NewYork NY USA 1982

[31] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New Jersey NJ USA 1962

[32] W Hackbusch ldquoA numerical method for solving parabolicequations with opposite orientationsrdquo Computing vol 20 no3 pp 229ndash240 1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


infectivity drunk states mathematical analyses establishedthat the global dynamics of themodel were determined by thebasic reproduction number In [17] the authors modified themodel from [16] that is they considered different infectivityof two drunk states and a SEIR-type model of alcoholismwas thus presented in which two alcohol related states wereinvolved namely no alcohol dependent consumers 119863(119905)and alcohol dependent consumers 119860(119905) In [18] the authorsformulated a deterministic model for evaluating the impactof heavy alcohol drinking on the reemerging gonorrheaepidemic and both analytical and numerical results wereprovided to ascertain whether heavy alcohol drinking hadan impact on the transmission dynamics of gonorrhea Theapproach of the literature [18] was very meaningful sinceit provided a new direction of thinking when the cross-infection between alcoholism and other pathological diseasesoccurs In recent monograph [19] the authors also proposeda SIR-type model to investigate alcohol abuse phenomenonand generated some useful insights for example the basicreproductive number was not always the key to controllingdrinking within the population For other papers that studythe model of giving up smoking or quitting drinking pleasesee [20 21] and references cited therein

As living standard and health awareness get improvedmore and more people who fall into binge drinking stateare actively seeking the quitting alcoholism measures andtreatment methods [1 11 22] In [22] treatment strategy wasintroduced into a simple SIR-type alcoholics quitting modelin which the authors used the bilinear incidence to depict theldquoinfectionrdquo between the occasional drinkers 119878 and problemdrinkers 119863 Motivated by some aforementioned documents[10 19 22] in this paper we will formulate a more reasonablealcoholics quitting model The fact that our model is morereasonable is embodied from the following three aspects

(1) Taking into account that alcoholism is a widespreadsocial phenomenon so the standard incidence issuperior to bilinear incidence when we portray therelationship between the alcoholism and the suscep-tibles during the course of infection While in [22]the authors adopted bilinear incidence we will adoptstandard incidence in this paper

(2) Since alcohol is harmful to health moreover aswe all know alcoholism is treatable if we can takeapproximate measure in time for example artificialisolation from alcoholisms medications persuasionand education programing on alcoholism So it isnecessary to take effective measures to avoid alcoholor to treat after alcoholism Documents [10 19] havenot considered these aspects

(3) Since there is effective prevention and treatment indescribing the phenomenon of alcoholism there aresome people who will never drink due to successfulprevention or some people who no longer drink aftersuccessful treatment Therefore when we formulatethe model in this paper itrsquos reasonable to introducea new compartment119876 the people in which will neverdrink for ever Obviously themodels of [10 19 22] arenot involving the quitting compartment 119876

Based on the above considerations we will premeditatetwo treating methods namely prevention of susceptiblesfrom alcoholism and treatment on alcoholism as controlvariables hence we will derive a SATQ-type model Wenote the fact that many authors are interested in solvingoptimal control problems such as cost minimization andoptimal control of various disease especially with biologicalbackground and various mathematical models [22ndash24] Inthis paper we will propose an objective functional whichconsiders not only alcohol quitting effects but also the costof controlling alcohol Then we consider a range of issuesrelated to the optimal control with themethod of PontryaginrsquosMaximum Principle including optimal control existenceuniqueness and characterization

The organization of this paper is as follows In thenext section the alcoholism model with prevention for thesusceptibles and treatment for alcoholism is formulated InSection 3 the basic reproduction number and the existence ofequilibria are investigatedThe stability of the disease free andendemic equilibria is proved in Section 4 Optimal controlstrategies by the classic method of PMP (Pontryaginrsquos Maxi-mum Principle) are discussed in Section 5 In Section 6 wegive some numerical simulations We give some discussionsand conclusions in the last section

2 The Model Formulation and SomeFundamental Properties

In this section we introduce a mathematical model withprevention and treatment for the alcoholism and then studysome important properties such as the boundness and posi-tivity of its solutions

21 Model Formulation and Parameter Explanation Thetotal population is partitioned into four compartments thesusceptible compartment 119878 which refers to the personswho never drink or drink moderately without affecting thephysical health the alcoholism compartment 119860 which refersto the persons who binge drink and affect the physical healthseriously the treatment compartment 119879 which refers to thepersons who have been receiving treatments by taking pills orothermedical interventions after alcoholism and the quittingcompartment 119876 which refers to the persons who recoverfrom alcoholism after treatment and stay off alcohol hereafterIn this paper we focus on a closed environment such as acommunity a university or a village So the total number ofpopulation to be considered is a constant we denote it as119873The population flow among those compartments is shown inthe following diagram (Figure 1)

The schematic diagram leads to the following system ofordinary differential equations

1198781015840= 120583119873 minus (1 minus 119906

1)120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860


A

S Q

T

120583N

120583S

120585T

120583T

120583A

120583Q

u2A

120575T


N


1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(1)


1is the fraction of






1198831015840= 119866 (119883) (2)


119866 (119883) = (

1198661(119883)

1198662(119883)

1198663(119883)

1198664(119883)

)

=(

120583119873 minus (1 minus 1199061)120573119878119860

119873minus 120583119878

(1 minus 1199061)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1199062119860 minus (120583 + 120585 + 120575) 119879

120575119879 minus 120583119876

)

(3)


119866119894(119883)

1003816100381610038161003816119883119894(119905)=0119883119905isin119862+ge 0 119894 = 1 2 3 4 (4)




119889119873 (119905)

119889119905= 0 (5)


Ω = (119878 119860 119879 119876) isin 1198774

+ 119878 + 119860 + 119879 + 119876 le 119873 (6)





0= (1198780 0 0 0) =



119889119909

119889119905= F (119909) minusV (119909) (7)

where

F (119909) = (

(1 minus 1199061)120573119878119860

1198730

0

0

)

V (119909) = (

(1199062+ 120583)119860 minus 120585119879

(120583 + 120585 + 120575) 119879 minus 1199062119860

120583119876 minus 120575119879

(1 minus 1199061)120573119878119860

119873+ 120583119878 minus 120583119873

)

(8)


0are respectively

119863F (1198640) = (

1198653times3

0

0 0)

119863V (1198640) = (

1198813times3

0

(1 minus 1199061) 120573 0 0 120583

)

(9)

where

119865 = (

(1 minus 1199061) 120573 0 0

0 0 0

0 0 0

) 119881 = (

1199062+ 120583 minus120585 0

minus1199062

120583 + 120585 + 120575 0

0 minus120575 120583

)

(10)


by

1198770= 120588 (119865119881

minus1) =

120573 (1 minus 1199061) (120583 + 120585 + 120575)

1199062(120583 + 120575) + 120583 (120583 + 120585 + 120575)

(11)





equations

120583119873 minus (1 minus 1199061)120573119878119860

119873minus 120583119878 = 0

(1 minus 1199061)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860 = 0

1199062119860 minus (120583 + 120585 + 120575) 119879 = 0

120575119879 minus 120583119876 = 0

(12)


119860 =120583 + 120585 + 120575

1199062

119879 (13)


119876 =120575119879

120583 (14)


119878 =120583119873

120583 + (((1 minus 1199061) 120573) 119873)119860

=12058311987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879 (120583 + 120585 + 120575)

(15)


120583119873 minus120583211987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879 (120583 + 120585 + 120575)

+ 120585119879 minus (120583 + 1199062)120583 + 120585 + 120575

1199062

119879 = 0

(16)


119879 [1199062120585 (1 minus 119906

1) 120573 (120583 + 120585 + 120575)

minus (1 minus 1199061) 120573(120583 + 120585 + 120575)

2

(120583 + 1199062)] 119879 + 120590 = 0

(17)

where120590 = 119906

2120583119873 (1 minus 119906

1) 120573 (120583 + 120585 + 120575)

+ (1199062)2

120583119873120585 minus 1205831198731199062(120583 + 119906

2) (120583 + 120585 + 120575)

(18)



1) 120573(120583 + 120585 + 120575)

2

(120583 + 1199062)

minus1199062120585 (1 minus 119906

1) 120573 (120583 + 120585 + 120575) )

minus1

=120590

(1 minus 1199061) 120573 (120583 + 120585 + 120575) [120583120585 + (120575 + 120583) (120583 + 119906

2)]

(19)




120590 = 1205831198731199062(120583 + 120585 + 120575) (1 minus 119906

1) 120573

minus [(120583 + 1199062) (120575 + 120583) + 120583120585]

(20)


120573 (1 minus 1199061) (120583 + 120585 + 120575) gt 119906

2(120583 + 120575) + 120583 (120583 + 120585 + 120575) (21)




0= (119873 0 0 0) When 119877




119878lowast=

12058311987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879lowast (120583 + 120585 + 120575)

119860lowast=120583 + 120585 + 120575

1199062

119879lowast

119876lowast=120575119879lowast

120583

119879lowast=

120590

(1 minus 1199061) 120573 (120583 + 120585 + 120575) [120583120585 + (120575 + 120583) (120583 + 119906

2)]

(22)



119879 and

119891 (119883) =(

(1 minus 1199061)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1199062119860 minus (120583 + 120585 + 120575) 119879

120575119879 minus 120583119876

120583119873 minus (1 minus 1199061)120573119878119860

119873minus 120583119878

) (23)


119869 =

120597119891 (119883)

120597119883

=(

(1 minus 1199061) 120573119878

119873

minus (120583 + 1199062) 120585 0

(1 minus 1199061) 120573119860

119873

1199062 minus (120583 + 120585 + 120575) 0 0

0 120575 minus120583 0

minus

(1 minus 1199061) 120573119878

119873


(1 minus 1199061) 120573119860

119873

)

(24)




Proof Since

119869 (1198640)

= (

(1 minus 1199061) 120573 minus (120583 + 119906

2) 120585 0 0

1199062

minus (120583 + 120585 + 120575) 0 0

0 120575 minus120583 0


)

(25)



1205822+ [2120583 + 120585 + 120575 + 119906


1) 120573] 120582

+ (120583 + 120585 + 120575) (120583 + 1199062minus (1 minus 119906

1) 120573) minus 119906

2120585 = 0

(26)

Thus

1205823+ 1205824= (1 minus 119906

1) 120573 minus (120583 + 120585 + 120575) minus (120583 + 119906

2) (27)


120573 (1 minus 1199061) (120583 + 120585 + 120575) lt 119906

2(120583 + 120575) + 120583 (120583 + 120585 + 120575)

lt (120583 + 1199062) (120583 + 120585 + 120575)

(28)

so

120573 (1 minus 1199061) lt 120583 + 119906

2 (29)

and then

1205823+ 1205824lt minus (120583 + 120585 + 120575) lt 0 (30)

while

12058231205824= (120583 + 120585 + 120575) (120583 + 119906


1) 120573) minus 119906

2120585 (31)



2(120583 + 120575) minus 120583 (120583 + 120585 + 120575) (32)

so12058231205824gt (120583 + 119906

2) (120583 + 120585 + 120575)


(33)

It reduces to

12058231205824gt 0 (34)

Hence Re 1205823lt 0 Re 120582






1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(35)


(

) = (119865 minus 119881)(

119860

119879

119876


119873)

times (

120573 (1 minus 1199061) 0 0

0 0 0

0 0 0

)(

119860

119879

119876

)

(36)


(

) le (119865 minus 119881)(

119860

119879

119876

) (37)


0lt 1 So (119860(119905) 119879(119905) 119876(119905)) rarr





0gt 1


1198781015840= 120583119873 minus (1 minus 119906

1)120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

(38)


1198781015840= 120583 minus (1 minus 119906

1) 120573119878119860 minus 120583119878

1199041198601015840= (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

(39)




lowast

1199062119860lowast= (120583 + 120585 + 120575) 119879

lowast

(40)


Let 119881 = 1199091(119878 minus 119878


2(119860 minus 119860

lowastminus

119860lowast ln(119860119860lowast)) + 119909

3(119879 minus 119879


119881101584010038161003816100381610038161003816(39)

= 1199091[120583 minus (1 minus 119906

1) 120573119878119860 minus 120583119878

minus119878lowast

119878120583 + (1 minus 119906

1) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860


119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091[ (1 minus 119906


1) 120573119878119860 minus 120583119878

minus119878lowast

119878((1 minus 119906


+ (1 minus 1199061) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860


119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091120583119878lowast(2 minus

119878


119878)

+ [1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast]

minus [1199091

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878+ 1199092(1 minus 119906

1) 120573119878119860

lowast

+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

+ 119878119860 [minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573]

+ 119860 [minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093]

+ 119879 [1205851199092minus (120583 + 120585 + 120575) 119909

3]

(41)


minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573 = 0

minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093= 0

1205851199092minus (120583 + 120585 + 120575) 119909

3= 0

(42)


1199091= 1 119909

2= 1

1199093=

120585

120583 + 120585 + 120575=1199062+ 120583 minus (1 minus 119906

1) 120573119878lowast

1199062

(43)

Next we let

1198811015840

1= 1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast

1198811015840

2= minus[119909

1

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ 1199092(1 minus 119906

1) 120573119878119860

lowast+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

(44)

and then

1198811015840= 1199091120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2 (45)

Due to

(1199062+ 120583)119860



1199093(120583 + 120585 + 120575) 119879


(46)


so

1198811015840

1= 2 (1 minus 119906

1) 120573119878lowast119860lowast+ 2120585119879

lowast

1198811015840

2= minus[

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ (1 minus 1199061) 120573119878119860

lowast+119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]


(1 minus 1199061) 120573119860lowast

119878sdot (1 minus 119906

1) 120573119878119860

lowast]

12

minus 2 [119860lowast120585119879

119860sdot 1199093

119879lowast

1198791199062119860]



lowast(11990931199062119860lowast)]12


minus 2120585119879lowast[(1199062+ 120583)119860


1) 120573119878lowast119860lowast]12




lowast

(47)

Hence

1198811015840= 120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2


119878


119878) + 2 (1 minus 119906

1) 120573119878lowast119860lowast



lowast

= 120583119878lowast(2 minus

119878


119878) le 0

(48)

1198811015840= 0 if and only if 119878 = 119878

lowast 119860 = 119860lowast 119879 = 119879







1(119905)




119869 (1199061 1199062) = int

119905119891

0

[119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)] 119889119905 (49)


1198781015840= 120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860

1198791015840= 1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(50)


119878 (0) = 1198780 119860 (0) = 119860

0

119879 (0) = 1198790 119876 (0) = 119876

0

(51)

Here 119906119894(119905) isin 119880 ≜ (119906


119906119894(119905) le 1 for all 119905 isin [0 119905






=

(119906lowast

1 119906lowast


119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) 119906

1(119905) 1199062(119905) isin 119880 (52)






1 1198892gt 0 and 120572 gt 1 such


12)1199062

1(119905) +

(11988822)1199062


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(53)



1198781015840le 120583119873 119860

1015840le (1 minus 119906

1(119905)) 120573119878 + 120585119879

1198791015840le 1199062(119905) 119860 119876

1015840le 120575119879

(54)


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(55)


12 11988822 and for all




119867 = 119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)

+ 1205821[120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878]

+ 1205822[(1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860]

+ 1205823[1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879] + 120582

4(120575119879 minus 120583119876)

minus 119908111199061(119905) minus 119908

12(1 minus 119906

1(119905))

minus 119908211199062(119905) minus 119908

22(1 minus 119906

2(119905))

(56)


11990811(119905) 1199061(119905) = 119908

12(119905) (1 minus 119906

1(119905))


11990821(119905) 1199062(119905) = 119908

22(119905) (1 minus 119906

2(119905))


(57)


2) and solutions


119894 119894 = 1 2 3 4 satisfying

1205821015840

1= 1205821(1 minus 119906

1(119905))

120573119860

119873+ 1205831205821minus 1205822(1 minus 119906

1(119905))

120573119860

119873

1205821015840

2= minus1 + 120582

1(1 minus 119906

1(119905))

120573119878

119873minus 1205822(1 minus 119906

1(119905))

120573119878

119873

+ 1205822(120583 + 119906

2(119905)) minus 120582

31199062(119905)

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

(58)


120582119894(119905119891) = 0 119894 = 1 2 3 4 (59)



119906lowast

1= min(1max(0

120573119878119860 (1205822minus 1205821)

1198881119873

))

119906lowast

2= min(1max(0

119860 (1205822minus 1205823)

1198882

))

(60)


1205821015840

1= minus

120597119867

120597119878 120582

1015840

2= minus

120597119867

120597119860

1205821015840

3= minus

120597119867

120597119879 120582

1015840

4= minus

120597119867

120597119876

(61)




120597119867

1205971199061

= 11988811199061(119905) + 120582

1

120573119878119860

119873minus 1205822

120573119878119860

119873minus 11990811+ 11990812= 0

120597119867

1205971199062

= 11988821199062(119905) minus 120582

2119860 + 120582

3119860 minus 119908

21+ 11990822= 0

(62)


119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811minus 11990812] (63)


11and119908




1(119905) lt 1 we have 119908

11(119905) =



119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873] (64)


11(119905) = 0 Hence

1 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873minus 11990812] (65)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873ge 1 since 119908

12(119905) ge 0 (66)


12(119905) = 0 Hence

0 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811] (67)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873le 0 since 119908

11(119905) ge 0 (68)


characterized as

119906lowast

1= min1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (69)


119906lowast

2= min1max0

(1205822minus 1205823) 119860

1198882

(70)





1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119878119860

119873minus 120583119878

1198601015840= (1minusmin1max0 1

1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

120573119878119860

119873


1198882

(119905) + 120583)119860

1198791015840= min1max0

(1205822minus 1205823) 119860

1198882

(119905) 119860

minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

1205821015840

1= 1205821(1 minusmin1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119860

119873+ 1205831205821


1198881

[(1205822minus1205821)120573119878119860

119873](119905))

times120573119860

119873

1205821015840

2= minus1


1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

times120573119878


(1205822minus 1205823) 119860

1198882

(119905)


1198881

[(1205822minus 1205821)120573119878119860

119873])

times120573119878

119873+ 1205822(120583+min1max0

(1205822minus 1205823) 119860

1198882

(119905))

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

119878 (0) = 1198780 119860 (0) = 119860

0 119879 (0) = 119879

0

119876 (0) = 1198760 120582

119894(119905119891) = 0 119894 = 1 2 3 4

(71)









119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

(72)


119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

(73)


120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902


119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902 1205821= 119890minus120582119905119903 1205822=



+ 120582119898 = 120583119873119890minus120582119905


120582119905

1198881119873

)

times120573119898119899119890

120582119905

119873minus 120583119898

(74)


119899 + 120582119899


120582119905

1198881119873

)120573119898119899119890

120582119905

119873


+ 120583) 119899

(75)


+ 120582119901 = min1max0 119899 (119904 minus 119908)1198882

119899

minus (120583 + 120585 + 120575) 119901

(76)


119902 + 120582119902 = 120575119901 minus 120583119902 (77)



120582119905

1198881119873

)120573119899119890120582119905

119873


120582119905

1198881119873

)

times120573119899119890120582119905

119873

(78)


119904 minus 120582119904 = minus119890120582119905


120582119905

1198881119873

)120573119898119890120582119905

119873


120582119905

1198881119873

)

times120573119898119890120582119905

119873+ 119904(min1max0 119899 (119904 minus 119908)

1198882

+ 120583)


)

(79)




V minus 120582V = 120583V (81)




1003816100381610038161003816119906lowast

1(119905) minus 119906

lowast

1(119905)1003816100381610038161003816 le

120573119890120582119905


1003816100381610038161003816119906lowast

2(119905) minus 119906

lowast

2(119905)1003816100381610038161003816 le

1

1198882


(82)




minus + (120583 + 120582) (119898 minus 119898)

=120573119890120582119905


120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

)119898119899


120582119905

1198881119873

)119898119899]

(83)


119891gives

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873


1)119898119899 + (1 minus 119906

lowast

1)119898119899] 119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873[(119906lowast


+ 119898119899 (119906lowast

1minus 119906lowast

1)] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [119906lowast

1minus 1

100381610038161003816100381610038161003816100381610038161003816

(119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

100381610038161003816100381610038161003816100381610038161003816


le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 (119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

|119898119899 (119904 minus 119904) + (119898119899 minus 119898119899) 119904


le120573119890120582119905119891

119873int

119905119891

0

[1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 |119898| |119898 minus 119898| |119899 minus 119899|

+ |119898 minus 119898|2

|119899| + |119861| |119904| |119899| |119898 minus 119898|2

+ |119861| |119898119899| |119898 minus 119898| |119903 minus 119903|

+ |119861| |119898119899| |119898 minus 119898| |119904 minus 119904|

+ |119861| |119904| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119899| |119898 minus 119898|2] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[

1003816100381610038161003816119906lowastminus 1

1003816100381610038161003816

2|119898| ((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119904 minus 119904)

2)

+|119861| |119904| |119898|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119904| |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119903 minus 119903)

2)

+|119861| |119898| |119903|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119903| |119899| (119898 minus 119898)2]119889119905

(84)


1(119905) minus 119906

lowast





2(119898 minus119898)

2 (119898 minus119898)2+ (119904 minus 119904)

2 (119898 minus119898)2+ (119899 minus 119899)

2 (119898 minus119898)2

(119898minus119898)2+(119903minus119903)



2such that

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905


le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)


1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)


(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)




1= 0 119906

2= 0 Before








0ge













0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A

S Q

T

120583N

120583S

120585T

120583T

120583A

120583Q

u2A

120575T


N


1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(1)


1is the fraction of






1198831015840= 119866 (119883) (2)


119866 (119883) = (

1198661(119883)

1198662(119883)

1198663(119883)

1198664(119883)

)

=(

120583119873 minus (1 minus 1199061)120573119878119860

119873minus 120583119878

(1 minus 1199061)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1199062119860 minus (120583 + 120585 + 120575) 119879

120575119879 minus 120583119876

)

(3)


119866119894(119883)

1003816100381610038161003816119883119894(119905)=0119883119905isin119862+ge 0 119894 = 1 2 3 4 (4)




119889119873 (119905)

119889119905= 0 (5)


Ω = (119878 119860 119879 119876) isin 1198774

+ 119878 + 119860 + 119879 + 119876 le 119873 (6)





0= (1198780 0 0 0) =



119889119909

119889119905= F (119909) minusV (119909) (7)

where

F (119909) = (

(1 minus 1199061)120573119878119860

1198730

0

0

)

V (119909) = (

(1199062+ 120583)119860 minus 120585119879

(120583 + 120585 + 120575) 119879 minus 1199062119860

120583119876 minus 120575119879

(1 minus 1199061)120573119878119860

119873+ 120583119878 minus 120583119873

)

(8)


0are respectively

119863F (1198640) = (

1198653times3

0

0 0)

119863V (1198640) = (

1198813times3

0

(1 minus 1199061) 120573 0 0 120583

)

(9)

where

119865 = (

(1 minus 1199061) 120573 0 0

0 0 0

0 0 0

) 119881 = (

1199062+ 120583 minus120585 0

minus1199062

120583 + 120585 + 120575 0

0 minus120575 120583

)

(10)


by

1198770= 120588 (119865119881

minus1) =

120573 (1 minus 1199061) (120583 + 120585 + 120575)

1199062(120583 + 120575) + 120583 (120583 + 120585 + 120575)

(11)





equations

120583119873 minus (1 minus 1199061)120573119878119860

119873minus 120583119878 = 0

(1 minus 1199061)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860 = 0

1199062119860 minus (120583 + 120585 + 120575) 119879 = 0

120575119879 minus 120583119876 = 0

(12)


119860 =120583 + 120585 + 120575

1199062

119879 (13)


119876 =120575119879

120583 (14)


119878 =120583119873

120583 + (((1 minus 1199061) 120573) 119873)119860

=12058311987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879 (120583 + 120585 + 120575)

(15)


120583119873 minus120583211987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879 (120583 + 120585 + 120575)

+ 120585119879 minus (120583 + 1199062)120583 + 120585 + 120575

1199062

119879 = 0

(16)


119879 [1199062120585 (1 minus 119906

1) 120573 (120583 + 120585 + 120575)

minus (1 minus 1199061) 120573(120583 + 120585 + 120575)

2

(120583 + 1199062)] 119879 + 120590 = 0

(17)

where120590 = 119906

2120583119873 (1 minus 119906

1) 120573 (120583 + 120585 + 120575)

+ (1199062)2

120583119873120585 minus 1205831198731199062(120583 + 119906

2) (120583 + 120585 + 120575)

(18)



1) 120573(120583 + 120585 + 120575)

2

(120583 + 1199062)

minus1199062120585 (1 minus 119906

1) 120573 (120583 + 120585 + 120575) )

minus1

=120590

(1 minus 1199061) 120573 (120583 + 120585 + 120575) [120583120585 + (120575 + 120583) (120583 + 119906

2)]

(19)




120590 = 1205831198731199062(120583 + 120585 + 120575) (1 minus 119906

1) 120573

minus [(120583 + 1199062) (120575 + 120583) + 120583120585]

(20)


120573 (1 minus 1199061) (120583 + 120585 + 120575) gt 119906

2(120583 + 120575) + 120583 (120583 + 120585 + 120575) (21)




0= (119873 0 0 0) When 119877




119878lowast=

12058311987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879lowast (120583 + 120585 + 120575)

119860lowast=120583 + 120585 + 120575

1199062

119879lowast

119876lowast=120575119879lowast

120583

119879lowast=

120590

(1 minus 1199061) 120573 (120583 + 120585 + 120575) [120583120585 + (120575 + 120583) (120583 + 119906

2)]

(22)



119879 and

119891 (119883) =(

(1 minus 1199061)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1199062119860 minus (120583 + 120585 + 120575) 119879

120575119879 minus 120583119876

120583119873 minus (1 minus 1199061)120573119878119860

119873minus 120583119878

) (23)


119869 =

120597119891 (119883)

120597119883

=(

(1 minus 1199061) 120573119878

119873

minus (120583 + 1199062) 120585 0

(1 minus 1199061) 120573119860

119873

1199062 minus (120583 + 120585 + 120575) 0 0

0 120575 minus120583 0

minus

(1 minus 1199061) 120573119878

119873


(1 minus 1199061) 120573119860

119873

)

(24)




Proof Since

119869 (1198640)

= (

(1 minus 1199061) 120573 minus (120583 + 119906

2) 120585 0 0

1199062

minus (120583 + 120585 + 120575) 0 0

0 120575 minus120583 0


)

(25)



1205822+ [2120583 + 120585 + 120575 + 119906


1) 120573] 120582

+ (120583 + 120585 + 120575) (120583 + 1199062minus (1 minus 119906

1) 120573) minus 119906

2120585 = 0

(26)

Thus

1205823+ 1205824= (1 minus 119906

1) 120573 minus (120583 + 120585 + 120575) minus (120583 + 119906

2) (27)


120573 (1 minus 1199061) (120583 + 120585 + 120575) lt 119906

2(120583 + 120575) + 120583 (120583 + 120585 + 120575)

lt (120583 + 1199062) (120583 + 120585 + 120575)

(28)

so

120573 (1 minus 1199061) lt 120583 + 119906

2 (29)

and then

1205823+ 1205824lt minus (120583 + 120585 + 120575) lt 0 (30)

while

12058231205824= (120583 + 120585 + 120575) (120583 + 119906


1) 120573) minus 119906

2120585 (31)



2(120583 + 120575) minus 120583 (120583 + 120585 + 120575) (32)

so12058231205824gt (120583 + 119906

2) (120583 + 120585 + 120575)


(33)

It reduces to

12058231205824gt 0 (34)

Hence Re 1205823lt 0 Re 120582






1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(35)


(

) = (119865 minus 119881)(

119860

119879

119876


119873)

times (

120573 (1 minus 1199061) 0 0

0 0 0

0 0 0

)(

119860

119879

119876

)

(36)


(

) le (119865 minus 119881)(

119860

119879

119876

) (37)


0lt 1 So (119860(119905) 119879(119905) 119876(119905)) rarr





0gt 1


1198781015840= 120583119873 minus (1 minus 119906

1)120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

(38)


1198781015840= 120583 minus (1 minus 119906

1) 120573119878119860 minus 120583119878

1199041198601015840= (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

(39)




lowast

1199062119860lowast= (120583 + 120585 + 120575) 119879

lowast

(40)


Let 119881 = 1199091(119878 minus 119878


2(119860 minus 119860

lowastminus

119860lowast ln(119860119860lowast)) + 119909

3(119879 minus 119879


119881101584010038161003816100381610038161003816(39)

= 1199091[120583 minus (1 minus 119906

1) 120573119878119860 minus 120583119878

minus119878lowast

119878120583 + (1 minus 119906

1) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860


119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091[ (1 minus 119906


1) 120573119878119860 minus 120583119878

minus119878lowast

119878((1 minus 119906


+ (1 minus 1199061) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860


119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091120583119878lowast(2 minus

119878


119878)

+ [1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast]

minus [1199091

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878+ 1199092(1 minus 119906

1) 120573119878119860

lowast

+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

+ 119878119860 [minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573]

+ 119860 [minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093]

+ 119879 [1205851199092minus (120583 + 120585 + 120575) 119909

3]

(41)


minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573 = 0

minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093= 0

1205851199092minus (120583 + 120585 + 120575) 119909

3= 0

(42)


1199091= 1 119909

2= 1

1199093=

120585

120583 + 120585 + 120575=1199062+ 120583 minus (1 minus 119906

1) 120573119878lowast

1199062

(43)

Next we let

1198811015840

1= 1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast

1198811015840

2= minus[119909

1

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ 1199092(1 minus 119906

1) 120573119878119860

lowast+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

(44)

and then

1198811015840= 1199091120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2 (45)

Due to

(1199062+ 120583)119860



1199093(120583 + 120585 + 120575) 119879


(46)


so

1198811015840

1= 2 (1 minus 119906

1) 120573119878lowast119860lowast+ 2120585119879

lowast

1198811015840

2= minus[

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ (1 minus 1199061) 120573119878119860

lowast+119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]


(1 minus 1199061) 120573119860lowast

119878sdot (1 minus 119906

1) 120573119878119860

lowast]

12

minus 2 [119860lowast120585119879

119860sdot 1199093

119879lowast

1198791199062119860]



lowast(11990931199062119860lowast)]12


minus 2120585119879lowast[(1199062+ 120583)119860


1) 120573119878lowast119860lowast]12




lowast

(47)

Hence

1198811015840= 120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2


119878


119878) + 2 (1 minus 119906

1) 120573119878lowast119860lowast



lowast

= 120583119878lowast(2 minus

119878


119878) le 0

(48)

1198811015840= 0 if and only if 119878 = 119878

lowast 119860 = 119860lowast 119879 = 119879







1(119905)




119869 (1199061 1199062) = int

119905119891

0

[119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)] 119889119905 (49)


1198781015840= 120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860

1198791015840= 1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(50)


119878 (0) = 1198780 119860 (0) = 119860

0

119879 (0) = 1198790 119876 (0) = 119876

0

(51)

Here 119906119894(119905) isin 119880 ≜ (119906


119906119894(119905) le 1 for all 119905 isin [0 119905






=

(119906lowast

1 119906lowast


119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) 119906

1(119905) 1199062(119905) isin 119880 (52)






1 1198892gt 0 and 120572 gt 1 such


12)1199062

1(119905) +

(11988822)1199062


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(53)



1198781015840le 120583119873 119860

1015840le (1 minus 119906

1(119905)) 120573119878 + 120585119879

1198791015840le 1199062(119905) 119860 119876

1015840le 120575119879

(54)


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(55)


12 11988822 and for all




119867 = 119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)

+ 1205821[120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878]

+ 1205822[(1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860]

+ 1205823[1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879] + 120582

4(120575119879 minus 120583119876)

minus 119908111199061(119905) minus 119908

12(1 minus 119906

1(119905))

minus 119908211199062(119905) minus 119908

22(1 minus 119906

2(119905))

(56)


11990811(119905) 1199061(119905) = 119908

12(119905) (1 minus 119906

1(119905))


11990821(119905) 1199062(119905) = 119908

22(119905) (1 minus 119906

2(119905))


(57)


2) and solutions


119894 119894 = 1 2 3 4 satisfying

1205821015840

1= 1205821(1 minus 119906

1(119905))

120573119860

119873+ 1205831205821minus 1205822(1 minus 119906

1(119905))

120573119860

119873

1205821015840

2= minus1 + 120582

1(1 minus 119906

1(119905))

120573119878

119873minus 1205822(1 minus 119906

1(119905))

120573119878

119873

+ 1205822(120583 + 119906

2(119905)) minus 120582

31199062(119905)

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

(58)


120582119894(119905119891) = 0 119894 = 1 2 3 4 (59)



119906lowast

1= min(1max(0

120573119878119860 (1205822minus 1205821)

1198881119873

))

119906lowast

2= min(1max(0

119860 (1205822minus 1205823)

1198882

))

(60)


1205821015840

1= minus

120597119867

120597119878 120582

1015840

2= minus

120597119867

120597119860

1205821015840

3= minus

120597119867

120597119879 120582

1015840

4= minus

120597119867

120597119876

(61)




120597119867

1205971199061

= 11988811199061(119905) + 120582

1

120573119878119860

119873minus 1205822

120573119878119860

119873minus 11990811+ 11990812= 0

120597119867

1205971199062

= 11988821199062(119905) minus 120582

2119860 + 120582

3119860 minus 119908

21+ 11990822= 0

(62)


119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811minus 11990812] (63)


11and119908




1(119905) lt 1 we have 119908

11(119905) =



119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873] (64)


11(119905) = 0 Hence

1 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873minus 11990812] (65)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873ge 1 since 119908

12(119905) ge 0 (66)


12(119905) = 0 Hence

0 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811] (67)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873le 0 since 119908

11(119905) ge 0 (68)


characterized as

119906lowast

1= min1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (69)


119906lowast

2= min1max0

(1205822minus 1205823) 119860

1198882

(70)





1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119878119860

119873minus 120583119878

1198601015840= (1minusmin1max0 1

1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

120573119878119860

119873


1198882

(119905) + 120583)119860

1198791015840= min1max0

(1205822minus 1205823) 119860

1198882

(119905) 119860

minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

1205821015840

1= 1205821(1 minusmin1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119860

119873+ 1205831205821


1198881

[(1205822minus1205821)120573119878119860

119873](119905))

times120573119860

119873

1205821015840

2= minus1


1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

times120573119878


(1205822minus 1205823) 119860

1198882

(119905)


1198881

[(1205822minus 1205821)120573119878119860

119873])

times120573119878

119873+ 1205822(120583+min1max0

(1205822minus 1205823) 119860

1198882

(119905))

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

119878 (0) = 1198780 119860 (0) = 119860

0 119879 (0) = 119879

0

119876 (0) = 1198760 120582

119894(119905119891) = 0 119894 = 1 2 3 4

(71)









119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

(72)


119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

(73)


120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902


119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902 1205821= 119890minus120582119905119903 1205822=



+ 120582119898 = 120583119873119890minus120582119905


120582119905

1198881119873

)

times120573119898119899119890

120582119905

119873minus 120583119898

(74)


119899 + 120582119899


120582119905

1198881119873

)120573119898119899119890

120582119905

119873


+ 120583) 119899

(75)


+ 120582119901 = min1max0 119899 (119904 minus 119908)1198882

119899

minus (120583 + 120585 + 120575) 119901

(76)


119902 + 120582119902 = 120575119901 minus 120583119902 (77)



120582119905

1198881119873

)120573119899119890120582119905

119873


120582119905

1198881119873

)

times120573119899119890120582119905

119873

(78)


119904 minus 120582119904 = minus119890120582119905


120582119905

1198881119873

)120573119898119890120582119905

119873


120582119905

1198881119873

)

times120573119898119890120582119905

119873+ 119904(min1max0 119899 (119904 minus 119908)

1198882

+ 120583)


)

(79)




V minus 120582V = 120583V (81)




1003816100381610038161003816119906lowast

1(119905) minus 119906

lowast

1(119905)1003816100381610038161003816 le

120573119890120582119905


1003816100381610038161003816119906lowast

2(119905) minus 119906

lowast

2(119905)1003816100381610038161003816 le

1

1198882


(82)




minus + (120583 + 120582) (119898 minus 119898)

=120573119890120582119905


120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

)119898119899


120582119905

1198881119873

)119898119899]

(83)


119891gives

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873


1)119898119899 + (1 minus 119906

lowast

1)119898119899] 119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873[(119906lowast


+ 119898119899 (119906lowast

1minus 119906lowast

1)] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [119906lowast

1minus 1

100381610038161003816100381610038161003816100381610038161003816

(119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

100381610038161003816100381610038161003816100381610038161003816


le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 (119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

|119898119899 (119904 minus 119904) + (119898119899 minus 119898119899) 119904


le120573119890120582119905119891

119873int

119905119891

0

[1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 |119898| |119898 minus 119898| |119899 minus 119899|

+ |119898 minus 119898|2

|119899| + |119861| |119904| |119899| |119898 minus 119898|2

+ |119861| |119898119899| |119898 minus 119898| |119903 minus 119903|

+ |119861| |119898119899| |119898 minus 119898| |119904 minus 119904|

+ |119861| |119904| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119899| |119898 minus 119898|2] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[

1003816100381610038161003816119906lowastminus 1

1003816100381610038161003816

2|119898| ((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119904 minus 119904)

2)

+|119861| |119904| |119898|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119904| |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119903 minus 119903)

2)

+|119861| |119898| |119903|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119903| |119899| (119898 minus 119898)2]119889119905

(84)


1(119905) minus 119906

lowast





2(119898 minus119898)

2 (119898 minus119898)2+ (119904 minus 119904)

2 (119898 minus119898)2+ (119899 minus 119899)

2 (119898 minus119898)2

(119898minus119898)2+(119903minus119903)



2such that

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905


le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)


1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)


(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)




1= 0 119906

2= 0 Before








0ge













0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













equations

120583119873 minus (1 minus 1199061)120573119878119860

119873minus 120583119878 = 0

(1 minus 1199061)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860 = 0

1199062119860 minus (120583 + 120585 + 120575) 119879 = 0

120575119879 minus 120583119876 = 0

(12)


119860 =120583 + 120585 + 120575

1199062

119879 (13)


119876 =120575119879

120583 (14)


119878 =120583119873

120583 + (((1 minus 1199061) 120573) 119873)119860

=12058311987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879 (120583 + 120585 + 120575)

(15)


120583119873 minus120583211987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879 (120583 + 120585 + 120575)

+ 120585119879 minus (120583 + 1199062)120583 + 120585 + 120575

1199062

119879 = 0

(16)


119879 [1199062120585 (1 minus 119906

1) 120573 (120583 + 120585 + 120575)

minus (1 minus 1199061) 120573(120583 + 120585 + 120575)

2

(120583 + 1199062)] 119879 + 120590 = 0

(17)

where120590 = 119906

2120583119873 (1 minus 119906

1) 120573 (120583 + 120585 + 120575)

+ (1199062)2

120583119873120585 minus 1205831198731199062(120583 + 119906

2) (120583 + 120585 + 120575)

(18)



1) 120573(120583 + 120585 + 120575)

2

(120583 + 1199062)

minus1199062120585 (1 minus 119906

1) 120573 (120583 + 120585 + 120575) )

minus1

=120590

(1 minus 1199061) 120573 (120583 + 120585 + 120575) [120583120585 + (120575 + 120583) (120583 + 119906

2)]

(19)




120590 = 1205831198731199062(120583 + 120585 + 120575) (1 minus 119906

1) 120573

minus [(120583 + 1199062) (120575 + 120583) + 120583120585]

(20)


120573 (1 minus 1199061) (120583 + 120585 + 120575) gt 119906

2(120583 + 120575) + 120583 (120583 + 120585 + 120575) (21)




0= (119873 0 0 0) When 119877




119878lowast=

12058311987321199062

1205831198731199062+ (1 minus 119906

1) 120573119879lowast (120583 + 120585 + 120575)

119860lowast=120583 + 120585 + 120575

1199062

119879lowast

119876lowast=120575119879lowast

120583

119879lowast=

120590

(1 minus 1199061) 120573 (120583 + 120585 + 120575) [120583120585 + (120575 + 120583) (120583 + 119906

2)]

(22)



119879 and

119891 (119883) =(

(1 minus 1199061)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1199062119860 minus (120583 + 120585 + 120575) 119879

120575119879 minus 120583119876

120583119873 minus (1 minus 1199061)120573119878119860

119873minus 120583119878

) (23)


119869 =

120597119891 (119883)

120597119883

=(

(1 minus 1199061) 120573119878

119873

minus (120583 + 1199062) 120585 0

(1 minus 1199061) 120573119860

119873

1199062 minus (120583 + 120585 + 120575) 0 0

0 120575 minus120583 0

minus

(1 minus 1199061) 120573119878

119873


(1 minus 1199061) 120573119860

119873

)

(24)




Proof Since

119869 (1198640)

= (

(1 minus 1199061) 120573 minus (120583 + 119906

2) 120585 0 0

1199062

minus (120583 + 120585 + 120575) 0 0

0 120575 minus120583 0


)

(25)



1205822+ [2120583 + 120585 + 120575 + 119906


1) 120573] 120582

+ (120583 + 120585 + 120575) (120583 + 1199062minus (1 minus 119906

1) 120573) minus 119906

2120585 = 0

(26)

Thus

1205823+ 1205824= (1 minus 119906

1) 120573 minus (120583 + 120585 + 120575) minus (120583 + 119906

2) (27)


120573 (1 minus 1199061) (120583 + 120585 + 120575) lt 119906

2(120583 + 120575) + 120583 (120583 + 120585 + 120575)

lt (120583 + 1199062) (120583 + 120585 + 120575)

(28)

so

120573 (1 minus 1199061) lt 120583 + 119906

2 (29)

and then

1205823+ 1205824lt minus (120583 + 120585 + 120575) lt 0 (30)

while

12058231205824= (120583 + 120585 + 120575) (120583 + 119906


1) 120573) minus 119906

2120585 (31)



2(120583 + 120575) minus 120583 (120583 + 120585 + 120575) (32)

so12058231205824gt (120583 + 119906

2) (120583 + 120585 + 120575)


(33)

It reduces to

12058231205824gt 0 (34)

Hence Re 1205823lt 0 Re 120582






1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(35)


(

) = (119865 minus 119881)(

119860

119879

119876


119873)

times (

120573 (1 minus 1199061) 0 0

0 0 0

0 0 0

)(

119860

119879

119876

)

(36)


(

) le (119865 minus 119881)(

119860

119879

119876

) (37)


0lt 1 So (119860(119905) 119879(119905) 119876(119905)) rarr





0gt 1


1198781015840= 120583119873 minus (1 minus 119906

1)120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

(38)


1198781015840= 120583 minus (1 minus 119906

1) 120573119878119860 minus 120583119878

1199041198601015840= (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

(39)




lowast

1199062119860lowast= (120583 + 120585 + 120575) 119879

lowast

(40)


Let 119881 = 1199091(119878 minus 119878


2(119860 minus 119860

lowastminus

119860lowast ln(119860119860lowast)) + 119909

3(119879 minus 119879


119881101584010038161003816100381610038161003816(39)

= 1199091[120583 minus (1 minus 119906

1) 120573119878119860 minus 120583119878

minus119878lowast

119878120583 + (1 minus 119906

1) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860


119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091[ (1 minus 119906


1) 120573119878119860 minus 120583119878

minus119878lowast

119878((1 minus 119906


+ (1 minus 1199061) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860


119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091120583119878lowast(2 minus

119878


119878)

+ [1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast]

minus [1199091

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878+ 1199092(1 minus 119906

1) 120573119878119860

lowast

+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

+ 119878119860 [minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573]

+ 119860 [minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093]

+ 119879 [1205851199092minus (120583 + 120585 + 120575) 119909

3]

(41)


minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573 = 0

minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093= 0

1205851199092minus (120583 + 120585 + 120575) 119909

3= 0

(42)


1199091= 1 119909

2= 1

1199093=

120585

120583 + 120585 + 120575=1199062+ 120583 minus (1 minus 119906

1) 120573119878lowast

1199062

(43)

Next we let

1198811015840

1= 1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast

1198811015840

2= minus[119909

1

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ 1199092(1 minus 119906

1) 120573119878119860

lowast+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

(44)

and then

1198811015840= 1199091120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2 (45)

Due to

(1199062+ 120583)119860



1199093(120583 + 120585 + 120575) 119879


(46)


so

1198811015840

1= 2 (1 minus 119906

1) 120573119878lowast119860lowast+ 2120585119879

lowast

1198811015840

2= minus[

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ (1 minus 1199061) 120573119878119860

lowast+119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]


(1 minus 1199061) 120573119860lowast

119878sdot (1 minus 119906

1) 120573119878119860

lowast]

12

minus 2 [119860lowast120585119879

119860sdot 1199093

119879lowast

1198791199062119860]



lowast(11990931199062119860lowast)]12


minus 2120585119879lowast[(1199062+ 120583)119860


1) 120573119878lowast119860lowast]12




lowast

(47)

Hence

1198811015840= 120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2


119878


119878) + 2 (1 minus 119906

1) 120573119878lowast119860lowast



lowast

= 120583119878lowast(2 minus

119878


119878) le 0

(48)

1198811015840= 0 if and only if 119878 = 119878

lowast 119860 = 119860lowast 119879 = 119879







1(119905)




119869 (1199061 1199062) = int

119905119891

0

[119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)] 119889119905 (49)


1198781015840= 120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860

1198791015840= 1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(50)


119878 (0) = 1198780 119860 (0) = 119860

0

119879 (0) = 1198790 119876 (0) = 119876

0

(51)

Here 119906119894(119905) isin 119880 ≜ (119906


119906119894(119905) le 1 for all 119905 isin [0 119905






=

(119906lowast

1 119906lowast


119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) 119906

1(119905) 1199062(119905) isin 119880 (52)






1 1198892gt 0 and 120572 gt 1 such


12)1199062

1(119905) +

(11988822)1199062


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(53)



1198781015840le 120583119873 119860

1015840le (1 minus 119906

1(119905)) 120573119878 + 120585119879

1198791015840le 1199062(119905) 119860 119876

1015840le 120575119879

(54)


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(55)


12 11988822 and for all




119867 = 119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)

+ 1205821[120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878]

+ 1205822[(1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860]

+ 1205823[1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879] + 120582

4(120575119879 minus 120583119876)

minus 119908111199061(119905) minus 119908

12(1 minus 119906

1(119905))

minus 119908211199062(119905) minus 119908

22(1 minus 119906

2(119905))

(56)


11990811(119905) 1199061(119905) = 119908

12(119905) (1 minus 119906

1(119905))


11990821(119905) 1199062(119905) = 119908

22(119905) (1 minus 119906

2(119905))


(57)


2) and solutions


119894 119894 = 1 2 3 4 satisfying

1205821015840

1= 1205821(1 minus 119906

1(119905))

120573119860

119873+ 1205831205821minus 1205822(1 minus 119906

1(119905))

120573119860

119873

1205821015840

2= minus1 + 120582

1(1 minus 119906

1(119905))

120573119878

119873minus 1205822(1 minus 119906

1(119905))

120573119878

119873

+ 1205822(120583 + 119906

2(119905)) minus 120582

31199062(119905)

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

(58)


120582119894(119905119891) = 0 119894 = 1 2 3 4 (59)



119906lowast

1= min(1max(0

120573119878119860 (1205822minus 1205821)

1198881119873

))

119906lowast

2= min(1max(0

119860 (1205822minus 1205823)

1198882

))

(60)


1205821015840

1= minus

120597119867

120597119878 120582

1015840

2= minus

120597119867

120597119860

1205821015840

3= minus

120597119867

120597119879 120582

1015840

4= minus

120597119867

120597119876

(61)




120597119867

1205971199061

= 11988811199061(119905) + 120582

1

120573119878119860

119873minus 1205822

120573119878119860

119873minus 11990811+ 11990812= 0

120597119867

1205971199062

= 11988821199062(119905) minus 120582

2119860 + 120582

3119860 minus 119908

21+ 11990822= 0

(62)


119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811minus 11990812] (63)


11and119908




1(119905) lt 1 we have 119908

11(119905) =



119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873] (64)


11(119905) = 0 Hence

1 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873minus 11990812] (65)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873ge 1 since 119908

12(119905) ge 0 (66)


12(119905) = 0 Hence

0 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811] (67)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873le 0 since 119908

11(119905) ge 0 (68)


characterized as

119906lowast

1= min1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (69)


119906lowast

2= min1max0

(1205822minus 1205823) 119860

1198882

(70)





1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119878119860

119873minus 120583119878

1198601015840= (1minusmin1max0 1

1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

120573119878119860

119873


1198882

(119905) + 120583)119860

1198791015840= min1max0

(1205822minus 1205823) 119860

1198882

(119905) 119860

minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

1205821015840

1= 1205821(1 minusmin1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119860

119873+ 1205831205821


1198881

[(1205822minus1205821)120573119878119860

119873](119905))

times120573119860

119873

1205821015840

2= minus1


1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

times120573119878


(1205822minus 1205823) 119860

1198882

(119905)


1198881

[(1205822minus 1205821)120573119878119860

119873])

times120573119878

119873+ 1205822(120583+min1max0

(1205822minus 1205823) 119860

1198882

(119905))

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

119878 (0) = 1198780 119860 (0) = 119860

0 119879 (0) = 119879

0

119876 (0) = 1198760 120582

119894(119905119891) = 0 119894 = 1 2 3 4

(71)









119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

(72)


119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

(73)


120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902


119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902 1205821= 119890minus120582119905119903 1205822=



+ 120582119898 = 120583119873119890minus120582119905


120582119905

1198881119873

)

times120573119898119899119890

120582119905

119873minus 120583119898

(74)


119899 + 120582119899


120582119905

1198881119873

)120573119898119899119890

120582119905

119873


+ 120583) 119899

(75)


+ 120582119901 = min1max0 119899 (119904 minus 119908)1198882

119899

minus (120583 + 120585 + 120575) 119901

(76)


119902 + 120582119902 = 120575119901 minus 120583119902 (77)



120582119905

1198881119873

)120573119899119890120582119905

119873


120582119905

1198881119873

)

times120573119899119890120582119905

119873

(78)


119904 minus 120582119904 = minus119890120582119905


120582119905

1198881119873

)120573119898119890120582119905

119873


120582119905

1198881119873

)

times120573119898119890120582119905

119873+ 119904(min1max0 119899 (119904 minus 119908)

1198882

+ 120583)


)

(79)




V minus 120582V = 120583V (81)




1003816100381610038161003816119906lowast

1(119905) minus 119906

lowast

1(119905)1003816100381610038161003816 le

120573119890120582119905


1003816100381610038161003816119906lowast

2(119905) minus 119906

lowast

2(119905)1003816100381610038161003816 le

1

1198882


(82)




minus + (120583 + 120582) (119898 minus 119898)

=120573119890120582119905


120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

)119898119899


120582119905

1198881119873

)119898119899]

(83)


119891gives

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873


1)119898119899 + (1 minus 119906

lowast

1)119898119899] 119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873[(119906lowast


+ 119898119899 (119906lowast

1minus 119906lowast

1)] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [119906lowast

1minus 1

100381610038161003816100381610038161003816100381610038161003816

(119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

100381610038161003816100381610038161003816100381610038161003816


le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 (119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

|119898119899 (119904 minus 119904) + (119898119899 minus 119898119899) 119904


le120573119890120582119905119891

119873int

119905119891

0

[1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 |119898| |119898 minus 119898| |119899 minus 119899|

+ |119898 minus 119898|2

|119899| + |119861| |119904| |119899| |119898 minus 119898|2

+ |119861| |119898119899| |119898 minus 119898| |119903 minus 119903|

+ |119861| |119898119899| |119898 minus 119898| |119904 minus 119904|

+ |119861| |119904| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119899| |119898 minus 119898|2] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[

1003816100381610038161003816119906lowastminus 1

1003816100381610038161003816

2|119898| ((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119904 minus 119904)

2)

+|119861| |119904| |119898|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119904| |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119903 minus 119903)

2)

+|119861| |119898| |119903|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119903| |119899| (119898 minus 119898)2]119889119905

(84)


1(119905) minus 119906

lowast





2(119898 minus119898)

2 (119898 minus119898)2+ (119904 minus 119904)

2 (119898 minus119898)2+ (119899 minus 119899)

2 (119898 minus119898)2

(119898minus119898)2+(119903minus119903)



2such that

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905


le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)


1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)


(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)




1= 0 119906

2= 0 Before








0ge













0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Proof Since

119869 (1198640)

= (

(1 minus 1199061) 120573 minus (120583 + 119906

2) 120585 0 0

1199062

minus (120583 + 120585 + 120575) 0 0

0 120575 minus120583 0


)

(25)



1205822+ [2120583 + 120585 + 120575 + 119906


1) 120573] 120582

+ (120583 + 120585 + 120575) (120583 + 1199062minus (1 minus 119906

1) 120573) minus 119906

2120585 = 0

(26)

Thus

1205823+ 1205824= (1 minus 119906

1) 120573 minus (120583 + 120585 + 120575) minus (120583 + 119906

2) (27)


120573 (1 minus 1199061) (120583 + 120585 + 120575) lt 119906

2(120583 + 120575) + 120583 (120583 + 120585 + 120575)

lt (120583 + 1199062) (120583 + 120585 + 120575)

(28)

so

120573 (1 minus 1199061) lt 120583 + 119906

2 (29)

and then

1205823+ 1205824lt minus (120583 + 120585 + 120575) lt 0 (30)

while

12058231205824= (120583 + 120585 + 120575) (120583 + 119906


1) 120573) minus 119906

2120585 (31)



2(120583 + 120575) minus 120583 (120583 + 120585 + 120575) (32)

so12058231205824gt (120583 + 119906

2) (120583 + 120585 + 120575)


(33)

It reduces to

12058231205824gt 0 (34)

Hence Re 1205823lt 0 Re 120582






1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(35)


(

) = (119865 minus 119881)(

119860

119879

119876


119873)

times (

120573 (1 minus 1199061) 0 0

0 0 0

0 0 0

)(

119860

119879

119876

)

(36)


(

) le (119865 minus 119881)(

119860

119879

119876

) (37)


0lt 1 So (119860(119905) 119879(119905) 119876(119905)) rarr





0gt 1


1198781015840= 120583119873 minus (1 minus 119906

1)120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1)120573119878119860

119873+ 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

(38)


1198781015840= 120583 minus (1 minus 119906

1) 120573119878119860 minus 120583119878

1199041198601015840= (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

1198791015840= 1199062119860 minus (120583 + 120585 + 120575) 119879

(39)




lowast

1199062119860lowast= (120583 + 120585 + 120575) 119879

lowast

(40)


Let 119881 = 1199091(119878 minus 119878


2(119860 minus 119860

lowastminus

119860lowast ln(119860119860lowast)) + 119909

3(119879 minus 119879


119881101584010038161003816100381610038161003816(39)

= 1199091[120583 minus (1 minus 119906

1) 120573119878119860 minus 120583119878

minus119878lowast

119878120583 + (1 minus 119906

1) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860


119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091[ (1 minus 119906


1) 120573119878119860 minus 120583119878

minus119878lowast

119878((1 minus 119906


+ (1 minus 1199061) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860


119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091120583119878lowast(2 minus

119878


119878)

+ [1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast]

minus [1199091

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878+ 1199092(1 minus 119906

1) 120573119878119860

lowast

+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

+ 119878119860 [minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573]

+ 119860 [minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093]

+ 119879 [1205851199092minus (120583 + 120585 + 120575) 119909

3]

(41)


minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573 = 0

minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093= 0

1205851199092minus (120583 + 120585 + 120575) 119909

3= 0

(42)


1199091= 1 119909

2= 1

1199093=

120585

120583 + 120585 + 120575=1199062+ 120583 minus (1 minus 119906

1) 120573119878lowast

1199062

(43)

Next we let

1198811015840

1= 1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast

1198811015840

2= minus[119909

1

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ 1199092(1 minus 119906

1) 120573119878119860

lowast+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

(44)

and then

1198811015840= 1199091120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2 (45)

Due to

(1199062+ 120583)119860



1199093(120583 + 120585 + 120575) 119879


(46)


so

1198811015840

1= 2 (1 minus 119906

1) 120573119878lowast119860lowast+ 2120585119879

lowast

1198811015840

2= minus[

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ (1 minus 1199061) 120573119878119860

lowast+119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]


(1 minus 1199061) 120573119860lowast

119878sdot (1 minus 119906

1) 120573119878119860

lowast]

12

minus 2 [119860lowast120585119879

119860sdot 1199093

119879lowast

1198791199062119860]



lowast(11990931199062119860lowast)]12


minus 2120585119879lowast[(1199062+ 120583)119860


1) 120573119878lowast119860lowast]12




lowast

(47)

Hence

1198811015840= 120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2


119878


119878) + 2 (1 minus 119906

1) 120573119878lowast119860lowast



lowast

= 120583119878lowast(2 minus

119878


119878) le 0

(48)

1198811015840= 0 if and only if 119878 = 119878

lowast 119860 = 119860lowast 119879 = 119879







1(119905)




119869 (1199061 1199062) = int

119905119891

0

[119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)] 119889119905 (49)


1198781015840= 120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860

1198791015840= 1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(50)


119878 (0) = 1198780 119860 (0) = 119860

0

119879 (0) = 1198790 119876 (0) = 119876

0

(51)

Here 119906119894(119905) isin 119880 ≜ (119906


119906119894(119905) le 1 for all 119905 isin [0 119905






=

(119906lowast

1 119906lowast


119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) 119906

1(119905) 1199062(119905) isin 119880 (52)






1 1198892gt 0 and 120572 gt 1 such


12)1199062

1(119905) +

(11988822)1199062


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(53)



1198781015840le 120583119873 119860

1015840le (1 minus 119906

1(119905)) 120573119878 + 120585119879

1198791015840le 1199062(119905) 119860 119876

1015840le 120575119879

(54)


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(55)


12 11988822 and for all




119867 = 119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)

+ 1205821[120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878]

+ 1205822[(1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860]

+ 1205823[1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879] + 120582

4(120575119879 minus 120583119876)

minus 119908111199061(119905) minus 119908

12(1 minus 119906

1(119905))

minus 119908211199062(119905) minus 119908

22(1 minus 119906

2(119905))

(56)


11990811(119905) 1199061(119905) = 119908

12(119905) (1 minus 119906

1(119905))


11990821(119905) 1199062(119905) = 119908

22(119905) (1 minus 119906

2(119905))


(57)


2) and solutions


119894 119894 = 1 2 3 4 satisfying

1205821015840

1= 1205821(1 minus 119906

1(119905))

120573119860

119873+ 1205831205821minus 1205822(1 minus 119906

1(119905))

120573119860

119873

1205821015840

2= minus1 + 120582

1(1 minus 119906

1(119905))

120573119878

119873minus 1205822(1 minus 119906

1(119905))

120573119878

119873

+ 1205822(120583 + 119906

2(119905)) minus 120582

31199062(119905)

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

(58)


120582119894(119905119891) = 0 119894 = 1 2 3 4 (59)



119906lowast

1= min(1max(0

120573119878119860 (1205822minus 1205821)

1198881119873

))

119906lowast

2= min(1max(0

119860 (1205822minus 1205823)

1198882

))

(60)


1205821015840

1= minus

120597119867

120597119878 120582

1015840

2= minus

120597119867

120597119860

1205821015840

3= minus

120597119867

120597119879 120582

1015840

4= minus

120597119867

120597119876

(61)




120597119867

1205971199061

= 11988811199061(119905) + 120582

1

120573119878119860

119873minus 1205822

120573119878119860

119873minus 11990811+ 11990812= 0

120597119867

1205971199062

= 11988821199062(119905) minus 120582

2119860 + 120582

3119860 minus 119908

21+ 11990822= 0

(62)


119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811minus 11990812] (63)


11and119908




1(119905) lt 1 we have 119908

11(119905) =



119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873] (64)


11(119905) = 0 Hence

1 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873minus 11990812] (65)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873ge 1 since 119908

12(119905) ge 0 (66)


12(119905) = 0 Hence

0 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811] (67)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873le 0 since 119908

11(119905) ge 0 (68)


characterized as

119906lowast

1= min1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (69)


119906lowast

2= min1max0

(1205822minus 1205823) 119860

1198882

(70)





1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119878119860

119873minus 120583119878

1198601015840= (1minusmin1max0 1

1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

120573119878119860

119873


1198882

(119905) + 120583)119860

1198791015840= min1max0

(1205822minus 1205823) 119860

1198882

(119905) 119860

minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

1205821015840

1= 1205821(1 minusmin1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119860

119873+ 1205831205821


1198881

[(1205822minus1205821)120573119878119860

119873](119905))

times120573119860

119873

1205821015840

2= minus1


1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

times120573119878


(1205822minus 1205823) 119860

1198882

(119905)


1198881

[(1205822minus 1205821)120573119878119860

119873])

times120573119878

119873+ 1205822(120583+min1max0

(1205822minus 1205823) 119860

1198882

(119905))

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

119878 (0) = 1198780 119860 (0) = 119860

0 119879 (0) = 119879

0

119876 (0) = 1198760 120582

119894(119905119891) = 0 119894 = 1 2 3 4

(71)









119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

(72)


119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

(73)


120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902


119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902 1205821= 119890minus120582119905119903 1205822=



+ 120582119898 = 120583119873119890minus120582119905


120582119905

1198881119873

)

times120573119898119899119890

120582119905

119873minus 120583119898

(74)


119899 + 120582119899


120582119905

1198881119873

)120573119898119899119890

120582119905

119873


+ 120583) 119899

(75)


+ 120582119901 = min1max0 119899 (119904 minus 119908)1198882

119899

minus (120583 + 120585 + 120575) 119901

(76)


119902 + 120582119902 = 120575119901 minus 120583119902 (77)



120582119905

1198881119873

)120573119899119890120582119905

119873


120582119905

1198881119873

)

times120573119899119890120582119905

119873

(78)


119904 minus 120582119904 = minus119890120582119905


120582119905

1198881119873

)120573119898119890120582119905

119873


120582119905

1198881119873

)

times120573119898119890120582119905

119873+ 119904(min1max0 119899 (119904 minus 119908)

1198882

+ 120583)


)

(79)




V minus 120582V = 120583V (81)




1003816100381610038161003816119906lowast

1(119905) minus 119906

lowast

1(119905)1003816100381610038161003816 le

120573119890120582119905


1003816100381610038161003816119906lowast

2(119905) minus 119906

lowast

2(119905)1003816100381610038161003816 le

1

1198882


(82)




minus + (120583 + 120582) (119898 minus 119898)

=120573119890120582119905


120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

)119898119899


120582119905

1198881119873

)119898119899]

(83)


119891gives

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873


1)119898119899 + (1 minus 119906

lowast

1)119898119899] 119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873[(119906lowast


+ 119898119899 (119906lowast

1minus 119906lowast

1)] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [119906lowast

1minus 1

100381610038161003816100381610038161003816100381610038161003816

(119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

100381610038161003816100381610038161003816100381610038161003816


le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 (119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

|119898119899 (119904 minus 119904) + (119898119899 minus 119898119899) 119904


le120573119890120582119905119891

119873int

119905119891

0

[1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 |119898| |119898 minus 119898| |119899 minus 119899|

+ |119898 minus 119898|2

|119899| + |119861| |119904| |119899| |119898 minus 119898|2

+ |119861| |119898119899| |119898 minus 119898| |119903 minus 119903|

+ |119861| |119898119899| |119898 minus 119898| |119904 minus 119904|

+ |119861| |119904| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119899| |119898 minus 119898|2] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[

1003816100381610038161003816119906lowastminus 1

1003816100381610038161003816

2|119898| ((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119904 minus 119904)

2)

+|119861| |119904| |119898|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119904| |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119903 minus 119903)

2)

+|119861| |119898| |119903|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119903| |119899| (119898 minus 119898)2]119889119905

(84)


1(119905) minus 119906

lowast





2(119898 minus119898)

2 (119898 minus119898)2+ (119904 minus 119904)

2 (119898 minus119898)2+ (119899 minus 119899)

2 (119898 minus119898)2

(119898minus119898)2+(119903minus119903)



2such that

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905


le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)


1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)


(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)




1= 0 119906

2= 0 Before








0ge













0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Let 119881 = 1199091(119878 minus 119878


2(119860 minus 119860

lowastminus

119860lowast ln(119860119860lowast)) + 119909

3(119879 minus 119879


119881101584010038161003816100381610038161003816(39)

= 1199091[120583 minus (1 minus 119906

1) 120573119878119860 minus 120583119878

minus119878lowast

119878120583 + (1 minus 119906

1) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860


119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091[ (1 minus 119906


1) 120573119878119860 minus 120583119878

minus119878lowast

119878((1 minus 119906


+ (1 minus 1199061) 120573119878lowast119860 + 120583119878

lowast]

+ 1199092[ (1 minus 119906

1) 120573119878119860 + 120585119879 minus (119906

2+ 120583)119860

minus (1 minus 1199061) 120573119878119860


119860120585119879 + (119906

2+ 120583)119860

lowast]

+ 1199093[1199062119860 minus (120583 + 120585 + 120575) 119879

minus119879lowast

1198791199062119860 + (120583 + 120585 + 120575) 119879

lowast]

= 1199091120583119878lowast(2 minus

119878


119878)

+ [1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast]

minus [1199091

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878+ 1199092(1 minus 119906

1) 120573119878119860

lowast

+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

+ 119878119860 [minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573]

+ 119860 [minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093]

+ 119879 [1205851199092minus (120583 + 120585 + 120575) 119909

3]

(41)


minus1199091(1 minus 119906

1) 120573 + 119909

2(1 minus 119906

1) 120573 = 0

minus (1199062+ 120583) 119909

2+ 1199091(1 minus 119906

1) 120573119878lowast+ 11990621199093= 0

1205851199092minus (120583 + 120585 + 120575) 119909

3= 0

(42)


1199091= 1 119909

2= 1

1199093=

120585

120583 + 120585 + 120575=1199062+ 120583 minus (1 minus 119906

1) 120573119878lowast

1199062

(43)

Next we let

1198811015840

1= 1199091(1 minus 119906

1) 120573119878lowast119860lowast+ 1199092(1199062+ 120583)119860

lowast

+ 1199093(120583 + 120585 + 120575) 119879

lowast

1198811015840

2= minus[119909

1

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ 1199092(1 minus 119906

1) 120573119878119860

lowast+ 1199092

119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]

(44)

and then

1198811015840= 1199091120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2 (45)

Due to

(1199062+ 120583)119860



1199093(120583 + 120585 + 120575) 119879


(46)


so

1198811015840

1= 2 (1 minus 119906

1) 120573119878lowast119860lowast+ 2120585119879

lowast

1198811015840

2= minus[

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ (1 minus 1199061) 120573119878119860

lowast+119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]


(1 minus 1199061) 120573119860lowast

119878sdot (1 minus 119906

1) 120573119878119860

lowast]

12

minus 2 [119860lowast120585119879

119860sdot 1199093

119879lowast

1198791199062119860]



lowast(11990931199062119860lowast)]12


minus 2120585119879lowast[(1199062+ 120583)119860


1) 120573119878lowast119860lowast]12




lowast

(47)

Hence

1198811015840= 120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2


119878


119878) + 2 (1 minus 119906

1) 120573119878lowast119860lowast



lowast

= 120583119878lowast(2 minus

119878


119878) le 0

(48)

1198811015840= 0 if and only if 119878 = 119878

lowast 119860 = 119860lowast 119879 = 119879







1(119905)




119869 (1199061 1199062) = int

119905119891

0

[119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)] 119889119905 (49)


1198781015840= 120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860

1198791015840= 1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(50)


119878 (0) = 1198780 119860 (0) = 119860

0

119879 (0) = 1198790 119876 (0) = 119876

0

(51)

Here 119906119894(119905) isin 119880 ≜ (119906


119906119894(119905) le 1 for all 119905 isin [0 119905






=

(119906lowast

1 119906lowast


119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) 119906

1(119905) 1199062(119905) isin 119880 (52)






1 1198892gt 0 and 120572 gt 1 such


12)1199062

1(119905) +

(11988822)1199062


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(53)



1198781015840le 120583119873 119860

1015840le (1 minus 119906

1(119905)) 120573119878 + 120585119879

1198791015840le 1199062(119905) 119860 119876

1015840le 120575119879

(54)


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(55)


12 11988822 and for all




119867 = 119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)

+ 1205821[120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878]

+ 1205822[(1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860]

+ 1205823[1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879] + 120582

4(120575119879 minus 120583119876)

minus 119908111199061(119905) minus 119908

12(1 minus 119906

1(119905))

minus 119908211199062(119905) minus 119908

22(1 minus 119906

2(119905))

(56)


11990811(119905) 1199061(119905) = 119908

12(119905) (1 minus 119906

1(119905))


11990821(119905) 1199062(119905) = 119908

22(119905) (1 minus 119906

2(119905))


(57)


2) and solutions


119894 119894 = 1 2 3 4 satisfying

1205821015840

1= 1205821(1 minus 119906

1(119905))

120573119860

119873+ 1205831205821minus 1205822(1 minus 119906

1(119905))

120573119860

119873

1205821015840

2= minus1 + 120582

1(1 minus 119906

1(119905))

120573119878

119873minus 1205822(1 minus 119906

1(119905))

120573119878

119873

+ 1205822(120583 + 119906

2(119905)) minus 120582

31199062(119905)

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

(58)


120582119894(119905119891) = 0 119894 = 1 2 3 4 (59)



119906lowast

1= min(1max(0

120573119878119860 (1205822minus 1205821)

1198881119873

))

119906lowast

2= min(1max(0

119860 (1205822minus 1205823)

1198882

))

(60)


1205821015840

1= minus

120597119867

120597119878 120582

1015840

2= minus

120597119867

120597119860

1205821015840

3= minus

120597119867

120597119879 120582

1015840

4= minus

120597119867

120597119876

(61)




120597119867

1205971199061

= 11988811199061(119905) + 120582

1

120573119878119860

119873minus 1205822

120573119878119860

119873minus 11990811+ 11990812= 0

120597119867

1205971199062

= 11988821199062(119905) minus 120582

2119860 + 120582

3119860 minus 119908

21+ 11990822= 0

(62)


119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811minus 11990812] (63)


11and119908




1(119905) lt 1 we have 119908

11(119905) =



119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873] (64)


11(119905) = 0 Hence

1 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873minus 11990812] (65)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873ge 1 since 119908

12(119905) ge 0 (66)


12(119905) = 0 Hence

0 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811] (67)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873le 0 since 119908

11(119905) ge 0 (68)


characterized as

119906lowast

1= min1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (69)


119906lowast

2= min1max0

(1205822minus 1205823) 119860

1198882

(70)





1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119878119860

119873minus 120583119878

1198601015840= (1minusmin1max0 1

1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

120573119878119860

119873


1198882

(119905) + 120583)119860

1198791015840= min1max0

(1205822minus 1205823) 119860

1198882

(119905) 119860

minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

1205821015840

1= 1205821(1 minusmin1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119860

119873+ 1205831205821


1198881

[(1205822minus1205821)120573119878119860

119873](119905))

times120573119860

119873

1205821015840

2= minus1


1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

times120573119878


(1205822minus 1205823) 119860

1198882

(119905)


1198881

[(1205822minus 1205821)120573119878119860

119873])

times120573119878

119873+ 1205822(120583+min1max0

(1205822minus 1205823) 119860

1198882

(119905))

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

119878 (0) = 1198780 119860 (0) = 119860

0 119879 (0) = 119879

0

119876 (0) = 1198760 120582

119894(119905119891) = 0 119894 = 1 2 3 4

(71)









119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

(72)


119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

(73)


120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902


119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902 1205821= 119890minus120582119905119903 1205822=



+ 120582119898 = 120583119873119890minus120582119905


120582119905

1198881119873

)

times120573119898119899119890

120582119905

119873minus 120583119898

(74)


119899 + 120582119899


120582119905

1198881119873

)120573119898119899119890

120582119905

119873


+ 120583) 119899

(75)


+ 120582119901 = min1max0 119899 (119904 minus 119908)1198882

119899

minus (120583 + 120585 + 120575) 119901

(76)


119902 + 120582119902 = 120575119901 minus 120583119902 (77)



120582119905

1198881119873

)120573119899119890120582119905

119873


120582119905

1198881119873

)

times120573119899119890120582119905

119873

(78)


119904 minus 120582119904 = minus119890120582119905


120582119905

1198881119873

)120573119898119890120582119905

119873


120582119905

1198881119873

)

times120573119898119890120582119905

119873+ 119904(min1max0 119899 (119904 minus 119908)

1198882

+ 120583)


)

(79)




V minus 120582V = 120583V (81)




1003816100381610038161003816119906lowast

1(119905) minus 119906

lowast

1(119905)1003816100381610038161003816 le

120573119890120582119905


1003816100381610038161003816119906lowast

2(119905) minus 119906

lowast

2(119905)1003816100381610038161003816 le

1

1198882


(82)




minus + (120583 + 120582) (119898 minus 119898)

=120573119890120582119905


120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

)119898119899


120582119905

1198881119873

)119898119899]

(83)


119891gives

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873


1)119898119899 + (1 minus 119906

lowast

1)119898119899] 119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873[(119906lowast


+ 119898119899 (119906lowast

1minus 119906lowast

1)] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [119906lowast

1minus 1

100381610038161003816100381610038161003816100381610038161003816

(119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

100381610038161003816100381610038161003816100381610038161003816


le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 (119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

|119898119899 (119904 minus 119904) + (119898119899 minus 119898119899) 119904


le120573119890120582119905119891

119873int

119905119891

0

[1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 |119898| |119898 minus 119898| |119899 minus 119899|

+ |119898 minus 119898|2

|119899| + |119861| |119904| |119899| |119898 minus 119898|2

+ |119861| |119898119899| |119898 minus 119898| |119903 minus 119903|

+ |119861| |119898119899| |119898 minus 119898| |119904 minus 119904|

+ |119861| |119904| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119899| |119898 minus 119898|2] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[

1003816100381610038161003816119906lowastminus 1

1003816100381610038161003816

2|119898| ((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119904 minus 119904)

2)

+|119861| |119904| |119898|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119904| |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119903 minus 119903)

2)

+|119861| |119898| |119903|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119903| |119899| (119898 minus 119898)2]119889119905

(84)


1(119905) minus 119906

lowast





2(119898 minus119898)

2 (119898 minus119898)2+ (119904 minus 119904)

2 (119898 minus119898)2+ (119899 minus 119899)

2 (119898 minus119898)2

(119898minus119898)2+(119903minus119903)



2such that

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905


le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)


1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)


(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)




1= 0 119906

2= 0 Before








0ge













0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










so

1198811015840

1= 2 (1 minus 119906

1) 120573119878lowast119860lowast+ 2120585119879

lowast

1198811015840

2= minus[

(119878lowast)2

(1 minus 1199061) 120573119860lowast

119878

+ (1 minus 1199061) 120573119878119860

lowast+119860lowast120585119879

119860+ 1199093

119879lowast

1198791199062119860]


(1 minus 1199061) 120573119860lowast

119878sdot (1 minus 119906

1) 120573119878119860

lowast]

12

minus 2 [119860lowast120585119879

119860sdot 1199093

119879lowast

1198791199062119860]



lowast(11990931199062119860lowast)]12


minus 2120585119879lowast[(1199062+ 120583)119860


1) 120573119878lowast119860lowast]12




lowast

(47)

Hence

1198811015840= 120583119878lowast(2 minus

119878


119878) + 119881

1015840

1+ 1198811015840

2


119878


119878) + 2 (1 minus 119906

1) 120573119878lowast119860lowast



lowast

= 120583119878lowast(2 minus

119878


119878) le 0

(48)

1198811015840= 0 if and only if 119878 = 119878

lowast 119860 = 119860lowast 119879 = 119879







1(119905)




119869 (1199061 1199062) = int

119905119891

0

[119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)] 119889119905 (49)


1198781015840= 120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878

1198601015840= (1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860

1198791015840= 1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

(50)


119878 (0) = 1198780 119860 (0) = 119860

0

119879 (0) = 1198790 119876 (0) = 119876

0

(51)

Here 119906119894(119905) isin 119880 ≜ (119906


119906119894(119905) le 1 for all 119905 isin [0 119905






=

(119906lowast

1 119906lowast


119869 (119906lowast

1 119906lowast

2) = min 119869 (119906

1 1199062) 119906

1(119905) 1199062(119905) isin 119880 (52)






1 1198892gt 0 and 120572 gt 1 such


12)1199062

1(119905) +

(11988822)1199062


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(53)



1198781015840le 120583119873 119860

1015840le (1 minus 119906

1(119905)) 120573119878 + 120585119879

1198791015840le 1199062(119905) 119860 119876

1015840le 120575119879

(54)


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(55)


12 11988822 and for all




119867 = 119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)

+ 1205821[120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878]

+ 1205822[(1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860]

+ 1205823[1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879] + 120582

4(120575119879 minus 120583119876)

minus 119908111199061(119905) minus 119908

12(1 minus 119906

1(119905))

minus 119908211199062(119905) minus 119908

22(1 minus 119906

2(119905))

(56)


11990811(119905) 1199061(119905) = 119908

12(119905) (1 minus 119906

1(119905))


11990821(119905) 1199062(119905) = 119908

22(119905) (1 minus 119906

2(119905))


(57)


2) and solutions


119894 119894 = 1 2 3 4 satisfying

1205821015840

1= 1205821(1 minus 119906

1(119905))

120573119860

119873+ 1205831205821minus 1205822(1 minus 119906

1(119905))

120573119860

119873

1205821015840

2= minus1 + 120582

1(1 minus 119906

1(119905))

120573119878

119873minus 1205822(1 minus 119906

1(119905))

120573119878

119873

+ 1205822(120583 + 119906

2(119905)) minus 120582

31199062(119905)

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

(58)


120582119894(119905119891) = 0 119894 = 1 2 3 4 (59)



119906lowast

1= min(1max(0

120573119878119860 (1205822minus 1205821)

1198881119873

))

119906lowast

2= min(1max(0

119860 (1205822minus 1205823)

1198882

))

(60)


1205821015840

1= minus

120597119867

120597119878 120582

1015840

2= minus

120597119867

120597119860

1205821015840

3= minus

120597119867

120597119879 120582

1015840

4= minus

120597119867

120597119876

(61)




120597119867

1205971199061

= 11988811199061(119905) + 120582

1

120573119878119860

119873minus 1205822

120573119878119860

119873minus 11990811+ 11990812= 0

120597119867

1205971199062

= 11988821199062(119905) minus 120582

2119860 + 120582

3119860 minus 119908

21+ 11990822= 0

(62)


119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811minus 11990812] (63)


11and119908




1(119905) lt 1 we have 119908

11(119905) =



119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873] (64)


11(119905) = 0 Hence

1 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873minus 11990812] (65)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873ge 1 since 119908

12(119905) ge 0 (66)


12(119905) = 0 Hence

0 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811] (67)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873le 0 since 119908

11(119905) ge 0 (68)


characterized as

119906lowast

1= min1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (69)


119906lowast

2= min1max0

(1205822minus 1205823) 119860

1198882

(70)





1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119878119860

119873minus 120583119878

1198601015840= (1minusmin1max0 1

1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

120573119878119860

119873


1198882

(119905) + 120583)119860

1198791015840= min1max0

(1205822minus 1205823) 119860

1198882

(119905) 119860

minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

1205821015840

1= 1205821(1 minusmin1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119860

119873+ 1205831205821


1198881

[(1205822minus1205821)120573119878119860

119873](119905))

times120573119860

119873

1205821015840

2= minus1


1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

times120573119878


(1205822minus 1205823) 119860

1198882

(119905)


1198881

[(1205822minus 1205821)120573119878119860

119873])

times120573119878

119873+ 1205822(120583+min1max0

(1205822minus 1205823) 119860

1198882

(119905))

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

119878 (0) = 1198780 119860 (0) = 119860

0 119879 (0) = 119879

0

119876 (0) = 1198760 120582

119894(119905119891) = 0 119894 = 1 2 3 4

(71)









119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

(72)


119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

(73)


120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902


119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902 1205821= 119890minus120582119905119903 1205822=



+ 120582119898 = 120583119873119890minus120582119905


120582119905

1198881119873

)

times120573119898119899119890

120582119905

119873minus 120583119898

(74)


119899 + 120582119899


120582119905

1198881119873

)120573119898119899119890

120582119905

119873


+ 120583) 119899

(75)


+ 120582119901 = min1max0 119899 (119904 minus 119908)1198882

119899

minus (120583 + 120585 + 120575) 119901

(76)


119902 + 120582119902 = 120575119901 minus 120583119902 (77)



120582119905

1198881119873

)120573119899119890120582119905

119873


120582119905

1198881119873

)

times120573119899119890120582119905

119873

(78)


119904 minus 120582119904 = minus119890120582119905


120582119905

1198881119873

)120573119898119890120582119905

119873


120582119905

1198881119873

)

times120573119898119890120582119905

119873+ 119904(min1max0 119899 (119904 minus 119908)

1198882

+ 120583)


)

(79)




V minus 120582V = 120583V (81)




1003816100381610038161003816119906lowast

1(119905) minus 119906

lowast

1(119905)1003816100381610038161003816 le

120573119890120582119905


1003816100381610038161003816119906lowast

2(119905) minus 119906

lowast

2(119905)1003816100381610038161003816 le

1

1198882


(82)




minus + (120583 + 120582) (119898 minus 119898)

=120573119890120582119905


120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

)119898119899


120582119905

1198881119873

)119898119899]

(83)


119891gives

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873


1)119898119899 + (1 minus 119906

lowast

1)119898119899] 119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873[(119906lowast


+ 119898119899 (119906lowast

1minus 119906lowast

1)] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [119906lowast

1minus 1

100381610038161003816100381610038161003816100381610038161003816

(119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

100381610038161003816100381610038161003816100381610038161003816


le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 (119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

|119898119899 (119904 minus 119904) + (119898119899 minus 119898119899) 119904


le120573119890120582119905119891

119873int

119905119891

0

[1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 |119898| |119898 minus 119898| |119899 minus 119899|

+ |119898 minus 119898|2

|119899| + |119861| |119904| |119899| |119898 minus 119898|2

+ |119861| |119898119899| |119898 minus 119898| |119903 minus 119903|

+ |119861| |119898119899| |119898 minus 119898| |119904 minus 119904|

+ |119861| |119904| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119899| |119898 minus 119898|2] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[

1003816100381610038161003816119906lowastminus 1

1003816100381610038161003816

2|119898| ((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119904 minus 119904)

2)

+|119861| |119904| |119898|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119904| |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119903 minus 119903)

2)

+|119861| |119898| |119903|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119903| |119899| (119898 minus 119898)2]119889119905

(84)


1(119905) minus 119906

lowast





2(119898 minus119898)

2 (119898 minus119898)2+ (119904 minus 119904)

2 (119898 minus119898)2+ (119899 minus 119899)

2 (119898 minus119898)2

(119898minus119898)2+(119903minus119903)



2such that

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905


le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)


1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)


(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)




1= 0 119906

2= 0 Before








0ge













0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198781015840le 120583119873 119860

1015840le (1 minus 119906

1(119905)) 120573119878 + 120585119879

1198791015840le 1199062(119905) 119860 119876

1015840le 120575119879

(54)


119871 (119905 1199061 1199062) ge 1198891(100381610038161003816100381611990611003816100381610038161003816

2

+100381610038161003816100381611990621003816100381610038161003816

2

)1205722

minus 1198892

(55)


12 11988822 and for all




119867 = 119860 (119905) +1198881

21199062

1(119905) +

1198882

21199062

2(119905)

+ 1205821[120583119873 minus (1 minus 119906

1(119905))

120573119878119860

119873minus 120583119878]

+ 1205822[(1 minus 119906

1(119905))

120573119878119860

119873+ 120585119879 minus (119906

2(119905) + 120583)119860]

+ 1205823[1199062(119905) 119860 minus (120583 + 120585 + 120575) 119879] + 120582

4(120575119879 minus 120583119876)

minus 119908111199061(119905) minus 119908

12(1 minus 119906

1(119905))

minus 119908211199062(119905) minus 119908

22(1 minus 119906

2(119905))

(56)


11990811(119905) 1199061(119905) = 119908

12(119905) (1 minus 119906

1(119905))


11990821(119905) 1199062(119905) = 119908

22(119905) (1 minus 119906

2(119905))


(57)


2) and solutions


119894 119894 = 1 2 3 4 satisfying

1205821015840

1= 1205821(1 minus 119906

1(119905))

120573119860

119873+ 1205831205821minus 1205822(1 minus 119906

1(119905))

120573119860

119873

1205821015840

2= minus1 + 120582

1(1 minus 119906

1(119905))

120573119878

119873minus 1205822(1 minus 119906

1(119905))

120573119878

119873

+ 1205822(120583 + 119906

2(119905)) minus 120582

31199062(119905)

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

(58)


120582119894(119905119891) = 0 119894 = 1 2 3 4 (59)



119906lowast

1= min(1max(0

120573119878119860 (1205822minus 1205821)

1198881119873

))

119906lowast

2= min(1max(0

119860 (1205822minus 1205823)

1198882

))

(60)


1205821015840

1= minus

120597119867

120597119878 120582

1015840

2= minus

120597119867

120597119860

1205821015840

3= minus

120597119867

120597119879 120582

1015840

4= minus

120597119867

120597119876

(61)




120597119867

1205971199061

= 11988811199061(119905) + 120582

1

120573119878119860

119873minus 1205822

120573119878119860

119873minus 11990811+ 11990812= 0

120597119867

1205971199062

= 11988821199062(119905) minus 120582

2119860 + 120582

3119860 minus 119908

21+ 11990822= 0

(62)


119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811minus 11990812] (63)


11and119908




1(119905) lt 1 we have 119908

11(119905) =



119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873] (64)


11(119905) = 0 Hence

1 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873minus 11990812] (65)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873ge 1 since 119908

12(119905) ge 0 (66)


12(119905) = 0 Hence

0 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811] (67)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873le 0 since 119908

11(119905) ge 0 (68)


characterized as

119906lowast

1= min1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (69)


119906lowast

2= min1max0

(1205822minus 1205823) 119860

1198882

(70)





1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119878119860

119873minus 120583119878

1198601015840= (1minusmin1max0 1

1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

120573119878119860

119873


1198882

(119905) + 120583)119860

1198791015840= min1max0

(1205822minus 1205823) 119860

1198882

(119905) 119860

minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

1205821015840

1= 1205821(1 minusmin1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119860

119873+ 1205831205821


1198881

[(1205822minus1205821)120573119878119860

119873](119905))

times120573119860

119873

1205821015840

2= minus1


1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

times120573119878


(1205822minus 1205823) 119860

1198882

(119905)


1198881

[(1205822minus 1205821)120573119878119860

119873])

times120573119878

119873+ 1205822(120583+min1max0

(1205822minus 1205823) 119860

1198882

(119905))

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

119878 (0) = 1198780 119860 (0) = 119860

0 119879 (0) = 119879

0

119876 (0) = 1198760 120582

119894(119905119891) = 0 119894 = 1 2 3 4

(71)









119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

(72)


119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

(73)


120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902


119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902 1205821= 119890minus120582119905119903 1205822=



+ 120582119898 = 120583119873119890minus120582119905


120582119905

1198881119873

)

times120573119898119899119890

120582119905

119873minus 120583119898

(74)


119899 + 120582119899


120582119905

1198881119873

)120573119898119899119890

120582119905

119873


+ 120583) 119899

(75)


+ 120582119901 = min1max0 119899 (119904 minus 119908)1198882

119899

minus (120583 + 120585 + 120575) 119901

(76)


119902 + 120582119902 = 120575119901 minus 120583119902 (77)



120582119905

1198881119873

)120573119899119890120582119905

119873


120582119905

1198881119873

)

times120573119899119890120582119905

119873

(78)


119904 minus 120582119904 = minus119890120582119905


120582119905

1198881119873

)120573119898119890120582119905

119873


120582119905

1198881119873

)

times120573119898119890120582119905

119873+ 119904(min1max0 119899 (119904 minus 119908)

1198882

+ 120583)


)

(79)




V minus 120582V = 120583V (81)




1003816100381610038161003816119906lowast

1(119905) minus 119906

lowast

1(119905)1003816100381610038161003816 le

120573119890120582119905


1003816100381610038161003816119906lowast

2(119905) minus 119906

lowast

2(119905)1003816100381610038161003816 le

1

1198882


(82)




minus + (120583 + 120582) (119898 minus 119898)

=120573119890120582119905


120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

)119898119899


120582119905

1198881119873

)119898119899]

(83)


119891gives

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873


1)119898119899 + (1 minus 119906

lowast

1)119898119899] 119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873[(119906lowast


+ 119898119899 (119906lowast

1minus 119906lowast

1)] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [119906lowast

1minus 1

100381610038161003816100381610038161003816100381610038161003816

(119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

100381610038161003816100381610038161003816100381610038161003816


le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 (119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

|119898119899 (119904 minus 119904) + (119898119899 minus 119898119899) 119904


le120573119890120582119905119891

119873int

119905119891

0

[1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 |119898| |119898 minus 119898| |119899 minus 119899|

+ |119898 minus 119898|2

|119899| + |119861| |119904| |119899| |119898 minus 119898|2

+ |119861| |119898119899| |119898 minus 119898| |119903 minus 119903|

+ |119861| |119898119899| |119898 minus 119898| |119904 minus 119904|

+ |119861| |119904| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119899| |119898 minus 119898|2] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[

1003816100381610038161003816119906lowastminus 1

1003816100381610038161003816

2|119898| ((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119904 minus 119904)

2)

+|119861| |119904| |119898|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119904| |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119903 minus 119903)

2)

+|119861| |119898| |119903|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119903| |119899| (119898 minus 119898)2]119889119905

(84)


1(119905) minus 119906

lowast





2(119898 minus119898)

2 (119898 minus119898)2+ (119904 minus 119904)

2 (119898 minus119898)2+ (119899 minus 119899)

2 (119898 minus119898)2

(119898minus119898)2+(119903minus119903)



2such that

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905


le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)


1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)


(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)




1= 0 119906

2= 0 Before








0ge













0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906lowast

1=1

1198881

[(1205822minus 1205821)120573119878119860

119873] (64)


11(119905) = 0 Hence

1 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873minus 11990812] (65)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873ge 1 since 119908

12(119905) ge 0 (66)


12(119905) = 0 Hence

0 = 119906lowast

1(119905) =

1

1198881

[(1205822minus 1205821)120573119878119860

119873+ 11990811] (67)

This implies that

1

1198881

(1205822minus 1205821)120573119878119860

119873le 0 since 119908

11(119905) ge 0 (68)


characterized as

119906lowast

1= min1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (69)


119906lowast

2= min1max0

(1205822minus 1205823) 119860

1198882

(70)





1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119878119860

119873minus 120583119878

1198601015840= (1minusmin1max0 1

1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

120573119878119860

119873


1198882

(119905) + 120583)119860

1198791015840= min1max0

(1205822minus 1205823) 119860

1198882

(119905) 119860

minus (120583 + 120585 + 120575) 119879

1198761015840= 120575119879 minus 120583119876

1205821015840

1= 1205821(1 minusmin1max0 1

1198881

[(1205822minus 1205821)120573119878119860

119873] (119905))

times120573119860

119873+ 1205831205821


1198881

[(1205822minus1205821)120573119878119860

119873](119905))

times120573119860

119873

1205821015840

2= minus1


1198881

[(1205822minus1205821)120573119878119860

119873] (119905))

times120573119878


(1205822minus 1205823) 119860

1198882

(119905)


1198881

[(1205822minus 1205821)120573119878119860

119873])

times120573119878

119873+ 1205822(120583+min1max0

(1205822minus 1205823) 119860

1198882

(119905))

1205821015840

3= minus1205822120585 + 1205823(120583 + 120585 + 120575) minus 120582

4120575

1205821015840

4= 1205831205824

119878 (0) = 1198780 119860 (0) = 119860

0 119879 (0) = 119879

0

119876 (0) = 1198760 120582

119894(119905119891) = 0 119894 = 1 2 3 4

(71)









119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

(72)


119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

(73)


120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902


119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902 1205821= 119890minus120582119905119903 1205822=



+ 120582119898 = 120583119873119890minus120582119905


120582119905

1198881119873

)

times120573119898119899119890

120582119905

119873minus 120583119898

(74)


119899 + 120582119899


120582119905

1198881119873

)120573119898119899119890

120582119905

119873


+ 120583) 119899

(75)


+ 120582119901 = min1max0 119899 (119904 minus 119908)1198882

119899

minus (120583 + 120585 + 120575) 119901

(76)


119902 + 120582119902 = 120575119901 minus 120583119902 (77)



120582119905

1198881119873

)120573119899119890120582119905

119873


120582119905

1198881119873

)

times120573119899119890120582119905

119873

(78)


119904 minus 120582119904 = minus119890120582119905


120582119905

1198881119873

)120573119898119890120582119905

119873


120582119905

1198881119873

)

times120573119898119890120582119905

119873+ 119904(min1max0 119899 (119904 minus 119908)

1198882

+ 120583)


)

(79)




V minus 120582V = 120583V (81)




1003816100381610038161003816119906lowast

1(119905) minus 119906

lowast

1(119905)1003816100381610038161003816 le

120573119890120582119905


1003816100381610038161003816119906lowast

2(119905) minus 119906

lowast

2(119905)1003816100381610038161003816 le

1

1198882


(82)




minus + (120583 + 120582) (119898 minus 119898)

=120573119890120582119905


120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

)119898119899


120582119905

1198881119873

)119898119899]

(83)


119891gives

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873


1)119898119899 + (1 minus 119906

lowast

1)119898119899] 119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873[(119906lowast


+ 119898119899 (119906lowast

1minus 119906lowast

1)] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [119906lowast

1minus 1

100381610038161003816100381610038161003816100381610038161003816

(119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

100381610038161003816100381610038161003816100381610038161003816


le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 (119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

|119898119899 (119904 minus 119904) + (119898119899 minus 119898119899) 119904


le120573119890120582119905119891

119873int

119905119891

0

[1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 |119898| |119898 minus 119898| |119899 minus 119899|

+ |119898 minus 119898|2

|119899| + |119861| |119904| |119899| |119898 minus 119898|2

+ |119861| |119898119899| |119898 minus 119898| |119903 minus 119903|

+ |119861| |119898119899| |119898 minus 119898| |119904 minus 119904|

+ |119861| |119904| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119899| |119898 minus 119898|2] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[

1003816100381610038161003816119906lowastminus 1

1003816100381610038161003816

2|119898| ((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119904 minus 119904)

2)

+|119861| |119904| |119898|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119904| |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119903 minus 119903)

2)

+|119861| |119898| |119903|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119903| |119899| (119898 minus 119898)2]119889119905

(84)


1(119905) minus 119906

lowast





2(119898 minus119898)

2 (119898 minus119898)2+ (119904 minus 119904)

2 (119898 minus119898)2+ (119899 minus 119899)

2 (119898 minus119898)2

(119898minus119898)2+(119903minus119903)



2such that

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905


le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)


1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)


(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)




1= 0 119906

2= 0 Before








0ge













0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

119878 = 119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901

119876 = 119890120582119905119902 120582

1= 119890minus120582119905119903 120582

2= 119890minus120582119905119904

1205823= 119890minus120582119905119908 120582

4= 119890minus120582119905V

(72)


119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

119906lowast

1(119905) = min1max0

120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

119906lowast

2(119905) = min1max0 (119904 minus 119908) 119899

1198882

(73)


120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902


119890120582119905119898 119860 = 119890

120582119905119899 119879 = 119890

120582119905119901 119876 = 119890

120582119905119902 1205821= 119890minus120582119905119903 1205822=



+ 120582119898 = 120583119873119890minus120582119905


120582119905

1198881119873

)

times120573119898119899119890

120582119905

119873minus 120583119898

(74)


119899 + 120582119899


120582119905

1198881119873

)120573119898119899119890

120582119905

119873


+ 120583) 119899

(75)


+ 120582119901 = min1max0 119899 (119904 minus 119908)1198882

119899

minus (120583 + 120585 + 120575) 119901

(76)


119902 + 120582119902 = 120575119901 minus 120583119902 (77)



120582119905

1198881119873

)120573119899119890120582119905

119873


120582119905

1198881119873

)

times120573119899119890120582119905

119873

(78)


119904 minus 120582119904 = minus119890120582119905


120582119905

1198881119873

)120573119898119890120582119905

119873


120582119905

1198881119873

)

times120573119898119890120582119905

119873+ 119904(min1max0 119899 (119904 minus 119908)

1198882

+ 120583)


)

(79)




V minus 120582V = 120583V (81)




1003816100381610038161003816119906lowast

1(119905) minus 119906

lowast

1(119905)1003816100381610038161003816 le

120573119890120582119905


1003816100381610038161003816119906lowast

2(119905) minus 119906

lowast

2(119905)1003816100381610038161003816 le

1

1198882


(82)




minus + (120583 + 120582) (119898 minus 119898)

=120573119890120582119905


120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

)119898119899


120582119905

1198881119873

)119898119899]

(83)


119891gives

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873


1)119898119899 + (1 minus 119906

lowast

1)119898119899] 119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873[(119906lowast


+ 119898119899 (119906lowast

1minus 119906lowast

1)] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [119906lowast

1minus 1

100381610038161003816100381610038161003816100381610038161003816

(119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

100381610038161003816100381610038161003816100381610038161003816


le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 (119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

|119898119899 (119904 minus 119904) + (119898119899 minus 119898119899) 119904


le120573119890120582119905119891

119873int

119905119891

0

[1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 |119898| |119898 minus 119898| |119899 minus 119899|

+ |119898 minus 119898|2

|119899| + |119861| |119904| |119899| |119898 minus 119898|2

+ |119861| |119898119899| |119898 minus 119898| |119903 minus 119903|

+ |119861| |119898119899| |119898 minus 119898| |119904 minus 119904|

+ |119861| |119904| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119899| |119898 minus 119898|2] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[

1003816100381610038161003816119906lowastminus 1

1003816100381610038161003816

2|119898| ((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119904 minus 119904)

2)

+|119861| |119904| |119898|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119904| |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119903 minus 119903)

2)

+|119861| |119898| |119903|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119903| |119899| (119898 minus 119898)2]119889119905

(84)


1(119905) minus 119906

lowast





2(119898 minus119898)

2 (119898 minus119898)2+ (119904 minus 119904)

2 (119898 minus119898)2+ (119899 minus 119899)

2 (119898 minus119898)2

(119898minus119898)2+(119903minus119903)



2such that

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905


le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)


1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)


(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)




1= 0 119906

2= 0 Before








0ge













0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003816100381610038161003816119906lowast

1(119905) minus 119906

lowast

1(119905)1003816100381610038161003816 le

120573119890120582119905


1003816100381610038161003816119906lowast

2(119905) minus 119906

lowast

2(119905)1003816100381610038161003816 le

1

1198882


(82)




minus + (120583 + 120582) (119898 minus 119898)

=120573119890120582119905


120573119898119899 (119904 minus 119903) 119890120582119905

1198881119873

)119898119899


120582119905

1198881119873

)119898119899]

(83)


119891gives

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873


1)119898119899 + (1 minus 119906

lowast

1)119898119899] 119889119905

= int

119905119891

0

(119898 minus 119898)120573119890120582119905

119873[(119906lowast


+ 119898119899 (119906lowast

1minus 119906lowast

1)] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [119906lowast

1minus 1

100381610038161003816100381610038161003816100381610038161003816

(119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

100381610038161003816100381610038161003816100381610038161003816


le120573119890120582119905119891

119873int

119905119891

0

(119898 minus 119898)

times [1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 (119898 |119899 minus 119899| + |119898 minus 119898| 119899) + 119898119899120573119890120582119905

1198881119873

|119898119899 (119904 minus 119904) + (119898119899 minus 119898119899) 119904


le120573119890120582119905119891

119873int

119905119891

0

[1003816100381610038161003816119906lowast

1minus 1

1003816100381610038161003816 |119898| |119898 minus 119898| |119899 minus 119899|

+ |119898 minus 119898|2

|119899| + |119861| |119904| |119899| |119898 minus 119898|2

+ |119861| |119898119899| |119898 minus 119898| |119903 minus 119903|

+ |119861| |119898119899| |119898 minus 119898| |119904 minus 119904|

+ |119861| |119904| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119898| |119898 minus 119898| |119899 minus 119899|

+ |119861| |119903| |119899| |119898 minus 119898|2] 119889119905

le120573119890120582119905119891

119873int

119905119891

0

[

1003816100381610038161003816119906lowastminus 1

1003816100381610038161003816

2|119898| ((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119904 minus 119904)

2)

+|119861| |119904| |119898|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119904| |119899| (119898 minus 119898)2

+|119861| |119898119899|

2((119898 minus 119898)

2+ (119903 minus 119903)

2)

+|119861| |119898| |119903|

2((119898 minus 119898)

2+ (119899 minus 119899)

2)

+ |119861| |119903| |119899| (119898 minus 119898)2]119889119905

(84)


1(119905) minus 119906

lowast





2(119898 minus119898)

2 (119898 minus119898)2+ (119904 minus 119904)

2 (119898 minus119898)2+ (119899 minus 119899)

2 (119898 minus119898)2

(119898minus119898)2+(119903minus119903)



2such that

1

2(119898 minus 119898)

2(119905119891) + (120583 + 120582)int

119905119891

0

(119898 minus 119898)2119889119905


le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)


1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)


(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)




1= 0 119906

2= 0 Before








0ge













0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 1198882

120573119890120582119905119891

119873int

119905119891

0

((119898 minus 119898)2+ (119899 minus 119899)

2

+ (119904 minus 119904)2+ (119903 minus 119903)

2) 119889119905

(85)


1

2(119898 minus 119898) (119905

119891) +

1

2(119899 minus 119899) (119905

119891) +

1

2(119901 minus 119901) (119905

119891)

+1

2(119902 minus 119902) (119905

119891) +

1

2(119903 minus 119903) (0) +

1

2(119904 minus 119904) (0)

+1

2(119908 minus 119908) (0) +

1

2(V minus V) (0) + (120583 + 120582)

times int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119904 minus 119904)2

+ (119903 minus 119903)2+ (119908 minus 119908)

2+ (V minus V)2) 119889119905

le 119861int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2+ (119901 minus 119901)

2

+ (119902 minus 119902)2

+ (119903 minus 119903)2+ (119904 minus 119904)

2

+ (119908 minus 119908)2+ (V minus V)2 119889119905

(86)


(120583 + 120582 minus 119861)int

119905119891

0

(119898 minus 119898)2+ (119899 minus 119899)

2

+ (119901 minus 119901)2

+ (119902 minus 119902)2

+ (119903 minus 119903)2

+ (119904 minus 119904)2+(119908 minus 119908)

2+ (V minus V)2 119889119905 le 0

(87)




1= 0 119906

2= 0 Before








0ge













0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50

0

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e

S(t)

A(t)

T(t)

Q(t)


0is globally


S(t)

A(t)

T(t)

Q(t)

0 10 20 30 40

48

2520

303540

49 50

500

1000

2000

3000

4000

5000

6000

7000

Time

Popu

latio

n siz

e



=


0= 108511)










0

02

04

06

08

1

0

05

1

0 02 04 06 08 1

u1

u2

R0


1

and 1199062









1= 102 1198882= 104 From



2is better than




1 1199062



0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 10

2 1198882= 10


1=

1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 102 1198882= 104 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2


0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re




0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)






2 it




0 10 20 30 40 504500

5000

5500

6000

6500

7000

S(n

umbe

r of s

usce

ptib

les)





Optimal control




(2) single control 1199061= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2


2=





7 Conclusions


0and


0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50





Optimal control

0

500

1000

1500

A(n

umbe

r of t

he al

coho

lism

)

t (unit week)




control 1199061= 05 119906


1= 0 119906

2= 05 (4)


1= 119906lowast

1 1199062= 119906lowast

2

0

50

100

150

200

250

300

350

400

T(n

umbe

r of t

reat

men

t)

0 10 20 30 40 50





Optimal control


1= 104 1198882= 102 and under


1199062= 0 (2) single control 119906

1= 05 119906


1= 0

1199062= 05 (4) middle control 119906


1199061= 119906lowast

1 1199062= 119906lowast

2

0 10 20 30 40 50





Optimal control

0

50

100

150

200

250

300

350

400

Q(n

umbe

r of q

uitti

ng)


1= 104 1198882= 102 and


1= 05 119906


1199061= 0 119906


1= 1199062= 05 (5) optimal


1 1199062= 119906lowast

2

0 10 20 30 40 50t (unit week)

0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re







0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

02

04

06

08

1

12

Deg

ree o

f the

cont

rol m

easu

re

0 10 20 30 40 50t (unit week)







Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







