Research ArticleModelling the Influence of Awareness Programs by Media onthe Drinking Dynamics

Hai-Feng Huo and Qian Wang

Department of Applied Mathematics Lanzhou University of Technology Lanzhou Gansu 730050 China

Correspondence should be addressed to Hai-Feng Huo huohf1970gmailcom

Received 25 January 2014 Accepted 17 March 2014 Published 16 April 2014

Academic Editor Yongli Song

Copyright copy 2014 H-F Huo and Q Wang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We develop a nonlinear mathematical model with the effect of awareness programs on the binge drinking Due to the fact thatawareness programs are capable of inducing behavioral changes in nondrinkers we introduce a separate class by avoiding contactswith the heavy drinkers Furthermore we assume that cumulative density of awareness programs increases at a rate proportional tothe number of heavy drinkers We establish some sufficient conditions for the stability of the alcohol free and the alcohol presentequilibria and give some numerical simulations to explain ourmain result Our results show that awareness programs is an effectivemeasure in reducing alcohol problems

1 Introduction

Alcohol consumption particularly heavier drinking is animportant risk factor formany health problems and is amajorcontributor to the global burden of disease [1] It has beenidentified as an important risk factor for chronic disease andinjury [2] Drinking alcohol causes the liver to become evenmore damaged in people infected with the hepatitis B virus(HBV) [3] The CAS (Central Authentication Service) find-ings have shown that alcohol consumption at binge levels andbeyond has a significant impact on college studentsrsquo academicperformance social relationships risk taking behaviors andhealthThis form of drinking is associated with missing classfalling behind in schoolwork and lower grade point averagea relationship mediated by fewer hours spent in studying [45] In addition students who attended schools with high ratesof binge drinking experienced a greater number of second-hand effects including disruption of sleep or study propertydamage and verbal physical or sexual violence than theirpeers attending schools with low binge drinking rates [6]Prior studies have indicated that heavy alcohol drinkers arelikely to engage in risky sexual behaviours and thus morelikely to get sexually transmitted infections (STIs) than socialdrinkers [7] The rapid consumption of large amounts of

alcohol especially by young people leads to serious antisocialand criminal behavior in urban centres This phenomenonhas grown very rapidly [8] Since mathematical model is apredictive tool which can mimic the process of drinking andprovide useful tools to analyze the spread and control ofdrinking behavior several different mathematical models fordrinking have been formulated and studied recently Simplemodels for alcohol treatment are presented by Sanchez etal [9] which were based on studying binge drinking in acollege system and assumed the same ldquoleaving raterdquo Muloneand Straughan [10] investigated a possible model for bingedrinking taking into account admitting and nonadmittingdrinkers But the global analysis of binge drinking model isnot discussed in the literature Huo and Song [11] introduceda more realistic two-stage model for binge drinking problemwhere the youths with alcohol problems are divided intothose who admit the problem and those who do not admit itFor the other mathematical models for drinking or smokingwe refer to [12ndash14] and the references cited therein

The media may be the most important source of healthinformation for the general public it can play a special rolein providing a voice for people to express their experiencesof illness [15] Jaramillo [16] studied a model presenting anevaluation of the impact on case finding of a mass media

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 938080 8 pageshttpdxdoiorg1011552014938080

2 Abstract and Applied Analysis

health education campaign for TB control in Cali ColombiaLiu et al [17] incorporate the psychological impacts ofmedia coverage on disease transmission and emphasize theimportance of refining classical mathematical models toreflect the partially self-limiting nature of infectious diseaseoutbreaks due to the effects of widespread news coverageand fast information flow Web-based screening and briefinterventions that include personalized feedback about theiralcohol use have proven to be particularly promising forreducing hazardous drinking among university students [18]Mass media (television radio newspapers billboards andbooklets) have been used as a way of delivering preventivehealth messages as they have the potential to influencepeoplersquos behavior Further mass media have been deployedin the effort to control and eliminate epidemic diseases [19]Xiang et al [20] studied a drinking model with public healtheducational campaigns which can be an efficient optionfor reducing the spread of disease Misra et al [21 22]considered the effects of awareness programs driven bymediaon the spread of infectious diseases their results showedthat though awareness programs cannot eradicate infectionthey help in controlling the prevalence of disease The mostsuccessful approaches concentrate not on trying to change thedrinkerrsquos behavior directly but on influencing the communityenvironment around the nondrinker For instance educationand persuasionmay increase knowledge and change attitudesbut have no effect on drinking [23 24] so we only considerthe interaction from awareness programs between awareindividual and nondrinker

Motivated by the above works we introduce a mathe-matical model with the effect of awareness programs on thebinge drinking in this paper Because awareness programsare capable of inducing behavioral changes in susceptibleindividuals we introduce a separate class 119883(119905) of those whoare aware of risk and refrain from drinking by avoidingcontact with the heavy drinkers This newly formed awareclass is assumed to be fully protected from the heavy drinkersdue to media-induced isolation and individuals of this classmay contact heavy drinkers only if they lose awareness[21 22] Furthermore we assume that cumulative density ofawareness programs119872(119905) increases at a rate proportional tothe number of heavy drinkers We establish some sufficientconditions for the global stability of equilibria and give somenumerical simulations to explain ourmain result Our resultsshow that awareness programs are an effective measure inreducing alcohol problems

The organization of this paper is as follows In the nextsection the effect of awareness programs by media on bingedrinking model is formulated In Section 3 the existence andthe stability of equilibria are investigated Some numericalsimulations are given in Section 4 Somediscussions are givenin Section 5

2 The Model

21 System Description The total population is divided intofive compartments namely the susceptible compartmentwho do not drink or drink only moderately denoted by 119878(119905)those who are aware of risk and avoid drinking denoted by

120583 120573SA pA

qR

120583R120583S 120583A

120582SM120590X

120583X

mA

120574M

S A R

X M

Figure 1 Transfer diagram of model (2)

119883(119905) those who drink heavily denoted by 119860(119905) and thosewho are in treatment denoted by 119877(119905) Also let 119872(119905) bethe cumulative density of awareness programs driven bymedia The growth rate of density of awareness programsis assumed to be proportional to the number of heavydrinkers It is considered that due to the awareness programsnondrinkers form a different class and avoid contacting withthe heavy drinkers And we only consider the interactionfrom awareness programs between aware individual andnondrinkerThe total number of population at time 119905 is givenby

119873(119905) = 119878 (119905) + 119883 (119905) + 119860 (119905) + 119877 (119905) (1)

The model structure is shown in Figure 1 The transferdiagram leads to the following system of ordinary differentialequations

119889119878 (119905)

119889119905= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

119889119883 (119905)

119889119905= 120582119878119872 minus 120590119883 minus 120583119883

119889119860 (119905)

119889119905= 120573119878119860 + 119902119877 minus 119901119860 minus 120583119860

119889119877 (119905)

119889119905= 119901119860 minus 119902119877 minus 120583119877

119889119872 (119905)

119889119905= 119898119860 minus 120574119872

(2)

where 120583 is the rate of individuals entering into the system ina given time interval (say each year) Since we are dealingwith youths we assume that the death rate is negligible andso the leaving rate is also 120583 This is in accordance withan assumption by Sanchez et al [9] where ldquoleaving raterdquopresents the departure rate from drinking environment 120573is the transmission coefficient for the nondrinkers turningto heavy drinkers through peer pressure 120582 represents thedissemination rate of awareness among nondrinkersdue tothe awareness programs nondrinkers form a different class119883(119905) and avoid contacting with the heavy drinkers Theconstant 120590 denotes the rate of transfer of aware individualsto nondrinker class 119901 is a proportionality coefficient for 119860going into treatment 119902 is a proportionality coefficient for

Abstract and Applied Analysis 3

119877 relapsing into drinkers 119898 represents the growth rate ofdensity of awareness programs which is assumed to be pro-portional to the number of heavy drinkers and 120574 representsthe depletion rate of these programs due to ineffectivenesssocial problems and so forth For instance as time passes bysome of the campaigns by media lose their influence or fail toattract people

22 Basic Properties

221 Invariant Region

Lemma 1 All feasible regions Ω defined by

Ω = (119878 (119905) 119883 (119905) 119860 (119905) 119877 (119905) 119872 (119905)) isin 1198775

+

119878 (119905) + 119883 (119905) + 119860 (119905) + 119877 (119905) le 1 119872 (119905) le119898

120574

(3)

with the initial conditions 119878(0) ge 0119883(0) ge 0119860(0) ge 0119877(0) ge0 and119872(0) ge 0 are positively invariant for system (2)

Proof Adding the first four equations of (2) we have119889119873

119889119905= 120583 minus 120583119873 (4)

It follows that

119873(119905) = 1 + 119873 (0) 119890minus120583119905 (5)

where119873(0) is the initial total number of people Thus

lim119905rarrinfin

119873(119905) = 1 (6)

Then 119878(119905) + 119883(119905) + 119860(119905) + 119877(119905) le 1 the subpopulation isrepresented in proportion to the total number of population

Thus since the system (2) monitors human populationit is plausible to assume that all its state variables andparameters are nonnegative for all 119905 ge 0 Further from thelast equation of system (2) we have

119889119872

119889119905= 119898119860 minus 120574119872

le 119898 minus 120574119872

(7)

It follows that

0 le 119872 (119905) le119898

120574+119872(0) 119890

minus120574119905 (8)

where119872(0) represents the initial value of cumulative densityof awareness programs Thus

lim119905rarrinfin

sup119872(119905) le 119898120574 (9)

It implies that the region

Ω = (119878 (119905) 119883 (119905) 119860 (119905) 119877 (119905) 119872 (119905))

isin 1198775

+ 119878 (119905) + 119883 (119905) + 119860 (119905) + 119877 (119905) le 1 119872 (119905) le

119898

120574

(10)

is a positively invariant set for (2) So we consider dynamicsof system (2) on the setΩ in this paper

222 Positivity of Solutions For system (2) to ensure that thesolutions of the systemwith positive initial conditions remainpositive for all 119905 gt 0 it is necessary to prove that all the statevariables are nonnegative so we have the following lemma

Lemma 2 If 119878(0) gt 0 119883(0) gt 0 119860(0) gt 0 119877(0) gt 0and119872(0) gt 0 then the solutions 119878(119905) 119883(119905) 119860(119905) 119877(119905) and119872(119905) of system (2) are positive for all 119905 ge 0

Proof Under the given initial conditions it is easy to provethat the solutions of the system (2) are positive if not weassume a contradiction that there exists a first time 119905

1such

that119878 (0) gt 0 119878 (119905

1) = 0 119878

1015840(1199051) le 0

119883 (119905) gt 0 119860 (119905) gt 0 119877 (119905) gt 0

119872 (119905) gt 0 0 le 119905 lt 1199051

(11)

In that case from the first equation of system (2) we have

1198781015840(1199051) = 120583 + 120590119883 gt 0 (12)

which is a contradiction meaning that 119878(119905) gt 0 119905 gt 0Or there exists a 119905

2such that

119883(0) gt 0 119883 (1199052) = 0 119883

1015840(1199052) le 0

119878 (119905) gt 0 119860 (119905) gt 0 119877 (119905) gt 0

119872 (119905) gt 0 0 le 119905 lt 1199052

(13)

In that case from the second equation of system (2) we have

1198831015840(1199052) = 120582119878119872 gt 0 (14)

which is a contradiction meaning that119883(119905) gt 0 119905 ge 0Or there exists a 119905

3such that

119860 (0) gt 0 119860 (1199053) le 0 119860

1015840(1199053) = 0

119878 (119905) gt 0 119883 (119905) gt 0 119877 (119905) gt 0

119872 (119905) gt 0 0 le 119905 lt 1199053

(15)

In that case from the third equation of system (2) we have

1198601015840(1199053) = 119902119877 gt 0 (16)

which is a contradiction meaning that 119860(119905) gt 0 119905 ge 0Similarly it can be shown that 119877(119905) gt 0 and 119872(119905) gt 0 forall 119905 ge 0 Thus the solutions 119878(119905) 119883(119905) 119860(119905) 119877(119905) and119872(119905)of system (2) remain positive for all 119905 gt 0

3 Analysis of the Model

There are one alcohol free equilibrium 1198640and one alcohol

present equilibrium 119864lowast for system (2)

31 Alcohol Free Equilibrium and the Basic ReproductionNumber Themodel has an alcohol free equilibrium given by

1198640= (1 0 0 0 0) (17)

4 Abstract and Applied Analysis

In the following the basic reproduction number of system(2) will be obtained by the next generation matrix methodformulated in [25]

Let 119909 = (119860 119877119872 119878 119883)119879 then system (2) can be writtenas

119889119909

119889119905= F (119909) minusV (119909) (18)

where

F (119909) = (

120573119878119860

0

0

0

0

)

V (119909) =((((

(

minus119902119877 + (119901 + 120583)119860

minus119901119860 + (119902 + 120583) 119877

120574119872 minus 119898119860

minus120583 + 120573119878119860 + 120582119878119872 minus 120590119883 + 120583119878

minus120582119878119872 + (120590 + 120583)119883

))))

)

(19)

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

0are respectively

119863F (1198640) = (1198652times20

0 0) (20)

119863V (1198640) = (

1198812times20 0 0

minus119898 0 120574 0 0

120573 0 120582 120583 minus120590

0 0 minus120582 0 120590 + 120583

) (21)

where

119865 = (120573 0

0 0) 119881 = (

119901 + 120583 minus119902

minus119901 119902 + 120583) (22)

Themodel reproduction number denoted by 1198770is thus given

by

1198770=

120573 (119902 + 120583)

(119901 + 120583) (119902 + 120583) minus 119901119902 (23)

32 Global Stability of 1198640

Theorem 3 For system (2) the alcohol free equilibrium 1198640is

globally asymptotically stable if 1198770le 1

Proof We introduce the following Lyapunov function [26]

119881 (119878 (119905) 119883 (119905) 119860 (119905) 119877 (119905) 119872 (119905))

= 119878 minus ln 119878 + 119883 + 119860 +119902

119902 + 120583119877

(24)

The derivative of 119881 is given by

= 119878 minus

119878

119878+ +

119902

119902 + 120583 +

= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

minus1

119878(120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878)

+ 120582119878119872 minus 120590119883 minus 120583119883 + 120573119878119860 + 119902119877 minus 119901119860

minus 120583119860 +119902

119902 + 120583(119901119860 minus 119902119877 minus 120583119877)

= 120583 minus 120583119878 minus1

119878120583 + 120573119860 + 120582119872 minus

1

119878120590119883 + 120583

minus 120583119883 + 119902119877 minus 119901119860 minus 120583119860 +119902

119902 + 120583119901119860 minus 119902119877

= 120583(2 minus 119878 minus1

119878) + 120573(1 minus

1

1198770

)119860 + (1 minus1

119878) 120590119883

(25)

If 1198770le 1 then 120573(1 minus (1119877

0)) le 0 As we know

2 minus 119878 minus (1119878) le 0 and 1 minus (1119878)120590119883 le 0 so we obtain le 0 Furthermore = 0 only if 119878 = 1 or 119877

0= 1

The maximum invariant set in (119878 119883 119860 119877119872) = 0 isthe singleton 119864

0 By Lasallersquos invariance principle [27] 119864

0is

globally asymptotically stable inΩ

33 Alcohol Present Equilibrium

331 Existence of the Alcohol Present Equilibrium

Theorem 4 If 1198770gt 1 system (2) has a unique alcohol present

equilibrium 119864lowast(119878lowast 119883lowast 119860lowast 119877lowast119872lowast) where

119878lowast=1

120573(119901 + 120583 minus

119901119902

119902 + 120583)

119883lowast=120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860lowast

119877lowast=119901

119902 + 120583119860lowast

119872lowast=119898

120574119860lowast

(26)

Proof It follows from system (2) that

120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878 = 0

120582119878119872 minus 120590119883 minus 120583119883 = 0

120573119878119860 + 119902119877 minus 119901119860 minus 120583119860 = 0

119901119860 minus 119902119877 minus 120583119877 = 0

119898119860 minus 120574119872 = 0

(27)

Abstract and Applied Analysis 5

From the fourth and fifth equations of (27) we obtain

119877 =119901

119902 + 120583119860 (28)

119872 =119898

120574119860 (29)

Substituting 119877 into the third equation of (27) we have

119878 =1

120573(119901 + 120583 minus

119901119902

119902 + 120583) (30)

Then substituting (29) and (30) into the second equation of(27) we get

119883 =120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860 (31)

For 119860 = 0 substituting (29) (30) and (31) into the firstequation of (27) gives

120583 minus 120583119878

= (119901 + 120583 minus119901119902

119902 + 120583)119860 +120582119898

120573120574(119901 + 120583 minus

119901119902

119902 + 120583)119860

minus120590120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860

(32)

Substituting (30) into (32) we get

119860 = (120583 minus120583

120573(119901 + 120583 minus

119901119902

119902 + 120583))

times ((119901 + 120583 minus119901119902

119902 + 120583)[1 +120582119898

120573120574minus120590120582119898

120573120574 (120590 + 120583)])

minus1

= 120583(1 minus1

1198770

)

times ((119901 + 120583 minus119901119902

119902 + 120583)[1 +120582119898

120573120574minus120590120582119898

120573120574 (120590 + 120583)])

minus1

(33)

Therefore there exists a unique positive root in theinterval (0 1) when 119877

0gt 1 there is no positive root in the

interval [0 1] when 1198770le 1

For1198770gt 1 we carry out the simulation (see Figure 4) and

give the following conjecture

Conjecture If 1198770gt 1 the alcohol present equilibrium 119864lowast of

(2) is globally asymptotically stable

Remark Since the global stability of (2) is a hard problem weonly carry out simulation (see Figure 4) How to prove theglobal stability of the alcohol present equilibrium 119864lowast of (2)and give the conditions based on model parameters is still anopen problem

4 Numerical Simulation

To illustrate the analytic results obtained above we give somesimulations using the parameter values in Table 1 Numericalresults are displayed in the following figures It is reportedthat binge drinking occurs in up to 30 percent of adolescentsand alcohol use disorders occur in about 6 percent of thisage group approximately 50 to 60 percent of them remainabstinent at the end of a yearrsquos treatment and a majority ofthose stay dry permanently [29 30] So the initial condition istaken as the lowest by119860(0) = 03 and119877(0) = 015 in additionwe assume that119883(0) = 01 119878(0) = 045 and119872(0) = 0002

First we choose 119875 = 03 and the numerical simulationgives 119877

0= 05691 lt 1 then the alcohol free equilibrium 119864

0is

globally asymptotically stable (Figure 2) Second we choose119875 = 021 and the numerical simulation gives 119877

0= 1 the

alcohol free equilibrium 1198640is globally asymptotically stable

(Figure 3) Third we choose 119875 = 0032 and the numericalsimulation gives 119877

0= 1165 gt 1 the alcohol present

equilibrium 119864lowast is globally asymptotically stable (Figure 4)Finally for showing that awareness programs by media

can help to control the population of binge drinking wecompare our model with the model which has been studiedby Mulone and Straughan [10] the model in [10] is governedby the following system of nonlinear ordinary differentialequations

119889119878 (119905)

119889119905= 120583119873 minus

120573119860119878

119873minus 120583119878

119889119860 (119905)

119889119905=120573119860119878

119873+ 120588119877 minus (120601 + 120583)119860

119889119877 (119905)

119889119905= 120601119860 minus (120588 + 120583) 119877

(34)

We choose 119875 = 0032 and the numerical simulation gives1198770= 11645 gt 1 the alcohol present equilibrium 119875lowast of

(34) is globally asymptotically stableThenumber of the bingedrinking population is generally on the rise without effects ofawareness programs by media

For showing the distinction between (2) and (34) we take119875 = 0032 and compare the variations of the binge drinkingpopulation 119860(119905) We can see that because of awarenessprograms by media (Figure 5) the number of people whohave alcohol problems decreases due to awareness-inducedisolation of susceptible people so making more people avoidcontacting with the heavy drinkers due to the fact thatthe awareness programs by media can reduce the alcoholproblems

Further the variations of the binge drinking population119860(119905) with respect to time 119905 for different values of growth rateof density of awareness programs 119898 are shown in Figure 6From these figures it is apparent that as the rate of density ofawareness programs 119898 increases the binge drinking popula-tion 119860(119905) decreases Then the awareness programs by mediaare an effective measure in alcohol problems

6 Abstract and Applied Analysis

Table 1 Description and estimation of parameters

120583 Recruitment rate of the population 025 yearminus1 [10]120573 Transmission coefficient 03 yearminus1 [10]120582 Dissemination rate of aware individuals 002 yearminus1 estimate120590 Rate of transfer of aware individuals to susceptible class 0001 yearminus1 estimate119901 The fraction of 119860 go in to treatment Variable Variable119902 The fraction of 119877 who relapse into 119860 08 yearminus1 [10]119898 Growth rate of density of awareness programs 005 yearminus1 [28]120574 Depletion rate of awareness programs 005 yearminus1 estimate

050 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 2 If 1198770lt 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 200 250 3000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 3 If 1198770= 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 4 If 1198770gt 1 the alcohol present equilibrium 119864lowast is globally

asymptotically stable

0 10 20 30 40 50015

02

025

03

035

04

Bing

e drin

king

pop

ulat

ion

size

A(t) of (21)A(t) of (41)

Time

Figure 5 Variations of the binge drinking population 119860(119905) of (2)and (34)

Abstract and Applied Analysis 7

024

026

028

03

032

034

Bing

e drin

king

pop

ulat

ion

size

m = 002

m = 0008m = 0

0 10 20 30 40 50Time

Figure 6 Variations of binge drinking population with time fordifferent values of119898

5 Discussion

We have formulated the effect of awareness programs onthe binge drinking and analyzed their dynamical behaviorsThe model exhibits two equilibria explicitly the alcohol freeequilibrium and alcohol present equilibrium By constructingLyapunov function we establish some sufficient conditionsfor the global stability of the alcohol free and the alcoholpresent equilibria and give some numerical simulations toexplain our main result In our model we have assumed thatthe cumulative density of awareness programs by media isproportional to the population of binge drinking heavily Itmay be noted that the number of alcohol consumption casesknown to the policy makers is sometimes old and thus thenumber of awareness depends on data So it is more plausibleto consider a time delay in the growth rate of awarenessprograms [21]We canmodify (2) to the followingmodel withdelay

119889119878 (119905)

119889119905= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

119889119883 (119905)

119889119905= 120582119878119872 minus 120590119883 minus 120583119883

119889119860 (119905)

119889119905= 120573119878119860 + 119902119877 minus 119901119860 minus 120583119860

119889119877 (119905)

119889119905= 119901119860 minus 119902119877 minus 120583119877

119889119872 (119905)

119889119905= 119898119860 (119905 minus 120591) minus 120574119872

(35)

We leave these works for the future

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 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] J Rehm ldquoThe risks associated with alcohol use and alcoholismrdquoAlcohol Research amp Health vol 34 no 2 pp 135ndash143 2011

[2] J Rehm C Mathers S Popova M Thavorncharoensap YTeerawattananon and J Patra ldquoGlobal burden of disease andinjury and economic cost attributable to alcohol use andalcohol-use disordersrdquoThe Lancet vol 373 no 9682 pp 2223ndash2233 2009

[3] J Ribes R Cleries L Esteban V Moreno and F X BoschldquoThe influence of alcohol consumption and hepatitis B and Cinfections on the risk of liver cancer in Europerdquo Journal ofHepatology vol 49 no 2 pp 233ndash242 2008

[4] H Wechsler and T F Nelson ldquoFocusing attention on collegestudent alcohol consumption and the environmental conditionsthat promote itrdquo Journal of Studies on Alcohol andDrugs vol 69no 4 pp 481ndash490 2008

[5] L M Powell J Williams and H Wechsler ldquoStudy habits andthe level of alcohol use among college studentsrdquo EducationEconomics vol 12 no 2 pp 135ndash149 2004

[6] H Wechsler B Moeykens A Davenport S Castillo and JHansen ldquoThe adverse impact of heavy episodic drinkers onother college studentsrdquo Journal of Studies on Alcohol and Drugsvol 56 no 6 pp 628ndash634 1995

[7] 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

[8] P Ormerod and G Wiltshire ldquolsquoBingerdquo drinking in the UK asocial network phenomenonrdquoMind amp Society vol 8 no 2 pp135ndash152 2009

[9] F Sanchez X HWang C Castillo-Chavez DM Gorman andP J Gruenewald ldquoDrinking as an epidemica simple mathemat-ical model with recovery and relapserdquo in Therapists Guide toEvidence-Based Relapse Prevention Practical Resources for theMental Health Professional K A Witkiewitz and G A MarlattEds pp 353ndash368 Academic Press Burlington Canada 2007

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

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

[12] H F Huo and C C Zhu ldquoModeling the effect of constantimmigration on drinking behaviourrdquo Submitted

8 Abstract and Applied Analysis

[13] H FHuo andC C Zhu ldquoStability of a quit drinkingmodel withrelapserdquo Submitted

[14] 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

[15] G Schwitzer G Mudur D Henry et al ldquoWhat are theroles and responsibilities of the media in disseminating healthinformationrdquo PLoS Medicine vol 2 no 7 article e215 2005

[16] E Jaramillo ldquoThe impact of media-based health education ontuberculosis diagnosis in Cali Colombiardquo Health Policy andPlanning vol 16 no 1 pp 68ndash73 2001

[17] R Liu JWu andH Zhu ldquoMediapsychological impact onmul-tiple outbreaks of emerging infectious diseasesrdquo Computationaland Mathematical Methods in Medicine vol 8 no 3 pp 153ndash164 2007

[18] T P Palfai R Zisserson and R Saitz ldquoUsing personalizedfeedback to reduce alcohol use among hazardous drinking col-lege students the moderating effect of alcohol-related negativeconsequencesrdquo Addictive Behaviors vol 36 no 5 pp 539ndash5422011

[19] J M Tchuenche and C T Bauch ldquoDynamics of an infectiousdisease where media coverage influences transmissionrdquo ISRNBiomathematics vol 2012 Article ID 581274 10 pages 2012

[20] H Xiang N N Song and H F Huo ldquoModelling effects ofpublic health educational campaigns on drinking dynamicsrdquoSubmitted

[21] A K Misra A Sharma and V Singh ldquoEffect of awarenessprograms in controlling the prevalence of an epidemic withtime delayrdquo Journal of Biological Systems vol 19 no 2 pp 389ndash402 2011

[22] A KMisra A Sharma and J B Shukla ldquoModeling and analysisof effects of awareness programs by media on the spread ofinfectious diseasesrdquoMathematical andComputerModelling vol53 no 5-6 pp 1221ndash1228 2011

[23] ldquoWHO Report of Alcohol in developing societies summaryrdquohttpwwwwhoint

[24] D Chisholm J Rehm M van Ommeren and M MonteiroldquoReducing the global burden of hazardous alcohol use acomparative cost-effectiveness analysisrdquo Journal of Studies onAlcohol vol 65 no 6 pp 782ndash793 2004

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

[26] J-Q Li Y-L Yang and Y-C Zhou ldquoGlobal stability of anepidemic model with latent stage and vaccinationrdquo NonlinearAnalysis Real World Applications vol 12 no 4 pp 2163ndash21732011

[27] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[28] L B Snyder ldquoHealth communication campaigns and theirimpact on behaviorrdquo Journal of Nutrition Education and Behav-ior vol 39 no 2 pp S32ndashS40 2007

[29] D B ClarkO Bukstein and J Cornelius ldquoAlcohol use disordersin adolescents epidemiology diagnosis psychosocial interven-tions and pharmacological treatmentrdquo Pediatric Drugs vol 4no 8 pp 493ndash502 2002

[30] ldquoWebMDReport of Alcohol Abuse Health Centerrdquo httpwwwwebmdcom

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

health education campaign for TB control in Cali ColombiaLiu et al [17] incorporate the psychological impacts ofmedia coverage on disease transmission and emphasize theimportance of refining classical mathematical models toreflect the partially self-limiting nature of infectious diseaseoutbreaks due to the effects of widespread news coverageand fast information flow Web-based screening and briefinterventions that include personalized feedback about theiralcohol use have proven to be particularly promising forreducing hazardous drinking among university students [18]Mass media (television radio newspapers billboards andbooklets) have been used as a way of delivering preventivehealth messages as they have the potential to influencepeoplersquos behavior Further mass media have been deployedin the effort to control and eliminate epidemic diseases [19]Xiang et al [20] studied a drinking model with public healtheducational campaigns which can be an efficient optionfor reducing the spread of disease Misra et al [21 22]considered the effects of awareness programs driven bymediaon the spread of infectious diseases their results showedthat though awareness programs cannot eradicate infectionthey help in controlling the prevalence of disease The mostsuccessful approaches concentrate not on trying to change thedrinkerrsquos behavior directly but on influencing the communityenvironment around the nondrinker For instance educationand persuasionmay increase knowledge and change attitudesbut have no effect on drinking [23 24] so we only considerthe interaction from awareness programs between awareindividual and nondrinker

Motivated by the above works we introduce a mathe-matical model with the effect of awareness programs on thebinge drinking in this paper Because awareness programsare capable of inducing behavioral changes in susceptibleindividuals we introduce a separate class 119883(119905) of those whoare aware of risk and refrain from drinking by avoidingcontact with the heavy drinkers This newly formed awareclass is assumed to be fully protected from the heavy drinkersdue to media-induced isolation and individuals of this classmay contact heavy drinkers only if they lose awareness[21 22] Furthermore we assume that cumulative density ofawareness programs119872(119905) increases at a rate proportional tothe number of heavy drinkers We establish some sufficientconditions for the global stability of equilibria and give somenumerical simulations to explain ourmain result Our resultsshow that awareness programs are an effective measure inreducing alcohol problems

The organization of this paper is as follows In the nextsection the effect of awareness programs by media on bingedrinking model is formulated In Section 3 the existence andthe stability of equilibria are investigated Some numericalsimulations are given in Section 4 Somediscussions are givenin Section 5

2 The Model

21 System Description The total population is divided intofive compartments namely the susceptible compartmentwho do not drink or drink only moderately denoted by 119878(119905)those who are aware of risk and avoid drinking denoted by

120583 120573SA pA

qR

120583R120583S 120583A

120582SM120590X

120583X

mA

120574M

S A R

X M

Figure 1 Transfer diagram of model (2)

119883(119905) those who drink heavily denoted by 119860(119905) and thosewho are in treatment denoted by 119877(119905) Also let 119872(119905) bethe cumulative density of awareness programs driven bymedia The growth rate of density of awareness programsis assumed to be proportional to the number of heavydrinkers It is considered that due to the awareness programsnondrinkers form a different class and avoid contacting withthe heavy drinkers And we only consider the interactionfrom awareness programs between aware individual andnondrinkerThe total number of population at time 119905 is givenby

119873(119905) = 119878 (119905) + 119883 (119905) + 119860 (119905) + 119877 (119905) (1)

The model structure is shown in Figure 1 The transferdiagram leads to the following system of ordinary differentialequations

119889119878 (119905)

119889119905= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

119889119883 (119905)

119889119905= 120582119878119872 minus 120590119883 minus 120583119883

119889119860 (119905)

119889119905= 120573119878119860 + 119902119877 minus 119901119860 minus 120583119860

119889119877 (119905)

119889119905= 119901119860 minus 119902119877 minus 120583119877

119889119872 (119905)

119889119905= 119898119860 minus 120574119872

(2)

where 120583 is the rate of individuals entering into the system ina given time interval (say each year) Since we are dealingwith youths we assume that the death rate is negligible andso the leaving rate is also 120583 This is in accordance withan assumption by Sanchez et al [9] where ldquoleaving raterdquopresents the departure rate from drinking environment 120573is the transmission coefficient for the nondrinkers turningto heavy drinkers through peer pressure 120582 represents thedissemination rate of awareness among nondrinkersdue tothe awareness programs nondrinkers form a different class119883(119905) and avoid contacting with the heavy drinkers Theconstant 120590 denotes the rate of transfer of aware individualsto nondrinker class 119901 is a proportionality coefficient for 119860going into treatment 119902 is a proportionality coefficient for

Abstract and Applied Analysis 3

119877 relapsing into drinkers 119898 represents the growth rate ofdensity of awareness programs which is assumed to be pro-portional to the number of heavy drinkers and 120574 representsthe depletion rate of these programs due to ineffectivenesssocial problems and so forth For instance as time passes bysome of the campaigns by media lose their influence or fail toattract people

22 Basic Properties

221 Invariant Region

Lemma 1 All feasible regions Ω defined by

Ω = (119878 (119905) 119883 (119905) 119860 (119905) 119877 (119905) 119872 (119905)) isin 1198775

+

119878 (119905) + 119883 (119905) + 119860 (119905) + 119877 (119905) le 1 119872 (119905) le119898

120574

(3)

with the initial conditions 119878(0) ge 0119883(0) ge 0119860(0) ge 0119877(0) ge0 and119872(0) ge 0 are positively invariant for system (2)

Proof Adding the first four equations of (2) we have119889119873

119889119905= 120583 minus 120583119873 (4)

It follows that

119873(119905) = 1 + 119873 (0) 119890minus120583119905 (5)

where119873(0) is the initial total number of people Thus

lim119905rarrinfin

119873(119905) = 1 (6)

Then 119878(119905) + 119883(119905) + 119860(119905) + 119877(119905) le 1 the subpopulation isrepresented in proportion to the total number of population

Thus since the system (2) monitors human populationit is plausible to assume that all its state variables andparameters are nonnegative for all 119905 ge 0 Further from thelast equation of system (2) we have

119889119872

119889119905= 119898119860 minus 120574119872

le 119898 minus 120574119872

(7)

It follows that

0 le 119872 (119905) le119898

120574+119872(0) 119890

minus120574119905 (8)

where119872(0) represents the initial value of cumulative densityof awareness programs Thus

lim119905rarrinfin

sup119872(119905) le 119898120574 (9)

It implies that the region

Ω = (119878 (119905) 119883 (119905) 119860 (119905) 119877 (119905) 119872 (119905))

isin 1198775

+ 119878 (119905) + 119883 (119905) + 119860 (119905) + 119877 (119905) le 1 119872 (119905) le

119898

120574

(10)

is a positively invariant set for (2) So we consider dynamicsof system (2) on the setΩ in this paper

222 Positivity of Solutions For system (2) to ensure that thesolutions of the systemwith positive initial conditions remainpositive for all 119905 gt 0 it is necessary to prove that all the statevariables are nonnegative so we have the following lemma

Lemma 2 If 119878(0) gt 0 119883(0) gt 0 119860(0) gt 0 119877(0) gt 0and119872(0) gt 0 then the solutions 119878(119905) 119883(119905) 119860(119905) 119877(119905) and119872(119905) of system (2) are positive for all 119905 ge 0

Proof Under the given initial conditions it is easy to provethat the solutions of the system (2) are positive if not weassume a contradiction that there exists a first time 119905

1such

that119878 (0) gt 0 119878 (119905

1) = 0 119878

1015840(1199051) le 0

119883 (119905) gt 0 119860 (119905) gt 0 119877 (119905) gt 0

119872 (119905) gt 0 0 le 119905 lt 1199051

(11)

In that case from the first equation of system (2) we have

1198781015840(1199051) = 120583 + 120590119883 gt 0 (12)

which is a contradiction meaning that 119878(119905) gt 0 119905 gt 0Or there exists a 119905

2such that

119883(0) gt 0 119883 (1199052) = 0 119883

1015840(1199052) le 0

119878 (119905) gt 0 119860 (119905) gt 0 119877 (119905) gt 0

119872 (119905) gt 0 0 le 119905 lt 1199052

(13)

In that case from the second equation of system (2) we have

1198831015840(1199052) = 120582119878119872 gt 0 (14)

which is a contradiction meaning that119883(119905) gt 0 119905 ge 0Or there exists a 119905

3such that

119860 (0) gt 0 119860 (1199053) le 0 119860

1015840(1199053) = 0

119878 (119905) gt 0 119883 (119905) gt 0 119877 (119905) gt 0

119872 (119905) gt 0 0 le 119905 lt 1199053

(15)

In that case from the third equation of system (2) we have

1198601015840(1199053) = 119902119877 gt 0 (16)

which is a contradiction meaning that 119860(119905) gt 0 119905 ge 0Similarly it can be shown that 119877(119905) gt 0 and 119872(119905) gt 0 forall 119905 ge 0 Thus the solutions 119878(119905) 119883(119905) 119860(119905) 119877(119905) and119872(119905)of system (2) remain positive for all 119905 gt 0

3 Analysis of the Model

There are one alcohol free equilibrium 1198640and one alcohol

present equilibrium 119864lowast for system (2)

31 Alcohol Free Equilibrium and the Basic ReproductionNumber Themodel has an alcohol free equilibrium given by

1198640= (1 0 0 0 0) (17)

4 Abstract and Applied Analysis

In the following the basic reproduction number of system(2) will be obtained by the next generation matrix methodformulated in [25]

Let 119909 = (119860 119877119872 119878 119883)119879 then system (2) can be writtenas

119889119909

119889119905= F (119909) minusV (119909) (18)

where

F (119909) = (

120573119878119860

0

0

0

0

)

V (119909) =((((

(

minus119902119877 + (119901 + 120583)119860

minus119901119860 + (119902 + 120583) 119877

120574119872 minus 119898119860

minus120583 + 120573119878119860 + 120582119878119872 minus 120590119883 + 120583119878

minus120582119878119872 + (120590 + 120583)119883

))))

)

(19)

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

0are respectively

119863F (1198640) = (1198652times20

0 0) (20)

119863V (1198640) = (

1198812times20 0 0

minus119898 0 120574 0 0

120573 0 120582 120583 minus120590

0 0 minus120582 0 120590 + 120583

) (21)

where

119865 = (120573 0

0 0) 119881 = (

119901 + 120583 minus119902

minus119901 119902 + 120583) (22)

Themodel reproduction number denoted by 1198770is thus given

by

1198770=

120573 (119902 + 120583)

(119901 + 120583) (119902 + 120583) minus 119901119902 (23)

32 Global Stability of 1198640

Theorem 3 For system (2) the alcohol free equilibrium 1198640is

globally asymptotically stable if 1198770le 1

Proof We introduce the following Lyapunov function [26]

119881 (119878 (119905) 119883 (119905) 119860 (119905) 119877 (119905) 119872 (119905))

= 119878 minus ln 119878 + 119883 + 119860 +119902

119902 + 120583119877

(24)

The derivative of 119881 is given by

= 119878 minus

119878

119878+ +

119902

119902 + 120583 +

= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

minus1

119878(120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878)

+ 120582119878119872 minus 120590119883 minus 120583119883 + 120573119878119860 + 119902119877 minus 119901119860

minus 120583119860 +119902

119902 + 120583(119901119860 minus 119902119877 minus 120583119877)

= 120583 minus 120583119878 minus1

119878120583 + 120573119860 + 120582119872 minus

1

119878120590119883 + 120583

minus 120583119883 + 119902119877 minus 119901119860 minus 120583119860 +119902

119902 + 120583119901119860 minus 119902119877

= 120583(2 minus 119878 minus1

119878) + 120573(1 minus

1

1198770

)119860 + (1 minus1

119878) 120590119883

(25)

If 1198770le 1 then 120573(1 minus (1119877

0)) le 0 As we know

2 minus 119878 minus (1119878) le 0 and 1 minus (1119878)120590119883 le 0 so we obtain le 0 Furthermore = 0 only if 119878 = 1 or 119877

0= 1

The maximum invariant set in (119878 119883 119860 119877119872) = 0 isthe singleton 119864

0 By Lasallersquos invariance principle [27] 119864

0is

globally asymptotically stable inΩ

33 Alcohol Present Equilibrium

331 Existence of the Alcohol Present Equilibrium

Theorem 4 If 1198770gt 1 system (2) has a unique alcohol present

equilibrium 119864lowast(119878lowast 119883lowast 119860lowast 119877lowast119872lowast) where

119878lowast=1

120573(119901 + 120583 minus

119901119902

119902 + 120583)

119883lowast=120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860lowast

119877lowast=119901

119902 + 120583119860lowast

119872lowast=119898

120574119860lowast

(26)

Proof It follows from system (2) that

120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878 = 0

120582119878119872 minus 120590119883 minus 120583119883 = 0

120573119878119860 + 119902119877 minus 119901119860 minus 120583119860 = 0

119901119860 minus 119902119877 minus 120583119877 = 0

119898119860 minus 120574119872 = 0

(27)

Abstract and Applied Analysis 5

From the fourth and fifth equations of (27) we obtain

119877 =119901

119902 + 120583119860 (28)

119872 =119898

120574119860 (29)

Substituting 119877 into the third equation of (27) we have

119878 =1

120573(119901 + 120583 minus

119901119902

119902 + 120583) (30)

Then substituting (29) and (30) into the second equation of(27) we get

119883 =120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860 (31)

For 119860 = 0 substituting (29) (30) and (31) into the firstequation of (27) gives

120583 minus 120583119878

= (119901 + 120583 minus119901119902

119902 + 120583)119860 +120582119898

120573120574(119901 + 120583 minus

119901119902

119902 + 120583)119860

minus120590120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860

(32)

Substituting (30) into (32) we get

119860 = (120583 minus120583

120573(119901 + 120583 minus

119901119902

119902 + 120583))

times ((119901 + 120583 minus119901119902

119902 + 120583)[1 +120582119898

120573120574minus120590120582119898

120573120574 (120590 + 120583)])

minus1

= 120583(1 minus1

1198770

)

times ((119901 + 120583 minus119901119902

119902 + 120583)[1 +120582119898

120573120574minus120590120582119898

120573120574 (120590 + 120583)])

minus1

(33)

Therefore there exists a unique positive root in theinterval (0 1) when 119877

0gt 1 there is no positive root in the

interval [0 1] when 1198770le 1

For1198770gt 1 we carry out the simulation (see Figure 4) and

give the following conjecture

Conjecture If 1198770gt 1 the alcohol present equilibrium 119864lowast of

(2) is globally asymptotically stable

Remark Since the global stability of (2) is a hard problem weonly carry out simulation (see Figure 4) How to prove theglobal stability of the alcohol present equilibrium 119864lowast of (2)and give the conditions based on model parameters is still anopen problem

4 Numerical Simulation

To illustrate the analytic results obtained above we give somesimulations using the parameter values in Table 1 Numericalresults are displayed in the following figures It is reportedthat binge drinking occurs in up to 30 percent of adolescentsand alcohol use disorders occur in about 6 percent of thisage group approximately 50 to 60 percent of them remainabstinent at the end of a yearrsquos treatment and a majority ofthose stay dry permanently [29 30] So the initial condition istaken as the lowest by119860(0) = 03 and119877(0) = 015 in additionwe assume that119883(0) = 01 119878(0) = 045 and119872(0) = 0002

First we choose 119875 = 03 and the numerical simulationgives 119877

0= 05691 lt 1 then the alcohol free equilibrium 119864

0is

globally asymptotically stable (Figure 2) Second we choose119875 = 021 and the numerical simulation gives 119877

0= 1 the

alcohol free equilibrium 1198640is globally asymptotically stable

(Figure 3) Third we choose 119875 = 0032 and the numericalsimulation gives 119877

0= 1165 gt 1 the alcohol present

equilibrium 119864lowast is globally asymptotically stable (Figure 4)Finally for showing that awareness programs by media

can help to control the population of binge drinking wecompare our model with the model which has been studiedby Mulone and Straughan [10] the model in [10] is governedby the following system of nonlinear ordinary differentialequations

119889119878 (119905)

119889119905= 120583119873 minus

120573119860119878

119873minus 120583119878

119889119860 (119905)

119889119905=120573119860119878

119873+ 120588119877 minus (120601 + 120583)119860

119889119877 (119905)

119889119905= 120601119860 minus (120588 + 120583) 119877

(34)

We choose 119875 = 0032 and the numerical simulation gives1198770= 11645 gt 1 the alcohol present equilibrium 119875lowast of

(34) is globally asymptotically stableThenumber of the bingedrinking population is generally on the rise without effects ofawareness programs by media

For showing the distinction between (2) and (34) we take119875 = 0032 and compare the variations of the binge drinkingpopulation 119860(119905) We can see that because of awarenessprograms by media (Figure 5) the number of people whohave alcohol problems decreases due to awareness-inducedisolation of susceptible people so making more people avoidcontacting with the heavy drinkers due to the fact thatthe awareness programs by media can reduce the alcoholproblems

Further the variations of the binge drinking population119860(119905) with respect to time 119905 for different values of growth rateof density of awareness programs 119898 are shown in Figure 6From these figures it is apparent that as the rate of density ofawareness programs 119898 increases the binge drinking popula-tion 119860(119905) decreases Then the awareness programs by mediaare an effective measure in alcohol problems

6 Abstract and Applied Analysis

Table 1 Description and estimation of parameters

120583 Recruitment rate of the population 025 yearminus1 [10]120573 Transmission coefficient 03 yearminus1 [10]120582 Dissemination rate of aware individuals 002 yearminus1 estimate120590 Rate of transfer of aware individuals to susceptible class 0001 yearminus1 estimate119901 The fraction of 119860 go in to treatment Variable Variable119902 The fraction of 119877 who relapse into 119860 08 yearminus1 [10]119898 Growth rate of density of awareness programs 005 yearminus1 [28]120574 Depletion rate of awareness programs 005 yearminus1 estimate

050 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 2 If 1198770lt 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 200 250 3000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 3 If 1198770= 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 4 If 1198770gt 1 the alcohol present equilibrium 119864lowast is globally

asymptotically stable

0 10 20 30 40 50015

02

025

03

035

04

Bing

e drin

king

pop

ulat

ion

size

A(t) of (21)A(t) of (41)

Time

Figure 5 Variations of the binge drinking population 119860(119905) of (2)and (34)

Abstract and Applied Analysis 7

024

026

028

03

032

034

Bing

e drin

king

pop

ulat

ion

size

m = 002

m = 0008m = 0

0 10 20 30 40 50Time

Figure 6 Variations of binge drinking population with time fordifferent values of119898

5 Discussion

We have formulated the effect of awareness programs onthe binge drinking and analyzed their dynamical behaviorsThe model exhibits two equilibria explicitly the alcohol freeequilibrium and alcohol present equilibrium By constructingLyapunov function we establish some sufficient conditionsfor the global stability of the alcohol free and the alcoholpresent equilibria and give some numerical simulations toexplain our main result In our model we have assumed thatthe cumulative density of awareness programs by media isproportional to the population of binge drinking heavily Itmay be noted that the number of alcohol consumption casesknown to the policy makers is sometimes old and thus thenumber of awareness depends on data So it is more plausibleto consider a time delay in the growth rate of awarenessprograms [21]We canmodify (2) to the followingmodel withdelay

119889119878 (119905)

119889119905= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

119889119883 (119905)

119889119905= 120582119878119872 minus 120590119883 minus 120583119883

119889119860 (119905)

119889119905= 120573119878119860 + 119902119877 minus 119901119860 minus 120583119860

119889119877 (119905)

119889119905= 119901119860 minus 119902119877 minus 120583119877

119889119872 (119905)

119889119905= 119898119860 (119905 minus 120591) minus 120574119872

(35)

We leave these works for the future

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 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] J Rehm ldquoThe risks associated with alcohol use and alcoholismrdquoAlcohol Research amp Health vol 34 no 2 pp 135ndash143 2011

[2] J Rehm C Mathers S Popova M Thavorncharoensap YTeerawattananon and J Patra ldquoGlobal burden of disease andinjury and economic cost attributable to alcohol use andalcohol-use disordersrdquoThe Lancet vol 373 no 9682 pp 2223ndash2233 2009

[3] J Ribes R Cleries L Esteban V Moreno and F X BoschldquoThe influence of alcohol consumption and hepatitis B and Cinfections on the risk of liver cancer in Europerdquo Journal ofHepatology vol 49 no 2 pp 233ndash242 2008

[4] H Wechsler and T F Nelson ldquoFocusing attention on collegestudent alcohol consumption and the environmental conditionsthat promote itrdquo Journal of Studies on Alcohol andDrugs vol 69no 4 pp 481ndash490 2008

[5] L M Powell J Williams and H Wechsler ldquoStudy habits andthe level of alcohol use among college studentsrdquo EducationEconomics vol 12 no 2 pp 135ndash149 2004

[6] H Wechsler B Moeykens A Davenport S Castillo and JHansen ldquoThe adverse impact of heavy episodic drinkers onother college studentsrdquo Journal of Studies on Alcohol and Drugsvol 56 no 6 pp 628ndash634 1995

[7] 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

[8] P Ormerod and G Wiltshire ldquolsquoBingerdquo drinking in the UK asocial network phenomenonrdquoMind amp Society vol 8 no 2 pp135ndash152 2009

[9] F Sanchez X HWang C Castillo-Chavez DM Gorman andP J Gruenewald ldquoDrinking as an epidemica simple mathemat-ical model with recovery and relapserdquo in Therapists Guide toEvidence-Based Relapse Prevention Practical Resources for theMental Health Professional K A Witkiewitz and G A MarlattEds pp 353ndash368 Academic Press Burlington Canada 2007

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

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

[12] H F Huo and C C Zhu ldquoModeling the effect of constantimmigration on drinking behaviourrdquo Submitted

8 Abstract and Applied Analysis

[13] H FHuo andC C Zhu ldquoStability of a quit drinkingmodel withrelapserdquo Submitted

[14] 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

[15] G Schwitzer G Mudur D Henry et al ldquoWhat are theroles and responsibilities of the media in disseminating healthinformationrdquo PLoS Medicine vol 2 no 7 article e215 2005

[16] E Jaramillo ldquoThe impact of media-based health education ontuberculosis diagnosis in Cali Colombiardquo Health Policy andPlanning vol 16 no 1 pp 68ndash73 2001

[17] R Liu JWu andH Zhu ldquoMediapsychological impact onmul-tiple outbreaks of emerging infectious diseasesrdquo Computationaland Mathematical Methods in Medicine vol 8 no 3 pp 153ndash164 2007

[18] T P Palfai R Zisserson and R Saitz ldquoUsing personalizedfeedback to reduce alcohol use among hazardous drinking col-lege students the moderating effect of alcohol-related negativeconsequencesrdquo Addictive Behaviors vol 36 no 5 pp 539ndash5422011

[19] J M Tchuenche and C T Bauch ldquoDynamics of an infectiousdisease where media coverage influences transmissionrdquo ISRNBiomathematics vol 2012 Article ID 581274 10 pages 2012

[20] H Xiang N N Song and H F Huo ldquoModelling effects ofpublic health educational campaigns on drinking dynamicsrdquoSubmitted

[21] A K Misra A Sharma and V Singh ldquoEffect of awarenessprograms in controlling the prevalence of an epidemic withtime delayrdquo Journal of Biological Systems vol 19 no 2 pp 389ndash402 2011

[22] A KMisra A Sharma and J B Shukla ldquoModeling and analysisof effects of awareness programs by media on the spread ofinfectious diseasesrdquoMathematical andComputerModelling vol53 no 5-6 pp 1221ndash1228 2011

[23] ldquoWHO Report of Alcohol in developing societies summaryrdquohttpwwwwhoint

[24] D Chisholm J Rehm M van Ommeren and M MonteiroldquoReducing the global burden of hazardous alcohol use acomparative cost-effectiveness analysisrdquo Journal of Studies onAlcohol vol 65 no 6 pp 782ndash793 2004

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

[26] J-Q Li Y-L Yang and Y-C Zhou ldquoGlobal stability of anepidemic model with latent stage and vaccinationrdquo NonlinearAnalysis Real World Applications vol 12 no 4 pp 2163ndash21732011

[27] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[28] L B Snyder ldquoHealth communication campaigns and theirimpact on behaviorrdquo Journal of Nutrition Education and Behav-ior vol 39 no 2 pp S32ndashS40 2007

[29] D B ClarkO Bukstein and J Cornelius ldquoAlcohol use disordersin adolescents epidemiology diagnosis psychosocial interven-tions and pharmacological treatmentrdquo Pediatric Drugs vol 4no 8 pp 493ndash502 2002

[30] ldquoWebMDReport of Alcohol Abuse Health Centerrdquo httpwwwwebmdcom

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

119877 relapsing into drinkers 119898 represents the growth rate ofdensity of awareness programs which is assumed to be pro-portional to the number of heavy drinkers and 120574 representsthe depletion rate of these programs due to ineffectivenesssocial problems and so forth For instance as time passes bysome of the campaigns by media lose their influence or fail toattract people

22 Basic Properties

221 Invariant Region

Lemma 1 All feasible regions Ω defined by

Ω = (119878 (119905) 119883 (119905) 119860 (119905) 119877 (119905) 119872 (119905)) isin 1198775

+

119878 (119905) + 119883 (119905) + 119860 (119905) + 119877 (119905) le 1 119872 (119905) le119898

120574

(3)

with the initial conditions 119878(0) ge 0119883(0) ge 0119860(0) ge 0119877(0) ge0 and119872(0) ge 0 are positively invariant for system (2)

Proof Adding the first four equations of (2) we have119889119873

119889119905= 120583 minus 120583119873 (4)

It follows that

119873(119905) = 1 + 119873 (0) 119890minus120583119905 (5)

where119873(0) is the initial total number of people Thus

lim119905rarrinfin

119873(119905) = 1 (6)

Then 119878(119905) + 119883(119905) + 119860(119905) + 119877(119905) le 1 the subpopulation isrepresented in proportion to the total number of population

Thus since the system (2) monitors human populationit is plausible to assume that all its state variables andparameters are nonnegative for all 119905 ge 0 Further from thelast equation of system (2) we have

119889119872

119889119905= 119898119860 minus 120574119872

le 119898 minus 120574119872

(7)

It follows that

0 le 119872 (119905) le119898

120574+119872(0) 119890

minus120574119905 (8)

where119872(0) represents the initial value of cumulative densityof awareness programs Thus

lim119905rarrinfin

sup119872(119905) le 119898120574 (9)

It implies that the region

Ω = (119878 (119905) 119883 (119905) 119860 (119905) 119877 (119905) 119872 (119905))

isin 1198775

+ 119878 (119905) + 119883 (119905) + 119860 (119905) + 119877 (119905) le 1 119872 (119905) le

119898

120574

(10)

is a positively invariant set for (2) So we consider dynamicsof system (2) on the setΩ in this paper

222 Positivity of Solutions For system (2) to ensure that thesolutions of the systemwith positive initial conditions remainpositive for all 119905 gt 0 it is necessary to prove that all the statevariables are nonnegative so we have the following lemma

Lemma 2 If 119878(0) gt 0 119883(0) gt 0 119860(0) gt 0 119877(0) gt 0and119872(0) gt 0 then the solutions 119878(119905) 119883(119905) 119860(119905) 119877(119905) and119872(119905) of system (2) are positive for all 119905 ge 0

Proof Under the given initial conditions it is easy to provethat the solutions of the system (2) are positive if not weassume a contradiction that there exists a first time 119905

1such

that119878 (0) gt 0 119878 (119905

1) = 0 119878

1015840(1199051) le 0

119883 (119905) gt 0 119860 (119905) gt 0 119877 (119905) gt 0

119872 (119905) gt 0 0 le 119905 lt 1199051

(11)

In that case from the first equation of system (2) we have

1198781015840(1199051) = 120583 + 120590119883 gt 0 (12)

which is a contradiction meaning that 119878(119905) gt 0 119905 gt 0Or there exists a 119905

2such that

119883(0) gt 0 119883 (1199052) = 0 119883

1015840(1199052) le 0

119878 (119905) gt 0 119860 (119905) gt 0 119877 (119905) gt 0

119872 (119905) gt 0 0 le 119905 lt 1199052

(13)

In that case from the second equation of system (2) we have

1198831015840(1199052) = 120582119878119872 gt 0 (14)

which is a contradiction meaning that119883(119905) gt 0 119905 ge 0Or there exists a 119905

3such that

119860 (0) gt 0 119860 (1199053) le 0 119860

1015840(1199053) = 0

119878 (119905) gt 0 119883 (119905) gt 0 119877 (119905) gt 0

119872 (119905) gt 0 0 le 119905 lt 1199053

(15)

In that case from the third equation of system (2) we have

1198601015840(1199053) = 119902119877 gt 0 (16)

which is a contradiction meaning that 119860(119905) gt 0 119905 ge 0Similarly it can be shown that 119877(119905) gt 0 and 119872(119905) gt 0 forall 119905 ge 0 Thus the solutions 119878(119905) 119883(119905) 119860(119905) 119877(119905) and119872(119905)of system (2) remain positive for all 119905 gt 0

3 Analysis of the Model

There are one alcohol free equilibrium 1198640and one alcohol

present equilibrium 119864lowast for system (2)

31 Alcohol Free Equilibrium and the Basic ReproductionNumber Themodel has an alcohol free equilibrium given by

1198640= (1 0 0 0 0) (17)

4 Abstract and Applied Analysis

In the following the basic reproduction number of system(2) will be obtained by the next generation matrix methodformulated in [25]

Let 119909 = (119860 119877119872 119878 119883)119879 then system (2) can be writtenas

119889119909

119889119905= F (119909) minusV (119909) (18)

where

F (119909) = (

120573119878119860

0

0

0

0

)

V (119909) =((((

(

minus119902119877 + (119901 + 120583)119860

minus119901119860 + (119902 + 120583) 119877

120574119872 minus 119898119860

minus120583 + 120573119878119860 + 120582119878119872 minus 120590119883 + 120583119878

minus120582119878119872 + (120590 + 120583)119883

))))

)

(19)

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

0are respectively

119863F (1198640) = (1198652times20

0 0) (20)

119863V (1198640) = (

1198812times20 0 0

minus119898 0 120574 0 0

120573 0 120582 120583 minus120590

0 0 minus120582 0 120590 + 120583

) (21)

where

119865 = (120573 0

0 0) 119881 = (

119901 + 120583 minus119902

minus119901 119902 + 120583) (22)

Themodel reproduction number denoted by 1198770is thus given

by

1198770=

120573 (119902 + 120583)

(119901 + 120583) (119902 + 120583) minus 119901119902 (23)

32 Global Stability of 1198640

Theorem 3 For system (2) the alcohol free equilibrium 1198640is

globally asymptotically stable if 1198770le 1

Proof We introduce the following Lyapunov function [26]

119881 (119878 (119905) 119883 (119905) 119860 (119905) 119877 (119905) 119872 (119905))

= 119878 minus ln 119878 + 119883 + 119860 +119902

119902 + 120583119877

(24)

The derivative of 119881 is given by

= 119878 minus

119878

119878+ +

119902

119902 + 120583 +

= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

minus1

119878(120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878)

+ 120582119878119872 minus 120590119883 minus 120583119883 + 120573119878119860 + 119902119877 minus 119901119860

minus 120583119860 +119902

119902 + 120583(119901119860 minus 119902119877 minus 120583119877)

= 120583 minus 120583119878 minus1

119878120583 + 120573119860 + 120582119872 minus

1

119878120590119883 + 120583

minus 120583119883 + 119902119877 minus 119901119860 minus 120583119860 +119902

119902 + 120583119901119860 minus 119902119877

= 120583(2 minus 119878 minus1

119878) + 120573(1 minus

1

1198770

)119860 + (1 minus1

119878) 120590119883

(25)

If 1198770le 1 then 120573(1 minus (1119877

0)) le 0 As we know

2 minus 119878 minus (1119878) le 0 and 1 minus (1119878)120590119883 le 0 so we obtain le 0 Furthermore = 0 only if 119878 = 1 or 119877

0= 1

The maximum invariant set in (119878 119883 119860 119877119872) = 0 isthe singleton 119864

0 By Lasallersquos invariance principle [27] 119864

0is

globally asymptotically stable inΩ

33 Alcohol Present Equilibrium

331 Existence of the Alcohol Present Equilibrium

Theorem 4 If 1198770gt 1 system (2) has a unique alcohol present

equilibrium 119864lowast(119878lowast 119883lowast 119860lowast 119877lowast119872lowast) where

119878lowast=1

120573(119901 + 120583 minus

119901119902

119902 + 120583)

119883lowast=120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860lowast

119877lowast=119901

119902 + 120583119860lowast

119872lowast=119898

120574119860lowast

(26)

Proof It follows from system (2) that

120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878 = 0

120582119878119872 minus 120590119883 minus 120583119883 = 0

120573119878119860 + 119902119877 minus 119901119860 minus 120583119860 = 0

119901119860 minus 119902119877 minus 120583119877 = 0

119898119860 minus 120574119872 = 0

(27)

Abstract and Applied Analysis 5

From the fourth and fifth equations of (27) we obtain

119877 =119901

119902 + 120583119860 (28)

119872 =119898

120574119860 (29)

Substituting 119877 into the third equation of (27) we have

119878 =1

120573(119901 + 120583 minus

119901119902

119902 + 120583) (30)

Then substituting (29) and (30) into the second equation of(27) we get

119883 =120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860 (31)

For 119860 = 0 substituting (29) (30) and (31) into the firstequation of (27) gives

120583 minus 120583119878

= (119901 + 120583 minus119901119902

119902 + 120583)119860 +120582119898

120573120574(119901 + 120583 minus

119901119902

119902 + 120583)119860

minus120590120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860

(32)

Substituting (30) into (32) we get

119860 = (120583 minus120583

120573(119901 + 120583 minus

119901119902

119902 + 120583))

times ((119901 + 120583 minus119901119902

119902 + 120583)[1 +120582119898

120573120574minus120590120582119898

120573120574 (120590 + 120583)])

minus1

= 120583(1 minus1

1198770

)

times ((119901 + 120583 minus119901119902

119902 + 120583)[1 +120582119898

120573120574minus120590120582119898

120573120574 (120590 + 120583)])

minus1

(33)

Therefore there exists a unique positive root in theinterval (0 1) when 119877

0gt 1 there is no positive root in the

interval [0 1] when 1198770le 1

For1198770gt 1 we carry out the simulation (see Figure 4) and

give the following conjecture

Conjecture If 1198770gt 1 the alcohol present equilibrium 119864lowast of

(2) is globally asymptotically stable

Remark Since the global stability of (2) is a hard problem weonly carry out simulation (see Figure 4) How to prove theglobal stability of the alcohol present equilibrium 119864lowast of (2)and give the conditions based on model parameters is still anopen problem

4 Numerical Simulation

To illustrate the analytic results obtained above we give somesimulations using the parameter values in Table 1 Numericalresults are displayed in the following figures It is reportedthat binge drinking occurs in up to 30 percent of adolescentsand alcohol use disorders occur in about 6 percent of thisage group approximately 50 to 60 percent of them remainabstinent at the end of a yearrsquos treatment and a majority ofthose stay dry permanently [29 30] So the initial condition istaken as the lowest by119860(0) = 03 and119877(0) = 015 in additionwe assume that119883(0) = 01 119878(0) = 045 and119872(0) = 0002

First we choose 119875 = 03 and the numerical simulationgives 119877

0= 05691 lt 1 then the alcohol free equilibrium 119864

0is

globally asymptotically stable (Figure 2) Second we choose119875 = 021 and the numerical simulation gives 119877

0= 1 the

alcohol free equilibrium 1198640is globally asymptotically stable

(Figure 3) Third we choose 119875 = 0032 and the numericalsimulation gives 119877

0= 1165 gt 1 the alcohol present

equilibrium 119864lowast is globally asymptotically stable (Figure 4)Finally for showing that awareness programs by media

can help to control the population of binge drinking wecompare our model with the model which has been studiedby Mulone and Straughan [10] the model in [10] is governedby the following system of nonlinear ordinary differentialequations

119889119878 (119905)

119889119905= 120583119873 minus

120573119860119878

119873minus 120583119878

119889119860 (119905)

119889119905=120573119860119878

119873+ 120588119877 minus (120601 + 120583)119860

119889119877 (119905)

119889119905= 120601119860 minus (120588 + 120583) 119877

(34)

We choose 119875 = 0032 and the numerical simulation gives1198770= 11645 gt 1 the alcohol present equilibrium 119875lowast of

(34) is globally asymptotically stableThenumber of the bingedrinking population is generally on the rise without effects ofawareness programs by media

For showing the distinction between (2) and (34) we take119875 = 0032 and compare the variations of the binge drinkingpopulation 119860(119905) We can see that because of awarenessprograms by media (Figure 5) the number of people whohave alcohol problems decreases due to awareness-inducedisolation of susceptible people so making more people avoidcontacting with the heavy drinkers due to the fact thatthe awareness programs by media can reduce the alcoholproblems

Further the variations of the binge drinking population119860(119905) with respect to time 119905 for different values of growth rateof density of awareness programs 119898 are shown in Figure 6From these figures it is apparent that as the rate of density ofawareness programs 119898 increases the binge drinking popula-tion 119860(119905) decreases Then the awareness programs by mediaare an effective measure in alcohol problems

6 Abstract and Applied Analysis

Table 1 Description and estimation of parameters

120583 Recruitment rate of the population 025 yearminus1 [10]120573 Transmission coefficient 03 yearminus1 [10]120582 Dissemination rate of aware individuals 002 yearminus1 estimate120590 Rate of transfer of aware individuals to susceptible class 0001 yearminus1 estimate119901 The fraction of 119860 go in to treatment Variable Variable119902 The fraction of 119877 who relapse into 119860 08 yearminus1 [10]119898 Growth rate of density of awareness programs 005 yearminus1 [28]120574 Depletion rate of awareness programs 005 yearminus1 estimate

050 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 2 If 1198770lt 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 200 250 3000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 3 If 1198770= 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 4 If 1198770gt 1 the alcohol present equilibrium 119864lowast is globally

asymptotically stable

0 10 20 30 40 50015

02

025

03

035

04

Bing

e drin

king

pop

ulat

ion

size

A(t) of (21)A(t) of (41)

Time

Figure 5 Variations of the binge drinking population 119860(119905) of (2)and (34)

Abstract and Applied Analysis 7

024

026

028

03

032

034

Bing

e drin

king

pop

ulat

ion

size

m = 002

m = 0008m = 0

0 10 20 30 40 50Time

Figure 6 Variations of binge drinking population with time fordifferent values of119898

5 Discussion

We have formulated the effect of awareness programs onthe binge drinking and analyzed their dynamical behaviorsThe model exhibits two equilibria explicitly the alcohol freeequilibrium and alcohol present equilibrium By constructingLyapunov function we establish some sufficient conditionsfor the global stability of the alcohol free and the alcoholpresent equilibria and give some numerical simulations toexplain our main result In our model we have assumed thatthe cumulative density of awareness programs by media isproportional to the population of binge drinking heavily Itmay be noted that the number of alcohol consumption casesknown to the policy makers is sometimes old and thus thenumber of awareness depends on data So it is more plausibleto consider a time delay in the growth rate of awarenessprograms [21]We canmodify (2) to the followingmodel withdelay

119889119878 (119905)

119889119905= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

119889119883 (119905)

119889119905= 120582119878119872 minus 120590119883 minus 120583119883

119889119860 (119905)

119889119905= 120573119878119860 + 119902119877 minus 119901119860 minus 120583119860

119889119877 (119905)

119889119905= 119901119860 minus 119902119877 minus 120583119877

119889119872 (119905)

119889119905= 119898119860 (119905 minus 120591) minus 120574119872

(35)

We leave these works for the future

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 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] J Rehm ldquoThe risks associated with alcohol use and alcoholismrdquoAlcohol Research amp Health vol 34 no 2 pp 135ndash143 2011

[2] J Rehm C Mathers S Popova M Thavorncharoensap YTeerawattananon and J Patra ldquoGlobal burden of disease andinjury and economic cost attributable to alcohol use andalcohol-use disordersrdquoThe Lancet vol 373 no 9682 pp 2223ndash2233 2009

[3] J Ribes R Cleries L Esteban V Moreno and F X BoschldquoThe influence of alcohol consumption and hepatitis B and Cinfections on the risk of liver cancer in Europerdquo Journal ofHepatology vol 49 no 2 pp 233ndash242 2008

[4] H Wechsler and T F Nelson ldquoFocusing attention on collegestudent alcohol consumption and the environmental conditionsthat promote itrdquo Journal of Studies on Alcohol andDrugs vol 69no 4 pp 481ndash490 2008

[5] L M Powell J Williams and H Wechsler ldquoStudy habits andthe level of alcohol use among college studentsrdquo EducationEconomics vol 12 no 2 pp 135ndash149 2004

[6] H Wechsler B Moeykens A Davenport S Castillo and JHansen ldquoThe adverse impact of heavy episodic drinkers onother college studentsrdquo Journal of Studies on Alcohol and Drugsvol 56 no 6 pp 628ndash634 1995

[7] 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

[8] P Ormerod and G Wiltshire ldquolsquoBingerdquo drinking in the UK asocial network phenomenonrdquoMind amp Society vol 8 no 2 pp135ndash152 2009

[9] F Sanchez X HWang C Castillo-Chavez DM Gorman andP J Gruenewald ldquoDrinking as an epidemica simple mathemat-ical model with recovery and relapserdquo in Therapists Guide toEvidence-Based Relapse Prevention Practical Resources for theMental Health Professional K A Witkiewitz and G A MarlattEds pp 353ndash368 Academic Press Burlington Canada 2007

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

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

[12] H F Huo and C C Zhu ldquoModeling the effect of constantimmigration on drinking behaviourrdquo Submitted

8 Abstract and Applied Analysis

[13] H FHuo andC C Zhu ldquoStability of a quit drinkingmodel withrelapserdquo Submitted

[14] 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

[15] G Schwitzer G Mudur D Henry et al ldquoWhat are theroles and responsibilities of the media in disseminating healthinformationrdquo PLoS Medicine vol 2 no 7 article e215 2005

[16] E Jaramillo ldquoThe impact of media-based health education ontuberculosis diagnosis in Cali Colombiardquo Health Policy andPlanning vol 16 no 1 pp 68ndash73 2001

[17] R Liu JWu andH Zhu ldquoMediapsychological impact onmul-tiple outbreaks of emerging infectious diseasesrdquo Computationaland Mathematical Methods in Medicine vol 8 no 3 pp 153ndash164 2007

[18] T P Palfai R Zisserson and R Saitz ldquoUsing personalizedfeedback to reduce alcohol use among hazardous drinking col-lege students the moderating effect of alcohol-related negativeconsequencesrdquo Addictive Behaviors vol 36 no 5 pp 539ndash5422011

[19] J M Tchuenche and C T Bauch ldquoDynamics of an infectiousdisease where media coverage influences transmissionrdquo ISRNBiomathematics vol 2012 Article ID 581274 10 pages 2012

[20] H Xiang N N Song and H F Huo ldquoModelling effects ofpublic health educational campaigns on drinking dynamicsrdquoSubmitted

[21] A K Misra A Sharma and V Singh ldquoEffect of awarenessprograms in controlling the prevalence of an epidemic withtime delayrdquo Journal of Biological Systems vol 19 no 2 pp 389ndash402 2011

[22] A KMisra A Sharma and J B Shukla ldquoModeling and analysisof effects of awareness programs by media on the spread ofinfectious diseasesrdquoMathematical andComputerModelling vol53 no 5-6 pp 1221ndash1228 2011

[23] ldquoWHO Report of Alcohol in developing societies summaryrdquohttpwwwwhoint

[24] D Chisholm J Rehm M van Ommeren and M MonteiroldquoReducing the global burden of hazardous alcohol use acomparative cost-effectiveness analysisrdquo Journal of Studies onAlcohol vol 65 no 6 pp 782ndash793 2004

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

[26] J-Q Li Y-L Yang and Y-C Zhou ldquoGlobal stability of anepidemic model with latent stage and vaccinationrdquo NonlinearAnalysis Real World Applications vol 12 no 4 pp 2163ndash21732011

[27] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[28] L B Snyder ldquoHealth communication campaigns and theirimpact on behaviorrdquo Journal of Nutrition Education and Behav-ior vol 39 no 2 pp S32ndashS40 2007

[29] D B ClarkO Bukstein and J Cornelius ldquoAlcohol use disordersin adolescents epidemiology diagnosis psychosocial interven-tions and pharmacological treatmentrdquo Pediatric Drugs vol 4no 8 pp 493ndash502 2002

[30] ldquoWebMDReport of Alcohol Abuse Health Centerrdquo httpwwwwebmdcom

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

In the following the basic reproduction number of system(2) will be obtained by the next generation matrix methodformulated in [25]

Let 119909 = (119860 119877119872 119878 119883)119879 then system (2) can be writtenas

119889119909

119889119905= F (119909) minusV (119909) (18)

where

F (119909) = (

120573119878119860

0

0

0

0

)

V (119909) =((((

(

minus119902119877 + (119901 + 120583)119860

minus119901119860 + (119902 + 120583) 119877

120574119872 minus 119898119860

minus120583 + 120573119878119860 + 120582119878119872 minus 120590119883 + 120583119878

minus120582119878119872 + (120590 + 120583)119883

))))

)

(19)

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

0are respectively

119863F (1198640) = (1198652times20

0 0) (20)

119863V (1198640) = (

1198812times20 0 0

minus119898 0 120574 0 0

120573 0 120582 120583 minus120590

0 0 minus120582 0 120590 + 120583

) (21)

where

119865 = (120573 0

0 0) 119881 = (

119901 + 120583 minus119902

minus119901 119902 + 120583) (22)

Themodel reproduction number denoted by 1198770is thus given

by

1198770=

120573 (119902 + 120583)

(119901 + 120583) (119902 + 120583) minus 119901119902 (23)

32 Global Stability of 1198640

Theorem 3 For system (2) the alcohol free equilibrium 1198640is

globally asymptotically stable if 1198770le 1

Proof We introduce the following Lyapunov function [26]

119881 (119878 (119905) 119883 (119905) 119860 (119905) 119877 (119905) 119872 (119905))

= 119878 minus ln 119878 + 119883 + 119860 +119902

119902 + 120583119877

(24)

The derivative of 119881 is given by

= 119878 minus

119878

119878+ +

119902

119902 + 120583 +

= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

minus1

119878(120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878)

+ 120582119878119872 minus 120590119883 minus 120583119883 + 120573119878119860 + 119902119877 minus 119901119860

minus 120583119860 +119902

119902 + 120583(119901119860 minus 119902119877 minus 120583119877)

= 120583 minus 120583119878 minus1

119878120583 + 120573119860 + 120582119872 minus

1

119878120590119883 + 120583

minus 120583119883 + 119902119877 minus 119901119860 minus 120583119860 +119902

119902 + 120583119901119860 minus 119902119877

= 120583(2 minus 119878 minus1

119878) + 120573(1 minus

1

1198770

)119860 + (1 minus1

119878) 120590119883

(25)

If 1198770le 1 then 120573(1 minus (1119877

0)) le 0 As we know

2 minus 119878 minus (1119878) le 0 and 1 minus (1119878)120590119883 le 0 so we obtain le 0 Furthermore = 0 only if 119878 = 1 or 119877

0= 1

The maximum invariant set in (119878 119883 119860 119877119872) = 0 isthe singleton 119864

0 By Lasallersquos invariance principle [27] 119864

0is

globally asymptotically stable inΩ

33 Alcohol Present Equilibrium

331 Existence of the Alcohol Present Equilibrium

Theorem 4 If 1198770gt 1 system (2) has a unique alcohol present

equilibrium 119864lowast(119878lowast 119883lowast 119860lowast 119877lowast119872lowast) where

119878lowast=1

120573(119901 + 120583 minus

119901119902

119902 + 120583)

119883lowast=120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860lowast

119877lowast=119901

119902 + 120583119860lowast

119872lowast=119898

120574119860lowast

(26)

Proof It follows from system (2) that

120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878 = 0

120582119878119872 minus 120590119883 minus 120583119883 = 0

120573119878119860 + 119902119877 minus 119901119860 minus 120583119860 = 0

119901119860 minus 119902119877 minus 120583119877 = 0

119898119860 minus 120574119872 = 0

(27)

Abstract and Applied Analysis 5

From the fourth and fifth equations of (27) we obtain

119877 =119901

119902 + 120583119860 (28)

119872 =119898

120574119860 (29)

Substituting 119877 into the third equation of (27) we have

119878 =1

120573(119901 + 120583 minus

119901119902

119902 + 120583) (30)

Then substituting (29) and (30) into the second equation of(27) we get

119883 =120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860 (31)

For 119860 = 0 substituting (29) (30) and (31) into the firstequation of (27) gives

120583 minus 120583119878

= (119901 + 120583 minus119901119902

119902 + 120583)119860 +120582119898

120573120574(119901 + 120583 minus

119901119902

119902 + 120583)119860

minus120590120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860

(32)

Substituting (30) into (32) we get

119860 = (120583 minus120583

120573(119901 + 120583 minus

119901119902

119902 + 120583))

times ((119901 + 120583 minus119901119902

119902 + 120583)[1 +120582119898

120573120574minus120590120582119898

120573120574 (120590 + 120583)])

minus1

= 120583(1 minus1

1198770

)

times ((119901 + 120583 minus119901119902

119902 + 120583)[1 +120582119898

120573120574minus120590120582119898

120573120574 (120590 + 120583)])

minus1

(33)

Therefore there exists a unique positive root in theinterval (0 1) when 119877

0gt 1 there is no positive root in the

interval [0 1] when 1198770le 1

For1198770gt 1 we carry out the simulation (see Figure 4) and

give the following conjecture

Conjecture If 1198770gt 1 the alcohol present equilibrium 119864lowast of

(2) is globally asymptotically stable

Remark Since the global stability of (2) is a hard problem weonly carry out simulation (see Figure 4) How to prove theglobal stability of the alcohol present equilibrium 119864lowast of (2)and give the conditions based on model parameters is still anopen problem

4 Numerical Simulation

To illustrate the analytic results obtained above we give somesimulations using the parameter values in Table 1 Numericalresults are displayed in the following figures It is reportedthat binge drinking occurs in up to 30 percent of adolescentsand alcohol use disorders occur in about 6 percent of thisage group approximately 50 to 60 percent of them remainabstinent at the end of a yearrsquos treatment and a majority ofthose stay dry permanently [29 30] So the initial condition istaken as the lowest by119860(0) = 03 and119877(0) = 015 in additionwe assume that119883(0) = 01 119878(0) = 045 and119872(0) = 0002

First we choose 119875 = 03 and the numerical simulationgives 119877

0= 05691 lt 1 then the alcohol free equilibrium 119864

0is

globally asymptotically stable (Figure 2) Second we choose119875 = 021 and the numerical simulation gives 119877

0= 1 the

alcohol free equilibrium 1198640is globally asymptotically stable

(Figure 3) Third we choose 119875 = 0032 and the numericalsimulation gives 119877

0= 1165 gt 1 the alcohol present

equilibrium 119864lowast is globally asymptotically stable (Figure 4)Finally for showing that awareness programs by media

can help to control the population of binge drinking wecompare our model with the model which has been studiedby Mulone and Straughan [10] the model in [10] is governedby the following system of nonlinear ordinary differentialequations

119889119878 (119905)

119889119905= 120583119873 minus

120573119860119878

119873minus 120583119878

119889119860 (119905)

119889119905=120573119860119878

119873+ 120588119877 minus (120601 + 120583)119860

119889119877 (119905)

119889119905= 120601119860 minus (120588 + 120583) 119877

(34)

We choose 119875 = 0032 and the numerical simulation gives1198770= 11645 gt 1 the alcohol present equilibrium 119875lowast of

(34) is globally asymptotically stableThenumber of the bingedrinking population is generally on the rise without effects ofawareness programs by media

For showing the distinction between (2) and (34) we take119875 = 0032 and compare the variations of the binge drinkingpopulation 119860(119905) We can see that because of awarenessprograms by media (Figure 5) the number of people whohave alcohol problems decreases due to awareness-inducedisolation of susceptible people so making more people avoidcontacting with the heavy drinkers due to the fact thatthe awareness programs by media can reduce the alcoholproblems

Further the variations of the binge drinking population119860(119905) with respect to time 119905 for different values of growth rateof density of awareness programs 119898 are shown in Figure 6From these figures it is apparent that as the rate of density ofawareness programs 119898 increases the binge drinking popula-tion 119860(119905) decreases Then the awareness programs by mediaare an effective measure in alcohol problems

6 Abstract and Applied Analysis

Table 1 Description and estimation of parameters

120583 Recruitment rate of the population 025 yearminus1 [10]120573 Transmission coefficient 03 yearminus1 [10]120582 Dissemination rate of aware individuals 002 yearminus1 estimate120590 Rate of transfer of aware individuals to susceptible class 0001 yearminus1 estimate119901 The fraction of 119860 go in to treatment Variable Variable119902 The fraction of 119877 who relapse into 119860 08 yearminus1 [10]119898 Growth rate of density of awareness programs 005 yearminus1 [28]120574 Depletion rate of awareness programs 005 yearminus1 estimate

050 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 2 If 1198770lt 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 200 250 3000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 3 If 1198770= 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 4 If 1198770gt 1 the alcohol present equilibrium 119864lowast is globally

asymptotically stable

0 10 20 30 40 50015

02

025

03

035

04

Bing

e drin

king

pop

ulat

ion

size

A(t) of (21)A(t) of (41)

Time

Figure 5 Variations of the binge drinking population 119860(119905) of (2)and (34)

Abstract and Applied Analysis 7

024

026

028

03

032

034

Bing

e drin

king

pop

ulat

ion

size

m = 002

m = 0008m = 0

0 10 20 30 40 50Time

Figure 6 Variations of binge drinking population with time fordifferent values of119898

5 Discussion

We have formulated the effect of awareness programs onthe binge drinking and analyzed their dynamical behaviorsThe model exhibits two equilibria explicitly the alcohol freeequilibrium and alcohol present equilibrium By constructingLyapunov function we establish some sufficient conditionsfor the global stability of the alcohol free and the alcoholpresent equilibria and give some numerical simulations toexplain our main result In our model we have assumed thatthe cumulative density of awareness programs by media isproportional to the population of binge drinking heavily Itmay be noted that the number of alcohol consumption casesknown to the policy makers is sometimes old and thus thenumber of awareness depends on data So it is more plausibleto consider a time delay in the growth rate of awarenessprograms [21]We canmodify (2) to the followingmodel withdelay

119889119878 (119905)

119889119905= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

119889119883 (119905)

119889119905= 120582119878119872 minus 120590119883 minus 120583119883

119889119860 (119905)

119889119905= 120573119878119860 + 119902119877 minus 119901119860 minus 120583119860

119889119877 (119905)

119889119905= 119901119860 minus 119902119877 minus 120583119877

119889119872 (119905)

119889119905= 119898119860 (119905 minus 120591) minus 120574119872

(35)

We leave these works for the future

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 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] J Rehm ldquoThe risks associated with alcohol use and alcoholismrdquoAlcohol Research amp Health vol 34 no 2 pp 135ndash143 2011

[2] J Rehm C Mathers S Popova M Thavorncharoensap YTeerawattananon and J Patra ldquoGlobal burden of disease andinjury and economic cost attributable to alcohol use andalcohol-use disordersrdquoThe Lancet vol 373 no 9682 pp 2223ndash2233 2009

[3] J Ribes R Cleries L Esteban V Moreno and F X BoschldquoThe influence of alcohol consumption and hepatitis B and Cinfections on the risk of liver cancer in Europerdquo Journal ofHepatology vol 49 no 2 pp 233ndash242 2008

[4] H Wechsler and T F Nelson ldquoFocusing attention on collegestudent alcohol consumption and the environmental conditionsthat promote itrdquo Journal of Studies on Alcohol andDrugs vol 69no 4 pp 481ndash490 2008

[5] L M Powell J Williams and H Wechsler ldquoStudy habits andthe level of alcohol use among college studentsrdquo EducationEconomics vol 12 no 2 pp 135ndash149 2004

[6] H Wechsler B Moeykens A Davenport S Castillo and JHansen ldquoThe adverse impact of heavy episodic drinkers onother college studentsrdquo Journal of Studies on Alcohol and Drugsvol 56 no 6 pp 628ndash634 1995

[7] 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

[8] P Ormerod and G Wiltshire ldquolsquoBingerdquo drinking in the UK asocial network phenomenonrdquoMind amp Society vol 8 no 2 pp135ndash152 2009

[9] F Sanchez X HWang C Castillo-Chavez DM Gorman andP J Gruenewald ldquoDrinking as an epidemica simple mathemat-ical model with recovery and relapserdquo in Therapists Guide toEvidence-Based Relapse Prevention Practical Resources for theMental Health Professional K A Witkiewitz and G A MarlattEds pp 353ndash368 Academic Press Burlington Canada 2007

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

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

[12] H F Huo and C C Zhu ldquoModeling the effect of constantimmigration on drinking behaviourrdquo Submitted

8 Abstract and Applied Analysis

[13] H FHuo andC C Zhu ldquoStability of a quit drinkingmodel withrelapserdquo Submitted

[14] 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

[15] G Schwitzer G Mudur D Henry et al ldquoWhat are theroles and responsibilities of the media in disseminating healthinformationrdquo PLoS Medicine vol 2 no 7 article e215 2005

[16] E Jaramillo ldquoThe impact of media-based health education ontuberculosis diagnosis in Cali Colombiardquo Health Policy andPlanning vol 16 no 1 pp 68ndash73 2001

[17] R Liu JWu andH Zhu ldquoMediapsychological impact onmul-tiple outbreaks of emerging infectious diseasesrdquo Computationaland Mathematical Methods in Medicine vol 8 no 3 pp 153ndash164 2007

[18] T P Palfai R Zisserson and R Saitz ldquoUsing personalizedfeedback to reduce alcohol use among hazardous drinking col-lege students the moderating effect of alcohol-related negativeconsequencesrdquo Addictive Behaviors vol 36 no 5 pp 539ndash5422011

[19] J M Tchuenche and C T Bauch ldquoDynamics of an infectiousdisease where media coverage influences transmissionrdquo ISRNBiomathematics vol 2012 Article ID 581274 10 pages 2012

[20] H Xiang N N Song and H F Huo ldquoModelling effects ofpublic health educational campaigns on drinking dynamicsrdquoSubmitted

[21] A K Misra A Sharma and V Singh ldquoEffect of awarenessprograms in controlling the prevalence of an epidemic withtime delayrdquo Journal of Biological Systems vol 19 no 2 pp 389ndash402 2011

[22] A KMisra A Sharma and J B Shukla ldquoModeling and analysisof effects of awareness programs by media on the spread ofinfectious diseasesrdquoMathematical andComputerModelling vol53 no 5-6 pp 1221ndash1228 2011

[23] ldquoWHO Report of Alcohol in developing societies summaryrdquohttpwwwwhoint

[24] D Chisholm J Rehm M van Ommeren and M MonteiroldquoReducing the global burden of hazardous alcohol use acomparative cost-effectiveness analysisrdquo Journal of Studies onAlcohol vol 65 no 6 pp 782ndash793 2004

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

[26] J-Q Li Y-L Yang and Y-C Zhou ldquoGlobal stability of anepidemic model with latent stage and vaccinationrdquo NonlinearAnalysis Real World Applications vol 12 no 4 pp 2163ndash21732011

[27] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[28] L B Snyder ldquoHealth communication campaigns and theirimpact on behaviorrdquo Journal of Nutrition Education and Behav-ior vol 39 no 2 pp S32ndashS40 2007

[29] D B ClarkO Bukstein and J Cornelius ldquoAlcohol use disordersin adolescents epidemiology diagnosis psychosocial interven-tions and pharmacological treatmentrdquo Pediatric Drugs vol 4no 8 pp 493ndash502 2002

[30] ldquoWebMDReport of Alcohol Abuse Health Centerrdquo httpwwwwebmdcom

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

From the fourth and fifth equations of (27) we obtain

119877 =119901

119902 + 120583119860 (28)

119872 =119898

120574119860 (29)

Substituting 119877 into the third equation of (27) we have

119878 =1

120573(119901 + 120583 minus

119901119902

119902 + 120583) (30)

Then substituting (29) and (30) into the second equation of(27) we get

119883 =120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860 (31)

For 119860 = 0 substituting (29) (30) and (31) into the firstequation of (27) gives

120583 minus 120583119878

= (119901 + 120583 minus119901119902

119902 + 120583)119860 +120582119898

120573120574(119901 + 120583 minus

119901119902

119902 + 120583)119860

minus120590120582119898

120573120574 (120590 + 120583)(119901 + 120583 minus

119901119902

119902 + 120583)119860

(32)

Substituting (30) into (32) we get

119860 = (120583 minus120583

120573(119901 + 120583 minus

119901119902

119902 + 120583))

times ((119901 + 120583 minus119901119902

119902 + 120583)[1 +120582119898

120573120574minus120590120582119898

120573120574 (120590 + 120583)])

minus1

= 120583(1 minus1

1198770

)

times ((119901 + 120583 minus119901119902

119902 + 120583)[1 +120582119898

120573120574minus120590120582119898

120573120574 (120590 + 120583)])

minus1

(33)

Therefore there exists a unique positive root in theinterval (0 1) when 119877

0gt 1 there is no positive root in the

interval [0 1] when 1198770le 1

For1198770gt 1 we carry out the simulation (see Figure 4) and

give the following conjecture

Conjecture If 1198770gt 1 the alcohol present equilibrium 119864lowast of

(2) is globally asymptotically stable

Remark Since the global stability of (2) is a hard problem weonly carry out simulation (see Figure 4) How to prove theglobal stability of the alcohol present equilibrium 119864lowast of (2)and give the conditions based on model parameters is still anopen problem

4 Numerical Simulation

To illustrate the analytic results obtained above we give somesimulations using the parameter values in Table 1 Numericalresults are displayed in the following figures It is reportedthat binge drinking occurs in up to 30 percent of adolescentsand alcohol use disorders occur in about 6 percent of thisage group approximately 50 to 60 percent of them remainabstinent at the end of a yearrsquos treatment and a majority ofthose stay dry permanently [29 30] So the initial condition istaken as the lowest by119860(0) = 03 and119877(0) = 015 in additionwe assume that119883(0) = 01 119878(0) = 045 and119872(0) = 0002

First we choose 119875 = 03 and the numerical simulationgives 119877

0= 05691 lt 1 then the alcohol free equilibrium 119864

0is

globally asymptotically stable (Figure 2) Second we choose119875 = 021 and the numerical simulation gives 119877

0= 1 the

alcohol free equilibrium 1198640is globally asymptotically stable

(Figure 3) Third we choose 119875 = 0032 and the numericalsimulation gives 119877

0= 1165 gt 1 the alcohol present

equilibrium 119864lowast is globally asymptotically stable (Figure 4)Finally for showing that awareness programs by media

can help to control the population of binge drinking wecompare our model with the model which has been studiedby Mulone and Straughan [10] the model in [10] is governedby the following system of nonlinear ordinary differentialequations

119889119878 (119905)

119889119905= 120583119873 minus

120573119860119878

119873minus 120583119878

119889119860 (119905)

119889119905=120573119860119878

119873+ 120588119877 minus (120601 + 120583)119860

119889119877 (119905)

119889119905= 120601119860 minus (120588 + 120583) 119877

(34)

We choose 119875 = 0032 and the numerical simulation gives1198770= 11645 gt 1 the alcohol present equilibrium 119875lowast of

(34) is globally asymptotically stableThenumber of the bingedrinking population is generally on the rise without effects ofawareness programs by media

For showing the distinction between (2) and (34) we take119875 = 0032 and compare the variations of the binge drinkingpopulation 119860(119905) We can see that because of awarenessprograms by media (Figure 5) the number of people whohave alcohol problems decreases due to awareness-inducedisolation of susceptible people so making more people avoidcontacting with the heavy drinkers due to the fact thatthe awareness programs by media can reduce the alcoholproblems

Further the variations of the binge drinking population119860(119905) with respect to time 119905 for different values of growth rateof density of awareness programs 119898 are shown in Figure 6From these figures it is apparent that as the rate of density ofawareness programs 119898 increases the binge drinking popula-tion 119860(119905) decreases Then the awareness programs by mediaare an effective measure in alcohol problems

6 Abstract and Applied Analysis

Table 1 Description and estimation of parameters

120583 Recruitment rate of the population 025 yearminus1 [10]120573 Transmission coefficient 03 yearminus1 [10]120582 Dissemination rate of aware individuals 002 yearminus1 estimate120590 Rate of transfer of aware individuals to susceptible class 0001 yearminus1 estimate119901 The fraction of 119860 go in to treatment Variable Variable119902 The fraction of 119877 who relapse into 119860 08 yearminus1 [10]119898 Growth rate of density of awareness programs 005 yearminus1 [28]120574 Depletion rate of awareness programs 005 yearminus1 estimate

050 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 2 If 1198770lt 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 200 250 3000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 3 If 1198770= 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 4 If 1198770gt 1 the alcohol present equilibrium 119864lowast is globally

asymptotically stable

0 10 20 30 40 50015

02

025

03

035

04

Bing

e drin

king

pop

ulat

ion

size

A(t) of (21)A(t) of (41)

Time

Figure 5 Variations of the binge drinking population 119860(119905) of (2)and (34)

Abstract and Applied Analysis 7

024

026

028

03

032

034

Bing

e drin

king

pop

ulat

ion

size

m = 002

m = 0008m = 0

0 10 20 30 40 50Time

Figure 6 Variations of binge drinking population with time fordifferent values of119898

5 Discussion

We have formulated the effect of awareness programs onthe binge drinking and analyzed their dynamical behaviorsThe model exhibits two equilibria explicitly the alcohol freeequilibrium and alcohol present equilibrium By constructingLyapunov function we establish some sufficient conditionsfor the global stability of the alcohol free and the alcoholpresent equilibria and give some numerical simulations toexplain our main result In our model we have assumed thatthe cumulative density of awareness programs by media isproportional to the population of binge drinking heavily Itmay be noted that the number of alcohol consumption casesknown to the policy makers is sometimes old and thus thenumber of awareness depends on data So it is more plausibleto consider a time delay in the growth rate of awarenessprograms [21]We canmodify (2) to the followingmodel withdelay

119889119878 (119905)

119889119905= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

119889119883 (119905)

119889119905= 120582119878119872 minus 120590119883 minus 120583119883

119889119860 (119905)

119889119905= 120573119878119860 + 119902119877 minus 119901119860 minus 120583119860

119889119877 (119905)

119889119905= 119901119860 minus 119902119877 minus 120583119877

119889119872 (119905)

119889119905= 119898119860 (119905 minus 120591) minus 120574119872

(35)

We leave these works for the future

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 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] J Rehm ldquoThe risks associated with alcohol use and alcoholismrdquoAlcohol Research amp Health vol 34 no 2 pp 135ndash143 2011

[2] J Rehm C Mathers S Popova M Thavorncharoensap YTeerawattananon and J Patra ldquoGlobal burden of disease andinjury and economic cost attributable to alcohol use andalcohol-use disordersrdquoThe Lancet vol 373 no 9682 pp 2223ndash2233 2009

[3] J Ribes R Cleries L Esteban V Moreno and F X BoschldquoThe influence of alcohol consumption and hepatitis B and Cinfections on the risk of liver cancer in Europerdquo Journal ofHepatology vol 49 no 2 pp 233ndash242 2008

[4] H Wechsler and T F Nelson ldquoFocusing attention on collegestudent alcohol consumption and the environmental conditionsthat promote itrdquo Journal of Studies on Alcohol andDrugs vol 69no 4 pp 481ndash490 2008

[5] L M Powell J Williams and H Wechsler ldquoStudy habits andthe level of alcohol use among college studentsrdquo EducationEconomics vol 12 no 2 pp 135ndash149 2004

[6] H Wechsler B Moeykens A Davenport S Castillo and JHansen ldquoThe adverse impact of heavy episodic drinkers onother college studentsrdquo Journal of Studies on Alcohol and Drugsvol 56 no 6 pp 628ndash634 1995

[7] 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

[8] P Ormerod and G Wiltshire ldquolsquoBingerdquo drinking in the UK asocial network phenomenonrdquoMind amp Society vol 8 no 2 pp135ndash152 2009

[9] F Sanchez X HWang C Castillo-Chavez DM Gorman andP J Gruenewald ldquoDrinking as an epidemica simple mathemat-ical model with recovery and relapserdquo in Therapists Guide toEvidence-Based Relapse Prevention Practical Resources for theMental Health Professional K A Witkiewitz and G A MarlattEds pp 353ndash368 Academic Press Burlington Canada 2007

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

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

[12] H F Huo and C C Zhu ldquoModeling the effect of constantimmigration on drinking behaviourrdquo Submitted

8 Abstract and Applied Analysis

[13] H FHuo andC C Zhu ldquoStability of a quit drinkingmodel withrelapserdquo Submitted

[14] 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

[15] G Schwitzer G Mudur D Henry et al ldquoWhat are theroles and responsibilities of the media in disseminating healthinformationrdquo PLoS Medicine vol 2 no 7 article e215 2005

[16] E Jaramillo ldquoThe impact of media-based health education ontuberculosis diagnosis in Cali Colombiardquo Health Policy andPlanning vol 16 no 1 pp 68ndash73 2001

[17] R Liu JWu andH Zhu ldquoMediapsychological impact onmul-tiple outbreaks of emerging infectious diseasesrdquo Computationaland Mathematical Methods in Medicine vol 8 no 3 pp 153ndash164 2007

[18] T P Palfai R Zisserson and R Saitz ldquoUsing personalizedfeedback to reduce alcohol use among hazardous drinking col-lege students the moderating effect of alcohol-related negativeconsequencesrdquo Addictive Behaviors vol 36 no 5 pp 539ndash5422011

[19] J M Tchuenche and C T Bauch ldquoDynamics of an infectiousdisease where media coverage influences transmissionrdquo ISRNBiomathematics vol 2012 Article ID 581274 10 pages 2012

[20] H Xiang N N Song and H F Huo ldquoModelling effects ofpublic health educational campaigns on drinking dynamicsrdquoSubmitted

[21] A K Misra A Sharma and V Singh ldquoEffect of awarenessprograms in controlling the prevalence of an epidemic withtime delayrdquo Journal of Biological Systems vol 19 no 2 pp 389ndash402 2011

[22] A KMisra A Sharma and J B Shukla ldquoModeling and analysisof effects of awareness programs by media on the spread ofinfectious diseasesrdquoMathematical andComputerModelling vol53 no 5-6 pp 1221ndash1228 2011

[23] ldquoWHO Report of Alcohol in developing societies summaryrdquohttpwwwwhoint

[24] D Chisholm J Rehm M van Ommeren and M MonteiroldquoReducing the global burden of hazardous alcohol use acomparative cost-effectiveness analysisrdquo Journal of Studies onAlcohol vol 65 no 6 pp 782ndash793 2004

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

[26] J-Q Li Y-L Yang and Y-C Zhou ldquoGlobal stability of anepidemic model with latent stage and vaccinationrdquo NonlinearAnalysis Real World Applications vol 12 no 4 pp 2163ndash21732011

[27] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[28] L B Snyder ldquoHealth communication campaigns and theirimpact on behaviorrdquo Journal of Nutrition Education and Behav-ior vol 39 no 2 pp S32ndashS40 2007

[29] D B ClarkO Bukstein and J Cornelius ldquoAlcohol use disordersin adolescents epidemiology diagnosis psychosocial interven-tions and pharmacological treatmentrdquo Pediatric Drugs vol 4no 8 pp 493ndash502 2002

[30] ldquoWebMDReport of Alcohol Abuse Health Centerrdquo httpwwwwebmdcom

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Table 1 Description and estimation of parameters

120583 Recruitment rate of the population 025 yearminus1 [10]120573 Transmission coefficient 03 yearminus1 [10]120582 Dissemination rate of aware individuals 002 yearminus1 estimate120590 Rate of transfer of aware individuals to susceptible class 0001 yearminus1 estimate119901 The fraction of 119860 go in to treatment Variable Variable119902 The fraction of 119877 who relapse into 119860 08 yearminus1 [10]119898 Growth rate of density of awareness programs 005 yearminus1 [28]120574 Depletion rate of awareness programs 005 yearminus1 estimate

050 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 2 If 1198770lt 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 200 250 3000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 3 If 1198770= 1 the alcohol free equilibrium 119864

0is globally

asymptotically stable

0 50 100 150 2000

02

04

06

08

1

Hum

ans p

opul

atio

n siz

e

Time t

S(t)

X(t)

A(t)

R(t)M(t)

Figure 4 If 1198770gt 1 the alcohol present equilibrium 119864lowast is globally

asymptotically stable

0 10 20 30 40 50015

02

025

03

035

04

Bing

e drin

king

pop

ulat

ion

size

A(t) of (21)A(t) of (41)

Time

Figure 5 Variations of the binge drinking population 119860(119905) of (2)and (34)

Abstract and Applied Analysis 7

024

026

028

03

032

034

Bing

e drin

king

pop

ulat

ion

size

m = 002

m = 0008m = 0

0 10 20 30 40 50Time

Figure 6 Variations of binge drinking population with time fordifferent values of119898

5 Discussion

We have formulated the effect of awareness programs onthe binge drinking and analyzed their dynamical behaviorsThe model exhibits two equilibria explicitly the alcohol freeequilibrium and alcohol present equilibrium By constructingLyapunov function we establish some sufficient conditionsfor the global stability of the alcohol free and the alcoholpresent equilibria and give some numerical simulations toexplain our main result In our model we have assumed thatthe cumulative density of awareness programs by media isproportional to the population of binge drinking heavily Itmay be noted that the number of alcohol consumption casesknown to the policy makers is sometimes old and thus thenumber of awareness depends on data So it is more plausibleto consider a time delay in the growth rate of awarenessprograms [21]We canmodify (2) to the followingmodel withdelay

119889119878 (119905)

119889119905= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

119889119883 (119905)

119889119905= 120582119878119872 minus 120590119883 minus 120583119883

119889119860 (119905)

119889119905= 120573119878119860 + 119902119877 minus 119901119860 minus 120583119860

119889119877 (119905)

119889119905= 119901119860 minus 119902119877 minus 120583119877

119889119872 (119905)

119889119905= 119898119860 (119905 minus 120591) minus 120574119872

(35)

We leave these works for the future

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 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] J Rehm ldquoThe risks associated with alcohol use and alcoholismrdquoAlcohol Research amp Health vol 34 no 2 pp 135ndash143 2011

[2] J Rehm C Mathers S Popova M Thavorncharoensap YTeerawattananon and J Patra ldquoGlobal burden of disease andinjury and economic cost attributable to alcohol use andalcohol-use disordersrdquoThe Lancet vol 373 no 9682 pp 2223ndash2233 2009

[3] J Ribes R Cleries L Esteban V Moreno and F X BoschldquoThe influence of alcohol consumption and hepatitis B and Cinfections on the risk of liver cancer in Europerdquo Journal ofHepatology vol 49 no 2 pp 233ndash242 2008

[4] H Wechsler and T F Nelson ldquoFocusing attention on collegestudent alcohol consumption and the environmental conditionsthat promote itrdquo Journal of Studies on Alcohol andDrugs vol 69no 4 pp 481ndash490 2008

[5] L M Powell J Williams and H Wechsler ldquoStudy habits andthe level of alcohol use among college studentsrdquo EducationEconomics vol 12 no 2 pp 135ndash149 2004

[6] H Wechsler B Moeykens A Davenport S Castillo and JHansen ldquoThe adverse impact of heavy episodic drinkers onother college studentsrdquo Journal of Studies on Alcohol and Drugsvol 56 no 6 pp 628ndash634 1995

[7] 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

[8] P Ormerod and G Wiltshire ldquolsquoBingerdquo drinking in the UK asocial network phenomenonrdquoMind amp Society vol 8 no 2 pp135ndash152 2009

[9] F Sanchez X HWang C Castillo-Chavez DM Gorman andP J Gruenewald ldquoDrinking as an epidemica simple mathemat-ical model with recovery and relapserdquo in Therapists Guide toEvidence-Based Relapse Prevention Practical Resources for theMental Health Professional K A Witkiewitz and G A MarlattEds pp 353ndash368 Academic Press Burlington Canada 2007

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

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

[12] H F Huo and C C Zhu ldquoModeling the effect of constantimmigration on drinking behaviourrdquo Submitted

8 Abstract and Applied Analysis

[13] H FHuo andC C Zhu ldquoStability of a quit drinkingmodel withrelapserdquo Submitted

[14] 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

[15] G Schwitzer G Mudur D Henry et al ldquoWhat are theroles and responsibilities of the media in disseminating healthinformationrdquo PLoS Medicine vol 2 no 7 article e215 2005

[16] E Jaramillo ldquoThe impact of media-based health education ontuberculosis diagnosis in Cali Colombiardquo Health Policy andPlanning vol 16 no 1 pp 68ndash73 2001

[17] R Liu JWu andH Zhu ldquoMediapsychological impact onmul-tiple outbreaks of emerging infectious diseasesrdquo Computationaland Mathematical Methods in Medicine vol 8 no 3 pp 153ndash164 2007

[18] T P Palfai R Zisserson and R Saitz ldquoUsing personalizedfeedback to reduce alcohol use among hazardous drinking col-lege students the moderating effect of alcohol-related negativeconsequencesrdquo Addictive Behaviors vol 36 no 5 pp 539ndash5422011

[19] J M Tchuenche and C T Bauch ldquoDynamics of an infectiousdisease where media coverage influences transmissionrdquo ISRNBiomathematics vol 2012 Article ID 581274 10 pages 2012

[20] H Xiang N N Song and H F Huo ldquoModelling effects ofpublic health educational campaigns on drinking dynamicsrdquoSubmitted

[21] A K Misra A Sharma and V Singh ldquoEffect of awarenessprograms in controlling the prevalence of an epidemic withtime delayrdquo Journal of Biological Systems vol 19 no 2 pp 389ndash402 2011

[22] A KMisra A Sharma and J B Shukla ldquoModeling and analysisof effects of awareness programs by media on the spread ofinfectious diseasesrdquoMathematical andComputerModelling vol53 no 5-6 pp 1221ndash1228 2011

[23] ldquoWHO Report of Alcohol in developing societies summaryrdquohttpwwwwhoint

[24] D Chisholm J Rehm M van Ommeren and M MonteiroldquoReducing the global burden of hazardous alcohol use acomparative cost-effectiveness analysisrdquo Journal of Studies onAlcohol vol 65 no 6 pp 782ndash793 2004

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

[26] J-Q Li Y-L Yang and Y-C Zhou ldquoGlobal stability of anepidemic model with latent stage and vaccinationrdquo NonlinearAnalysis Real World Applications vol 12 no 4 pp 2163ndash21732011

[27] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[28] L B Snyder ldquoHealth communication campaigns and theirimpact on behaviorrdquo Journal of Nutrition Education and Behav-ior vol 39 no 2 pp S32ndashS40 2007

[29] D B ClarkO Bukstein and J Cornelius ldquoAlcohol use disordersin adolescents epidemiology diagnosis psychosocial interven-tions and pharmacological treatmentrdquo Pediatric Drugs vol 4no 8 pp 493ndash502 2002

[30] ldquoWebMDReport of Alcohol Abuse Health Centerrdquo httpwwwwebmdcom

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

024

026

028

03

032

034

Bing

e drin

king

pop

ulat

ion

size

m = 002

m = 0008m = 0

0 10 20 30 40 50Time

Figure 6 Variations of binge drinking population with time fordifferent values of119898

5 Discussion

We have formulated the effect of awareness programs onthe binge drinking and analyzed their dynamical behaviorsThe model exhibits two equilibria explicitly the alcohol freeequilibrium and alcohol present equilibrium By constructingLyapunov function we establish some sufficient conditionsfor the global stability of the alcohol free and the alcoholpresent equilibria and give some numerical simulations toexplain our main result In our model we have assumed thatthe cumulative density of awareness programs by media isproportional to the population of binge drinking heavily Itmay be noted that the number of alcohol consumption casesknown to the policy makers is sometimes old and thus thenumber of awareness depends on data So it is more plausibleto consider a time delay in the growth rate of awarenessprograms [21]We canmodify (2) to the followingmodel withdelay

119889119878 (119905)

119889119905= 120583 minus 120573119878119860 minus 120582119878119872 + 120590119883 minus 120583119878

119889119883 (119905)

119889119905= 120582119878119872 minus 120590119883 minus 120583119883

119889119860 (119905)

119889119905= 120573119878119860 + 119902119877 minus 119901119860 minus 120583119860

119889119877 (119905)

119889119905= 119901119860 minus 119902119877 minus 120583119877

119889119872 (119905)

119889119905= 119898119860 (119905 minus 120591) minus 120574119872

(35)

We leave these works for the future

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 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] J Rehm ldquoThe risks associated with alcohol use and alcoholismrdquoAlcohol Research amp Health vol 34 no 2 pp 135ndash143 2011

[2] J Rehm C Mathers S Popova M Thavorncharoensap YTeerawattananon and J Patra ldquoGlobal burden of disease andinjury and economic cost attributable to alcohol use andalcohol-use disordersrdquoThe Lancet vol 373 no 9682 pp 2223ndash2233 2009

[3] J Ribes R Cleries L Esteban V Moreno and F X BoschldquoThe influence of alcohol consumption and hepatitis B and Cinfections on the risk of liver cancer in Europerdquo Journal ofHepatology vol 49 no 2 pp 233ndash242 2008

[4] H Wechsler and T F Nelson ldquoFocusing attention on collegestudent alcohol consumption and the environmental conditionsthat promote itrdquo Journal of Studies on Alcohol andDrugs vol 69no 4 pp 481ndash490 2008

[5] L M Powell J Williams and H Wechsler ldquoStudy habits andthe level of alcohol use among college studentsrdquo EducationEconomics vol 12 no 2 pp 135ndash149 2004

[6] H Wechsler B Moeykens A Davenport S Castillo and JHansen ldquoThe adverse impact of heavy episodic drinkers onother college studentsrdquo Journal of Studies on Alcohol and Drugsvol 56 no 6 pp 628ndash634 1995

[7] 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

[8] P Ormerod and G Wiltshire ldquolsquoBingerdquo drinking in the UK asocial network phenomenonrdquoMind amp Society vol 8 no 2 pp135ndash152 2009

[9] F Sanchez X HWang C Castillo-Chavez DM Gorman andP J Gruenewald ldquoDrinking as an epidemica simple mathemat-ical model with recovery and relapserdquo in Therapists Guide toEvidence-Based Relapse Prevention Practical Resources for theMental Health Professional K A Witkiewitz and G A MarlattEds pp 353ndash368 Academic Press Burlington Canada 2007

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

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

[12] H F Huo and C C Zhu ldquoModeling the effect of constantimmigration on drinking behaviourrdquo Submitted

8 Abstract and Applied Analysis

[13] H FHuo andC C Zhu ldquoStability of a quit drinkingmodel withrelapserdquo Submitted

[14] 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

[15] G Schwitzer G Mudur D Henry et al ldquoWhat are theroles and responsibilities of the media in disseminating healthinformationrdquo PLoS Medicine vol 2 no 7 article e215 2005

[16] E Jaramillo ldquoThe impact of media-based health education ontuberculosis diagnosis in Cali Colombiardquo Health Policy andPlanning vol 16 no 1 pp 68ndash73 2001

[17] R Liu JWu andH Zhu ldquoMediapsychological impact onmul-tiple outbreaks of emerging infectious diseasesrdquo Computationaland Mathematical Methods in Medicine vol 8 no 3 pp 153ndash164 2007

[18] T P Palfai R Zisserson and R Saitz ldquoUsing personalizedfeedback to reduce alcohol use among hazardous drinking col-lege students the moderating effect of alcohol-related negativeconsequencesrdquo Addictive Behaviors vol 36 no 5 pp 539ndash5422011

[19] J M Tchuenche and C T Bauch ldquoDynamics of an infectiousdisease where media coverage influences transmissionrdquo ISRNBiomathematics vol 2012 Article ID 581274 10 pages 2012

[20] H Xiang N N Song and H F Huo ldquoModelling effects ofpublic health educational campaigns on drinking dynamicsrdquoSubmitted

[21] A K Misra A Sharma and V Singh ldquoEffect of awarenessprograms in controlling the prevalence of an epidemic withtime delayrdquo Journal of Biological Systems vol 19 no 2 pp 389ndash402 2011

[22] A KMisra A Sharma and J B Shukla ldquoModeling and analysisof effects of awareness programs by media on the spread ofinfectious diseasesrdquoMathematical andComputerModelling vol53 no 5-6 pp 1221ndash1228 2011

[23] ldquoWHO Report of Alcohol in developing societies summaryrdquohttpwwwwhoint

[24] D Chisholm J Rehm M van Ommeren and M MonteiroldquoReducing the global burden of hazardous alcohol use acomparative cost-effectiveness analysisrdquo Journal of Studies onAlcohol vol 65 no 6 pp 782ndash793 2004

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

[26] J-Q Li Y-L Yang and Y-C Zhou ldquoGlobal stability of anepidemic model with latent stage and vaccinationrdquo NonlinearAnalysis Real World Applications vol 12 no 4 pp 2163ndash21732011

[27] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[28] L B Snyder ldquoHealth communication campaigns and theirimpact on behaviorrdquo Journal of Nutrition Education and Behav-ior vol 39 no 2 pp S32ndashS40 2007

[29] D B ClarkO Bukstein and J Cornelius ldquoAlcohol use disordersin adolescents epidemiology diagnosis psychosocial interven-tions and pharmacological treatmentrdquo Pediatric Drugs vol 4no 8 pp 493ndash502 2002

[30] ldquoWebMDReport of Alcohol Abuse Health Centerrdquo httpwwwwebmdcom

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

[13] H FHuo andC C Zhu ldquoStability of a quit drinkingmodel withrelapserdquo Submitted

[14] 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

[15] G Schwitzer G Mudur D Henry et al ldquoWhat are theroles and responsibilities of the media in disseminating healthinformationrdquo PLoS Medicine vol 2 no 7 article e215 2005

[16] E Jaramillo ldquoThe impact of media-based health education ontuberculosis diagnosis in Cali Colombiardquo Health Policy andPlanning vol 16 no 1 pp 68ndash73 2001

[17] R Liu JWu andH Zhu ldquoMediapsychological impact onmul-tiple outbreaks of emerging infectious diseasesrdquo Computationaland Mathematical Methods in Medicine vol 8 no 3 pp 153ndash164 2007

[18] T P Palfai R Zisserson and R Saitz ldquoUsing personalizedfeedback to reduce alcohol use among hazardous drinking col-lege students the moderating effect of alcohol-related negativeconsequencesrdquo Addictive Behaviors vol 36 no 5 pp 539ndash5422011

[19] J M Tchuenche and C T Bauch ldquoDynamics of an infectiousdisease where media coverage influences transmissionrdquo ISRNBiomathematics vol 2012 Article ID 581274 10 pages 2012

[20] H Xiang N N Song and H F Huo ldquoModelling effects ofpublic health educational campaigns on drinking dynamicsrdquoSubmitted

[21] A K Misra A Sharma and V Singh ldquoEffect of awarenessprograms in controlling the prevalence of an epidemic withtime delayrdquo Journal of Biological Systems vol 19 no 2 pp 389ndash402 2011

[22] A KMisra A Sharma and J B Shukla ldquoModeling and analysisof effects of awareness programs by media on the spread ofinfectious diseasesrdquoMathematical andComputerModelling vol53 no 5-6 pp 1221ndash1228 2011

[23] ldquoWHO Report of Alcohol in developing societies summaryrdquohttpwwwwhoint

[24] D Chisholm J Rehm M van Ommeren and M MonteiroldquoReducing the global burden of hazardous alcohol use acomparative cost-effectiveness analysisrdquo Journal of Studies onAlcohol vol 65 no 6 pp 782ndash793 2004

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

[26] J-Q Li Y-L Yang and Y-C Zhou ldquoGlobal stability of anepidemic model with latent stage and vaccinationrdquo NonlinearAnalysis Real World Applications vol 12 no 4 pp 2163ndash21732011

[27] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[28] L B Snyder ldquoHealth communication campaigns and theirimpact on behaviorrdquo Journal of Nutrition Education and Behav-ior vol 39 no 2 pp S32ndashS40 2007

[29] D B ClarkO Bukstein and J Cornelius ldquoAlcohol use disordersin adolescents epidemiology diagnosis psychosocial interven-tions and pharmacological treatmentrdquo Pediatric Drugs vol 4no 8 pp 493ndash502 2002

[30] ldquoWebMDReport of Alcohol Abuse Health Centerrdquo httpwwwwebmdcom

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of