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Journal of Biological Dynamics
ISSN: 1751-3758 (Print) 1751-3766 (Online) Journal homepage: https://www.tandfonline.com/loi/tjbd20
Campus drinking: an epidemiological model
J. L. Manthey , A. Y. Aidoo & K. Y. Ward
To cite this article: J. L. Manthey , A. Y. Aidoo & K. Y. Ward (2008) Campus drinking: an epidemiological model, Journal of Biological Dynamics, 2:3, 346-356, DOI:
10.1080/17513750801911169
To link to this article: https://doi.org/10.1080/17513750801911169
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Published online: 27 Jan 2009.
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Journal of Biological Dynamics
Vol. 2, No. 3, July 2008, 346–356
Campus drinking: an epidemiological model
J.L. Mantheya*, A.Y. Aidooband K.Y. WardbaDepartment of Mathematical Sciences, Saint Joseph College, West Hartford, CT 06117, USA; bDepartment of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic,
CT 06226, USA
(Received 23 August 2007; final version received 11 January 2008 )
The drinking behaviours of college students have posed significant public health concerns for several generations. However, the dynamics of campus drinking have not been analysed using mathematical models. An epidemiological model capturing the dynamics of campus drinking is used to study how the ‘disease’ of drinking is spread on campus. The model suggests that the reproductive numbers are not sufficient to predict whether drinking behaviour will persist on campus and that the pattern of recruiting new members plays a significant role in the reduction of campus alcohol problems. In particular, campus alcohol abuse may be reduced by minimizing the ability of problem drinkers to directly recruit non-drinkers.
Keywords: alcohol, epidemiology, campus drinking, SIS model
AMS 2000 Mathematics Subject Classification: 92D25 population dynamics (general); 92D30
epidemiology
1. Introduction
The drinking habits of college students pose a significant public health concern. Recent US surveys indicate that approximately 90% of college students have consumed alcohol at least once [8] and more than 40% of college students have engaged in binge drinking [12,20]. Alcohol abuse by college students leads to a range of negative consequences. For example, college students who abuse alcohol are more likely to have a lower-grade point average, drive under the influence of alcohol, engage in vandalism, have unplanned and/or unprotected sex and cause injury to themselves and/or others [6,7,19]. Although there have been many attempts to reduce the problem, alcohol abuse by college students has persisted and in some cases increased over the past several decades [21].
Patterns of alcohol usage by college students have been studied extensively for over half a century beginning with the classical work of Straus and Bacon [17]. Since then there have been numerous studies investigating campus drinking and the associated consequences. Many studies focus on the drinking patterns of students at specific campuses [11,13,16], whereas
*Corresponding author. Email: [email protected] ISSN 1751-3758 print/ISSN 1751-3766 online © 2008 Taylor & Francis
DOI: 10.1080/17513750801911169 http://www.informaworld.com
others are national and longitudinal in scope (see, e.g. [12] and references therein). Several other studies highlight the consequences of alcohol abuse by students on campus [6]. In addi-tion, there has been a significant research effort devoted to reducing the extent of alcohol abuse on college campuses [18]. All of these studies utilized mainly statistical analyses. The use of mathematical models to analyse the dynamics of the spread of social problems is a relatively recent development and to the best of our knowledge such models have not been used to analyse the problem of campus drinking. However, since the work of Kermack and McKendrick [9], variations of their susceptible, infectious, recovered (SIR) model have been successfully applied to a variety of settings. Recently, applications involving social problems such as drug use [2], the spread of computer viruses [14], as well as the global spread of ideas [3] have emerged. In each case, the method has proved sufficiently powerful to shed light on these problems. Of particular note is the work of Sanchez et al. who used a mathe-matical model to study drinking epidemics in the general population [15]. Their focus was on the effect of a backward bifurcation with applications to recovery and relapse. Mathematical models focused on campus populations with their special interaction characteristics are virtually non-existent.
The transmission dynamics of alcohol abuse on campus are very complex. Factors influ-encing the dynamics include the delayed effects of parental drinking patterns, the influence of peers, and the proximity of liquor stores to campus. In this paper, drinking behaviour is modelled as a ‘disease’ which can be spread through social interaction [10]. Relation-ships between non-drinkers, social drinkers, and problem drinkers are represented using a system of ordinary differential equations. Our model is a variant of the classical suscepti-ble, infectious, susceptible (SIS) model as there are two levels of infectives, namely social drinkers and problem drinkers. Our analysis reveals that reproductive numbers do not com-pletely determine the dynamics of campus drinking and that recruitment patterns play a critical role.
2. Model description
Our model focuses on a college campus and divides the student population into three classes: non-drinkers (N ), social non-drinkers (S), and problem non-drinkers (P ). The student population is assumed to be constant. Although definitions of non-drinkers, social drinkers, and problem drinkers have not been consistently applied in the literature [5], we note that our modelling regime works in principle regardless of the particular definitions used. We have chosen definitions that would be of interest to college administrators and parents who are concerned with the negative consequences of campus drinking. Non-drinkers are the susceptibles which we define as students who have not consumed alcohol for an extended period of time, typically 1 year. Social drinkers are defined as students who consume alcohol occasionally and in moderation. Such students do not pose any danger to themselves or others. We define problem drinkers as students whose drinking habits and associated behaviours have negative consequences for themselves and/or others. As students are only on campus for a relatively short period of time (typically 5 years or less), it is not possible to assess whether or not a student who has stopped drinking has permanently recovered. Hence, the model does not include a recovered class and students who stop drinking will be regarded as non-drinkers. The interactions between the three categories of students are shown in Figure 1.
As seen in Figure 1, students join campus in any one of the three drinking states and mix randomly with the rest of the campus population. Once on campus, students may transition from any drinking state to any other drinking state. Transitions to higher drinking states are assumed to be the result of
Figure 1. Schematic of the relationships between the three drinking classes in a college campus. Students may enter or exit campus in any one of the three drinking states. Unlike the typical disease model, there is no recovered class and the infectious class is represented by two levels of the ‘disease’ (social drinking and problem drinking).
social interaction and transitions to lower drinking states are assumed to be the result of a recovery process. It is assumed that there is a net positive flow towards a higher drinking state, an effect that has been termed the ‘college effect’ in the literature [1]. The college effect is more pronounced in the first 2 years of college and declines in the last 2 years during which time students transition towards adult drinking patterns [4]. The student population is assumed to be homogeneous with respect to variables such as age, academic class rank, and socio-economic status. The concept of homogeneity is implicit in all compartmental models and in our case refers to the assumption that students within each compartment are similar with regards to their drinking and recruiting behaviours. In addition, our model does not consider the effects of breaks and vacations. The dynamics of campus drinking are modelled using the following system of nonlinear differential equations, where N , S, and P have been rescaled as proportions:
dN dt = η − ηN − αNS − κNP + βS + P, (1) dS dt = σS − (η + σ)S + αNS − βS − γ SP + δP, (2) dP dt = πP − (η + π)P + γ SP + κNP − δP − P, (3) 1= N + S + P. (4)
Note that conversion from non-drinkers to social drinkers, non-drinkers to problem drinkers, and social drinkers to problem drinkers are modelled using the interaction terms αN S, κN P , and
Table 1. Model parameters.
Parameter Description Typical value
α Transmission rate of non-drinkers to social drinkers 1.0
β Recovery rate of social drinkers 0.2
γ Transmission rate of social drinkers to problem drinkers 1.0
δ Recovery rate of problem drinkers to social drinkers 0.2
Recovery rate of problem drinkers to non-drinkers 0.2
η Departure rate from campus environment 0.25
κ Transmission rate of non-drinkers to problem drinkers 1.5
σ Entrance rate of social drinkers 0.25
π Entrance rate of problem drinkers 0.1
and P , and S and P depend linearly on the total campus population. However, the conversions from social drinkers to non-drinkers, problem drinkers to non-drinkers, and problem drinkers to social drinkers are assumed to be the result of a recovery process and are implemented via the terms βS, P , and δP , respectively. Parameters α, γ , and κ are the mean transmission rates and measure the effectiveness of the interactions between non-drinkers and social drinkers, social drinkers and problem drinkers, and non-drinkers and problem drinkers respectively. Recovery rate
βdenotes the mean rate at which social drinkers revert back to the non-drinking class. Recovery rates δ and are the mean rates at which problem drinkers transition back to the social drinking and the non-drinking classes, respectively. Parameter η represents the net departure rate from the campus environment. The parameters which are considered fixed for the model are summarized in Table 1.
3. Model analysis
3.1. Basic reproductive numbers
System (1)–(3) yield steady states of the form (N∗, S∗, P∗). The trivial equilibrium (1, 0, 0) represents a drinking-free environment. An analysis of the local stability of the trivial equilibrium enables us to identify conditions under which a culture of drinking can be established. The local stability may be determined from the eigenvalues of the Jacobian matrix
J (N, S, P )= ⎡ ⎢ ⎣ −η − αS − κP −αN + β −κN + αS −η + αN − β − γ P −γ S + δ κP γ P −η + γ S + κN − δ − ⎤ ⎥ ⎦. The eigenvalues at the ‘disease-free’ equilibrium are given by−η, α − η − β, and κ − η − δ − . As there are two levels of infectives, we consider two reproductive numbers based on the ‘disease-free’ equilibrium. The first reproductive number RS
0is defined as the average number of secondary
cases generated by a typical social drinker in a non-drinking campus environment. The secondary cases are cases of new social drinkers recruited from the non-drinking class. The reproductive num-ber RS
0may be computed by R0S= λ∗(infectious period)+ 1 where λ∗is the dominant eigenvalue
of J (1, 0, 0). Thus RS0 = (α − η − β) 1 η+ β + 1 = α η+ β (5)
Figure 2. Sensitivity analysis of the reproductive number RS
0. Changes in the parameters α (solid line), β (diamond), and η (open circle) are shown as factors of the parameter values contained in Table 1 and the corresponding changes in the reproductive number are shown as factors of the original reproductive number. The sensitivity analysis reveals that R0S is more sensitive to the transmission rate of non-drinkers to social drinkers (α) than the recovery rate of social drinkers (β) or the net departure rate from campus (η). For example, if the value of α is doubled, then R0Sis also doubled whereas doubling β decreases RS
0 by a factor of 0.69 and doubling η decreases RS0 by a factor of 0.64.
The second reproductive number RP
0 is defined as the average number of secondary cases generated
by a typical problem drinker in a non-drinking campus environment. The secondary cases are cases of new problem drinkers recruited from the non-drinking class. The reproductive number RP
0 is
given by
R0P = κ
η+ δ + . (6)
Hence the drinking-free equilibrium (1, 0, 0) is stable provided that RS
0 <1 and R P
0 <1, that
is, α < η+ β and κ < η + δ + . This means that the transmission rates α and κ relative to the recovery rates β, δ, and and the departure rate η play a significant role in determining whether or not a culture of drinking becomes established on campus. Note that the stability requirements involve two levels of infectives, social drinkers and problem drinkers, a distinct difference from the standard SIS model.
The parameter values used in the model will vary from campus to campus. A local sensitivity analysis is performed by varying the model input parameters to determine which parameter has the greatest impact on the reproductive number RS
0. As shown in Figure 2, R0Sis more sensitive
to changes in α than to changes in β or η.
3.2. Problem drinking-free equilibrium
A second equilibrium characterized by a lack of problem drinkers is given by ((η+ β)/α, 1− (η + β)/α, 0). If social drinking is not regarded as a problem, then we consider this as a second ‘disease-free’ equilibrium. Expressing this equilibrium in terms of the reproductive number RS
a steady state distinct from (1, 0, 0) provided RS
0 >1. The eigenvalues of the Jacobian
matrix evaluated at the problem drinking-free equilibrium are given by−η, −α + η + β, and
(−ηα + αγ − ηγ − βγ + ηκ + βκ − αδ − α)/α. These may be written in terms of RS0 and
RP 0 as−η, −α(1 − 1/R S 0), and γ (1− 1/R S 0)+ κ(1/R S
0 − 1/RP0). We note that the third
eigen-value is dominant and is used to define another reproductive number, R1, the average number
of secondary cases generated by a typical problem drinker in a campus environment consisting of only non-drinkers and social drinkers. Here, the secondary cases are cases of new problem drinkers recruited from the non-drinking and social drinking class. The reproductive number R1
is given by
R1=
γ (α− η − β) + κ(η + β)
α(η+ δ + ) . (7)
Using the following relationship between the reproductive numbers RS 0 and R1
R1=
γ (1− 1/RS
0)+ κ/RS0
η+ δ + , (8)
we further simplify the eigenvalues to obtain−η, −α(1 − 1/R0), and (η+ δ + )(R1− 1) from
which it can easily be deduced that the problem drinking-free equilibrium is stable if R1<1.
Hence it is possible for non-drinkers and social drinkers to coexist in equilibrium provided that
RS
0 >1 and R1<1. If R1is expressed in terms of R0Sand R0P as
R1 = RP
0[γ (R0S− 1) + κ] κRS0 ,
then the condition R1 <1 becomes
R0P < κR S 0 γ (R0S− 1) + κ.
This means that a campus free from problem drinkers is possible provided that problem drinkers are ineffective at recruiting non-drinkers and social drinkers. Furthermore, the relative proportion of non-drinkers depends on how effective the social drinkers are at recruiting non-drinkers.
3.3. Existence of endemic equilibrium
Alcohol abuse is a persistent problem in college campuses. At any point in time, all three drinking states are present in campus. This section captures this state of affairs and considers conditions under which all three drinking states can coexist in equilibrium. From Equations (1)–(3) we obtain
N =η− γ S + δ + κ , (9) P =αγ S 2+ (ηκ + βκ − αη − αδ − α)S κ(δ− γ S) , (10) 0= S2+ bS + c, (11)
where b= (−κγ + αδ + γ + ηα − δκ + α − βκ + ηγ + 2γ δ − ηκ)/(κγ − γ2− αγ ) and,
c= (−δ − ηδ − δ2+ δκ)/(κγ − γ2− αγ ). The following theorem provides conditions under
which an equilibrium proportion of social drinkers can exist.
THEOREM1 The quadratic equation (11) has at least one root in the interval (0, 1) provided any
one of the following conditions holds:
(1) b <−2 and 0 < c < −b − 1; (2) −2 < b < −1 and 0 < c < 14b2;
(3) −1 < b < 0 and −b − 1 < c < 14b2;
(4) b > 0 and−b − 1 < c < 0.
As the expressions for coefficients b and c in Equation (11) involve multiple parameters, a Monte Carlo simulation is used to determine whether or not the conditions of Theorem 1 are satisfied for the model parameters. To this end, the parameters were assigned uniform probability distributions based on the typical values contained in Table 1, for example, α∼ U(0, 1). The results of the simulation shown in Figure 3 demonstrate the conditions given in Theorem 1 are satisfied for many values of the parameters.
In addition to bounding S, an endemic equilibrium requires that N and P are also bounded. Using 0 < N < 1− S and Equation (9), we obtain the additional requirement
RP0 ≥ max 1, κ γ . (12)
The existence of an endemic equilibrium is clearly illustrated by our numerical simulations shown in Figure 4.
Figure 3. The shaded area in the b–c-plane represents the region where the conditions of Theorem 1 are satisfied. The model parameters were assigned uniform probability distributions based on the typical values given in Table 1. Corresponding values of the coefficients b and c in Equations (11) were plotted as points. Each point within the shaded region represents a possible value of S for the endemic equilibrium.
Figure 4. The establishment of a culture of drinking (endemic equilibrium) on campus over a period of 20 years. The model parameters used are α= 1.0, β = 0.20, γ = 1.0, δ = 0.2, = 0.2, η = 0.25, and κ = 1.5 and the corre-sponding reproductive numbers are RS0= 2.22, RP
0 = 2.31, and R1= 1.88. The solid curve represents the proportion of non-drinkers, the open circles represent the proportion of social drinkers, and the diamonds represent the proportion of problem drinkers.
4. Limit case: κ= 0
Conditions in many college campuses support rapid transition from the non-drinking to the prob-lem drinking state. Students are exposed to situations such as fraternity parties, bars, dance clubs, and drinking games, where very heavy drinking takes place. In these situations, a rapid progres-sion from a non-drinking state to a problem drinking state is almost inevitable. We considered this case in the previous section in which non-drinkers were allowed to transition to the problem drinking class directly and by progression. In this section, we consider the case where transition by progression via social drinking is the only avenue by which non-drinkers can become problem drinkers. This becomes possible if, for example, college administrators were to implement effec-tive control strategies which significantly reduces situations in which rapid progression thrives. We model the effects of this assumption on the dynamics of campus drinking as the limiting case κ → 0. As in the full model, the limiting case possesses three types of equilibrium includ-ing drinkinclud-ing-free, problem drinkinclud-ing-free, and endemic. A local stability analysis shows that the drinking-free equilibrium (1, 0, 0) is stable provided that R0S<1.
The problem drinking-free equilibrium ((η+ β)/α, 1 − (η + β)/α, 0) may be expressed in terms of RS
0 as (1/R S
0,1− 1/R S
0,0) and exists as a steady state distinct from (1, 0, 0)
pro-vided RS
0 >1. A local stability analysis of the problem drinking-free equilibrium leads to the
reproductive number
R∗1 =γ (α− η − β) α(η+ δ + ),
defined as the average number of secondary cases generated by a typical problem drinker in a campus environment consisting of only social drinkers. In this case, secondary cases refer to cases of new problem drinkers recruited from the social drinking class. The reproductive number R1∗is a simplified version of the reproductive number R1given in the full model, reflecting the fact that
Figure 5. Proportions of the three drinking classes under the assumption that problem drinkers cannot directly recruit from non-drinkers over a period of 20 years. Note that drinking behaviour declines even in the presence of effective recruiting by problem drinkers (R1∗>1). The model parameters used are
α= 1.0, β = 0.02, γ = 4.0, δ = 0.02, = 0.02, η = 0.25, and κ = 0. In this case, R0S= 3.70 and R∗1= 10.06. The solid curve represents the proportion of non-drinkers, the open circles represent the proportion of social drinkers, and the diamonds represent the proportion of problem drinkers.
problem drinkers are no longer able to directly recruit non-drinkers. The reproductive number R1∗ may be expressed in terms of RS0 as
R1∗= γ (1− 1/R S 0) η+ δ + .
From the eigenvalues of the Jacobian matrix, it can easily be shown that the problem drinking-free equilibrium is stable provided that R1∗<1. Up until this point, the differences between the limiting case and the full model have been relatively minor.An analysis of the endemic equilibrium, however, reveals important differences and provides insight into the problem of reducing drinking behaviour. In the limiting case, the endemic equilibrium is given by
S=η+ δ + γ , (13) N =(βS+ η)(δ − γ S) + (β + η)S (αS+ η)(δ − γ S) + αS , (14) P =(αS+ η)(β + η)S − αS(βS + η) (αS+ η)(δ − γ S) + αS . (15)
Using the reproductive numbers RS
0and R∗1, we express the endemic equilibrium value for social
drinkers [Equation (13)] as
S=R S 0 − 1
R0SR1∗ . (16)
Using Equations (13)–(16), an endemic equilibrium exists provided that RS
0 >1, R1∗>1− 1/R0S,
large and the third condition ensures that the net flow of social drinkers into the non-drinking class is small. From Equation (16), it is easy to see that S approaches zero for sufficiently large values of R1∗ and from Equations (14), and (15) it follows that N approaches 1 and P approaches 0. This means that the endemic equilibrium tends to the drinking-free equilibrium (1,0,0) for large values of the reproductive number R1∗in the limiting case κ = 0. This represents a departure from standard epidemiological models and illustrates that reproductive numbers do not completely determine the dynamics. The result demonstrates that a culture of drinking can decline even in the presence of effective recruiting by problem drinkers. The key is the new recruiting pattern in which problem drinkers cannot directly recruit non-drinkers. As problem drinkers recruit social drinkers, the population of social drinkers declines and the ability of social drinkers to recruit is no longer adequate to maintain the pool of social drinkers. This leads to a decline in the population of problem drinkers because they cannot recruit directly from the non-drinking class. A numerical simulation illustrating this behaviour is shown in Figure 5.
5. Discussion
We introduced a simple mathematical model capturing the dynamics of campus drinking. In the full model, we assume that problem drinkers can recruit both non-drinkers and social drinkers. This allows non-drinkers to transition to the problem drinking state both directly and via pro-gression through social drinking. In the limit case, we modified this assumption and allowed problem drinkers to only recruit social drinkers. In this case, non-drinkers may transition to the problem drinking class only after progressing through social drinking. As a result, the dynamics of campus drinking are significantly impacted. Surprisingly, in the limiting case, even with a large reproductive number R1∗, the proportion of both social and problems drinkers is reduced. This suggests that one possible strategy for reducing drinking problems on campus is to modify the recruitment patterns. Specifically, our research suggests that campus alcohol abuse may be reduced by minimizing the ability of problem drinkers to directly recruit non-drinkers. This may be accomplished by designing effective control strategies which limit the exposure of students to heavy drinking environments which facilitate rapid progression from the non-drinking to the problem drinking state.
References
[1] J.G Bachman, et al., Smoking, drinking, and Drug Use in Young Adulthood: The Impacts of New Freedoms and New
Responsibilities, Mahwah, NJ: Lawrence Erlbaum Associates (1997).
[2] B. Benedict, Modeling alcohol as a contagious disease: how infected drinking buddies spread problem drinking, SIAM News 40 (2007), p. 8.
[3] L.M.A. Bettencourt, et al., The power of a good idea: quantitative modeling of ideas from epidemiological models, Physica A 364 (2006), pp. 513–536.
[4] P.J. Gruenewald, et al., Understanding college drinking: assessing dose response from survey self-reports, J. Stud. Alcohol 64 (2003), pp. 500–515.
[5] L.S. Ham and D.A. Hope, College students and problematic drinking: a review of literature, Clin. Psychol. Rev. 23 (2003), pp. 719–759.
[6] R.W. Hingston, et al., Magnitude of alcohol-related mortality and morbidity among U.S. college students ages 18–24, J. Stud. Alcohol. 63 (2002), pp. 136–144.
[7] K.M. Jackson, K.J. Sher, and A. Park, Drinking among college students – consumption and consequences, in Recent
Developments in Alcoholism: Research on Alcohol Problems in Adolescents and Young Adults, M. Galanter, ed.,
Vol. XVII, Kluwer Academic/Plenum Publishers, New York, (2005) pp. 85–171.
[8] L.D. Johnston, P.M. O’Malley, and J.G. Bachman, National survey results on drug use from the monitoring the future
study, 1975–1992. National Institute on Drug Abuse: Rockville, MD.
[9] W.O. Kermack and A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London. Ser. A 115 (1927), pp. 700–721.
[10] C.D. Mohr, et al., Moving the keg party: a daily process study of college student drinking motivations, Psychol. Addict. Behav. 19 (2005), pp. 392–403.
[11] J.G. Murphy and J. MacKillop, Relative reinforcing efficacy of Alcohol among student drinkers, Exp. Clin. Psychopharmacol. 14 (2006), pp. 219–227.
[12] P.M. O’Malley and L.D. Johnston, Epidemiology of alcohol and other drug use among American college students, J. Stud. Alcohol. 63 (suppl. 14) (2002), pp. 23–39.
[13] T.P. Palfai and J. Weafer, College drinking and meaning in the pursuit of life goals, 2006, Psychol. Addict. Behav. 20 (2006), pp. 131–134.
[14] J.R.C. Piqueira, B.F. Navarro, and L.H.A. Monteiro, Epidemiological models applied to viruses in computer networks, J. Comput. Sci. 1 (2005), pp. 31–34.
[15] F. Sánchez, et al., Drinking as an epidemic – a simple mathematical model with recovery and relapse, in Katie Witkiewitz and G.A. Marlatt, eds. Therapist’s Guide to Evidence Based Relapse Prevention, Burlington, MA: Academic Press, (2007) pp. 353–368.
[16] J.K. Sher and P.C. Rutledge, Heavy drinking across transition to college: predicting first-semester heavy drinking
from precollege variables, Addict. Behav. 32 (2007), pp. 819–835.
[17] R. Straus and S.D. Bacon, Drinking in College, Yale University Press, 1953.
[18] S. Walters, M.E. Bennett, and J.V. Noto, Drinking on campus: what do we know about reducing alcohol use among
college students, J. Substance Abuse Treat. 19 (2000), pp. 223–228.
[19] H. Wechsler, et al., Health and behavioral consequences of binge drinking in college, J. Am. Med. Assoc. 272 (1994), pp. 1672–1677.
[20] ———, College binge drinking in the 1990’s: a continuing problem. Results of the Harvard School of Public Health
1999 College Alcohol Study, J. Am. Coll. Health. 49 (2000), pp. 199–210.
[21] ———, Underage college students drinking behavior, access to alcohol, and the influence of deterrence policies, J. Am. Coll. Health. 50 (issue 5) (2002), pp. 223–236.
