Drinking as an epidemic A simple mathematical model with recovery ...

16 Drinking as an Epidemic—A Simple Mathematical Model with Recovery and Relapse Fabio Sánchez Biological Statistics and Computational Biology Cornell University Ithaca, New York

Xiaohong Wang, and Carlos Castillo-Chávez Department of Mathematics Arizona State University Tempe, Arizona

Dennis M. Gorman School of Rural Public Health Texas A&M Health Science Center Bryan, Texas

Paul J. Gruenewald Prevention Research Center Berkeley, California

Therapist’s Guide to Evidence-Based Relapse Prevention

Ch016-P369429.indd 353

353

Copyright © 2007 by Elsevier, Inc. All rights reserved.

1/12/2007 12:15:24 PM

354

Specific Populations and Treatment Settings

INTRODUCTION The outcomes (patterns) associated with various biological and sociological processes are often the result of interactions or contacts between individuals, groups, subpopulations, or populations. For example, some aspects associated with the process of language acquisition can be thought of as the result of nonspecified contacts between those who speak the language and those who have yet to acquire it. Although contacts between individuals in different states are at the heart of these processes, the definition of “contact” (effective contact) is highly dependent on context, population structure, and other factors. Starting with the pioneering work of Ross (1911), researchers who conduct studies of social and health problems in which data are scarce have often relied on simple mean field deterministic models to generate insights and understanding on the relative role of various mechanisms of disease spread. The analysis of such mathematical models is used to generate hypotheses or to gain insights (with limited data) on the conversion process (change in epidemiological status) and ways of controlling disease dynamics under clearly specified assumptions (Chowell, Castillo-Chávez, Fenimore et al., 2004; Chowell, Fenimore, CastilloGarsow & Castillo-Chávez, 2003; Chowell, Hengartner, Castillo-Chávez, Fenimore & Hyman, 2004). In the case of social and behavioral processes, challenges arise from the fact that the dynamics of these processes at the population level are the result of nonlinear interactions between individuals in different states (that is, the whole is not the sum of its parts). Despite such complexity, epidemiological contact models have also been applied to the study of the dynamics of social and behavioral processes such as eating disorders (González, Huerta-Sánchez, Ortiz-Nieves, Vázquez & KribsZaleta, 2003), drug addictions, treatment, and prevention (Behrens, Caulkins, Tragler, Haunschmied & Feichtinger, 1999; Epstein, 1997; Song, Castillo-Garsow, Castillo-Chávez, et al., 2006; Winkler, Caulkins, Behrens & Tragler, 2004), violence (Patton & Arboleda-Florez, 2004), and even the diffusion of knowledge (Bettencourt, Cintron-Arias, Kaiser & Castillo-Chávez, 2006). Naturally, there are differences in the generation of addictive behaviors and the transmission of infectious diseases but the fact remains that the acquisition of both can be modeled (in the context of specific questions) as the likely result of contacts between individuals in different states in specified environments. For example, the dynamics of alcohol use among young people and the role of “supportive environments” on the development and maintenance of heavy drinking, alcohol abuse and dependence, and alcohol-related problems among adults are predicated upon the combined effects of social influence and access to alcohol (Cox, Yeates, Gilligan & Hosier, 2001; Gorman, 2001; Stockwell & Gruenewald, 2001; Stockwell, Gruenewald, Toumbourou & Loxley, 2005; Treno & Holder, 2001). Thus, additional understanding of the dynamics of drinking behaviors may result from the use of a perspective that models problem drinking as the result of contacts of “at-risk” individuals with individuals in distinct drinking states.

Ch016-P369429.indd 354

1/12/2007 12:15:25 PM

A Simple Mathematical Model with Recovery and Relapse

355

MATHEMATICAL MODEL OF DRINKING LAPSE Problem drinking is modeled as an “acquired” state, the result of frequent (i.e., high number of contacts) or intense (i.e., high likelihood of conversion) interactions between individuals in three drinking states (susceptible, problem drinkers, and temporarily recovered) within a specified drinking environment. The goal of the model is to identify mechanisms (quantitatively speaking) that facilitate or limit the conversion of a population of nondrinkers to one of drinkers within prespecified environments. The process of quantification helps to understand the role of social forces on the time evolution of drinking. Knowledge of these factors may be useful in the development of effective drinking control policies, in the evaluation of treatment interventions, and in the identification and ranking of mechanisms that facilitate the survival of a culture of drinking. We describe the dynamics of drinking within the context of the classic SIR (Susceptible-Infected-Recovered) epidemiological framework (Brauer & Castillo-Chávez, 2001). The population in question is divided into the following drinking classes: occasional and moderate drinkers (S); problem drinkers (D); and temporarily recovered (R). It is assumed that the population size remains constant; that is, that the time scale of interest is such that the size of the population under consideration, N, does not change significantly over the length of the study. New recruits join the population as occasional and moderate drinkers (S) and mix at random (i.e., homogeneous mixing) with the rest of the members of the population (see appendix for mathematical details). The dynamics of the model support two distinct states: the outcomes of the nonlinear interactions between problem drinkers and others in their shared drinking environment. The nature of these distinct outcomes depends in general on the size of the initial proportion of drinkers, the overall average residence time in the drinking environment, and the intensity of the interactions (in this environment) between problem drinkers and the rest of the residents (regulars to this environment). The “invasion” of the environment of a population, composed of moderate drinkers, by a small number of problem drinkers, may increase or decrease the proportion of problem drinkers in the invaded populations. The success or failure of such an invasion process depends on the average total time spent by problem drinkers and residents in this environment and the impact that the interactions between moderate and problem drinkers have on the drinking status of residents. It is assumed that individuals that change their status only do it by moving from moderate to problem drinker or from temporarily recovered back to problem drinker (a class with no members prior to invasion). The social outcomes that facilitate transmission are modeled by the functions, the incidence “infection” rates, B(S,D) and G(D,R) of the state variables (see appendix for mathematical details). It is assumed that individuals who share a common drinking environment do interact and that such interactions are only frequency-dependent; that is, the larger the proportion of one type, the more likely it is that one will drink with “him” or “her.”

Ch016-P369429.indd 355

1/12/2007 12:15:25 PM

356


TABLE 16.1:

Description of Parameters and Parameter Distribution Functions

Parameter

Description

Min

Max

Median

Std

Distribution Function

β

Transmission rate Per-person departure rate from drinking environments Per-person recovery rate Per-person relapse rate

0.0012

1.3

0.6434

0.3710

Unif(0.0001, 0.4)

0.0001

0.5

0.2553

0.1437

Unif(0.5, 1)

5.43e−4

0.4646

0.1033

0.0736

Beta(a = 2, b = 15)

5e−8

3.39e−4

3.83e−5

5.46e−5

exp(5.48e−3)

µ

φ ρ

From the analysis of the “drinking-free” equilibrium—that is, the state where a drinking culture does not exist—we compute the model’s basic reproductive number (!0), which provides a measure of the resilience of this state to the invasion of problem drinkers. This ratio gives the average number of secondary cases generated by a “typical” problem drinker in a population of moderate drinkers. That is, a population where problem drinkers are so rare that their numbers are insignificant (treatment is not yet needed). !0, computed in the situation when the R-class does not yet exist, provides a measure of the capacity for initial growth of drinking behaviors per generation in a population with few problem drinkers. !0 = bL where b is a measure of the influence of problem drinkers on 1 S-residents and L = denotes the average overall total average residence time µ of individuals in this environment. !0 decreases if either L (at-risk window) or b (transmission rate) or both decrease and it increases if either or both increase. Treatment will affect the ability of problem drinkers to reproduce. Conseβ , quently, the reproductive number with recovery via treatment is !φ = µ +φ 1 where b is the influence rate, and represents the average time a problem µ +φ drinker spends on the drinking class D (at-risk window). We observe that !0 > !f if f > 0. SDR dynamics are rather simple. If !0 < 1 and the number of problem drinkers in the community is initially low then it is not possible to establish a culture of drinking that is D(t) → 0 as t → ∞. However, if !0 > 1 even the introduction of a single problem drinker will lead (within this framework) to the establishment of a culture of problem drinkers. The results here are not typical

Ch016-P369429.indd 356

1/12/2007 12:15:25 PM


357

in epidemic models because the outcomes depend not only on the value of !f but on the size of the initial population of problem drinkers. Specifically, !f < 1 does not guarantee the eradication of drinking communities. Why? Because although treatment reduces the residence time of D-individuals in the system as regular drinkers, in the process it can create a second pool of susceptible whose level of susceptibility is measured by the strength of environmentally induced relapse rates. Hence, even when the number of conversions generated by an average typical problem drinker (!f < 1) is less than one in a community, with treatment a drinking culture can still be established as long as the effectiveness of treatment is low (high relapse rates). However, when !f > 1 (treatment also not that effective) only one outcome is possible—a drinking community becomes established, that is, the persistence of a regular drinking class over time is guaranteed.

UNCERTAINTY AND SENSITIVITY ANALYSIS The value of !f and the initial conditions determine whether or not a community of problem drinkers can be established in a population under treatment regime f. There is a lot of uncertainty associated with the parameters that enter !f . Thus, in order to better estimate the impact of variation in parameter ranges on dynamical outcomes, we conduct an uncertainty analysis on three dimension! less quantities, !f , c , and !r . !r is the number of secondary drinkers gener!φ ated by a typical problem drinker in the population of temporarily recovered individuals and !c is the critical value of !f that leads to the elimination of a drinking community experiencing treatment regime f (see Figure 16.1). We assigned probability distribution functions1 to each of the parameters (see Table ! 16.1) in !f , c , and !r based on our reading of the relevant literature pertaining !φ to the initiation, maintenance, and cessation of alcohol use, and proceeded to ! study their impact on the corresponding !f and c distributions. The level of !φ uncertainty in the model’s parameter values is explored via Monte Carlo2 simulations (based on 1000 realizations). Figures 16.2 and 16.3 show the resulting ! histograms for !f and c . !φ 1 A graph of some precise quantitative measure of a character against its frequency of occurrence (helios.bto.ed.ac.uk/bto/glossary/d.htm). 2 A trial-and-error method of repeated calculations to discover the best solution of a problem. Often used when a great number of variables are present, with interrelationships so extremely complex as to forestall straightforward analytical handling (www.control.co.kr/dic/dic-m.htm).

Ch016-P369429.indd 357

1/12/2007 12:15:25 PM

358


!c FIGURE 16.1:

!f Threshold Conditions: !c and !f

!φ FIGURE 16.2: 71% of !f > 1

Histogram for !f . The mean is 3.12 with a standard deviation of 7.39 and

!c < 1 then the possibility of having two permanent problem drinking !φ ! states exists. That is if c < 1 , then the initial number of problem drinkers plays !φ a major role on the dynamics of drinking. That is, for prespecified sets of param! eter values that satisfy c < 1 , there is a critical mass of problem drinkers that !φ would lead to the establishment of a subpopulation of problem drinkers. If

Ch016-P369429.indd 358

1/12/2007 12:15:25 PM


359

!c / !φ FIGURE 16.3:

Histogram for !c /!f . The mean is 0.16 with a standard deviation of 0.11 and

the median is 0.14

!c , and !r to parameter variations; !φ that is, we look at parameters that have a positive (increase) or negative (decrease) impact on these ratios. Using the partial rank correlation coefficient (PRCC3) we determined the qualitative relationship between the parameters and the threshold quantities previously described. The analysis showed that the alcohol recovery rate was the most significant (sensitive) parameter. Furthermore, if f (recovery rate) is not small and the relapse rate r is high enough then the situation can actually worsen despite treatment effectiveness. In other words, whenever the initial rate of recovery from treatment and the subsequent relapse rate are high, then a critical mass of at-risk (highly susceptible) individuals that can reenter the problem drinker class is created. From Table 16.2 we see that the transmission rate (b) has a negative effect on ! !f and c . In other words, it decreases both ratios but it has a greater effect !φ ! on !f . The relapse rate also (r) has a negative effect on !r and c , although !φ it has a relatively strong effect on both ratios. The treatment rate (f) has the Next, we analyze the sensitivity of !f ,

3

The PRCC value indicates the effect (positive or negative) of the parameter in the quantities !c !f , , and !r . !φ

Ch016-P369429.indd 359

1/12/2007 12:15:25 PM

360


TABLE 16.2:

Partial Rank Correlation Coefficient of !f , !c /!f , and !r with Their Respective p-Valuesa !f

!c /!f

!r

Parameter

PRCC

p-Value

Parameter

PRCC

p-Value

Parameter

PRCC

p-Value

b f m r

−0.263 0.213 0.066 —

0.000 0.000 0.036 —

r f m b

−0.673 0.079 −0.031 —

0.000 0.013 0.036 —

b r f m

−0.085 −0.909 0.984 0.021

0.007 0.000 0.000 0.500

a p-Value indicates the probability that the result attained is due to chance rather than a true relationship between measures. Small p-values indicate that it is highly unlikely that the results are due to chance.

biggest (positive) effect on nity becomes established.

!c ! . In other words, if c > 1 the drinking commu!φ !φ

NUMERICAL SIMULATIONS Numerical simulations are used to illustrate our model results on drinking dynamics. The most general model can support two permanent prevalent states when treatment only has short-term effect and relapse rates are high. If the relapse rate is high, then the probability that we enter the region where we have two prevalent states in the problem drinking class is high and, consequently, the development of successful treatment programs is unlikely. In other words, once a drinking culture is established it is difficult to bring it to a low enough level to completely eliminate it. As soon as a drinking culture is established, the effectiveness of treatment becomes a critical factor in limiting and curtailing its influence. High rates of recovery with high relapse rates will not affect reductions in problem drinking. In fact, a new pool of high-risk “susceptible” previous problem drinkers can become part of such a drinking community. Drinking communities become difficult to eradicate under these circumstances. In Figure 16.4a we illustrate the scenario where drinking behavior can become quickly established, and getting it under control would require a tremendous effort. We used a range of parameter values that allows for the possibility of two permanent prevalent states. If recovery and relapse rates are equal, !f < 1, then it is possible to bring the drinking culture under control (see Figure 16.4b). The number of initial problem drinkers introduced into the population plays a crucial role in the establishment of the drinking community. Figure 16.4c illustrates the role of initial conditions (initial number of occasional and moderate

Ch016-P369429.indd 360

1/12/2007 12:15:25 PM

Estimate of !f , !c /!f , and !r from 10 Monte Carlo Simulations !c /!f

!f

!r

Realization

Mean

Std

Pr(!f > 1)

Realization

Mean

Std

Median

Realization

Mean

Std

Pr(!r > 1)

1 2 3 4 5 6 7 8 9 10 Mean SE CV

3.71 3.27 3.11 2.79 3.00 3.00 2.93 2.95 3.20 3.23 3.12 0.0812 0.0823

22.24 7.80 6.11 3.86 4.47 7.23 4.07 4.96 6.43 6.73 7.39 1.7049 0.7296

0.72 0.70 0.69 0.70 0.73 0.70 0.69 0.71 0.73 0.72 0.71 0.0053 0.0235

1 2 3 4 5 6 7 8 9 10 Mean SE CV

0.1549 0.1665 0.1677 0.1662 0.1687 0.1657 0.1642 0.1659 0.1620 0.1597 0.1642 0.0042 0.0255

0.1079 0.1166 0.1145 0.1133 0.1168 0.1161 0.1149 0.1130 0.1115 0.1094 0.1134 0.0003 0.0267

0.1278 0.1382 0.1399 0.1403 0.1413 0.1373 0.1364 0.1397 0.1363 0.1354 0.1373 0.0039 0.0281

1 2 3 4 5 6 7 8 9 10 Mean SE CV

3.68 4.26 4.07 3.89 4.07 4.30 3.71 4.22 3.78 3.73 3.97 0.0771 0.0614

5.56 8.43 12.22 6.13 9.98 12.23 6.74 7.58 7.53 5.74 8.21 0.7880 0.3034

0.77 0.79 0.76 0.78 0.76 0.75 0.75 0.76 0.75 0.77 0.76 0.0042 0.0173


Ch016-P369429.indd 361

TABLE 16.3:

361

1/12/2007 12:15:25 PM

362


!φ

!φ

Backward Bifurcation and Time Series of the d-class. Parameter values: m = 0.0000548, f = 0.2, r = 0.21, and b = [0.001,1.3]. A time series plot of the system with different initial conditions. Lower left: s 0 = 0.97, d 0 = 0.03, and r 0 = 0. Lower right: s 0 = 0.99, d 0 = 0.01, and r 0 = 0. Parameter values: m = 0.0000548, b = 0.19, f = 0.2, and r = 0.21

FIGURE 16.4:

drinkers, problem drinkers, and recovered individuals) in the presence of two drinking endemic states (a backward bifurcation). Setting the initial parameter for problem drinkers within the population at just 3 percent is sufficient to establish a community of drinkers. Such a situation might occur, for example, when a new class of freshmen arrives at college, especially if this is comprised of a large number of drinkers with the opportunity to selectively interact with one another in high-risk environments (Duncan, Boisjoly, Kremer, Levy & Eccles, 2005; Weschler & Kuo, 2003). The critical proportion of problem drinkers may determine whether or not a drinking culture becomes endemic (established) even under unfavorable conditions). In contrast, when we start with less than 3 percent of the population as problem drinkers then the drinking community is eliminated (see Figure 16.4d).

DISCUSSION We introduced a simple mathematical model to describe the dynamics of drinking behaviors generated from contacts between individuals in shared drink-

Ch016-P369429.indd 362

1/12/2007 12:15:25 PM


363

ing environments. This simple model, despite its limitations, has generated some useful insights. Specifically, the basic reproductive number (as a function of treatment), is not always the key to controlling drinking within the population. This is especially the case when recovery and relapse rates are high. High relapse rates will occur when treatment programs have only short-term positive effects. This finding is in line with the idea that the initial lapse that follows treatment is best represented as a nonlinear dynamic system with abrupt transitions between sobriety and heavy drinking that can be triggered by relatively small changes in risk (Hufford, Witkiewitz, Shields, Kodya & Caruso, 2003; Witkiewitz, 2005). Of course, what is of central importance here is the type of drinking to which individuals return following this initial lapse. One possible outcome is a return to the problematic level of drinking that led them into treatment (relapse), and another is to more moderate and less problematic patterns of consumption (prolapse) (Witkiewitz & Marlatt, 2004). As noted by Witkiewitz (2005), engaging in any form of drinking following treatment is nearly always considered as a mark failure, given the abstinence orientation of the vast majority of treatment modalities. However, from a harm reduction perspective, it is the consequences associated with post-treatment drinking that is important rather than drinking per se (Marlatt & Witkiewitz, 2002; Witkiewitz, 2005). Model results and analyses also show that the propagation of drinking behaviors is the result of two conversion processes: susceptible to drinker and recovered to drinker. Furthermore, in contrast to classic epidemiology, outbreaks (sudden growth in the number of problem drinkers) are possible. In this last situation, initial conditions (the number of problem drinkers initially introduced into the population) play an essential role in the establishment of drinking communities. The case when the drinking community is established due to the large number of problem drinkers in the population is enhanced by intervention programs with high relapse rates. Under this scenario the control of drinking communities via treatment may be extremely difficult. It may be more effective to try to limit the average residence times of susceptible individuals in drinking environments (i.e., the average time they spend in places in which alcohol is available and drinking is commonplace). Indeed, this may be the most efficient way to proceed until treatments with more sustained effects are identified and widely implemented. This approach would require the extension and refinement of existing local control measures and community-based interventions (Stockwell & Gruenewald, 2001; Treno & Holder, 2001). Recent findings by Witkiewitz (2006) have shown that within the first year the majority of drinkers that receive treatment lapse to a lighter drinking state (lowrisk on S-class). However, some lapse to either moderate or heavy drinking. In the current setting, we have explored a simple model where lapses occur to what we call problem drinkers. Though our current model does not incorporate lapses to different drinking classes we are currently looking at a new model where this is taking into consideration using the recent findings by Witkiewitz (2006). Finally, it should be noted that a key limitation of our model is that it does not incorporate relapse to different drinking classes following treatment. Rather

Ch016-P369429.indd 363

1/12/2007 12:15:25 PM

364


we explored a simple model in which all the relapses that occurred among problem drinkers were treated as equivalent. However, empirical evidence shows that recovery from problem drinking among both treated and untreated individuals can vary considerably both in terms of the level of drinking involved and types of problems experienced (Dawson, Grant, Stinson, Chou, Huang & Ruan, 2005; Sobell, Sobell, Connors & Agrawal, 2003; Witkiewitz, 2005). For example, recent findings by Witkiewitz (2006) indicate that the majority of drinkers that receive treatment relapse to lighter drinking within the first year. In future models we will take into account this range of possible outcomes among the recovered class and explore the effects of this heterogeneity on the population dynamics.

ACKNOWLEDGEMENTS This work was conducted under National Institute of Alcohol Abuse and Alcoholism Contract #HHSN281200410012C (P.J. Gruenewald—PI). The first author acknowledges the support of the Alfred P. Sloan Foundation and Cornell University.

REFERENCES Behrens, D.A., Caulkins, J.P., Tragler, G., Haunschmied, J.L., Feichtinger, G. (1999). A dynamic model of drug initiation: Implications for treatment and drug control. Mathematical Biosciences 159:1–20. Bettencourt, L.M.A., Cintron-Arias, A., Kaiser, D.I., Castillo-Chávez, C. (2006). The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models. Physica A 364:513–536. Booth, B.M., Fortney, S.M., Fortney, J.C., Curran, G.M., Kirchner, J.E. (2001). Short-term course of drinking in an untreated sample of at-risk drinkers. Journal of Studies on Alcohol 62: 580–588. Borsari, B., Carey, K.B. (2001). Peer influences on college drinking: A review of the research. Journal of Substance Abuse 13:391–424. Brauer, F., Castillo-Chávez, C. (2001). Mathematical Models in Population Biology and Epidemiology (Texts in Applied Mathematics 40). New York: Springer-Verlag. Chowell, G., Castillo-Chávez, C., Fenimore, P.W., Kribs-Zaleta, C.M., Arriola, L., Hyman, J.M. (2004). Model parameters and outbreak control for SARS, Emerging Infectious Diseases 10: 1258–1263. Chowell, G., Fenimore, P.W., Castillo-Garsow, M.A., Castillo-Chávez, C. (2003). SARS outbreaks in Ontario, Hong Kong and Singapore: The role of diagnosis and isolation as a control mechanism. Journal of Theoretical Biology 224:1–8. Chowell, G., Hengartner, N.W., Castillo-Chávez, C., Fenimore, P.W., Hyman, J.M. (2004). The basic reproductive number of Ebola and the effects of public health measures: The cases of Congo and Uganda. Journal of Theoretical Biology 229:119–126. Cox, W.M., Yeates, G.N., Gilligan, A.T., Hosier, S.G. (2001). Individual differences. In Heather, N., Peters, T.J., Stockwell, T., eds., International Handbook of Alcohol Dependence and Problems, 357–374. New York: Wiley.

Ch016-P369429.indd 364

1/12/2007 12:15:25 PM


365

Daido, K. (2004). Risk-averse agents with peer pressure. Applied Economics Letters 11:383–386. Dawson, D.A., Grant, B.F., Stinson, F.S., Chou, P.S., Huang, B., Ruan, W.J. (2005). Recovery from DSM-IV alcohol dependence: United States, 2001–2002. Addiction 100:281–292. Dufour, M.C. (1999). What is moderate drinking? Alcohol Research & Health 23:5–14. Duncan, G.J., Boisjoly, J., Kremer, M., Levy, D.M., Eccles, J. (2005). Peer effects in drug use and sex among college students. Journal of Abnormal Child Psychology 33:375–385. Epstein, J.M. (1997). Nonlinear dynamics. Mathematical Biology, and Social Science. Boulder, CO: Westview Press. González, B., Huerta-Sánchez, E., Ortiz-Nieves, A., Vázquez-Alvarez, T., Kribs-Zaleta, C. (2003). Am I too fat? Bulimia as an epidemic. Journal of Mathematical Psychology 47:515–526. Gorman, D.M. (2001). Developmental processes. In Heather, N., Peters, T.J., Stockwell, T., eds., International Handbook of Alcohol Dependence and Problems, 339–355. New York: Wiley. Hufford, M.H., Witkiewitz, K., Shields, A.L., Kodya, S., Caruso, J.C. (2003). Relapse as a nonlinear dynamic system: Application to patients with alcohol use disorders. Journal of Abnormal Psychology 112:219–227. Kerr, W.C., Fillmore, K.M., Bostrom, A. (2002). Stability of alcohol consumption over time: Evidence from three longitudinal surveys from the United States. Journal of Studies on Alcohol 63:325–333. Marlatt, G.A., Witkiewitz, K. (2002). Harm reduction approaches to alcohol use: Health promotion, prevention, and treatment. Addictive Behaviors 27:867–886. McEvoy, P.M., Stritzke, W.G.K., French, D.J., Lang, A.R., Ketterman, R.L. (2003). Comparison of three models of alcohol craving in young adults: A cross-validation. Addiction 99:482–497. McKay, J.R. (1999). Studies of factors in relapse to alcohol, drug and nicotine use: A critical review of methodologies and findings. Journal of Studies on Alcohol 60:566–576. Miller, W.R., Walters, S.T., Bennett, M.E. (2001). How effective is alcoholism treatment in the United States. Journal of Studies of Alcohol 62:211–220. National Institute of Alcohol Abuse and Alcoholism. (2001). Frequently Asked Questions. November, 2001 (http://www.niaaa.nih.gov). Accessed April 2006. Orford, J., Krishnan, M., Balaam, M., Everitt, M., Van der Graaf, K. (2004). University student drinking: The role of motivational and social factors. Drugs: Education, Prevention & Policy 11:407–421. Patten, S.D.B., Arboleda-Florez, J.A. (2004). Epidemic theory and group violence. Social Psychiatry and Psychiatric Epidemiology 39:853–856. Ross, R. (1911). The Prevention of Malaria. London: John Murray, London. Sayette, M.A. (1999). Does drinking reduce stress? Alcohol Research & Health 23:250–255. Schutte, K.K., Nichols, K.A., Brennan, P.L., Moos, R.H. (2003). A ten-year follow-up of older former problem drinkers: Risk of relapse and implications of successfully sustained remission. Journal of Studies on Alcohol 64:367–374. Sobell, L.C., Sobell, M.B., Connors, G.J., Agrawal, S. (2003). Assessing drinking outcomes in alcohol treatment efficacy studies: Selecting a yardstick of success. Alcohol: Clinical & Experimental Research 27:1661–1666. Song, B., Castillo-Garsow, M., Castillo-Chávez, C., Rios Soto, K.R., Mejran, M., Henso, L. (2006). Raves, clubs, and ecstasy: the impact of peer pressure. Mathematical Biosciences & Engineering 3:249–266. Stockwell, T., Gruenewald, P.G. (2001). Controls on the physical availability of alcohol. In Heather, N., Peters, T.J., Stockwell, T., eds., International handbook of alcohol dependence and problems, 699–719. New York: Wiley. Stockwell, T., Gruenewald, P.J., Toumbourou, J., Loxley, W. (2005). Prevention of harmful substance use: The evidence base for policy and practice. New York: Wiley. Treno, A., Holder, H.D. (2001). Prevention at the local level. In Heather, N., Peters, T.J., Stockwell, T., eds., International handbook of alcohol dependence and problems, 771–783. New York: Wiley.

Ch016-P369429.indd 365

1/12/2007 12:15:26 PM

366


Vaillant, G.E. (2003). A 60-year follow-up of alcoholic men. Addiction 98:1043–1051. Walton, M.A., Reischl, T.M., Ramanathan, C.S. (1995). Social settings and addiction relapse. Journal of Substance Abuse 7:223–233. Weisner, C., Matzger, H., Kaskutas, L.A. (2003). How important is treatment? One-year outcomes of treated and untreated alcohol-dependent individuals. Addiction 98:901–911. Weschler, H., Kuo, K. (2003). Watering down the drinks: The moderating effect of college demographics on alcohol use of high-risk groups. American Journal of Public Health 93:1929– 1933. Winkler, D., Caulkins, J.P., Behrens, D.A., Tragler, G. (2004). Estimating the relative efficiency of various forms of prevention at different stages of a drug epidemic. Socio-Economic Planning Sciences 38:43–56. Witkiewitz, K. (2005). Defining relapse from a harm reduction perspective. Journal of EvidenceBased Social Work 2:191–206. Witkiewitz, K. (2006). Lapses following community alcohol treatment: Modeling the falls from the wagon. Manuscript under review. Witkiewitz, K., Marlatt, G.A. (2004). Relapse prevention for alcohol and drug problems: That was Zen, this is Tao. American Psychologist 59:224–235.

APPENDIX DETERMINISTIC MODEL

Uniform or homogeneous mixing means that the likelihood of coming into S D R contact with members of each class is either = s, = d , or = r , where N N N N = S + D + R. Under these assumptions, the process of transmitting drinking behaviors is modeled via the following rescaled (that is, we work with proportions) system of nonlinear differential equations: sY´ = m − bsd − ms, dY´ = bsd + rrd − (m + f)d, rY´ = fd − rrd − mr,

(1)

1 = s + d + r. The rate of conversion from the susceptible state (occasional drinker) to the regular drinking state is assumed to be proportional to the size of the susceptible population, the likelihood of interacting with a randomly selected drinking partner, and both the magnitude and intensity of contacts. The rate of relapse is the result of contacts between R and D individuals. The fact that individuals can transition to the D class from the S and R classes suggests that conversion is the result of multiple processes; that is, there is not just one way to move into the D-class. The first rate of transfer of individuals to the drinking class is the result of a nonlinear process modeled via a function of S and D, B(S, D). This function must satisfy the following conditions: B(0, D) = B(S, 0) = 0 (in the absence of susceptible drinkers there is no transition). D Homogeneous mixing means that B(S, D) is modeled as β S or B(s, d) = bsd N

Ch016-P369429.indd 366

1/12/2007 12:15:26 PM


367

(in rescaled variables), where b is a measure of the average number of effective interactions between susceptible and drinkers per unit of time. The rate of transfer from R to D is the result of a nonlinear process modeled via the function G(D, R) with G(0, R) = G(D, 0) = 0. That is, in the absence of recovered or problem drinkers transitions from R to D are not possible. In other words, we D or rrd (in rescaled variables), model the total nonlinear relapse rate by ρ R N where the parameter r is a measure of the average number of effective contacts per unit of time between drinkers and temporarily recovered individuals. This nonlinear process assumes that R and D individuals (as well as S individuals) share the same environments. POPULATION DYNAMICS OF DRINKING UNDER HIGH RELAPSE RATES

Here the relationship between recovery (f) and relapse (r) rates are explored. Ideally, effective treatments should increase recovery and reduce relapse rates to the extent that an epidemic of heavy drinking is reduced or stopped. However, it appears from an analysis of the basic drinking reproductive number (with recovery), !f , that this may be a difficult task. We have the trivial equilibrium (no drinking state) given (in proportions) by (s*, d*, r*) = (1, 0, 0). Positive solutions (s* > 0, d* > 0, r* > 0), that is, solutions where drinking may become established, are solutions of the quadratic equation f(d) = d2 − Bd + C = 0, where 1 1 1  1 β ρ . Two positive soluand C = − −  with ! ρ =  !0 ! ρ !0  ! ρ ρ  µ +φ tions d*1, d*2 in (0, 1) exist whenever B > 0, C > 0, f ′(1) > 0 and B2 − 4C > 0. From the definition of C it follows that C > 0 whenever !f < 1. The positivity of the discriminant (B2 − 4C > 0) requires the following conditions: !r > 1 and ρ and 0 < !c < !f < 1 (!c > 0 whenever !f < 1) where ! ρ = µ +φ B = 1−

  ρ 1 1 µ !c =  −2 −  1 !0 ρ  β 1+  !0  If both 0 < !c < !f < 1 and !r < 1 then drinking dies out. However, whether or not a culture of drinking becomes established (0 < !c < !f < 1 and !r > 1) depends on initial conditions (see Figure 16.4c,d); that is, where the system ends up (including the rapid growth and establishment of a D clsass or its elimination) depends on the size of the initial proportion of problem drinkers. In fact, a rapid and large “outbreak” is possible whenever the number (or proportion) of initial drinkers is high. Such an outbreak, the model predicts, will result in the long-term

Ch016-P369429.indd 367

1/12/2007 12:15:26 PM

368


survival of a regular drinking culture despite the fact that !f < 1. Furthermore, in this last case a community of drinkers not only becomes established but may be nearly impossible to eliminate. In fact, parameters must be modified so that the value of !0 is lower than that of !c. This result is unexpected since the system has in place parameters that represent the effects of highly effective treatment programs (that is, !f < 1). Using current drinking literature pertaining to recovery, relapse, and the social interpersonal influences upon drinking behavior (Booth, Fortney, Fortney, Curran & Kirchner, 2001; Borsari & Carey, 2001; Daido, 2004; Dufour, 1999; Kerr, Fillmore & Bostrom, 2002; McEvoy, Stritzke, French, Lang & Ketterman, 2003; McKay, 1999; Miller, Walters & Bennett, 2001; National Institute of Alcohol Abuse and Alcohilsm, 2001; Orford, Krishnan, Balaam, Everitt & van der Graaf, 2004; Sayette, 1999; Schutte, Nichols, Brennan & Moos, 2003; Vaillant, 2002; Walton, Reischl & Ramanathan, 1995; Weisner, Matzger & Kaskutas, 2003), we have estimated several of the parameters necessary to the initial specification of a simple SDR model of drinking behavior (see Table 16.1). The results of the analysis of the model using these parameter values are discussed further in the numerical simulations section.

REFERENCE Alcohol Epidemiologic Data System Division of Biometry and Epidemiology, National Institute on Alcohol Abuse and Alcoholism. Alcohol Epidemiologic Data Directory. SAMHSA, Office of Applied Studies, National Household Survey on Drug Abuse, 1999–2001. http://www.samhsa. gov/oas/nhsda.htm. Accessed 2005.

Ch016-P369429.indd 368

1/12/2007 12:15:26 PM