D. A. BEHRENS
Department of Economics, University of Klagenfurt, Klagenfurt, Austria
Department of Operations Research and Systems Theory, Vienna University of Technology, Vienna, Austria
A dynamic model of the cocaine epidemic in the United States, including endogenous feedback on initiation
An optimal conrol model
Modelling a memory of drug abuse
Refining the modelling of the memory of heavy use
Summary and conclusions
In the present article, the authors review several recent dynamic models of the current cocaine epidemic in the United States of America (both uncontrolled and optimally controlled), which differentiate between two levels of use (“light” and “heavy”). Even though all the models have their origin in a study carried out at the RAND Corporation's Drug Policy Research Center in the early 1990s, each has been developed by extending or refining another. In addition to pointing to interesting policy conclusions drawn from the analysis of those models, the authors also demonstrate that the development of dynamic models of illicit drug consumption is itself a dynamic process where subsequent refinements lead to increased quality and reliability of the resulting policy conclusions.
Illicit drug use and related crime have imposed significant costs on the United States of America and various source and trans-shipment countries for a number of years. A variety of control strategies exist, including prevention, treatment and various forms of enforcement, so a fundamental question in drug policy is how scarce resources should be allocated between the various programmes. Analysts have sought to inform that decision by estimating the cost-effectiveness of different interventions. The greater part of that work has made estimates only for a particular point in time, concluding, for example, that in 1992 domestic enforcement was three times more cost-effective than border interdiction . Earlier studies that used dynamic models have not focused on cost-effectiveness [2-6].
Behrens and others  contributed to the effort of understanding drug use and how it responds to drug control interventions by introducing a simple continuous time model of drug demand that incorporates a feedback effect of the current prevalence, or level, of use on initiation into new use. Analysing the model generates important new insights into how epidemics of drug use should be studied and, to the extent that such a simple model can be trusted, how they should be controlled. The model has been subject to refinements along different lines.
Firstly, in the original model, the deterrent effect on initiation was governed by the current number of heavy users. That approach kept the model's complexity low enough to simplify the analysis, while featuring the rather unrealistic assumption that when a heavy user exited the user population, his or her contribution to deterring initiation went immediately to zero. In other words, there was absolutely no memory of past bad experiences with heavy use even though in reality not all knowledge of a heavy user's bad experiences disappears the moment the individual exits the population, in particular if the exit is by death from drug use. More recent models avoid that problem by introducing a third category (in addition to the numbers of light and heavy users) that reflects some sort of memory of drug abuse.
In the present paper the model of Behrens and others  is reviewed, as well as the family of models stemming from its refinements [8, 9]. In addition to a description of those dynamic models, major policy conclusions are drawn on the basis of the analyses made. Hence, the purpose here is twofold. In an application-oriented vein, concrete examples are provided of dynamic models that have been parameterized with data concerning the current cocaine epidemic in the United States. In a more philosophical vein, reference is made to the fact that the development of dynamic models is itself a highly dynamic process owing to the fact that data sources are continuously reaching higher quality, understanding of the underlying epidemic processes is growing and the recent development of the tools needed for the analysis of dynamic models (both hard- and software) has led to subsequent improvements of the models (and in turn to higher quality/reliability of the resulting policy conclusions).
Since there is enormous heterogeneity between drug users with respect to rates of consumption and since the average rate of consumption for a population can change over time, tracking trends in total consumption (which is a reliable mea sure for the size of a drug problem, according to Rydell and others ) requires separate modelling of the numbers of users at different levels, or intensities, of drug use. Ideally, the whole spectrum of consumption behaviour would be modelled, from occasional use in small amounts up to frequent use in large amounts, but data limitations make that infeasible.
Everingham and Rydell  recognized that tension and suggested that, at least for cocaine, a simple dichotomous distinction between “light” and “heavy” users was useful. They operationalized the distinction using data from the National Household Survey of Drug Abuse , which measure the prevalence of cocaine use among the household population in the United States. In particular, people who reported using cocaine “at least weekly” were defined as “heavy” users, while those who had consumed at least once within the last year but had done so less than weekly were called “light” users. The average heavy user consumes cocaine at a rate approximately seven times that of an average light user and exhibits substantially greater adverse consequences associated with that drug use.
A significant limitation of Everingham and Rydell's model  was that initiation was scripted. Future projections and policy simulation exercises were predicated on a fixed projection of future initiation that was insensitive to the course of the drug epidemic. This is problematic because the current prevalence of use significantly influences initiation rates. In particular, most people who start using drugs do so through contact with a friend or sibling who is already using drugs. Indeed, the metaphor of a drug “epidemic” is commonly used precisely because of that tendency for current users to “recruit” new users. If that were the only mechanism by which current use affected initiation, initiation might be expected to increase monotonically. Musto  has argued that, in addition, knowledge of the possible adverse effects of drug use deters or slows down initiation. He has hypo the sized that drug epidemics eventually burn out when a new generation of potential users becomes aware of the dangers of drug abuse and, as a result, does not start to use drugs. Whereas many light users work, carry family responsibilities and generally do not manifest obvious adverse effects of drug use, a significant proportion of heavy users are visible reminders of the dangers of using addictive substances. Hence, it might be expected that large numbers of heavy users would suppress rates of initiation into drug use. It seems plausible that any reasonable model of an endogenous initiation might have the following properties:
(a) The rate at which current users recruit initiates is proportional to the number of light users. It is assumed that heavy users do not recruit initiates because they manifest ill effects of drug use and/or because they have been using drugs so long that they are older and socially distant from youth in the prime initiation ages;
(b) The rate at which current light users recruit initiates is moderated by the “reputation” or image the drug has and that reputation is governed by the relative number of heavy and light users, not the absolute number of heavy users. Even if there were a number of heavy users, the drug might appear benign if they were buried in a mass of (relatively happy) light users;
(c) Although most new users are recruited, for others the impetus to use drugs is internal. In the language of diffusion models , those individuals are “innovators” who initiate on their own for the sake of curiosity, by shifting from other drugs, or for some other reason, but not through the urging of someone who is already a user.
About 60 functional forms incorporating those features have been investi gated, where Behrens and others  choose one of those five functional forms which give the best system performance with respect to minimization of the squared differences between modelled and observed initiation data from 1970 to 1991.
The rest of the so-called LH model  is essentially a continuous time analogue of Everingham and Rydell's model . In that model, the population is divided into three groups: non-users, light users and heavy users (see the beginning of the present section). The number of non-users is assumed to be large enough compared with the number of users to behave like a constant and does not need to be modelled explicitly . The flow rates from one state to another are assumed to be proportional to the source states and are computed as the time-continuous equivalents of the Everingham-Rydell estimates .
In addition to the endogenously modelled initiation, another difference between the model of Behrens and others  (illustrated in figure I) and that of Everingham and Rydell  is that, in the latter, the outflow from heavy use is divided into a flow out of use altogether (currently denoted by the “rate of desistance”) and a flow back into light use. Behrens and others  dropped the latter flow for both theoretical and practical considerations. Theoretically, a flow from heavy to light use coupled with the Markov assumption implies that former heavy users who have de-escalated to light use and light users who had never been heavy users are indistinguishable. It is probably easier to relapse into heavy use than to enter the state for the first time, however. Hence, Behrens and others  prefer to have only a flow from heavy use to non-use and view that rate as net of relapse.
Before policy conclusions can be drawn from such a dynamic model of illicit drug use, it is necessary to make sure that the observed drug epidemic can be replicated by that model. (In the terminology of a mathematician, this is a “necessary” but not “sufficient” condition for further analyses.) As figure II shows, the fit of the modelled epidemic is not perfect; the historical data reflect a higher, sharper peak in light use. Nevertheless, the similarity is striking, given that the actual epidemic was subject to a varying set of drug control interventions over time that could be responsible for deviations from the model's uncontrolled path. Likewise, idiosyncratic historical events, such as Len Bias' death and the sharp increases in prices in late 1989, could account for some of the differences between historical and modelled data. And, of course, a perfect fit cannot be expected for a relatively simple model of a very complicated process such as the current cocaine epidemic in the United States.
|Figure II. Time paths of the continuously modelled
cocaine epidemic in the United States of America and the smoothed
historical data 
In their extensive analyses, Behrens and others  found that omitting the feedback effects of prevalence on initiation was of relatively little consequence for the analysis of the effectiveness of treatment and enforcement at a particular point in time, as Rydell and Everingham did . It is of enormous consequence, however, for understanding how effective prevention programmes are or for understanding how the effectiveness of an intervention such as treatment might vary over the course of an epidemic. Although Behrens and others  did not investigate an optimal control model but a purely descriptive model, they did derive a number of interesting results with respect to the nature of drug control interventions by means of simple sensitivity analysis. For example, they found that different strategies were most effective at different stages of an epidemic, and one would expect the optimal mix of interventions to depend significantly on the course and status of the epidemic. More precisely, they hypothesized that prevention programmes might be most effective at early stages of the epidemic, when most users were light users, whereas treatment programmes might be most effective when a greater proportion of users are heavy users, as is typical for later stages of an epidemic.
Inasmuch as it makes sense to vary the mix of interventions over the course of an epidemic, the authors intended to apply optimal control theory to their modelled epidemic—as a topic of further research in that area. Even though the extension seems to be the next logical step, it should be noticed that, generally, the derivation of the optimal choice of one or several controls in a dynamic model is a very complex and sophisticated task—also for models like the one created by Behrens and others , which in other respects are quite simple. The section below deals with that optimal control undertaking.
From the descriptive model developed by Behrens and others , one may conclude that drug control interventions should change over time—especially over the course of a drug epidemic. The way they do so depends to a crucial extent on the choice of the interventions, on the objective and finally on the choice of restrictions on the drug control budget.
Behrens and others  formulated and solved an optimal control model to derive optimal intertemporal treatment and prevention spending decisions under three different assumptions with respect to restrictions on the drug control bud get. In particular, the original model (presented in the previous section and illustrated in figure I) has been extended in that two of the flows are influenced (“controlled”) by suitable control instruments: (primary) prevention decreases initiation by a certain percentage, while treatment of heavy users increases their rate of desistance, as illustrated in figure III. The objective chosen by Behrens and others  was to minimize the total social costs rather than to maximize social welfare in the sense that drug users' consumer surplus was excluded from the objective functional. The total social costs included both the social costs caused by illicit drug use and the additional monetary costs of the control measures (i.e. treatment and prevention spending). An alternative objective for the LH model has recently been presented by Kaya , namely, to reach some predetermined target in optimal time.a
As mentioned above, Behrens and others  considered three different assumptions for restricting the drug control budget. These three cases can be described as follows:
(C1) The budget is constrained to be proportional to the size of the cocaine problemb and the proportions of that budget going towards treatment and prevention, respectively, are chosen once and fixed for all time;
(C2) The budget is chosen as in case (C1), but its allocation between treatment and prevention can be varied over time;
(C3) The budget is unconstrained in that both treatment and prevention spending can be chosen to be any non-negative number at all times. Theoretically, this case is the most reasonable one, because at some stages of the epidemic, high expenditures may be useful, while at other stages spending less money may be preferable. Practically, however, the implementation of the optimal solution to this problem may cause problems because the optimal expenditures can be considerably high or vary significantly over time.c It should also be noted that this model is more appropriate than one with a constrained budget if treatment and prevention resources are not allocated from a single source.
Comparing the results of these three constrained and unconstrained optimization models sheds light on how different forms of political constraints affect drug control. Insights of the optimally controlled LH model  include:
(a) Applying static interventions to a dynamic process may be counterproductive. This means that control measures, such as treatment and prevention, are most appropriate for specific stages of a drug epidemic and budget allocations across those measures should change over time. For instance, prevention works best when there are relatively few heavy users, that is, at the beginning of an epidemic. Treatment, on the other hand, is relatively more efficient at supporting the decline of drug abuse later in the epidemic (see figure IV);
|Figure IV. Prevention and treatment spending for the
unrestricted optimal control model (described in case (C3) above)
(b) The transition period, when it is optimal to use both prevention and treatment extensively, is brief (see figure IV);
(c) Some control, even a “dumb” one in the case where not only is the budget constrained in total size, but also the shares of that budget being spent on treatment and prevention are chosen once and fixed for all time, does better than no control at all (see figure V);
|Figure V. Total quantity consumed during the current
United States cocaine epidemic as well as controlled quantities for the
different budget rules
(d) People who perceive drug use to be costly for society should favour greater drug control spending per gram consumed and allocate a greater share of that spending to prevention. Generally, it would be most effective to provide very large financial resources for control measures right from the onset of an epidemic (for prevention programmes), even if it might be difficult to justify doing so by the magnitude of the problem at that time;
(e) Total social costs increase dramatically if control is delayed (see figure VI).
|Figure VI. Total costs as a function of T, where T
denotes the time when government starts to control (with T=0 representing
1970 conditions) (Billions of United States dollars)
Looking at these insights from the study by Behrens and others  more carefully, it is evident that most of the results are as could have been expected (e.g. controlling the epidemic is good, delaying the control is bad, the controls should vary over the course of an epidemic and so on). On the other hand, the conclusions to be drawn from figure IV are to some extent awkward: it follows that it is optimal to stop prevention when the epidemic is still in its early stages, whereas treatment should not be implemented before the epidemic has “matured” somewhat. That result follows from the implicit model assumption that large numbers of heavy users are not only bad in the sense that they consume at high rates and hence impose large costs on society, but also good in the sense that they tend to discourage initiation. In other words, heavy users do impose costs in the near term, but they also generate a perverse sort of “benefit” for the future by reducing current initiation and thus future use. Since the timing of treatment and the reputation of the drug strongly interact in the framework of the model described above, the reputation mechanism deserves reconsideration. One possible refinement of the LH model can be obtained by assuming that the reputation, which influences initiation, is not a function of the current number of heavy users but rather of the memory of past heavy users; that framework is discussed in the next section.
The extension of the reputation function to be a function of the decaying memory of heavy users carried out by Behrens and others  has obvious appeal. As mentioned already, all knowledge of the negative experiences of a heavy user is unlikely to disappear the moment that the individual exits the heavy-user population, in particular if the exit is by death from drug use (as opposed to ceasing use or moving out of the area). In the LH models described above, removing a heavy user immediately erased all memory of that individual, so it sometimes appeared preferable to allow a person to suffer rather than to help that person to recover. In other words, the benefits of helping heavy users directly were outweighed by the cost of not being able to set an example with the help of their suffering. Such inhumane policies are most likely to disappear in a framework where past users can be remembered.
From a technical point of view, this refinement of the LH model requires a third category, E (the number of so-called “ever-heavy” users), so the analysis becomes more complicated (see figure VII for an illustration of the LHE model). This extension of the model with an additional category does not significantly improve the system performance in the descriptive (i.e. uncontrolled) case. That is, the time paths of the numbers of light and heavy users look very much like those for the original LH model (see figure II). However, in the optimally controlled LHE model, the memory removes the seemingly perverse results of the LH model that the presence of a heavy user can be so valuable as a deterrent that successfully treating such users actually increases consumption in the long run. In particular, treatment is no longer counterproductive unless one fails to keep up the memory of the adverse consequences of abuse in an adequate way .
An interesting question investigated in the paper by Behrens and others  is the fascinating interaction between a society's present-orientation, its ability to remember the past and the occurrence of cycles in the future.d They prove once again the old adage that “those who forget the past are condemned to repeat it”. More precisely, the greater the deterrent power of memories of drug abuse, the less likely society is to wind up with a chain of drug epidemics. Additionally, they verify that it can be desirable to relive past epidemics—at least for myopic decision makers. Or, to put it in simple terms, “for those who forget the past and over-value the present, it may be optimal to have their future recreate the past”. Finally, it is shown that it is optimal to apply prevention throughout the epidemic because moderating the contagious aspect of initiation reduces the likelihood of cycles and instability.
The results derived from the LHE model and their comparison with the conclusions of the LH study suggest that the extension to include a memory of people who have been ever-heavy users significantly improves the model performance. Nonetheless, the LHE model is not the ultimate model for including the implementation of a memory of drug abuse. The following section deals with another refinement of the model.
The major drawback of the LHE approach is that three individuals who use drugs heavily for, say, one day, one year and one decade, would all contribute the same amount to the memory of heavy use. In reality, however, the longer an individual is addicted, the more problems he or she experiences, the greater the costs imposed on others, the more people there are who witness the behaviour and so on. So an appealing alternative is to base the negative reputation of an addictive substance not on the memory of the number of people who ever used drugs heavily, but instead on the memory of the number of heavy-user years, that is, the number of years spent in heavy use.
Behrens and others  have started to investigate such a model, where the number of ever-heavy users (E) is replaced by the number of heavy-user years (Y). That model, which is referred to as the LHY model, is illustrated in figure VIII.
From the preceding discussion it is clear that the LHE and the LHY models differ and that the LHY model provides a more realistic model formulation. A comparison of the respective flow diagrams (figures VII and VIII), however, does not reveal the fact that the analysis of the LHY model is also more difficult.e
What both models have in common, however, is the difficulty of estimating the parameters pertaining to the categories E and Y, respectively, for which there are no tangible quantities.f Nevertheless, at least for the LHY model, that problem has been resolved recently in a parameter estimation study with data on the current cocaine epidemic in the United States .
Behrens and others  make a thorough stability analysis of the uncontrolled LHY model's dynamics. The results obtained so far suggest, among other things, that drug prevention can temper drug prevalence and consumption and can avoid the reoccurrence of a drug epidemic. Furthermore, the results show the correlation between the deterrent power of negative experiences with drug abuse and the rate of forgetting them by providing a functional form for the phenomenon. The insights derived so far are general enough to allow a detailed characterization of what types of drugs—in terms of the probability of escalating to heavy use and the length of a typical addiction career—are most prone to generate cyclic, that is, reoccurring, drug epidemics. In addition, the LHY model even allows fairly general statements on epidemics of delinquent behaviour with a feedback effect of prevalence on initiation.
Four dynamic models of the current cocaine epidemic in the United States have been reviewed in the present article. All differentiate between two levels of use (“light” and “heavy”) and are based on Everingham and Rydell's 1994 model , but each has been developed by extending or refining another.
Firstly, the LH model by Behrens and others  is presented, which extends Everingham and Rydell's model  by introducing an endogenous function where light and heavy users feed back on initiation into light use (“infection” by light users versus “deterrence” by heavy users). As demonstrated in figure II, even such a “simple” model as the LH model is to some extent capable of reproducing such a “complicated” process as the current cocaine epidemic in the United States.
The optimally controlled LH model by Behrens and others  was then reviewed. The results derived from that study are interesting and most of them are not extremely surprising (e.g. controlling the epidemic is good, delaying the control is bad, the controls should vary over the course of an epidemic and so on). Still, one of the results (figure IV) suggested that the model needed some improvement: there are times in an epidemic when it is more desirable for a person to suffer than to help him or her, since the physical existence of a heavy user significantly diminishes initiation. Behrens and others  recognized that those seemingly “perverse” results were caused by the specific choice of the feedback function in the LH model, in particular, the assumption that heavy users would only contribute to a bad reputation of the drug as long as they were heavy users.
By introducing a third category (the “ever-heavy” users), Behrens and others  were able to formulate a refined model. One of the interesting questions investigated for that model is the fascinating interaction between a society's present- orientation, its ability to remember the past and the occurrence of cycles in the future.
In the LHE model, for the (negative) reputation of the drug it did not matter how long a heavy user was using drugs heavily. That problem was resolved by replacing the number of ever-heavy users with the number of heavy-user years. A short introduction to the LHY framework  was provided.
In conclusion, firstly, models such as the LHY model may be general enough to be applied to other drugs (both licit and illicit) and even to other forms of delinquent behaviour, provided they include a feedback effect of prevalence on initiation.
Secondly, it is clear from the studies reviewed here that dynamic models provide an important contribution to the problem of designing better drug control policies, because drug epidemics are, by definition, dynamic.
Thirdly, as has been shown here, the development of dynamic models of illicit drug consumption is itself a dynamic process, in which existing models are extended or refined. That improvement, however, depends to a crucial extent on improvement of the data sources, a better understanding of the underlying epidemic processes or new development of the tools needed for the analysis of dynamic models (both hard- and software).
*The present research was financed in part by the Austrian Science Foundation (FWF).
aIn particular, Kaya  considered the problem of finding a time-optimal control to get from some initial state (i.e. initial numbers of light and heavy users) to a target state.
bAccording to Rydell and others , total consumption is a reliable measure for the size of a drug problem.
cTragler and others  derived the optimal solution for an alternative optimal control model of the United States cocaine epidemic. Their model is different in that they consider "average" users (i.e. there is no distinction between different levels of use) and the controls are treatment and price-raising enforcement. They show that, if initiation into drug use is an increasing function of the current number of users and control measures are implemented early in the drug epidemic, then it is optimal to use very large amounts of both enforcement and treatment to eradicate the epidemic. In other words, one would need a very large budget to pursue an optimal control of a drug epidemic that is still in its early stages, that is, small. In such a case, the per-user budget is enormous and the optimal policy will probably not be implemented because the public would not accept very large expenditures for a problem that is hardly visible.
dNote that cycles (i.e. repeated drug epidemics) may also occur in the LH model presented above; for details, see Behrens and others .
eIn contrast to the LHY model, the LHE model breaks into two parts (the LE and the H part) and, hence, the relevant dynamics for the descriptive case (but not for the optimally controlled case) can be transformed into a two-dimensional dynamical system, that is, the model may be investigated in the plane allowing a more thorough analysis, for example, with respect to cycles. This simplifies the analysis of the LHE model significantly.
fObviously, this problem does not arise in the original LH models.