The author of the present article makes three points. Firstly, drug-related measures, such as the number of users, have changed rapidly over time, suggesting that they are not merely symptoms of underlying trends in the economy, demographics or other aggregates that change more slowly over time. Secondly, drug markets are subject to a wide range of feedback effects that can induce non-linearity into dynamic behaviour. Thirdly, there are at least five classes of drug epidemic models that reflect such non-linear dynamic behaviour. Some of those classes tend to be optimistic about the ability of drug control interventions to reduce use; others are pessimistic. It is hoped that the present article and, in particular, the typology, will inform and elevate the debate about drug policy, though it is unlikely to resolve that debate because of the inability to demonstrate empirically which classes of model is (are) more accurate.
The thesis of the present article is that drug problems are dynamic phenomena characterized by non-linearity and feedback. To the extent that this is true, it is important to analyse drug problems with tools that recognize and handle that complexity. Regrettably, most of the literature on drug problems and policies applies linear, static and/or imprecise models.
The present article contains three sections. The first section examines empirical evidence for the thesis that drug problems are dynamic. The next section lists some of the principal sources of non-linearity and feedback in drug systems. The final section offers a typology of five models of drug epidemic. Which type is considered by a person to be most accurate can be an important determinant of that person's view about the potential role for policy.
It is uncontroversial and uninteresting merely to assert that drug use and drug problems evolve over time. A stronger and more interesting argument is that they change more quickly and in more fundamental ways than do most other social phenomena. Such arguments cannot be quantified precisely, but it is clear that the variation in drug-related variables over the last 25-30 years has been quite substantial. Figures I-V illustrate this point by comparing variation in drug indicators (mostly pertaining to cocaine in the United States of America) with variation in series that are conventionally viewed as having been far from stable.
Violence is often described as having an epidemic component and the United States has witnessed sharp changes in the level of violence. Yet figure I shows that variation in the best-measured indicator of violence (the homicide rate) is much smaller than the variation in the number of Americans who are using cocaine (self-reported past-year prevalence as measured by the National Household Survey on Drug Abuse).
|Figure I. Variation in past-year cocaine use among the household population and homicide, 1977-1997|
Likewise, much is made of the baby boom and baby boom echo. Rapid changes in the number of youth have stressed social institutions, have been blamed for variation in crime rates and are correlated with rates of youthful drug use. Yet figure II shows that the magnitude of those variations pale in comparison with the variation in the rate of cocaine use by youth (past-year and lifetime prevalence as assessed by the Monitoring the Future survey).
|Figure II. Variation in youthful cocaine use exceeds variation in number of youth, 1975-1995|
Initiation of illicit drugs also varies much more than does initiation of licit drugs. For instance, as estimated by the National Household Survey of Drug Abuse , between 1962 and 1992 the ratios of maximum to minimum numbers of initiates into regular alcohol and cigarette use were just a little over 2:1. For marijuana the ratio was about 15:1, and for cocaine it was almost 150:1 (see figure III.)
|Figure III. Trends in incidence of cocaine, marijuana and regular cigarette use in the United States of America (thousands/year), 1962-1992|
Drug prices have also been unstable. Among licit goods, oil prices are notoriously unstable, being driven to a sharp peak during the oil crisis associated with the fall of the Shah of Iran and falling sharply in subsequent years. Yet the decline in retail cocaine prices was every bit as steep (figure IV).
|Figure IV. Declines in the price of cocaine and crude oil, 1978-1996 (1996 United States dollars)|
And the consequences of drug use have grown sharply over time. It is well known that the rate of new acquired immunodeficiency syndrome (AIDS) cases has grown from nearly zero before 1980 to epidemic proportions. The growth in the number of emergency room mentions for cocaine has been comparably swift (figure V).
|Figure V. Growth in emergency room mentions for cocaine and incidence of AIDS, 1978-1998|
These figures are meant to convey one simple point. Drug epidemics can, and the cocaine epidemic in the United States did, undergo very rapid change over time. None of the axes in the figures are false axes drawn to exaggerate the magnitude of modest changes. More such graphs could be drawn. Everingham and Rydell , for example, produced an often-cited chart showing how the mix of light and heavy cocaine users changed dramatically over time. In 1980 light users (those using cocaine less than once monthly) were responsible for about 60 per cent of cocaine demand in the United States; by 1990 that proportion had fallen to 30 per cent. In many ways, the cocaine problem of the 1980s was not the same as that of the 1970s, and the cocaine problem of the 1990s was not the same as that of the 1980s.
Examining a much larger period in history, Musto  notes alternating periods of greater and lesser drug use. In particular, a cycle of quiescence, rapid escalation, plateau and gradual decline has been observed for a number of drugs, including powder cocaine in the late nineteenth and early twentieth centuries  and crack in more recent times .
Some systems vary over time, primarily in response to variation in some exogenous forcing function. For example, the number of flowers in temperate regions varies around the year because of seasonal variation in temperature and amount of daylight. Other systems generate variation over time because of the character of their internal structure. For example, predator-prey models can gene rate cycles in levels of both the predator and the prey populations because of internal dynamics. When prey are plentiful, predator populations grow until they drive down the prey population. That can lead to starvation for predators, which allows the prey population to recover and the cycle to repeat itself.
An important question is whether cycles of drug use are driven primarily by exogenous factors (e.g. the business cycle) or internal structures (as with predator-prey models). The fact that drug-related phenomena vary so much more than do many other phenomena hints that there may be interesting internal dynamics to drug markets and patterns of drug use. What some of those dynamics may be is explored in the next section.
Drug-related phenomena seem capable of changing much faster than underlying societal characteristics such as economic well-being, demographic variables or other health-related behaviours. Systems that change quickly often do so because they contain some feedback or non-linearity. Some likely sources of feedback or non-linearity in drug systems are identified below.
Drug market participants, like people generally, respond to incentives. One important incentive is the risk of enforcement. That risk, in rough terms, is determined by the amount of effort expended by law enforcement agencies relative to the size of the market. For example, compare a small city that arrests 100 of its 500 drug sellers per year with a much larger city that has four times as many sellers (2,000) but makes only twice as many arrests (200) per year. The level of law enforcement activity is higher in the second city (200 arrests vs 100), but the enforcement pressure or intensity is greater in the first because 100 out of 500 is a greater proportion than 200 out of 2,000. In some sense individual market participants do not care how many people are arrested. They care selfishly only about their individual probability of arrest.
Responses to law enforcement pressure include reducing the frequency of offending and displacing the activity to another location, drug, or time of day . In either case, increased law enforcement pressure can reduce the number of offenders who are subject to that pressure.
Together these two observations are sufficient to create a powerful feedback effect, which has been dubbed “enforcement swamping” . Suppose the number of drug market participants increases for some exogenous reason, such as a shift in tastes. The expanded market dilutes the given level of enforcement over a larger denominator, reducing the enforcement pressure experienced by any given participant. That reduced enforcement risk makes it more appealing for others to join the market, which further dilutes enforcement pressure. Depending on the specific circumstances, the feedback effect could grow out of control (possibly “tipping” the market to a new, higher-level equilibrium) or it could merely amplify the effect of the original exogenous change.
The same feedback effect can operate in reverse. Suppose the authorities decide to increase the number of arrests. That increases the intensity of arrests, which might induce some people to cease or relocate their drug activities. If so, the resulting reduction in market size further increases the enforcement pressure borne by those who remain, which might in turn encourage still others to exit. Again, this feedback might push the market to a new type of equilibrium (e.g. eliminating the market altogether) or it might merely amplify the effect of the original change in enforcement level, but in either event represents a non-linearity.
Economists are careful to distinguish two related concepts: demand and consumption. The technical definition of consumption is the same as the lay definition. It refers to the amount of a good produced, sold and consumed in a market. Demand is different. Demand is not a single quantity but a relationship between price and consumption. It describes how much consumers would want to purchase as a function of price. Typically, consumers would purchase more of a good if prices were low than if they were high. This relationship between the market price and the quantity consumed is often drawn on a graph and referred to as a demand curve.
For many goods, the quantity consumed varies over time with market conditions but the demand curve is stable, or, if it varies, it varies because of exogenous factors. The demand for luxury goods, for example, may be higher during strong economic periods.
For drugs, demand is not fixed. It is a function of past consumption because of addiction and tolerance. The economic interpretation of addiction is subtle and still evolving , but one interpretation is that past consumption increases the value of future drug consumption relative to the value of consumption of other goods. This manifests itself in various ways, including the observation that, as some people become addicted, they spend a larger and larger share of their disposable income on the drug. Thus, drug consumption is reinforcing in an economic as well as a psychological sense.
Tolerance can have a similarly reinforcing effect if users seek out ever-increasing doses to achieve the same effect. It can also have the opposite effect if it reduces the psychic effect or benefit of a given amount of consumption. Which effect dominates depends on a variety of circumstances, including the type of drug. Anecdotal evidence suggests that tolerance reinforces future consumption for heroin but constrains it for Ecstasy (methylenedioxyamphetamine (MDMA)).
Addiction and the positively reinforcing effects of tolerance can create a positive feedback effect. Suppose supply increases. That has no immediate effect on demand, but would increase consumption. For a conventional good, that would be the end of the story; however, for drugs, that increase in consumption can subsequently lead to an increase in demand, which increases consumption still further, which increases demand, and so on. Whether that positive feedback pushes the market to some qualitatively different equilibrium or merely amplifies the effect of the original shift in supply depends on the particular circumstances, but in either case represents a non-linearity.
These demand-amplifying effects are not unique to drugs. Goods that are an “acquired taste” (opera is a common example) have a similar character and network externalities can make demand a function of past consumption. For example, demand for electronic mail (e-mail) grew as subscriptions to e-mail did because the value of e-mail depends in part on how many other people use it. Nevertheless, the fact that there are other exceptions to the “standard” notion of stable demand does not undermine the importance of this feedback for drugs.
Drug use is often described as being “contagious”. The metaphor is appropriate, even though there is not a physical, pathogenic infection vector as with malaria because initiation rates are significantly influenced by the current prevalence, or level, of use. In particular, most people who start using drugs do so through contact with a friend or sibling who is already using them. Indeed, the metaphor of a drug “epidemic” is commonly used precisely because of that tendency for current users to “recruit” new users.
The feedback from current use to initiation is not necessarily uniformly positive. Musto  has argued that, in addition, knowledge of the possible adverse effects of drug use acts as a deterrent or brake on initiation. He hypothesizes that drug epidemics eventually die out when a new generation of potential users be comes 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 fraction of heavy users are visible reminders of the dangers of using addictive substances. Hence, large numbers of heavy users might be expected to suppress rates of initiation into drug use.
Economists also distinguish between the quantity supplied in a market (which is the same as the quantity consumed) and the supply curve. The supply curve, like the demand curve, is not a single quantity, but a schedule or relation that describes how much suppliers would be willing to sell as a function of the market price. Again, as with demand, for the typical good, the supply curve is usually thought of as stable or as varying only in response to exogenous factors; however, for drugs, the supply curve can itself be a function of past production because of what Kleiman calls “learning by doing” .
“Learning by doing” refers to the idea that the supply curve is directly affected by the cost of production, and production costs decline as suppliers get more experience. The more a supplier organization has produced, the more chances it has had to discover more efficient means of production.
Again, this type of feedback is not unique to drugs. It occurs with many emerging industries; for example, prices for the electronic calculator collapsed as production volumes led to innovation. Even though drug use has occurred for millennia, the existing illicit drug markets are relatively new. High-volume cocaine production is less than 30 years old.
This phenomenon is more pervasive for illicit drug markets, however, because enforcement generates a constant turnover among drug suppliers and supply tactics, so at any given time many individual suppliers may be working up a learning curve even as the industry as a whole matures. Incapacitation drives some of that turnover. When experienced suppliers are incarcerated they are replaced by novice sellers. Avoidance plays a role as well. When improvements in law enforcement force suppliers to change smuggling routes or tactics, suppliers start over on a new learning curve for that route or tactic.
Such effects can operate at the market level as well as the organizational level. If interdiction forces smugglers to use a new trans-shipment country, initially smuggling may be costly, but if law enforcement agencies in the new trans-shipment country become corrupt over time, smuggling costs may decline.
The potential for positive feedback loops with “learning-by-doing” effects is clear. The more that is sold, the more efficient suppliers become. The more efficient suppliers become, the lower prices will be, and lower prices induce greater consumption, which leads to further “learning by doing”, and so on.
These and other feedback effects permit an almost infinite number of models to be created. Classifying them by mathematical structure (e.g. discrete vs continuous time models) is of limited value, but five classes of model may be identified on the basis of their explanation of the one empirical regularity concerning drug use about which there is little debate. Levels of use have been observed to rise rapidly from relatively low levels to much higher levels.
As yet, it is less clear what happens to drug use after that rise. Some evidence suggests that drug use remains at the new higher levels for an extended period of time; for example, the number of heroin addicts in the United States does not seem to have ebbed significantly after its rapid increase in the late 1960s and early 1970s. Some evidence suggests that use falls off from its peak but never returns to its original, low level; for example, marijuana use in the United States is well below peak levels, but remains far above levels of the period preceding the 1960s. And there is some evidence that the level of drug use can return to quite low levels, at least for a time; for example, the peak in cocaine use in the United States at the beginning of the twentieth century was separated from the current cocaine epidemic by a period (1930-1965) during which cocaine use was much less common.
The ambiguity about what happens after an explosion in drug use means that several types of epidemic models are consistent with the minimal facts about which there is clear consensus—namely, that drug use can rise rapidly from low to high levels. In particular, there are at least five broad classes of model of drug use.
The first class of model assumes that control drives everything. The internal dynamics of the drug epidemic play at most a secondary role. To those who subscribe to that view, if drug use is low, it is because drug policy is successful. Conversely, if drug use is high, that is clear evidence of a failure of policy. An explosion in drug use can easily be explained as a precipitous decline in the effectiveness of control efforts. Perhaps understandably, much of the debate among policy makers implicitly if not explicitly adopts this view that policy is central. It encourages evaluating control efforts with simple “before-and-after” comparisons. Counterfactuals are irrelevant.
A variation on this class of model assumes that drug use is always threatening to grow exponentially without bound and the only thing preventing every person or at least every child from using drugs is the control efforts that are in place. This version is a convenient one for agency administrators to adopt when seeking to justify their budgets. It is also not refutable. There is an old joke about a person snapping his fingers who, when asked why, replies “To keep the elephants away”. When informed that there are no elephants in the area, he triumphantly concludes that finger-snapping is an effective means of elephant control. Likewise, to those who subscribe to this model, the existence of non-users justifies continued funding of drug control efforts.
Given the foregoing discussion of drug epidemic dynamics and feedback, the reader can safely surmise that the author of the present article does not subscribe to this view.
The second class of model is as pessimistic about the power of drug control interventions as the first is optimistic. In it, the only stable level of use is a high level of use. Within the model, low levels of use are seen as unstable, transient periods. This transient character is reconciled with the observed persistence of low levels of drug use by invoking some exogenous shock. That is, the model is assumed to apply only after some exogenous structural change in conditions. Once that change occurs, the low levels of use are no longer sustainable and use explodes. For example, one might view high levels of cocaine use in the United States as inevitable if there is an efficient supply pipeline connecting the United States to source countries in South America. That supply line was tenuous before the 1970s, so use could remain at low levels. Once it was established, use rapidly expanded and, according to this view, there is little prospect for serious reduction in use without some other exogenous shock to the system (such as elimination of coca production in South America by some blight or fungus).
In this view of the world, routine drug control efforts are of little consequence once use has approached its high-level equilibrium. They might push use down a little, but unless some dramatic intervention manages to alter the structure of the system, control efforts will have only marginal effects.
The model of Tragler and colleagues  is of this character, with the exception that when drug use is low, control can suppress it. Thus, when use is very low, the model is like the first class of model. There would be explosive growth, but effective control prevents it. However, when use is high, “enforcement swamping” vitiates the power of controls.
In the third class of model, initially there are no drug users but there are many “susceptibles” who are vulnerable to drug initiation if the opportunity arises. When the drug is first introduced to this population, drug use rapidly infects everyone who is susceptible; however, not everyone is susceptible, so not everyone becomes a drug user. Furthermore, people do not use drugs for ever and “susceptibles” who quit using drugs are no longer vulnerable to “infection”. So, in the long run, initiation is restricted to people who are new arrivals in the system (e.g. new birth cohorts). Thus, an explosion in drug use can be seen when the drug is introduced to an unexposed population, but that explosion is followed by a decay to a lower endemic level as people mature out of drug use.
In these models (typified by Rossi ) the dimensions of the epidemic are determined by the proportion of the population that is susceptible and the typical duration of drug use. Everyone who is susceptible will get “infected” and continue to use drugs for however long people use them. Controls that operate on these two parameters are meaningful: prevention programmes that inoculate people against drug use or treatment programmes that shorten drug use careers can reduce the population of drug users. Other interventions, however, tend to have only modest effects, for example, slightly delaying the inevitable explosion in drug use.
The fourth class of model, so-called “tipping” models, suggests an inter mediate degree of optimism about the role of policy. “Tipping” models are characterized by (at least) two stable equilibria, one at a low level of drug use and one with a high level of drug use. Either low or high levels of drug use can persist indefinitely in the absence of some intervention or exogenous shock. These models view explosions in drug use as instances of “tipping” from the low- to the high-level equilibrium. Their implications for policy are twofold. Firstly, policy makers should do whatever they can to prevent the system from “tipping” from low to high levels of drug use. Typically that recommendation is of little value because the problem only attracts serious analysis after it has “tipped” into the high-level (problematic) state. Secondly, modest interventions are unlikely to have much effect, but a truly massive intervention might succeed in “tipping” the system back to a low-level equilibrium, at which point the level of intervention could be cut dramatically without having drug use return to its high levels. Hence, these models tend to suggest that it would be better to pursue a relatively modest control programme or to be very aggressive for a period long enough to “tip” the system back to its low-level equilibrium, at which point control can return to lower levels. Caulkins  and Baveja and colleagues [13, 14] offer examples of this type of model.
In the fifth class of model, drug use grows rapidly at first because of some positive feedback but, over time, negative feedback pushes drug use back down. Egan's journalistic description  of the ebbing of the crack epidemic in New York belongs to this class. Behrens and colleagues [16, 17] offer a more mathematical model in which drug use initially spreads exponentially, but prolonged drug use leads to adverse consequences that give the drug a negative reputation that suppresses initiation. By differentiating between light and heavy users, this model can endogenously create recurring epidemics of drug use separated by intervals of low drug use without invoking exogenous shocks or interventions to gene rate those cycles. Drug control interventions may or may not be highly valued in these models, depending on the details of the model and parameter values.
On the one hand this typology is very useful. When two people disagree fundamentally about the nature of drug epidemics or the efficacy of drug control interventions, it is often because their respective models (whether formal or intuitive) belong to different classes. Figuring out which classes people subscribe to can cut to the heart of the disagreement.
On the other hand, this typology is not very useful inasmuch as there is as yet no empirical means of validating one of these classes of model or disproving another. Model validation is tricky in general. In the drug policy domain one faces the added burden of a paucity of reliable data and an inability to run controlled experiments. So those people may simply have to agree to disagree about which class of model is most appropriate.
Drug use and associated phenomena change rapidly over time. In particular they change much more rapidly than do most other macro-level measures of social conditions, suggesting that such dynamics are, to a significant degree, driven by endogenous and not exogenous factors. This empirical observation is complemented by theoretical and qualitative depictions of drug market dynamics.
Particulars of the dynamic evolution of drug use vary by substance, time and location, and data characterizing those changes are relatively weak. Nevertheless, one empirical regularity stands out: drug use can and all too frequently does rise very rapidly from quite low to quite high levels.
The present article defines five broad classes of drug epidemic model that are consistent with such rapid escalation in drug use. They vary sharply in their implications for the ability of drug control interventions to materially influence drug use. A person who subscribes to the second class of model (high levels of drug use are the only stable condition) may disagree strongly with a person who subscribes to the third class (all “susceptibles” have a high probability of becoming “infected”) about whether a new drug epidemic will ebb of its own accord and will disagree with a person subscribing to any of the other classes of model about the benefits of drug control interventions.
Most people implicitly adopt one or another of these classes of model, but few consciously realize that they have done so. Bringing more explicit recognition of the models and their implications into discussions on drug policy may help resolve differences of opinion or at least concisely identify the sources of disagreement. In the longer run, a concerted effort to refine the models and collect data to support their validation and parameterization could elevate the precision and utility of drug policy analysis considerably.
*The research presented in this article was financed in part by the Austrian Science Foundation (FWF), the National Consortium on Violence Research and the National Science Foundation of the United States of America.