A dynamic model of drug initiation: implications for treatment and drug control

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Abstract

We set up a time-continuous version of the first-order difference equation model of cocaine use introduced by Everingham and Rydell [S.S. Everingham, C.P. Rydell, Modeling the Demand for Cocaine, MR-332-ONDCP/A/DPRC, RAND, Santa Monica, CA, 1994] and extend it by making initiation an endogenous function of prevalence. This function reflects both the epidemic spread of drug use as users `infect' non-users and Musto's [D.F. Musto, The American Disease: Origins of Narcotic Control, Oxford University, New York, 1987] hypothesis that drug epidemics die out when a new generation is deterred from initiating drug use by observing the ill effects manifest among heavy users. Analyzing the model's dynamics suggests that drug prevention can temper drug prevalence and consumption, but that drug treatment's effectiveness depends critically on the stage in the epidemic in which it is employed. Reducing the number of heavy users in the early stages of an epidemic can be counter-productive if it masks the risks of drug use and, thereby, removes a disincentive to initiation. This strong dependence of an intervention's effectiveness on the state of the dynamic system illustrates the pitfalls of applying a static control policy in a dynamic context.

Introduction

Illicit drug use and related crime have imposed significant costs on the US and various source and transshipment countries for a number of years. More recently, drug problems have grown in other industrialized countries to the point that, in Stares'terms [3], drugs have become a `global habit'. Hence it is important to understand drug use and how it responds to drug control interventions. This paper contributes to that effort 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. Analyzing the model generates important new insights for 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 is parameterized with data from the US cocaine epidemic because data on that epidemic are relatively good, but the qualitative conclusions may generalize to similar drugs in similar contexts.

Before describing the model, we provide a brief sketch of the US cocaine epidemic and the relevant literature we seek to extend. Cocaine use in the US started growing in the late 1960s. The number of users peaked in the early 1980s, but total consumption remained near its mid-1980s peak for more than a decade as the proportion of so-called `heavy' users grew. Thus trends in total consumption did not, and need not, mirror trends in overall prevalence. This raises the question of what summary measure best reflects trends in the magnitude of a drug problem. Although prevalence is often used as such a summary measure, it is not a particularly good one [4]. So, following Rydell et al. [5], we focus on the quantity or weight consumed.

Since there is enormous heterogeneity across users in rates of consumption and since the average rate of consumption for a population can change over time, tracking trends in total consumption requires modeling separately the numbers of users at different levels, or intensities, of drug use. Ideally one would model the whole spectrum of consumption behavior, from occasional use in small amounts up to frequent use in large amounts, but data limitations make that infeasible.

Everingham and Rydell [1] recognize this tension and suggest that, at least for cocaine, a simple dichotomous distinction between `light' and `heavy' users is sufficient. They operationalize the distinction using data from the National Household Survey of Drug Abuse (NHSDA), which measures the prevalence of cocaine use among the US household population. In particular, people who report using cocaine `at least weekly' are defined to be heavy users, while those who consumed at least once within the last year but used less than weekly are 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.

Using this definition and data on drug use among the household, homeless, and incarcerated populations, Everingham and Rydell set up a Markovian model of population flows from non-use into light use and then into and out of light and heavy use. The transition parameters that determine the flows between those states were selected to match the historical data. They used this model to understand what has happened to date in the current cocaine epidemic, to project the future under different initiation scenarios, and to compare the effectiveness of treating heavy users and three types of drug enforcement 1, 6.

A significant limitation of the Everingham and Rydell model is that initiation is scripted. Future projections and policy simulation exercises are predicated on a fixed projection of future initiation that is insensitive to the course of the drug epidemic. That is problematic because the current prevalence, or level, 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. Indeed, the metaphor of a drug `epidemic' is commonly used precisely because of this tendency for current users to `recruit' new users. If that were the only mechanism by which current use affected initiation one might expect initiation to increase monotonically. Musto [2] 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 becomes aware of the dangers of drug abuse and, as a result, does not start to use drugs. Whereas many light users work, uphold 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, one might expect large numbers of heavy users to suppress rates of initiation into drug use.

Thus, large numbers of heavy users is in some sense not only bad, because they consume at high rates and impose costs on society, but also good because they tend to discourage initiation. Heavy users impose costs in the near term, but they generate a perverse sort of `benefit' for the future by reducing current initiation and thus future use. This generates a tension or interplay between treatment programs that reduce the number of heavy users and prevention programs that try to discourage new (light) users. Furthermore, the impact of any given heavy user on initiation likely depends on the number of other heavy and light users, so the magnitude of the `benefit' of a heavy user depends on the stage of the drug epidemic. Thus, one would expect the effectiveness of treatment, and perhaps prevention, to depend on the stage of the epidemic.

Omitting these feedback effects is of relatively little consequence if the goal is to analyze 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 programs are or for understanding how the effectiveness of an intervention such as treatment might vary over the course of an epidemic. Hence, these issues are the focus of our analysis.

Section snippets

A model of the feedback effect of prevalence on initiation

From the above it is clear that the initiation rate should be increasing in the number of light users and decreasing in the number of heavy users, but many functions have these properties. It seems plausible that a reasonable model might have the following additional properties.

  • 1.

    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

Effects of changes in the flow rates on the equilibrium

In order to study the global qualitative properties of the solutions of system (2), we construct the phase diagram (Fig. 3) for the base case set of parameters a≅0.163, b≅0.024,g≅0.062,s=0.610,q=7.0,τ=50,000. The trajectories spiral counter-clockwise into a focus. This behavior is not obvious from Fig. 3 because the trajectories decay so quickly toward the equilibrium, but it is easy to see in an expanded view of the region surrounding the equilibrium (not shown).

The locus of points where the

Conclusion

Our analysis of a continuous analog to the Everingham and Rydell model of cocaine demand, augmented with an endogenous initiation function, generates a number of interesting observations.

Proposition 1. Modeled prevalence approaches a low-level (but positive) equilibrium state if and only if the change in incidence associated with a unit increase in the prevalence of light users, ÎL, minus the sum of all flow rates out of light use, a+b, is smaller than the flow rate out of heavy use, g. (This

Acknowledgements

This research was partly financed by the Austrian Science Foundation under contract No. P11711-SOZ, the National Consortium on Violence Research, and by the US National Science Foundation under Grant No. SBR-9357936. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.The helpful comments of two anonymous referees are gratefully acknowledged.

References (14)

  • S.S. Everingham, C.P. Rydell, Modeling the Demand for Cocaine, MR-332-ONDCP/A/DPRC, RAND, Santa Monica, CA,...
  • D.F. Musto, The American Disease: Origins of Narcotic Control, Oxford University, New York,...
  • P.B. Stares, Global Habit: The Drug Problem in a Borderless World, Brookings Institute, Washington DC,...
  • J.P. Caulkins et al.

    Setting goals for drug policy: harm reduction or use reduction

    Addiction

    (1997)
  • C.P. Rydell et al.

    Enforcement or treatment: modeling the relative efficacy of alternatives for controlling cocaine

    Oper. Res.

    (1996)
  • C.P. Rydell, S.S. Everingham, Controlling Cocaine. Supply Versus Demand Programs, MR-331-ONDCP/A/DPRC, RAND, Santa...
  • J. Gfroerer et al.

    The incidence of illicit drug use in the US, 1962–1989

    Br. J. Addict.

    (1992)
There are more references available in the full text version of this article.

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