Optimal Control of Drug Epidemics: Prevent and Treat

Optimal Control of Drug Epidemics: Prevent and Treat – But Not At the Same Time? Doris A. Behrens 1,2, Jonathan P. Caulkins 3,4, Gernot Tragler 2 Gustav Feichtinger 2 1

Department of Economics, University of Klagenfurt, Universitätsstraße 65-67, A-9020 Klagenfurt, Austria, email: [email protected] 2 Department of Operations Research and Systems Theory, Vienna University of Technology, Argentinierstraße 8, A-1040 Vienna, Austria 3 H. John Heinz III School of Public Policy Management, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, PA 15213, email: [email protected] 4 RAND, Drug Policy Research Center

Abstract. Drug use and related problems change substantially over time, so it seems plausible that drug interventions should vary too. To investigate this possibility, we set up a continuous time version of the first-order difference equation model of cocaine use introduced by Everingham and Rydell (1994), extended to make initiation an endogenous function of prevalence. We then formulate and solve drug treatment and prevention spending decisions in the framework of dynamic optimal control under different assumptions about how freely drug control budgets can be manipulated. Insights include: (1) The effectiveness of prevention and treatment depend critically on the stage in the epidemic in which they are employed. Prevention is most appropriate when there are relatively few heavy users, e.g. in the beginning of an epidemic. Treatment is more effective later. (2) Hence, the optimal mix of interventions varies over time. (3) The transition period when it is optimal to use extensively both prevention and treatment is brief. (4) Total social costs increase dramatically if control is delayed. Keywords: Illicit Drug Use, Demand Control, Nonlinear Dynamic Systems, Hopf-Bifurcations, Optimal Control, Public Policy.

1. Introduction Illicit drugs impose significant costs on the US, on source and transshipment countries and, increasingly, on other industrialized countries to the point that, in Stares’ (1996) terms, drugs have become a “global habit”. A variety of control strategies exist including prevention, treatment, and various forms of enforcement, so a fundamental question in drug policy is how should scarce resources be allocated between these programs. Analysts have sought to inform this decision by

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D.A. Behrens et al. / Optimal control of drug epidemics

estimating the cost-effectiveness of different interventions. The majority of this 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 is border interdiction (Rydell et al., 1996). Studies that used dynamic models have not focused on cost-effectiveness (e.g., Schlenger, 1973; Levin, Roberts, and Hirsch, 1975; Gardiner and Shreckengost, 1987; and Homer, 1993.) However, drug use and associated problems evolve over time, so it seems plausible that no single, static strategy is best but rather that the optimal mix of interventions should vary over time as well. This paper begins to address this problem by analyzing how the allocation of resources to drug treatment and prevention should vary over time because of the epidemic nature of drug use1. One can think of reasons why the allocation should vary besides this epidemic character. Inasmuch as we exclude those from the present analysis but still find benefits to varying the mix of interventions, we make an a fortiori argument for our general point that the benefits and design of dynamically varying policies deserves greater attention than it has received in the past. Comparing treatment and prevention is of interest for three reasons. First, prevention and treatment are two of the most promising controls (Rydell et al., 1996; Caulkins et al., 1999). Second, in some states treatment and prevention funding are administered through a single agency, so comparisons of their effectiveness are directly relevant to policy choices made by those agency administrators. Third, prevention and, to a lesser extent, treatment have been at the center of US national drug control strategy for the last two years (ONDCP, 1998 and 1999). This paper will not draw ultimate conclusions about the topic of dynamic drug control because many potentially relevant factors are excluded. For example, drug prices are assumed to be constant, and enforcement is not treated within the models presented here. Hence, this paper should be viewed as a first approach to be considered together with parallel efforts (e.g. Tragler et al., 1997) and subsequent research. The model’s parameters are based on data from the epidemic of cocaine use in the US that began in the late 1960s because data on that epidemic are relatively good. Even the best data on illicit activities are limited, though, so our parameter estimates are inevitably rough. The severity of this problem is mitigated, however, by the nature of our objective. We are not trying to “solve” the problem of how to manage the current US cocaine epidemic because it has already “matured”. The period of explosive growth ended by the late 1980s. The last ten years are best described as a stable plateau or the beginning of a slow decline from that plateau. There are important questions concerning how to manage this decline, but it is too late for prevention to play a decisive role (Caulkins et al., 1999). However, historically use of both licit and illicit drugs has been cyclic so a subsequent epidemic is 1

The classical reference for applications of mathematical modeling in epidemiology, which also provides a brief discussion of the fascinating behavior of non-linear epidemics, is Anderson and May (1991).

Prevent and treat – but not at the same time?

3

an ever present threat in the US (Behrens, 1998), and there are signs that Europe may still be in the building stages of a cocaine epidemic. So we de-emphasize specific quantitative conclusions that pertain to the past epidemic, and focus instead on insights that may pay dividends in the future.

2. The Model 2.1 Relationship to Past Models Our analysis is based on a continuous time version of the first-order difference equation model of cocaine use introduced by Everingham and Rydell (1994) and described in Everingham et al. (1995), extended to make initiation an endogenous function of prevalence as in Behrens et al. (1999). Everingham and Rydell (1994) distinguish between “light” and “heavy” users, where people who report using cocaine “at least weekly” are defined to be heavy users; those who consumed within the last year but used less than weekly are called “light users”. In Everingham and Rydell’s model initiation is scripted. Their future projections are predicated on a fixed scenario for future initiation that is insensitive to the course of the drug epidemic. That is problematic because initiation rates are influenced by the current prevalence, or level, of use. 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 used precisely because of this tendency for current users to “recruit” new users. In addition, Musto (1987) argues that knowledge of the possible adverse effects of drug use acts as a deterrent or brake on initiation. He hypothesizes that a drug epidemic dies out when a new generation becomes aware of the dangers and, as a result, does not start to use drugs. Whereas many light users 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 heavy users to suppress initiation into drug use. Thus heavy users are in some sense not only bad, because they consume at high rates and impose large costs on society, but also good because they discourage initiation. They impose costs in the near term, but generate a perverse sort of “benefit” for the future by reducing initiation. This generates a tension between treatment programs that reduce the number of heavy users and prevention programs that try to discourage new (light) users. Omitting these feedback effects of current prevalence on initiation 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 (1994) did. It is of enormous consequence, however, for understanding how the effectiveness of treatment and prevention vary over the course of an epidemic, which is our goal.

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2.2 Modeling Initiation and Use Behrens et al. (1999) extend Everingham and Rydell’s model to make initiation increasing in the number of light users and decreasing in the number of heavy users. In particular, they assume that (1) The rate at which current users “recruit” initiates is proportional to the number of light users. (2) This rate is moderated by the “reputation” or image the drug has, and the reputation is governed by the relative number of heavy and light users. (3) Although most new users are “recruited”, for others the impetus to use is internal. In the jargon of product diffusion models in the marketing literature, these individuals are “innovators” who initiate on their own, not through the urging of someone who is already a user. 2 These assumptions suggest the following functional form. (Since all variables depend on the current stage of the epidemic, we omit an explicit denotation of the dependence on time.) I ( L , H ) = τ + sL exp[− q H L ]

(1)

L = number of light users H = number of heavy users s = average rate at which light users attract non-users q = constant measuring the deterrent effect of heavy drug abuse τ = number of innovators per year The rest of our model is essentially a continuous time analogue of Everingham and Rydell’s model. The one difference is that as in Behrens et al. (1999) we have a flow from heavy use out of use altogether (g) but none back into light use ( f ). We omit 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. But probably it is easier to relapse to heavy use than to enter the state for the first time, so we prefer to have only a flow from heavy use to non-use and view that rate as net of relapse. The alternative of adding a third state for former heavy users would complicate the analysis, but may be worth pursuing as an extension. Practically, Everingham and Rydell found that the data did not identify f and g individually. Any combinations of these parameters such that f + g = 0.06 fit the historical data reasonably well.

2 The theory of the diffusion of innovations addresses how a new idea, a good, or a service is assimilated into a social system over time. This topic has been studied in depth by scientists from different disciplines, including sociologists, economists, and marketers. The diffusion process is the spread of an idea or the penetration of a market by a new product from its source of creation to its ultimate users or adopters, while the adoption process is the steps an individual goes through from the time he hears about an innovation until final adoption, the decision to use an innovation regularly (Lilien and Kotler, 1983). Bass’ (1969) model operationalized these concepts in a marketing framework. A detailed survey on the development of diffusion models in the context of optimal control theory is given in Feichtinger et al. (1994), where further references can be found.


5

Since reducing the number of flows simplifies the analysis, we choose f = 0 and g = – ln[1 – 0.06] = 0.06. Hence our model of use is described by Function 1 and: L(0 ) = L0

L& = I ( L , H ) − ( a + b )L , H& = bL − gH ,

(2)

H (0 ) = H 0

I(L,H) = initiation into light use a = average rate at which light users quit b = average rate at which light users escalate to heavy use g = average rate at which heavy users quit The phase diagram (Figure 1) illustrates the global qualitative properties of the solutions with our base case parameter values and depicts the trajectory associated with the data for which the parameters of the initiation function (Equation 1) where fitted. The trajectories spiral counter-clockwise into a stable focus that is the unique equilibrium. This structure holds for a range of parameter values, satisfying the condition

(a + b − sp − g )2 ≤ (spq b g )2 < (a + b − sp + g )2 , where

p := exp[− qb g ] ,

as is described in Behrens et al. (1999). Note, that stable limit cycles may emerge

H 2M 1.75M

...

H& = 0

historical trajectory modeled trajectories modeled 1970-trajectory

1996

1989

L& = 0

1987

1.5M 1985

1984

1.25M

1983 1982

1M

1981

750,000 1980 1979

500,000 1977

250,000

1975

1978

1976

1970

2M

4M

6M

8M

Figure 1. Phase portrait, including smoothed historical trajectory of the current US cocaine epidemic

L

6


in a so-called “supercritical Hopf bifurcation” if any of the parameters – except for the number of innovators, τ – is deviated sufficiently from its base case value.3 Figure 1 also shows the historical evolution of the cocaine epidemic. To date it has followed a spiral not so different from that which the model predicts, and the modeled and historical data move around their respective paths at nearly the same rate. The fit is not perfect; the historical data reflect a higher, sharper peak in light use. Nevertheless, the similarity is striking given that the actual epidemic has been subject to a varying set of drug control interventions and idiosyncratic historical events that could explain deviations from the model’s uncontrolled path.

2.3 Modeling Drug Prevention and Treatment Prevention is modeled as cutting initiation to a certain percentage, ψ, of its uncontrolled value. Prevention here includes all forms of primary prevention, including school-based prevention, after school programs, clubs, Red Ribbon weeks, broadcast media advertisements, etc. Prevention spending, w, is assumed to have a diminishing effect as additional resources are devoted to these programs. In particular, we model the effect of prevention spending by a simple exponential decay approaching an asymptotic value h: ψ (w) = h + (1 − h ) exp[− mw]

(3)

Treatment’s rehabilitative effect is captured by the parameter β, which denotes the increase above baseline in the rate at which heavy users exit use. In particular, β(H ,u ) = c (u (H + δ ))d

(4)

Following Rydell and Everingham (1994), we assume that treatment exhibits diminishing returns because of “cream skimming” so d < 1, that an average of $1,700 – $2,000 is spent per admission into treatment, and that this provides a 13% chance of ceasing to be a heavy user, over and above the baseline exit rates in the absence of treatment. The constant δ is a mathematical convenience that avoids the possibility of dividing by zero; its specific value has essentially no effect on the results. We ignore as second-order effects the possibility that treating dealers might reduce the labor supply of dealers, that education programs might be designed to prevent dealing (Kleiman, 1997), and treatment’s “incapacitation” effects – the temporary interruption of drug use while a client is physically in the treatment program.

3 The Hopf bifurcation is the most important two-dimensional bifurcation. It gives rise to and is the corresponding mathematical background for periodic oscillations (limit cycles). “Supercritical” refers to stable limit cycles. See e.g. Guckenheimer and Holmes, 1983, pp.151ff.


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2.4 The Objective Function We minimize social cost rather than maximize social welfare in the sense that we exclude drug users’ consumer surplus from the objective function. We think this makes sense both conceptually (counting as social gains the benefits of activities which society has prohibited seems perverse) and practically (in the current political climate, models that recognize benefits from drug use are unlikely to be considered by policy makers). If the question were whether drugs should be legal, then the loss of surplus should be considered. But we take prohibition as a given and ask how it should best be managed. Social cost includes both the social costs caused by illicit drug use and the monetary costs of the control measures. Quantity or weight consumed has advantages as a general purpose scalar measure of the size of the problem (Rydell et al., 1996; Caulkins and Reuter, 1997), so the objective is to minimize: ∞

∫0

J = e −r t (κ Q(t ) + u (t ) + w(t )) dt

(5)

The consumption rate in grams per year is a weighted sum of the number of light and heavy users, Q(t) = kL L(t) + kH H(t), with weights equal to the average annual consumption rates of light (kL) and heavy (kH) users. The constant κ represents the social cost per gram of consumption, u(t) and w(t) denote the expenditures on treatment and prevention programs at time t, respectively, and r > 0 is the discount rate. We model the government as a single decision maker and assume it seeks to minimize this objective function.

2.5 The Budget Constraints Ideally the government would be free to choose whatever controls u and w minimize total cost. We analyze this “unconstrained problem” but also recognize that the political process and the limitations of human institutions are such that optimal controls cannot always be pursued. In particular, public budgeting is sometimes reactive, responding to existing problems, not proactive. One simplistic image of this tendency to respond to the problem of the day is that society allocates drug control resources in proportion to the magnitude of the drug problem. Again using the quantity consumed as a general purpose scalar measure of the size of the problem, we operationalize this image by also considering the constrained control problem for which the drug control budget for prevention and treatment is proportional to total consumption, i.e. u (t ) + w(t ) = γQ(t ) = γ (k L L(t )+ k H H (t )) .

(6)

There can be constraints not only on the ability to reallocate resources between time periods but also to do so across programs. For example, bureaucratic intransigence might make it difficult to shift the proportion of resources going to pre-

8


vention or treatment. Before dismissing such a constraint as artificial, note that treatment’s share of the federal drug control budget in the US was never less than 18.4% nor more than 22.3% between 1987 and 1997 (ONDCP, 1996). Hence, we investigate three variants of the optimization model. First we consider the case in which the budget is not only constrained in total size (Equation 6), but also has the proportions going to treatment and prevention chosen once and fixed for all time. Then we consider the case when the total budget is fixed, but its allocation between treatment and prevention can be varied. Finally, we consider the unconstrained problem in which both treatment and prevention spending can be any non-negative number. When appropriate, we discuss as a foil the uncontrolled problem (treatment and prevention spending constrained to be zero).4 These are very simplistic representations of political and bureaucratic constraints. Nevertheless, analyzing them is of interest for two reasons. First, we might have somewhat greater confidence in findings that hold up both with and without these constraints. Second, comparing results of the constrained and unconstrained optimization models may shed light on how less stylized forms of political constraints would affect drug control. Note one peculiarity created by these constraints. In the model, spending money on prevention affects drug use immediately, but for school-based prevention programs there is about an eight-year lag between spending and effect. The lag arises because the median age of cocaine initiation is about 21.5 years, but school-based prevention programs are typically run in the 7th and 8th grades. For the unconstrained problem this essentially means that whatever the solution suggests spending on prevention at time t should actually be spent at time t−8 (or at least the amount corresponding to school-based programs should be). This lag is more problematic for the constrained problems. Theoretically one would like drug control spending constrained by u(t) + w(t+8) = γQ(t), but lagged differential optimal control problems are difficult to solve. Instead, we solve the constrained problem without lags, but remember that the answer would only be correct if the effects of prevention spending were felt immediately.

2.6 The Specification of the Parameters Parameter values governing use and initiation (a, b, g, s, q, and τ) are taken from Behrens et al. (1999). Average annual consumption rates for light and heavy users under base case conditions (kL and kH) are taken from Everingham and Rydell (1994). We discount at 4% per year to be consistent with other work in this area (e.g., Rydell and Everingham, 1994).

4 In all cases control programs which are not modeled (e.g., enforcement) are implicitly assumed to operate at levels comparable to that which pertained over the historical period upon which the model parameters were estimated. Dramatic changes in the nature of the market, e.g., legalization, could change the basic dynamics of the system.


9

Rydell and Everingham (1994, p.38) report a societal cost estimate (based on Rice et al., 1990) of $19.68 billion for cocaine in 1992. This cost is associated with 291 metric tons of consumption, yielding a value of κ = $67.6 per gram. This places very low estimates on costs associated with crime (e.g. no pain or suffering costs), so we take as our base estimate a figure that is two-thirds again as large, κ = $113 per gram. The ONDCP (1994) reports that federal prevention spending was $1.5565 billion versus $2.339 billion spent on treatment. If the ratio of federal spending for all drugs to national spending for cocaine is the same for treatment and prevention, this suggests that treatment and prevention spending was about $1.545 billion in our base year. Dividing by 291 metric tons suggests making the budget proportionality constant $5.31 spent on treatment and prevention per gram consumed. Table 1. Base case parameter values Parameter

Value

I

a b

0.163 0.024

0.152 – 0.348 0.018 – 0.038

g s q

0.062 0.610 7.000

0.043 – 0.093 0.166 – 0.728 6.578 – 11.42

τ

5x104

all

kL

16.42

all

kH

118.93 -4

all -9

5x10 – 1.33x10-3

c

5x10

d

0.600

0.001 – 0.735

δ

1x10-5

—

h

0.840

0.828 – 0.989

m

2.37x10-9

10-9 – 4x10-8

r

0.040

0.027 – 0.999

γ

5.310

0.000 – 171.0

κ

113

all

Description annual rate at which light users quit annual rate at which light users escalate to heavy use annual rate at which heavy users quit annual rate at which light users attract non-users constant measuring the deterrent effect of heavy use number of innovators per year average annual consumption rate for light users (grams per year) average annual consumption rate for heavy users (grams per year) constant measuring the efficiency of treatment constant measuring the marginal efficiency of treatment technical parameter minimum percentage of baseline to which initiation can be cut by prevention (i.e., prevention can cut initiation by 100(1−h)%) constant measuring efficiency of prevention spending annual discount rate (time preference rate) budget proportionality constant (total prevention and treatment budget per gram consumed) social cost of cocaine consumption (dollars per gram)

To be consistent with Tragler et al. (1997) we assume the diminishing returns to treatment parameter d = 0.6. Rydell et al. (1996) report the number of treatments in 1992 was 548,000, base year spending was $930 million per year, and

10


13% of those treated exited heavy use, with one-third of those exits being to a state of no use. Hence, the parameters of the function in Equation 4 must be chosen such that β($930 million / 548,000 heavy users) = 0.43, which implies c = 5 x 10-4. Caulkins et al. (1999) found that every student in a US birth cohort could receive a state of the art school-based drug prevention program for $600 million and that such a program would reduce cocaine consumption by about 10.6%. Botvin (1990) argues that community-based prevention programs have not even been shown to affect drug use. We think it is overly pessimistic to view such programs as completely ineffective, so we assume, with no particular basis, that they could, if pursued to their fullest extent, augment the effects of cutting-edge, school-based prevention programs by one-half. If we further assume that the school-based programs are employed first because they are the most cost-effective, these figures imply that h = 0.84 and m = 2.37 x 10–9. For each of the system parameters and each budget policy we derive intervals within which structural stability is guaranteed. The “stable focus range”, I, is determined as the intersection of these intervals yielding transient oscillations. In all but three cases the intersected intervals include deviations of at least 20% in both directions from the base case values. The three exceptions are from parameters a, q, and h, and even in these cases what changes is the nature of the admissible controls, not the system behavior.

3. Model Solution We sketch here how the three models were solved. Details can be found in Behrens et al. (1997). Interpretation of the solutions is deferred to the next section. For mathematical convenience, we maximize negative costs instead of minimizing costs.

3.1 The Constant Fraction Budget Allocation Model (Model #1) Suppose one constrained the budget not only so that u(t) + w(t) = γQ(t), but also so that u(t) = f γQ(t) and w(t) = (1 – f ) γQ(t ). I.e. f is the proportion of the budget going to treatment and (1 – f ) is the proportion going to prevention throughout the entire epidemic regardless of its course. The government can choose 0 ≤ f ≤ 1, optimally but just once and for all time in order to minimize Equation 5 subject to the constraints described by Equations 2*. Since the optimization is performed over a single parameter, f, the problem is easy to solve numerically. L& = I (L , H )ψ(w) − (a + b )L , H& = bL − (g + β(H , u ))H ,

L(0 ) = L0

H (0 ) = H 0

(2*)


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3.2 The Optimal Mix of Prevention and Treatment (Model #2) Now suppose that total spending on prevention and treatment is constrained to be proportional to the quantity consumed, but the mixture is allowed to vary over time. Because of the budget rule, we can replace prevention spending by w = γ Q – u, and the optimization problem becomes a nonlinear control problem with a single scalar control u and two states, L and H, where one seeks to minimize Equation 5 subject to the modified constraints (Equations 2*). The analysis proceeds in the usual manner (for the general theory see e.g. Léonard and Long, 1992; Feichtinger and Hartl, 1986), where H denotes the Hamiltonian and π1 and π2 denote the costate variables, all in current values. H = − (κ + γ )(16.42 L + 118.93H ) + π1 (Iψ − (a + b )L ) + π 2 (bL − ( g + β )H ) (7) Applying Pontryagin’s maximum principle to Equation 7, we derive the necessary condition for an optimal interior control u H u= π1 Iψ u − π 2 Hβ u = 0 .

(8)

A sufficient condition for the Hamiltonian H to be concave with respect to the control u is that the costate π1 is negative (see Behrens et al., 1997, Appendix A.3.2). Since π1 0 ) and u ≠ 0 (and σ = 0 ). For the base case parameter values listed in Table 1, we compute an equilibrium state at the border of the admissible range of controls, where the numerical values are listed below.


 Lˆ   Hˆ Eˆ 3 =  ˆ w  uˆ 

13

  356,958       130,997  =    0      4 ,779,340    

Though treatment receives the entire control budget in equilibrium, the resources spent on it are very modest. Each pair of initial values (L0,H0) defines a unique trajectory as described in Behrens et al. (1997), which is the optimal solution of the unrestricted optimal control problem.

4. Results 4.1 Steady State Results All three of the budget models and the uncontrolled model yield similar steady state levels of use and drug control spending, as is illustrated in Table 2.5 (There are large differences in spending levels in percentage but not absolute terms. In every case spending is quite small relative to current prevention and treatment spending.) Note, that the equilibrium budget for the unconstrained policy is lower than for the two allocation problems. Hence, in the final equilibrium it is optimal to spend less per user on prevention and treatment than is being spent now. Thus the first conclusion is that, with or without budget constraints, it is never optimal to use these policy instruments to push the final state of the epidemic to a place very different from where it would go “naturally”. That does not mean controls are unimportant. They can play a decisive role in how the system approaches this steady state (see Figure 2). Since it takes many decades to approach this steady state, this transient effect can have a dramatic effect on the net present

Table 2. Equilibrium levels of use and control spending ˆ ($B/yr.) w Lˆ (millions) Hˆ (millions)

Model of control

none constant budget allocation, f*=0 optimal budget allocation unconstrained policy rule

5

0.34 0.32 0.35 0.36

0.13 0.13 0.12 0.13

$0 $0.11 $0.10 $0

uˆ ($B/yr.) $0 $0 $0.01 $0.005

For the constant fraction model, f and, hence, the equilibrium, depend on the initial state. Table 2 describes the case f =0, because f*=0 for initial conditions such as those that pertained in 1967. The quantity consumed in the steady state may be as much as 121% of its uncontrolled value, if the epidemic is in an initial state which favors large budget-shares for treatment. This occurs because one is obligated to pursue that level of treatment throughout the course of the epidemic, even when treating and removing heavy users is counter-productive because they help deter initiation.

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value of social cost. However, once the feedback effect of heavy use on initiation is included, the system gravitates toward a relatively low use equilibrium on its own. That equilibrium is fairly “stubborn” (due in part to initiation by “innovators”) so it is not

300

Quantity consumed in tons

250

200

150

100

50

0 1967 1969 1971

1973

Uncontrolled Constant fraction

1975

1977

Year

Optimal allocation

1979

1981

1983

Unrestricted

1985

1987

1989

Figure 2. Total consumption over time, with and without control, starting with 1967 conditions.

worth trying to suppress it much further. We can examine how characteristics of the steady state depend on the model’s parameter values, but because little control is used in steady state the results are not so different from those Behrens et al. (1999) report for the uncontrolled model. (See Behrens et al., 1997 for details.)

4.2 Optimal Policies For the constrained budget, constant fraction model (Model #1), there is just one choice. One decides today and for all time, what fraction ( f) of the budget will be allocated to treatment, with the remainder going to prevention. Hence, the optimal allocation, f * (L0,H0), depends on the initial conditions, (L0,H0). Again, limiting oneself to fixed fraction rules is clearly not ideal, but those who perceive government budgeting to be inflexible might believe that such a constraint is realistic. For the base case parameter values the stability behavior of the equilibrium state, Eˆ1 = (Lˆ1 , Hˆ1 ) can be classified with the help of stability regions. (See Figure 3.)


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• The system dynamics approach a stable focus for initial values within Regions Ia and Ib. For initial values in Region Ia, only prevention is employed; for initial values in Region Ib, a mix of prevention and treatment is used. • The system dynamics approach a limit cycle for initial values from Region IIIa and IIIb. For initial values in Region IIIb, all drug control resources are invested in treatment. For initial values in Region IIIa, resources are split between prevention and treatment, but treatment receives the lion’s share (96% – 100%). For initial values along the curve labeled Region II ( f * (L0,H0)c ≅ 0.9596336), limit cycles emerge similar to the process observed for the uncontrolled model. Treatment should receive roughly 96% of the drug control budget at all times, leading to cycling levels of use (with a period of approximately 71 years). A salient feature of Figure 3 is that for many initial states, one should fund only prevention or only treatment. In particular, one should fund only prevention, not treatment, if initially there are many light users relative to heavy users and vice versa. Figure 4 is the comparable policy guide for Model #2. It informs, as a function of the current – not initial – state of the epidemic, how one should divide a fixed drug control budget between treatment and prevention. Given a current estimate of the number of light and heavy users, one can read off the fraction of the budget that should be allocated to treatment, f0. Over time, the number of light and heavy users will change, and that optimal fraction f0 will change too, but for any given starting point the resulting trajectory can be drawn. The all-prevention/all-treatment isoclines in Figure 4 are almost identical to the L& = 0 / H& = 0 isoclines. Assuming that the epidemic starts with relatively few heavy users, this implies the following strategy is optimal for Model #2 (ignoring any delay between prevention spending and its effects). One should do only prevention and no treatment until immediately after the peak in light use is observed. Then there is a relatively brief transition period during which resources should be divided between prevention and treatment. As soon as heavy use peaks, prevention should cease and one should invest all resources in treatment. Much later, one will switch back to prevention as the epidemic approaches its equilibrium. Starting with initial conditions that reflect US cocaine use in 1967 (the earliest year for which data are reliable), the timing of these peaks with control shows a curious regularity. The sequence of peaks, with initiation peaking in 1975, then light use, then the impact of prevention, then heavy use, then treatment is robust with respect to a wide range of parameter values, and the peaks are each separated by about three years.6 We can think of no obvious interpretation of the three-year lag or why the inter-peak times should be similar, but it is worth reiterating that the transition between funding only prevention and funding only treatment in Model #2 is brief, about six years. 6

Prevention spending is resumed in 2013, and it takes over nearly all of the (much smaller) drug control budget as the system approaches the equilibrium state.

16


H0

f *=1

2M Region IIIb: Treatment only

1.75M

1996

f * = 0.96

1989

1.5M

1987 1985

Region IIIa

1.25M

1984

Region II

1983 1982

1M

1981

0.75M 0.5M 1978

Region Ib

0.25M

1975

0.5M

1976

1979

1977

4M

2M

f *=0

Region Ia: Prevention only 6M

L0

8M

Figure 3. Stability regions of Model #1 for any initial combination of light and heavy users in millions

H0

f0 = 1

f0 = 0.8

f0 = 0.6

f0 = 0.4

2M 1.75M

f0 = 0.2

Treatment only

1996

1989

f0 = 0

1987

1.5M 1984

1985

1.25M

1983 1982

1M

1981

Prevention only

0.75M 8x10

1980 1979

0.5M 0.25M

1975

0.5M

2M

1976

4M

1977

1978

6M

Figure 4. Policy isocline chart for the constrained budget, optimal allocation problem (Model #2)

8M

L0


17

For the unrestricted control problem (Model #3) the projection of the optimal budget allocation into the (L0,H0)-plane looks similar to Figure 4. The main difference is that the transition from all prevention to 60% treatment (f0 = 0.6) is even quicker (taking about two years), and the transition from 20% prevention (f0 = 0.8) to all treatment (f0 = 1.0) is a little slower (taking about five years).

4.3 Spending Trajectories over Time Figure 5 shows how much money it is optimal to spend on each control over time with the unconstrained model starting with 1967 conditions (Model #3). Since, at least for school-based prevention programs, there is a delay of roughly eight years, we have shifted prevention spending back in time by eight years. We observe a rise and fall to zero in prevention spending followed by a rise and fall in treatment spending. I.e. at no time should one fund both prevention and treatment, and there is a very brief time when it is optimal to spend nothing – even though the epidemic would be raging in the sense that initiation would be near its peak. Hence when initiation is at its peak, the optimal policy is to spend nothing on either school-based prevention (because by the time it takes effect, initiation will already have fallen of its own accord) or treatment (because that would undermine the natural tendency of heavy users to suppress initiation)! (Some spending on prevention programs that take effect immediately may be appropriate, although evidence on the effectiveness of such programs is scant.) That is a rather interesting result. Drug policy debates often pit supply-side advocates (those favoring greater enforcement) against demand-side advocates (those favoring greater spending on treatment and prevention). But with the solution in this figure it does not makes sense to advocate demand-side measures generally. There is no point in the epidemic when it is optimal to fund both prevention and treatment. Conversely, if one does not specify a time period, it is somewhat nonsensical to say treatment is better or more important than prevention or vice versa. Over the course of the epidemic, the optimal policy uses both – but not at the same time. The extreme statement of this finding – that one should finish all prevention spending before beginning any treatment spending – is not robust with respect to parameter values or, in all likelihood, the structural assumptions embodied in this model. However, the general insights that the optimal policy funds prevention most generously when treatment is pursued more modestly and vice versa and that the transition from one period to the other is relatively rapid seems to be more robust, at least with respect to parameter values. Note that in the constrained models spending per unit use is flat at γ = $5.31 per gram consumed. In the unrestricted model it is optimal to spend much more than that in the first 35 years of the epidemic. After that, very little drug control spending is desirable. Despite these differences in total spending, the budget shares ( f ) for Models #2 and #3 evolve similarly over time. The only difference is that very late in the epidemic (after 2010) Model #3 allocates a little to treatment

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Control spending in billion dollars

1,2 1

prevention treatment

0,8 0,6 0,4 0,2

97

94

91

88

85

82

79

76

73

70

67

0 Year

Figure 5. Optimal prevention spending (shifted back by 8 years) and treatment spending for the unrestricted model; Initial data from 1967

and none to prevention. Model #2 is forced to spend more (γ dollars per gram of consumption); it is counter-productive to put it all in treatment, so it allocates some to prevention. Of course the pattern for Model #1 is completely different because the fraction going to treatment ( f) is fixed once and for all time. The number of kilograms of cocaine consumption averted per million dollars spent on control gives an aggregate measure of the efficiency of control. This ratio

Y (T ) =

∫

∞

0

∫

∞

0

e

e − rt ⋅ grams averted in tons(t ) dt

− rt

⋅ control spending in bill . $ (t ) dt

,

L0 = Luncontroled (T = year ), H 0 = H uncontroled (T = year )

was computed for control starting in each year from 1967 (T=0) to 1990 (T=23). Three insights are apparent from the resulting plot (Figure 6). First, control appears generally more cost-effective when started early. Second, the constant fraction control (model #1) is less cost-effective than the optimal allocation control (model #2). Finally, unrestricted control (model #3) is less cost-effective than either form of restricted control. That is because there are diminishing returns to control spending. The unrestricted controls are constrained to spend less than is optimal (until very late in the epidemic), but that constraint means the dollars that are spent have high marginal value.


19

200 180 160 140 Y(T)

120 100 80 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 T 21 22 23 Figure 6. Cocaine consumption averted in kilograms per million dollars spent on control measures for all three policies; Initial data from 1967

optimal allocation constant fraction unrestricted

4.4 Optimal Control Budget Allocation From Now On We have discussed what should have been done starting in 1967, which is intellectually interesting but irrelevant with respect to the current epidemic since it is not 1967 anymore. What should we do moving forward from now? Since the numbers of light and heavy users have not been changing very quickly in recent years, solving Model #2 for 1996 conditions (the most recent year for which data are available) is of interest. The best strategy (subject to a constrained budget) would have been to spend a few dollars on prevention in 1997 and 1998 (again, assuming no lag between spending and effect), but then allocate all drug control resources to treatment from 1999 through 2037. From 2037 to 2048 or so most of the (much smaller) level of resources would be shifted back into prevention. With two exceptions, the basic pattern is similar to that when starting with 1967 data. First, we are already close to the point when prevention should not be funded. Second, the period during which we should rely entirely on treatment is longer starting from 1996 conditions than it would have been if control started in 1967. Results for Model #3 starting with 1996 conditions are similar, except that there is no final shift back to prevention. Relative to no control, unrestricted con-

20


trol can reduce the value of the objective function by some 30% starting with initial conditions similar to those pertaining today. It is interesting to ask what level of prevention and treatment spending in Model #2 would be most appropriate starting with 1996 conditions. Currently it is about γ = $5.31 per gram. Whether that should be raised depends on one’s estimate of the social cost of drug use (κ). For any given κ there is a critical level of spending, γC, that minimizes total cost. For lower values of the social cost per gram (e.g., κ = $67/gram), that critical level is very close to the base value of γ = $5.31/gram. For larger values of κ, the peak occurs at larger values of γ. Thus, not surprisingly, people who perceive drug use to be costly to society should favor greater spending. For our base estimate (κ = $113/gram), the critical value of γ is about four times larger than its current value. This suggests the following conclusion if one takes seriously the quantitative results, not just their qualitative structure. (And again we are generally much more confident about qualitative conclusions, e.g., about the time sequence of funding preferences, than quantitative conclusions.) Either prevention and treatment have been under-funded, or decision-makers have adopted conservative estimates of the social costs associated with drug abuse, e.g. by excluding “pain and suffering” costs.

4.5 Importance of Starting Early Although the policies display similar patterns whether one starts with 1967 or 1996 conditions, there are large differences in the values of the objective function. Early intervention can shorten the epidemic and reduce its intensity drastically as Figure 7 shows for Model #2. This suggests that it might be worth investing in data systems that increase the likelihood of detecting an epidemic in its early stages, so control can begin promptly. Note: the budget in Model #2 is constrained to be proportional to the level of use, so this ability to cut short the epidemic is not entirely dependent on allocating huge amounts of resources in the early years of the epidemic. Of course unconstrained control can do even better, but implemented early in the epidemic, either allocation control can cut total social loss remarkably. The simple, constant fraction policy (Model #1) is also much better than no control if it is started early. Later, say around the peak in light use, one would be better off delaying the initiation of such a rigid control because the epidemic would ebb from its peak of its own accord. The benefits of hurrying the retreat in such a clumsy way would not justify the associated control costs. Although Model #1 might have some advantages, e.g. low implementation costs, its application to a dynamic process which is best managed through intertemporal controls is not advisable.


H

21

uncontrolled modeled epidemic

2M

controlled (optimally allocated budget)

1996

1.5M

1M 1979

0.5M 1975 1967

L 2M

4M

6M

Figure 7. Projections of Model #2’s controlled trajectories into the (L,H)-plane for initial data from various years of the uncontrolled modeled epidemic

5. Conclusions (1) The effectiveness of drug treatment and prevention vary over the course of a drug epidemic, so their budget allocations should too. Applying static interventions to a dynamic process may be counter-productive. In particular prevention does best when there are relatively few heavy users, e.g. in the beginning of an epidemic. Treatment, is relatively more efficient at supporting the decline of drug abuse later in the epidemic. (2) The transition between when all or most spending should be allocated to prevention and when it should be allocated to treatment is strikingly brief. Indeed, if one recognizes the time lag between prevention spending and its effect, with the present model it is essentially never optimal to fund prevention programs for youth and treatment simultaneously. The extreme version of this conclusion – that it is never optimal to fund both youth prevention and treatment programs simultaneously – is not robust with respect to modeling assumption, but the general finding that the optimal policy requires rapid changes in allocation may be. (3) Control can reduce total social loss substantially if implemented early in an epidemic. This is true even though with or without control the long-run equilibrium is similar; the differences show up during the transition to steady state. The benefits are much smaller if control begins later in the epidemic, suggesting that early detection of epidemics is valuable. But even starting

22


with 1996 conditions, optimal dynamic control could cut total social cost by almost one-third. (4) If spending is not constrained to be proportional to drug use, spending per unit of use should be very high early in the epidemic. This high level of spending minimizes the total social loss even though time discounting places a greater emphasis on costs that accrue early in the planning horizon. Later, as the epidemic recedes toward its steady state, it is optimal to spend very little on prevention or treatment. We close with three observations. First, historically there have been cycles of drug use, so the end of one epidemic may also be the beginning of another. (A possible scenario is described in Behrens, 1998.) Even top of the line schoolbased prevention is inexpensive. Given our limited ability to forecast new epidemics and the lag between prevention spending and its effect, running prevention continually might be cheap insurance against an as of yet undetected future epidemic, even if it can not dramatically affect the tail end of the current epidemic. Second, there are many avenues for further research, including modeling variants on how heavy use deters initiation; introducing prices and enforcement; contrasting this model’s single, low-quantity equilibrium with “tipping point” models; adding a delay between spending money on prevention and seeing its effects; capturing more of the heterogeneity in use patterns by having more types of users; modeling interaction between drugs; and/or modeling the number of innovators stochastically. Finally, our closing observation is that simple, plausible models of drug control suggest that the mix of drug control interventions should vary over time. This possibility has received little attention of any kind, mathematical or qualitative, in the past. Given the historical pattern of recurring epidemics, greater attention to this topic seems justified.

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.


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