Mathematical models for decision-making in planning drug abuse control programs

S&-&on. Plan. Sci., Vol. Il. pp. 213290. Pergamon Press 1977. Printed in Great Britain

MATHEMATICAL MODELS FOR DECISION-MAKING IN PLANNING DRUG ABUSE CONTROL PROGRAMS

HECTOR CORREA and JUDITH SANGL Graduate School of Public and International Maim, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

(Received 17 Feudal 1977)

Abstract-h this paper, a framework for systematizing the different applications of decision-making models to the control of drug abuse is first presented. Next, the available literature on the topic is used to exemplify as many as possible of the elements of the framework presented. Finally, extensions of the current state of the field in the direction of disaggregated and of dynamic models are presented.

1. INTRODUCTION

Agencies charged with the control of drug abuse face basically the same problem as any other agency, whether public or private; i.e. that of making the most efficient use of the resources available. Thus it is possible that mathematical methods for decision-making might be useful to them. The object of this paper is to explore that possibility systematically.

The first step in any attempt to apply the mathematical methods for decision-making is to list and clearly define the different benefits (or damages) of the process to be studied, as well as the alternative means of obtaining (or avoiding) them, and the costs of these means. This will be done in Section 2.

In Section 3, a framework for systematizing the different applications of decision-making methods to the control of drug abuse is presented.

As could be expected, limitations of space, and even more, limitations derived from the present state of the field do not allow a detailed discussion of each of the elements of the framework mentioned above. It will be used here, first, to organize the literature on the topic that we have been able to obtain (the studies by Hoiahanll], Jeffers and Leslie were not available), and second, to indicate gaps that need to be filled.

However, the object of this paper is not only to systematize the field. Some of the possible approaches to filling existing gaps will be developed and analyzed in detail. This will be done by means of a detailed discussion of a linear programming model to maximize the number of addicts abstaining from the use of heroin, and a dynamic model to specify the optimum policies for the control of the heroin supply.

2. ~H~C~~C ~ OF A DRUG ABUSE PROBLEM

As an introduction to the study of the control of drug abuse, the basic elements of the problem will be defined in this Section.

First, some clarification of the meaning of drug abuse is required. Frequently-used synonyms are overuse, misuse and chronic use. In most studies, these and similar terms are used without any indication of the dosage per use and the frequency of use. Attempts to give more precise references appear in a few cases. To collect the information used here, the statements of the authors of sources that referred to abuse, misuse,

overuse, chronic use, etc., regardless of whether they were supported with more specific information, were accepted.

In the presentation below, particular reference is made to the elements encountered in the control of heroin abuse. However, except for obvious modifications, similar elements are encountered in the control of any other drug. These elements are classified in two main groups: damages, and control ins~uments. This latter group is subdivided into elements that control supply of drugs, and those that affect demand.

The direct damages brought about by drug abuse usually considered are:

(a) Deterioration of the physical and mental health of the addicts;

(b) Production lost due to un- or underemployment of addicts;

(c) Personal and property losses as a consequence of the crimes committed by addicts; and

(d) Higher propensity of the addicts to use other drugs. The instruments that influence the supply of drugs can

be classified in two groups, depending on whether the government attempts to restrict the supply of a drug. If the government does not attempt to restrict the supply, the only instrument is that of legally unrestricted supply. In the opposite case, there are the non-coercive measures such as purchase and distruction of the entire production of a drug, and the coercive ones, such as penalties for production of the raw materials needed or the final elaborated drug, as well as for the transportation and distribution of either or both.

A first classification of the instruments that influence demand has as criteria the target population. Diierent formal and informal educational programs whose objective is to reduce the inclination or increase the resis- tance to drug use both by users and by non-users constitute the first set of instruments.

The second set affects only the population of addicts. These can be classified as:

(a) Law enforcement instruments that reduce to the prosecution and encarcelation of addicts, treating them like any other type of criminal, as a result of their addiction: and

(b) What can be called public health instruments, that can be classified into two basic types according to their objectives: (1) cure; or (2) continuous programs. Cure

273

274 HWII~R COKKEA and JUDITH SASGI

programs are directed toward treating the addict so that he no longer needs or wants the drug and will become a socially productive person as a result. Continuous programs treat the addict indefinitely while attempting to help him lead a reasonably normal life.

Treatment modalities that arc included as “cure” programs are the following:

(I) Detoxijicotion. An addict is helped through withdrawal from heroin. either without any drugs or using some drugs to ease his discomfort. in a hospital or clinic setting.

(2) Cicil commitment. Usually involves a period of incarceration or hospitali;r;ition, followed by a period of parole or outpatient supervision. During these periods, addicts are to be freed of their addiction by treatment and their other problems corrected with vocational and psychiatric therapy. At the Federal level, civil commitment was instituted in the 1930s and in the late l%Os, California and New York also began programs. Although these three are the largest programs, 32 other states have legal provisions for civil commitment. The data available suggest that civil commitment is initiated voluntarily by the addicts in about IO?& of the cases (DeI,ong[21, p. 185). When this occurs, the addict is free to leave the program at any time. More frequently, civil commitment is made compulsory in place of sentence or prosecution or at the initiative of relatives or other persons. In these cases, the addict is not free to leave, and during the aftercare period, certain violations. if detected, will result in additional confinement.

(3) Therapeutic community A volunteer treatment program which attempts to restructure the character and personality of the addict so that he will no longer require drugs in order to cope. Group therapy, a reward- punishment system and a highly structured communal life are the techniques employed to accomplish this objective. Although some communities view this treatment as continuing indefinitely, others aim to return the addict to society even though there may be a prolonged inresidence period.

(4) Outpatient abstinence. A drug-free modality where the addict remains in his home environment (some drugs may be used in short-term detoxification) in which the objective is to create a sense of personal worth which is contradictory to continual drug abuse. Group and individual therapy of various types: vocational and social counselling, vocational training and education and family counselling may all be utilized to achieve the objective.

Five continuous programs will be considered: (5) Antagonist treatment. Antagonists are drugs that

prevent opiates from having any effect, although they do not ease the addict’s hunger for heroin. Their effects are of short duration. Antagonist treatments, together with other supportive services, should cure drug addicts through lack of reinforcement. However, they are still in the experimental stage.

(6) Minimum methadone maintenance. Methadone is an addicting drug which, because of cross tolerance between opiates. will block heroin withdrawal and craving when a certain dose is used. If withdrawn from methadone. craving for heroin and not for methadone will return. The type of methadone now being used has an effect that lasts 24 hr. although for another type of methadone now being tested, the effect will last approx. 72 hr. Minimum methadone maintenance entails only administration of a daily dosage of the drug with few or no supportive services.

(7) Maximum methodone maintenance. This is the same as minimum methadone maintenance, but also includes supportive services.

(8) Heroin maintennnce. Continual administration of heroin through established clinics. usually for registered addicts only.

(9) Prescription heroin. Specified doctors will pre- scribe heroin for registered addicts.

It should be clear from the description above that the definitions of the different treatments are largely ar- bitrary, and that other. similar classifications could also be made.

3. A FRMfEWORK FOR THE CLA!!IFICATION OF MODELS

OF DRUG ARUSE AND CONTROL

As a starting point, it is useful to mention two classifications that can be applied to models of any process. The first subdivides the models on the basis of whether they explicitly include choice among alternatives. Models for description and for decision-making are included among this group. The second classification of the models subdivides them into deterministic and probabilistic models, depending on whether it is assumed that the values of variables and parameters are observed without error.

In the presentation below, deterministic models for decision-making are considered in Sections 4 and 5. Some brief observations with respect to the probabilistic models are presented in Section 5.

On intuitive grounds, the most acceptable objectives for mathematical models for decision-making, as applied to planning the control of drug abuse, are: (a) to minimize the damages of drug abuse by using the most appropriate combinations of the instruments described above, within the constraints set by the human. physical and financial resources available; and (b) to minimize the costs of these instruments required to achieve a fixed level of damages that is considered acceptable. Below, explicit attention will be paid to objective (a), although not all the observations are also valid for (b).

The models for decision-making can be classified according to the following criteria:

(1) CoDerage. Different extents of coverage can be used for either or both objectives and instruments. The coverage of the model varies according to whether the minimization of the damages of one or several drugs are included as objectives. On the basis of the classification above, the case in which the objective of the model is to minimize the effects of only one drug, while maintaining the abuses of the other drugs within exogenously dcter- mined constraints will also be considered one-drug models.

Coverage with respect to instruments varies according to whether the different types of supply and/or demand instruments are considered.

(2) Lecel of aggregation considered in the model. Diff- erent forms of aggregation can be considered in the models applied to drug control programs. The different damages of addiction can be considered apparent or can be aggregated into one total monetary value.

Aggregation is likely to be valid only when dealing with supply-oriented instruments or with demand for non-addicted populations. For instance, the aggregation of several supply instruments makes it possible to estimate the cost of, say, one ton of heroin kept off the streets.

Aggregation of several demand instruments or treat-

Mathematical models for decision-making in planning drug abuse control programs 275

ments is less meaningful. For example, the aggregation of prescription heroin and therapeutic communities is not likely to be of any use. However, it should be observed that the treatment defined above actually is an aggregation of several components. Their disaggregation would imply a separate consideration of the human, physical and financial resources required. Usually, all these aspects are aggregated into the monetary costs.

Finally, the population of addicts can be treated as one homogeneous group, or it can be classified by age, sex, educational level, etc. It is also possible either to consider all the addicts receiving a treatment as homogeneous, or to classify them according to whether they remain under treatment until it is completed, and if so, whether after completion they arc able to abstain from the drug.

(3) Time dimension assumed in the model. Two main possibilities can be considered in this case. The first is to assume away the influence of time and deal with multiperiod and/or dynamic models. A possibility showing that the classification above actually represents the two extremes of a continuum rather than two discrete classes is the use of some form of discounting of the future.

(4) The fourth and last criteria for classification is based on the mathematical methods used. For simplicity, two groups are defined, depending on whether only linear mathematical models are used.

4. SOME EXAMPLFS OF AGGREGATE MODEIS

4.1 Introduction

In this Section, four examples of deterministic, one- drug, aggregate models that have been studied in the literature of the subject are presented. Two of these are linear models, and two are non-linear.

4.2 The simplest linear programming model The simplest model available is implicit in reference

(Marks[3]). This study refers only to the control of heroin use.

Twelve instruments to reduce or eliminate these damages are examined. Four deal with the supply of heroin, and eight with its demand. It is argued that the four policy options regulating supply and the one policy option which is a deterrent (longer prison terms and increased arrests of addicts) are inefficient, especially if the demand for heroin is inelastic with respect to price. The remaining options. which are: (I) detoxification; (2) civil commitment; (3) therapeutic community; (4) outpatient abstinence; (5) antagonist; (6) methadone maintenance; (7) heroin maintenance; and (8) prescription heroin are examined in terms of several aspects of their effectiveness (abstinence rate from heroin, rise in employment, decrease in crime rates, etc.). Detailed analyses are made of maximum and minimum methadone maintenance and prescription heroin. since these three alternatives have well-documented results.

The objectives of the programs for the control of heroin abuse are assumed to be the maximization of “full employment social product benefit”, calculated as the productivity gain resulting from employment minus the costs. The benefits from crime decrease are not included, since they are the same for both methadone maintenance and prescription heroin programs. Psychological health and physical health of the addicts are considered to be good in both programs if properly administered, but are not explicitly included in the objective.

Only the government costs of the programs are con-

sidered for the instruments to be used. These are considered to be equal to direct monetary costs of running the program minus the tax increase resulting from employment.

The problem discussed reduces to the maximization of

Z=C(ai-ci)Y, ,-I

subject to

(2)

where ai, value of damages per addict avoided with treatment i = 1, 3; i = I, maximum methadone maintenance; i = 2, minimum methadone maintenance: i = 3, prescription heroin; ci. cost of treatment i per addict: N,, number of addicts treated with treatment i, i = I. 3 and R. total resources available.

Assuming that all the resources l2 are used, it can easily be shown that treatment j should be used if

adc, 6 a,lq (3)

which simply means that treatment j should be used if the returns per unit of cost in j are larger than those in i.

Using the data in Table 1, it is possible to rank the programs according to their cost effectiveness in the following order: Minimum methadone maintenance (az/c, = 23.33). Prescription heroin (17.02) and Maxi- mum methadone maintenance (1.63). Marks concludes that a combination of both heroin and methadone should be available to registered addicts and administered professionally if the government is concerned about the spread of addiction and the health of the addict.

4.3 A linear model including supply and demand instruments

In order to emphasize basic ideas before introducing the complexities of disaggregation. the model in eqns (I) and (2) will be extended to deal not only with the demand, but also with the supply instruments for the control of drug abuse.

Since very little in the way of simplicity is lost, instead of only three demand instruments, a general number n

will be considered. Also, m supply instruments will be assumed.

Let S be the total supply of a drug, measured, say. in weight. S is assumed to be an exogenously fixed, known constraint.

The total supply is divided into two parts, depending on whether the drug reaches the streets, i.e.

s=s*+s, (4)

Table I. Data used in Ref. [3I to apply model in eqns (I) and (2)

Maximum Minimum methadone methadone Prcccription

maintenance maintenance heroin

Full employment social product Tax receipts Gross govt. cost Ket govt. cost

3500 3500 3150 350 350 31.5

2500 500 500 2150 150 185

Source: Elaborated from data in Ref. [3]

276 HECTOR CORREA and JUDITH SANCL

where S1, supply reaches the streets; &, supply does not reach the streets.

Finally, let ci, the cost of keeping one unit of drug out of streets using instrument i, i = n t 2,. . . , n t m. (The supply instruments are numbered from n + 2 to II t m to simplify the notation below.) SZi, amount of drug kept off the street using instrument i.

With the notation above, we have that

n+m

..J+, &i (5)

is the total cost of keeping S, amount of drug out of the streets; and

*+1 UC YiIS, (6)

i=l

where the Y,+l is the number of addicts without treatment and u is the average use per addict, is the constraint imposed by the actual supply of the drug.

Finally, the cost-resources constraint in (2) now becomes

n fI+?VI CC~Y~+ C c+SisR. (7) i=l i=n+l

A basic limitation of the model above is the assumption that the total supply of the drug is fixed. In fact, it is likely to be, for all practical purposes, without limit (Marks [3], p. 73), increasing from one period to the next in response to the increments in price generated by new efforts in law enforcement. As a consequence, the distribution of resources between supply and demand instruments is more appropriately treated in a multiple period or dynamic model.

In summary, it can be said that the purpose of the extended model is to maximize (1) subject to constraints (3)-(7).

4.4 A simple non-linear model In the approach used by McGlothlin et al. [4], eqn (1) is

generalized to

Z=Cfi(YJ i= 1,7 (8)

where the seven treatments considered include all those presented as demand instruments in Section 2 with the exception of 4, 5 and 9. A ‘I-treatment combination of civil commitment and other modalities is added.

The main novelty introduced is that the fi( YJ are not assumed to be linear funttions as in Section 4.2, but rather functions with a negative second derivative.

In the estimation of the fi(Yi), the benefits derived from the reduction of value of property stolen and increasing employment are considered.

Equation (2) is generalized in a manner similar to (l), to

C=C (Ci(Yi)) i= 1,7 (9)

where it is assumed that functions Ci() have positive second derivatives, i.e. the marginal costs of treatment i are increasing.

The approach used by McGlothlin et al.[4] to specify the value that the Yr should have seems to be-mistaken. They define the profits of treatment i when Yr patients are treated as the value of

Pi =

The total profit of a_ll treatments is given by P = Pi. Assuming that the Yi are variable, McGlothlin et al. specify E as the value? that maximize the P. It can be easily shown that the Yi should satisfy

for all i.

fi( ri) = Ci( ET;,, (11)

The procedure described above is mistaken because fi(Y1) and Ci(Y1) represent total benefits and costs, so that the integral in (10) is the benefits that wo_uld be received if programs of all sizes for 0 I Yi I Yi were organized, instead of only one program with % addicts.

On the basis of the erroneous criteria mentioned above, McGlothlin et al. suggest the following ranking- in descending order-of the treatments: first, the treatment which combines civil commitment and other modalities; second, civil commitment; third, minimum methadone maintenance; fourth, prescription heroin; fifth, maximum methadone maintenance; and sixth, therapeutic community. Detoxification is not recom- mended.

It should be observed that the correct way to specify the Yi would be the maximization of the difference between the values in (8) and (9), i.e.

F=Z-C. (12)

It should be clear that, in this case, the first derivatives of fi and Ci would be equal for each i.

Even with the correct method for specifying the %, another limitation of the approach suggested by McGlothlin et a&one they are fully _aware of (p. 56)- remains. This limitation is that each Yi specified by the conditions in (ll), or the maximization of (12), is determined without considering-the values that the other Fj, j# i might take. That is, each treatment is examined in isolation, without considering that in reality it would be competing with other modalities. This limitation is a direct consequence of the assumption that both the benefits and the costs functions considered so far can be decomposed in additive elements. If this were not the case, the values specified for the Yi would be in- terdependent.

4.5 A non-linear model considering supply and demand instruments

The model in reference (Hannan[S]) can be considered an extension of the one presented in Section 4.2 and modified in Section 4.3. This is because it analyzes with non-linear functions the impact on addiction of both supply and demand oriented instruments, namely, law enforcement against pushers, law enforcement against addicts, and finally, methadone maintenance treatment for addicts.

To specify the values that the three instruments mentioned should have, it is assumed that the drug control agencies should maximize the net benefits of the pro-

Mathematical models for decision-ma~ng in planning drug abuse control programs 271

gram, i.e. the benefits derived from the reduction of addiction minus the costs of obtaining that reduction. The analysis is more general than that proposed by McGlothlin, because the objective function is not assumed to be additive separable.

5. A DISAGGREGATE LNEAR MODEL

5.1 Introduction In this Section, what basically is a d&aggregated and

extended version of the model in eqns (1) and (2) is discussed in detail, together with a numerical example using OOBE (Our Own Best Estimates) derived from a search of the literature available.

5.2 TJZZ model The population of addicts and the addicts in each

treatment are the first components of the model in (1) and (2) that are disaggregated in this Section. Next, each of the damages of drug addiction is considered se- parately. The model in (1) and (2) is also extended to include all the treatments influencing only drug addicts presented in Section 2.

At this point it should be observed that addicts could also be classified by sex. However, this subdivision is less important, because female addicts constitute only about 17% of the addict population (Ashley, p. 52).

Next, the different stages through which an addict who begins a treatment can pass will be considered. Making an abstraction of the different time periods that could be involved, it will be assumed here that addicts can drop out of a treatment, or if they remain, they can either do so for an indefinite period of time if the treatment is continuous, or graduate, i.e. complete the program, if it has an end. Those addicts that remain indefinitely or graduate can either abstain or not from using the drug. It is assumed here that addicts who are not in a program, or who drop out of one, do not abstain for any length of time. (Abstinence refers to long-term abstinence. Tem- porary abstinence, which an estimated 15% of all addicts adopt (McGlothlin et al., p. 9) is not considered.)

Below it will be assumed that, for each treatment, fixed known proportions of addicts take each of the three alternatives mentioned.

The three possibilities, and the notation for the fixed proportions, can be illustrated with the following branch diagram:

ABRijGRijY:) number that remain indefinitely or graduate and abstain

(1 - ABRii)G~ijYjj number that remain ind~fi~tely or graduate but do not abstain

(1 - GRij)Yij number that drop out of the program and do not abstain

In the model to be presented below, the number of addicts that should receive the different treatments are- as before-the control variables. However, in the present case, these addicts are classified in two groups: those 30 yr of age and over, and those under 30 yr of age. This classification takes account of differences that have been observed in some programs according to age (Winick[6], pp. 1-7; [7], pp. l-11), whether or not it may be due to “maturing out”. The age of 3.5 was found to be the mean ege at which an addict “matures out” in Winick’s[7] study, although it is not clear whether the process reflects the addict’s lie style or the life cycle of addiction. The average age of maturation may be lower now, due to the earlier age of initiation into narcotics (AshleytS], p. 52; McGlothlinL41, p. 10; Sells[9], p. 17). For this reason, our model will use thirty years as the dividing point.

Of the several estimates available of the number of addicts in the U.S.A. (Ashley[8], pp. 41-43; Brecher[lO], p. 62; Ref. fll], p. 141, the figure of 4~,0~ heroin addicts will be employed in this paper (Hunt[l2], p. 16). It will be proportioned according to age; approx. 70%, or 420,000 addicts under thirty years of age, and 30%, or 180,000 addicts, thirty years or older (Ashley, p. 194; Brecher, p. 18). Below, the symbol Z’j will be used to denote total number of addicts age j, j = 1 meaning under 30 yr of age, and j = 2, 30 yr of age or over.

To include the subdivision of addicts according to age, the symbol Yij will be used to denote number of addicts that begin treatment i, i = 1,. . . ,P of age j.

From the diagram it should be clear that GRii is the proportion of addicts age j who begin treatment i, and who remain in it indefinitely if the treatment is continuous, or graduate if the treatment ends; and ABRij is the propo~ion of abstinence for the GRfjYii addicts.

The values of these pammeters are presented in Table 2. It is possible to construct models in which the number of addicts in each of the three subdivisions considered above is not a fixed proportion of the total number of addicts beginning a program, but rather a control variable that, within constraints, can be determined taking into consideration the costs per addict of the different stages and the contribution they make to the desired objective. It should also be observed that additional classification of the addicts could be made according to whether they desire a specific treatment, apply for it, and are accepted. For example, a therapeutic community may accept as few as 10% of the addicts who apply, while a prescription heroin program would accept all those who apply, or 100% (Marks, Table 1, p. 77; Table 2, p. 80). The two approaches just mentioned are not considered in the model below.

In the models in Section 4.2, the objective is to maximize a weighted sum of the reduction in damages of drug addiction brought about by the use of different instruments. As observed, it is not possible to specify clearly the purpose of the drug control program with this aggregation procedure. The objective to be maximized in the model below is the number of addicts who abstain from heroin consumption. The reason for the choice is

218 HJXTOR CORREA and JUDITH SANGL

that it is the most common criteria for “success” in aftercare phase in the new program at Lexington, or about 90% heroin control programs, and data tiith respect to it are graduation rate (using overall ratio of 40% inpatient and 60% the most frequently collected and reported. outpatient).

Table 2. Graduation (CR) and abstinance (ABR) rates

~cGlothiin ei al. 141, p. 45 reports that during the initial years of the Cal~o~ia program (1962-68) only 17% were defined as successes and discharged after 3 consecutive years on satis- factory outpatient status. In one program with 327 parolees (outpatients), 71% either absconded or were returned to prison within 6 months after admission (thus, 29% of outpatients graduated). Assuming that 100% of inpatients complete treatment and weighting these graduation rates with the in- and outpatient rates of 40% and 60% (McGlot~in et al.[4], p. 6 and AlO), the overall ~adua~on rate is approx. 60%. McGlothlin et al. on p. 47 state that abscondence in New York program was 34% of those on outpatient basis (66% graduation rate). Again, assuming that 100% of inpatients complete treatment, and weighting it with its SO/SO ratio of inpatients to outpatients, the overall graduation rate is about 83%. In California, abscondence is about 20% (80% graduation). Assuming that 100% of the inpatients complete treatment, and we~ht~g it with its 2.5175 ratio of inpatients to outpatient, the overall graduation rate is about 85%. A 6-yr follow up study of 344 parolees showed that 43% absconded at some time (57% graduation rate), assuming that 100% of inpatients completed treatment and weighting it with the average inpatient/outpatient ratio of 40/60, the overall graduation rate is 74%.

Treatments Graduation Abstinance

i j rate G&j rate ABR,

Detoxification < 30 Civil commitment < 30 ~erapeut~c community < 30 Outpatient abstinance =Z 30 Antagonist treatment < 30 Min. methadone

maintenance < 30 Max. methadone

maintenance < 30 Heroin maintenance < 30 Prescription heroin 2 30 Detoxification 2 30 Civil commitment z 30 Therapeutic community 2 30 Outpatient abstinence z 30 Antagonist treatment z 30 Min. methadone

mainten~ce 2 30 Max. methadone

maintenance z 30 Heroin maintenance 2 30 Prescription heroin 2 30

1 1 2 1 3 1 4 1 5 1

6 1

I 1 a 1 9 1 1 2 2 2 3 2 4 2 5 2

6 2

I 2 8 2 9 2

0.1’” 0.6”’

;;$? 0.9’“’ o.2’*’

o.4C9’ 0.3’i”

0.7”” 0.8”*’

0.75”” 1 .o<13) $iZ? l.oo’“’ 0 o(14)

0.2”’

;:;::: $$

0.3”’ ;:$

o.4’9’ 0.C’“’

0.75”” o.9”2’

0.80(“’ 1.003’ ;:;?I?

1 .ooo3j o.oo4’

Notes for Table 2 (1) Graduation rates range from about 6 to 19%.

DeLong[2], p. 182: In one program using rne~~one for detoxification, 74% of the patients dropped out or relapsed within 48 hr. In a program that did not use opiates for withdrawal, only 10% successfully completed withdrawal, and in the Haight-Ash- bury outpatient Clinic in San Francisco, which was using anal- gesics for withdrawal, 56% dropped out before they were clean and 38% left after one visit, so only 6% completed treatment.

Il. Wilson, R. Elms and C. Thomson[l3], Table 1, summarize detoxi~cation studies. In the Berle and Nyswander study, (n = 268), onlv 53 subjects completed treatments, or 19%. (2) Abstinence rates range-from about 9 to 24%.

DeLonu121. D. 182-183: The New York Narcotics Addiction _. -. . Control Commission estimates that only 24% of those who are detoxified without any other services remain drug-free for some time. In a detoxification program that does not use opiates for wi~drawal, almost all of the 10% who completed it soon relapsed. The H~ht-Ashbury Clinic reported that only 5.5% were clean for one month or more.

B. Wilson, R. Elms and C. Thomson[l3], Table 1: In the Berle and Nyswander study (n = 268) only 10 patients, or 3.7%, abstained for 6-25 months. In the Canada study (n = 157), 10 patients, or 7%, were found to be abstaining 6 months after treatment.

Brecher[lO], p. 89: Of 62 “old style” addicts who were detoxified after Nov. 1969 at the Haight-Album Clinic, 94.8% were using heroin again in 1971, of 5.2% abstaining. Of 115 “new style” addicts, 93.3% were back on heroin, or 6.7% abstaining. Brecher on p. 70 reports that of 100 patients who underwent more than 200 voluntary hospitalizations, only 11 hospitalizations were followed by apparent abstinence for one year or longer, or approx. 5.5%. (3) Graduation rates range widely, as indicated in Marksf3]; p. 77, Table 1, being anywhere from 9 to 100%. It is assumed that older addicts have a higher graduation rate.

DeLong[2], p. 184, indicates that between 1935 and 1964 there were 87,000 admissions to Lexington and Fort Worth, of which 63,600 were voluntary. Among the voluntary patients, 70% left against medical advice (30% remained). He also reports on p. 190 that about 15% are dropouts who are in the civil committment

(4) Abstinence rates range from 0 to 80% again, it is assumed that older addicts will have a slinky higher abstinence rate.

G. Hunt and M. Odoroff [i4], p. 41: Only 6.6% of 1912 Lexing- ton graduates remained abstinent throughout the follow-up periods of one to 4$ yr.

H. Duvall, B. Locke and L. Brill[lS], p. 185: Less than 3% (12 of 453 addicts) were abstinent at all three follow-ups of 6 months, 2 yr and 5 yr after release.

G. V~llant[l6], p. 729: 90 of 100 patients were addicted within two years, even though provided with aftercare. Of the 10 unaddicted, 2 were alcoholic, 3 had not used narcotics more than once a day, 1 used drugs intermittently, and 3 died within 4yr.

DeLong[2], p. 184: Studies of addicts released from such abstinence facilities found that up to 90% of those followed up relapsed into heroin use in a few years, so abstinence rate is about 10%. On p. 185, DeLong states that, of 1209 addicts on release status in the California Civil Committment Program, 0nIy 35% were in good standing after 1 yr, and after 3 yr, only 16%. On P. 187: The New York program directed by the Narcotics Add&ion Control Commission (NACC) found that of those who had completed the program (“a relatively small number”) 25% were currently abstaining as verified by a physical follow-up. DeLong, pp. 189-190: In the first 27 months of operation under the new Federal Act of 1966-69, a study of 1200 patients at Lexington showed that those in aftercare for 3 months or more were, on the average, drug-free 80% of the time, and that 60% of the patients do not become readdicted during their first year in aftercare.

McGlothlin et al.[4], p. 45: In a study of 327 parolees, 87% exhibited narcotics usage within 12 weeks, as indicated by daily urine tests.

R. Trissell[l?], pp. 2-3: Of 247 adolescents admitted to Riverside Hospital, only 8 (3%) has not returned to heroin after 3 yr, but all of the 8 claimed that they had never been addicted in the first place, but were only caught for possession. So abstinence rate would be 0. (5) Graduation rates range from about 4 to 20%. In this case, younger addicts are assumed to have a slightly higher graduation rate.

~cGlo~l~n er ~1.141, p. 40: One study found that 20% of admissions were retained 12 months, while of 2110 admissions funded by the Addiction Services Agency to the Phoenix House in New York City, only 79 (3.7%) completed the program.

BrecherllOl, P. 79: In a study of Libertv Park Village, 272 addicts were admitted. After 2yr, 67 were still in the program, 183 had dropped out and 22 had graduated, or 11% of those who had left (22/205). Brecher also reports, on p. SO, that in a study of Phoenix House in 1968 of 157 residents, 40 had remained after


2 yr, 100 had dropped out and 17 had graduated, or 14% of those who had left (17/117).

DeLong[Z], p. 195: One multimodal program that also has a therapeutic community component found that of 122 persons who entered, 40% left within 3 weeks, 50% within 7 weeks, and 85% within 1 yr (15% graduation rate after 1 yr). (6) Abstinence rates vary from 10 to 8% and the younger addicts are assumed to show a slightly higher rate.

DeLong[2], p. 195: Of 113 graduates of one pro~am~ 13 had relapsed with 5 yr (8% abstinence rate).

Brecher[lO], p. 79: Of the 22 graduates of Liberty Park Vil- lage, 4 were back on heroin or in jail, or 18% relapsed (82% abstinence). He also reports, on p. 80, that of 17 graduates of Phoenix House, 2 had returned to heroin use or 1.2% (88% abstinence). On p. 78: Declerich, the founder of Synanon, estimated that about 10% stay clean outside the community for as long as 2 yr. (7) Graduation rates range from 15 to 50% and older addicts are assumed to have slightly higher rates.

DeLong[Z], p. 197 cites Narcotic Treatment Agency (NTA) study by R. DuPont in which only 15% of patients who entered the abstinence program remained for 6 months. On pp. 197-198 he reports another study of NTA facilities by the District of Columbia ~pa~ment of Corrections which showed that with 165 addicts, 50% were failures after 6 months (25% had escaped). This is a 50% graduation rate.

0. L. McCabe, A. Kurland and D. Sullivan[lS], p. 219: With 371 patients, the five year rate for retention for 12 months or longer was 2%. (8) Abstinence rate of about 35% has been reported.

0. L. McCabe, A. Kurland and D. Sullivan[l8], p. 215: Of 60 patients who completed the program over a 5 yr period, 21 (35%) abstained totally. (9) Graduation rates range from 5 to 59%.

M. Fink[l9], p. 1005: Trials with antagonists have shown an overall retention rate of 40%.

C. Chambers and L. Brill[20], p. 265: Attrition rates have been reported as hi as 83% in the cyclazoeine programs (17% retention).

J. Chappel and .I. Jaffe[21], p. 516, Table 5: 16 patients (N = 33) remained for 14 months (49.5%).

A. Kurland, T. Hanlon and 0. L. McCabe[22], p. 668: Of 39 who were in program, 17 completed it, or 43.5%.

A. Kurland, J. Krantz, J. Henderson and F. Kerman[23], p. 134: Of 23 patients who were transferred into the program, 8 (34.7%) were returned for 6 months and 1 (4.7%) for 12 months. Of those who were direct admissions, 31 were retained for 6 months (59.6%) and 7 for 12 months (13.4%). (10) Abstinence rates range from 28 to 39%.

A. Kurland, T. Hanlon and 0. L. McCabe[22], p. 668: Of 39 patients, 11 (28.2%) totally abstained and another 6 (15.4%) had some drug use.

Chappel and Jaffef211, p. 513: At 14 months, 13 out of 33 patients were abstaining, or 39.4%. At 20 months, 11 were abstaining, or 33.3%. (11) Graduation rates vary from 48 to 88%. Al~ou~ the minimum methadone maintenance program may have a lower retention rate than the maximum program, it might be just as cost- effective (see DeLong[2], p. 217-219).

DeLong[2], p. 205-206: A total of 4376 patients had been admitted -to the Methadone Maintenance Treatment Program (MMTP) in New York since it began in 1964. Of this arouu, 3485 (80%) were still in treatment as of 31 Oct. 1970. Of;his‘group, 2424 were originally inducted on an inpatient basis and 74% of these remained. The others in the total group, 1952, had been admitted as outpatients and 88% were still enrolled as of 31 Oct. 1970. Three special SOO-patient cohort studies were done. These showed an attrition rate of 10% in the first year. On each of the cohorts, about 23% were discharged at 21 months; 34% in two cohorts discharged after 33 months; and 42% of one cohort discharged after 48 months, inclu~ng both vo~u~ta~ and in- voluntary discharge.

Bourne[24], p. 5, Table 1, lists retention rates for three New York City programs at Beth Israel, Bronx State and Health

Services Administration between 1964-73. The overall average for the years 1964-73 is 72% at Beth Israel, 74% at Bronx State and 74% at Health Services Administration, with retention rates for dierent years in all the programs ranging from 63 to 87%. Bourne on p. 5 reports that eleven programs in Maryland with a sampIe of 467 patients admitted showed that 48% were in treatment 12 months later (the range was from 21 to 78% among the clinics). (12) Abstinence rates range from 70 to 100%.

Gearingj2S], p. 189 reports that “none of the patients who have remained in the program have become readdicted to heroin”, so there is 100% abstinence.

Wiimarth and Goldstein[26], p. 11: 30% of patients still have positive urine tests for heroin after 6 months of treatment and that after a year it remains between 5 and 10%. (13) Authors estimate that 100% of addicts would remain in either heroin maintenance or prescription heroin programs, because they are getting the drug of their choice. (14) Abstinence rate is 0, since in both a heroin maintenance and prescription heroin programs, addicts are using heroin, albeit legally.

With the no~tion introduced above, the objective becomes that of maximi~ng

03) j i

The proportions of abstinence and graduation in (13) were given in Table 2. In the specification of the values in this table it was taken into consideration that older addicts tend to show higher retention rates in continuous programs and higher graduation rates in programs having an end (Williams et d/27], pp. 439, 441; Knowles et al.[2q, pp. 61-74).

It is clear that the objective function (13) takes into consideration only part of the benefits of a drug control pro~am. In the model being presented, the benefits of employment and reduction in crime will be considered in the constraints as follows:

(a) ERACij X (1 - G&j) Y,i, i.e. the employmenf of the addicts wh’o drop out of the program; plus

(b) ERGRjj x (1 - ABRij)GRiiYii, i.e. the employment of those who graduate or remain in the program but do not abstain: and

(c) ERA& X ABRu X GRijyij, i.e. the employment of those who graduate or continue abstaining where the equations given are the definitions for ERAC, ERGR and ERAB.

The total of the coefficients of the Xi above will be called the employment rate of p~ticipants of age j in program i. It will be denoted with ERii.

The total employment of addicts of age j is equal to

ERAD~ ( rj - x Yi j) + z ERij Xi (14) i

where ERAQ is the employment rate of addicts of age j without any treatment.

To take into consideration the benefits of increased employment in the model below, the specification of the Yii is subject to the control that the sum in (14) must be larger than or equal to the number of addicts PERjT,, where PERi is the employment policy parameter for addicts age j.

Several definitions are used in the empirical estimates available in the literature of employment of addicts. Here the definition stating that an addict is employed if he has

280 HECTOR COMA and JVDITH SANGL

a full or part-time paying job will be used. Many programs expand the definition to include other socially- acceptable but unpaid occupations such as student or housewife.

The estimates to be used below for the employment rates in eqn (14) are presented in Table 3. In order to determine the impact of changes in PER,, several values will be assigned to these parameters in the numerical examples in Section 4.2.

When the benefits of employment are considered a part of the total benefits of a program of drug addiction, as is done in the models in Section 4, the policy-makers cannot set any specific ~nimum for the number of addicts that should be employed. The treatments selected could increase, decrease or leave that number without modi~cation. With the approach to be used in Section 5.2, the number of addicts employed is a clearly specified policy alternative that the agency in charge of the program can determine as it sees fit.

It is not possible to specify which of these approaches is superior. However, both may be used simultaneously in a model if desired.

In the presentation below, the hypothesis that the application of instruments for reducing addiction to heroin tends to reduce the number of crimes committed by addicts will be accepted. However, it should be observed that this hypothesis is not generalfy accepted. The National commission on ~arihuana and Drug Abuse ([32], p. 1771 stated that it did not find ~‘sufficient responsible research to conclude that any of the various treatment modalities, regardless of type, actually reduce crime”. On the other hand, Bourne (1331, p. 8) states that experts maintain that data only “tends to dispute claims regarding the magnitude of crime reduction”, rather than consider that they refute the basic hypothesis of crime reduction with methadone treatments.

Table 3. Employment ratesi

Graduates Treatments i j ER~R~j

Ah trea~ents < 30 1 , . . . , 1 0.25”’ All treatments 2 30 1 , . . . , 2 0.30”’ Detoxification < 30 1 1 0.25’” Civil commitment < 30 2 1 Therapeutic community < 30 3 1 Outpatient abstinence < 30 4 1

!!I?$

Antagonist treatment < 30 5 1 o.30’*’ Min. methadone maint. < 30 6 1 Max. methadone maint. < 30 7 1 ;::;:::

Heroin maintenance < 30 8 1 0.55”’ Prescription heroin < 30 9 1 o.50’4’ Detoxification 2 30 1 2 o.30’2’ Civil commitment 2 30 2 2 0.3scL’ Therapeutic community 2 30 3 Ou~atient abstinence 2 30 An~~onist treatment t 30 5 2 0.35@’ Min. methadone maint. 2 30 6 2 O.dO’*’ Max. methadone maint. 2 30 7 2 0.45”’ Heroin maintenance 2 30 Prescription heroin 2 30

Abstainers E~3~~

0.25(” 0 30”’ 0’30’5) 0:40’6’ o.84’7’

;:$

o.68”“’ 0.73”0’ 0.0 0.0

;:$

0.8do’ O.dO@’ 0.68’9’ 0.73”@ 0.78”” 0.0 0.0

iIncludes full and part time employment,

Notes on Table 3 (1) Maddux and Bowden[29], pp. 104-105. Five studies are cited which indicate employment rates at time of admission. They are

Perkins and Block, 27% employed; Gearing, 29% employed; Lexington and Fort Worth Clinicai Research Centers, 1%2,40% reported legal employment six months prior to admission; Lex- ington Center admissions in 1965, 30% employed; and 1971 admission to Bexas County Drug Dependence Program, 38% full of part-time employed. Authors assume net re-employment rate among general addict population will be approximately the same as net of addicts at time of admission. (2) Authors estimate that the employment rate of those who graduate or continue but do not abstain varies slightly from the employment rate of those accepted and those in general addict population, since they are not abstaining. However, the ditf- erences are likely to be greater with a therapeutic community or a “continuous” program like methadone maintenance. (3) Authors estimate that employment rate with heroin maintenance is at least equal to, if not greater than, employment rate with prescription heroin. Marks[3], p. 81 indicates that maintenance as opposed to prescription is preferable in terms of social costs and benefits for the physical health of the addict and for better supportive service and contact. (4) Employment rates range from 38 to 68%.

McGlothhn[4], p. 37: One English study reported 40% of heroin-maintained patients working full time, and another, 9% part-time.

Ashley[l], p. 162: 42% of patients at St. Clements’ Clinic employed by end of first year.

Ashley@], p. L62: Surveys by Addiction Research Unit at Maudsley Hospital report that 40% of all registered addicts are employed after one year.

Ashley@], p. 163: A study of 50 Canadian addicts who emmigrated to E&and showed that 19 were holding regular jobs, or 3%.

May[3OJ, p. 391: A 1970 study of Canadian addicts who emmigrated to England showed that 13 worked full time (53%) and 4 worked part-time (16%). Total employment 17125 or 68%. (5) Authors estimate that a detoxification program has little impact on employment rates because of its short duration. (6) Employment rates range from 14 to 65%.

McGlothlin[4], p. 49: California program shows 44% of outpatients employed full time and 13% employed part-time. New York outpatients showed 40% employed full time, 7% part-time and another 17% housewives or students. Federal program reports indicate that 65% of outpatients were employed at some time, although it was really the equivalent of 35 full time em- ployments (calculated as days worked/available working days for total sample). If 50% is assumed to be average on outpatient basis, then there is about 37.5% employment when weighted by inpatient and ou~a~ent ratio.

Maddux and Bowden[29], p. 104: Table 2 presents data from several studies. Baganz and Maddux report 63 of 100 men employed after one year follow-up from Fort Worth. Langenauer and Bowden report 41 of 97 patients employed (42%) with six month follow-up from Lexington. Maddux and Associates report 35 of 248 men emnloved (14%) at one-year follow-up from Fort Worth. These figures count students as employed, so estimates of “employment” are somewhat lower. (7) DeLong[2], p. 195: Out of 113 graduates over a 4-yr period of one therapeutic community, 95 were employed, or 84%. (8) DeLong[2], p, 197 states that many observers believe that the impact of outpatient abstinence programs on employment is minimal, so it may be slightly higher than the employment rate of the genera1 addict pop~ation. (9) Chapped and Jaffe[21], p. 519: At 2~month follow-up, 13 of 19 patients (68%) had jobs and another 3 (16%) were in socially acceptable activities. (10) Employment rates range from 41 to 90%.

Gearingl’LS], p. 180, Fig. 12: For 990 men, 29% were employed at time of a&&ion, 46% employed 6 months after, 61% 12 months after. 66% 18 months after, 70% 24 months after, 77% 30 months after, and 88% 36 months after.

Maddux and Bowden[29], p. 104, Table 2, shows data from several methadone maintenance programs. Wieland and Cham- bers report 59% employed with a mean follow-up period of I.5 months; Dole and Nyswander report 62% employed at follow-up


of 3 to 26 months. Bloom and Sudderth report 53% with a 1 to 11 month follow-up. Dole and Associates report 50% with a 7 to 10 month follow-up. Jaffe reports 43% employed with a 14 week follow-up. Perkins and Block report 41% employed with a mean follow-up of one year.

Bourne[24], p. 9 cites a study by Lofchie and Muskelly which reported that 90% of 188 patients were employed at the end of 1971, after varying periods of up to two years on methadone maintenance.

Wieland and Chambers[31], p. 652: In a study of 32 addicts, 34.4% were employed before admission to methadone maintenance, while 78.1% were employed after admission.

The benefits of the reduction of crime will be treated in the same way as those of the increase in employment. The total crime rate (CRii) of addicts age j in treatment i is defined as follows:

CRACri X (1 - GR,) Yij + CRGRij X (1 - ABR,)GRijYii

CRABij X ABRij X GRtj X Yij = CRij X Yij (15)

where CRACij, crime rate of addicts age j who are accepted in treatment i but do not graduate or continue; CRGRij, crime rate of addicts age j who are accepted in treatment i and who graduate or continue but do not abstain; and CRABi], crime rate of addicts age j who are accepted in treatment i and graduate or continue the treatment and abstain.

Adding to the number of crimes committed by addicts in the different treatments that of the addicts not treated, the following crime constraint is obtained:

CRADjc Tj - C Yij) + C CRij Yij % PCRjq (16)

where CRAQ, is the crime rate of untreated addicts; and PRCj, is the policy parameter of the acceptable proportion of crimes.

Arrest rates will be the proxy for addict criminality, even though it has been estimated that addicts are arrested for fewer than one percent of the crimes they commit (Hannan, @ 89). While it is an underestimation of real addict criminality, other measures, such as conviction rates and time spent in jail, would underestimate even more.

A second problem with arrest data given for a before and after treatment comparison is that, frequently, unequal time periods are compared. The before treatment arrest data may be in terms of the addict’s prior life history of arrests, while the after treatment arrest data may report arrests during a period of only 18 or 24 months. However, the model does incorporate data based on unequal time comparisons because of the scar- city of data with equal time periods and, as such, there is a bias toward an overestimation in the reduction in crime.

A third problem that must be taken into consideration is that the link between addiction and criminality is attributable in part to the fact that heroin possession and use is considered to be a crime. Data distinguishing between drug-related and non-drug-related crimes are extremely rare. As a consequence, crime reduction from the treatment modality might in some instances reflect only a reduction in arrests caused by drug-related offenses.

The data used to estimate the parameters of the crime constraint are presented in Table 4.

The crime policy parameters PCRj will receive the same treatment as the employment policy parameters PERi.

In addition to unemployment and crime, the addiction to one drug frequently brings as a consequence a higher than average propensity to use other drugs, particularly while abstaining from the first drug. The problem of substitution has caused the comment with respect to heroin addiction that the “cure may be worse than the disease”, since heroin has been ranked lower than barbituates, amphetamines and alcohol, both in terms of intrinsic hazard potential to the individual, and to the society (Irwin[34], Tables 1, 2, p. 9).

The monetary cost of the damage caused by the transfer of addiction of one drug to another could easily be introduced into the objective function of the models in Section 4. In this Section the use of other drugs will be treated in the same way as unemployment and crime; i.e. setting as a policy decision the maximum values it can reach. The following notation will be used for this purpose:

DRAD, the rate of drug h abuse by addicts age j not in treatment

DRACijh the rate of drug h abuse by addicts age j who are accepted in treatment i but do not graduate or continue

DRGR;jh the rate of drug h abuse by addicts age j who are accepted in treatment i and who graduate or continue but do not abstain

DRABijh the rate of drug h abuse by addicts age j who are accepted in treatment i, who graduate or continue and also abstain

DRi+ rate of drug h abuse by addicts age j in any stage of treatment i.

Table 4. Crime rates?

Treatments Addicts Dropouts

i j CRADi CRAC,

All treatment < 30 All treatment 2 30 Detoxification < 30 Civil commitment < 30 Theraaeutic communitv < 30 3 1 o.3o’5’ 0.10”’ Outpatient abstinence 2 30 4 1 O.65’5’ o.40”“’ Antagonist treatment < 30 5 1 0.60”’ 0.35”” Min. methadone maint. < 30 6 1 0.40@’ o.15”z’ Max. methadone maim. < 30 7 1 o.35’5’ 0.12”r’ Heroin maintenance < 30 8 1 0.15” 0.0 Prescription heroin < 30 9 1 0.20@ 0.0 Detoxification % 30 1 2 0.60”’ 0.60”’ Civil commitment 5 30 2 2 0.55’” 0.30@’ Therapeutic community 5 30 3 2 o.3o’5’ o.10’9’ Outpatient abstinence 5 30 4 2 o.55’5’ o.400°’ Antagonist treatment 5 30 5 2 o.50’5’ 0.30”” Min. methadone maint. 5 30 6 2 o.35C5’ o.12”2’ Max. methadone maint. 5 30 7 2 o.30’5’ o.10”2’ Heroin Maintenance 5 30 8 2 0.15” 0.0 Prescription heroin 5 30 9 2 o.20’8’ 0.0

tMeasured by arrest rates.

Notes on Table 4 (1) Brecher[lO], p. 133 cites study by O’Donnell showing that 59 of 82 male addicts who secured their heroin illegally committed crimes (72%). It is not clear whether these data were collected by reports of the addicts themselves or by some other evidence, such as arrest rates.

282 HECTOR CORREA and JUDITH SANGL

(2) Authors estimate that the crime rate would remain approximately the same as it was before acceptance for treatment, since, due to the fact that they are not abstaining, it is likely they would revert to former behavior patterns. (3) Hannan[S], p. 86, Tables 6-11. For patients in detoxification units, there were 131 arrests per 100 person years before admission, and 135 arrests per 100 person years after admission. Thus there is very little change in crime rates before and after admission. The crime rates used here are estimated on the basis of those mentioned in note (1) above. (4) DeLong[2], p. 188: In the New York NACC program, 50% were in jail as a result of new drug-related offenses or had returned to drugs. DeLong, on p. 186: 1209 outpatients took part in the California program at some time during the years 1962 through 1964. At the end of the period, only 35% were still in good standing. (5) Authors estimate that for most “cure” programs, crime rates would remain the same as those for general addict population, except for the therapeutic community, where it would be less tolerated and the person would likely to be expelled for com- mitting crimes. An addict in civil commitment would simply be returned to the inpatient phase for additional incarceration. For most of the continuous programs, one could except some decrease in the crime rate, but not as great as if they were abstaining. Rates for heroid maintenance and prescription heroin would definitely be lower, since heroin is obtained legally. (6) McGlothlin[4], p. 49: In the Federal civil commitment program, about 29% are arrested during an approximate one-year outpatient period. (7) Crime rates range from 9 to 36%. Brecher[lO], p. 133 cites a study by O’Donnell which reports that only 4 of 45 addicts on legal heroin (or 9%) committed crimes.

Ashley[l], p. 164 cites a study of 50 Canadian addicts who emmigrated to England. In England, with legalized heroin, 7 had criminal records (14%). In Canada, 9 had criminal records.

May[30], p. 381: In a 1970 study of 111 heroin addicts attend- ing London clinics, 51% had reported a conviction for a non-drug offense before they began using heroin. After using heroin, 36% reported convictions for a non-drug related offense. (8) Authors estimate that the crime rate with prescription heroin is at least equal to, if not slightly higher than, that with heroin maintenance. Marks [3], p. 81 indicates that maintenance,

as opposed to prescription, heroin is preferable in terms of social costs and benefits for the physical health of the addict, and for better supportive service and contact. See footnote 7. (9) McGlothlin[4], p. 40: A study of 104 continuing members with an average stay of 22 months found an arrest rate of 5% of the pre-admission level. (10) DeLong[2], p. 197 indicates that the impact of outpatient abstinence on criminal activity is minimal.

McGlothlin[Q], p. 24 cites a report by DuPont of patients in a Washington, D.C. outpatient abstinence program which found that 39% were arrested on a new charge during their first year of treatment. (11) Authors estimate that the crime rate would be similar to that of outpatient abstinence programs, and probably lower. (12) Arrest rates range from 2 to 19%.

McGlothlin[4], p. 24 cites a study by Gearing which compared arrest rates before and after methadone treatment for the first New York City admissions. The number of recorded arrests during the four years prior to admission to treatment was 0.6 per person per year. The reduction in arrests for patients in the first, second, third and fourth year of treatment was 81%, 92%, 96% and 98%, respectively. McGlothlin, on p. 24, also cites a study by Cushman (n = 248) where the pre-methadone arrest rate was 0.42 per yr in comparison with 0.10 and 0.05 during the first and second years of treatment, respectively.

McGlothlin[4], p. 24 cites a study by DuPont of methadone maintenance patients in Washington, D.C. which reported that 17% were arrested on a new charge during their first year of treatment.

Gearing[ZS], p. 184: A study of 1530 men in methadone maintenance showed a 6% arrest rate in the first year, 3% in the second year and 2% in the third year.

Wieland and Chambers[31], p. 655: A study of 32 patients showed that before methadone maintenance, 90.6% had been arrested, and after being in methadone maintenance a minimum of 20 months, the arrest rate was 18.8%.

The number of users of other drugs h by those not accepted in any treatment modality is

Table 5. Other drug abuse rates

Treatments Addicts Dropouts Addicts

i i DRADjl DRACijI DRADj,

All treatments < 30 All treatments 2 30

Detoxification < 30 1 1 Civil commitment < 30 2 1 Therap. comm. < 30 3 1 Outpatient abs. < 30 4 1 Antag. treat. < 30 5 1 Min. meth. treat. < 30 6 1 Max. meth. treat. < 30 7 1 Heroin maint. < 30 8 1 Prescrip. heroin < 30 9 1 Detoxification 2 30 1 2 Civil commitment 2 30 2 2 Therap. comm. 2 30 3 2 Outpatient abst. 2 30 4 2 Antag. treat. 2 30 5 2 Min. meth. treat. z 30 6 2 Max. meth. treat. z 30 7 2 Heroin maint. 2 30 8 2 Prescrip. heroin 2 30 9 2

1,...,9 1 1,...,9 2

Alcohol Amphet. & Barbit.

Dropouts DRACijz

-

0.2”’ 0.2Q’

Abstain DRABjjl

0.25@ 0.30'5' 0. lo@’ 0.25'"

g;:::

0.25"' 0.0 0.0 0.30'4

;:;::::

;$Z

f$!:

0.0 0.0

0.14@ 0. 14C8’

0.14’” 0.14@’

Grads. Abstain DRGR,-2 o.14’r0]

DRAB;.I 0.25'"j

0.14”0’ 0.25"" o.14”0’ o.07'13' 0.14”0’ 0.20"" 0.14”O’ 0.20"" 0.14”a’ 0.10”” o.14’r”’ 0. 1o04) 0.14”0’ 0.0 0. 14’r”’ 0.0 o.14”0’ 0.25"" 0.14”” 0.25"" 0.14”O’ 0.07"3' o.14’r”’ 0.20"" o.14”“’ 0.20"" 0.14”0’ 0. lo’r4’ 0.14”0’ 0.10”” o.14”“’ 0.0 o.14’r”’ 0.0


For those accepted in treatment i, the number of users that about 20% overused alcohol, both before and after ad-

of other drug h is mission into the methadone program.

DRAC,h( 1 - GRij) Yii t DRGRijh( 1 - ABR,)GRijxj

Bourne[24], p. 11 states that the incidence of heavy alcohol consumption may increase when the patients are on methadone maintenance, particularly among older addicts.

DRABtjh X ABRij X GRij X Yij = DRijh X Yij (17)

In the model, the constraint specifies that a certain number of users of drug h should be less than or equal to PDRjhYjh where PDR, is the policy parameter for the proportion of acceptable drug abuse in the addict population age j. This is stated in the following manner:

DRADjh( Tj - ,Yij) + iDRijhYij - PDRjhl’j. (18)

The “other” drugs to be considered in this model are

N. Scott, W. Winslow and D. Gorman[35], pp. 285-287: A point prevalence study with a 20% random sample (N = 120) showed 14% had a confirmed alcoholic diagnosis, 9% had suspected alcoholism, 2% had alcoholism symptomatic of psychosis, 3% had aicoholism in remission, and alcoholism was not present in 72% (about a 25% alcoholism rate). The same group had only 5% alcoholism upon admission. Another group (the original program group with 61 patients) had zero percent alcoholism upon admission, since it was a contraindication for acceptance. At a four-year follow-up of program survivors (N = 32), the alcoholism rate was 34% (11 cases).

alcohol (h = 1) and, jointly, amphetamines and barbituates (h = 2). The data to estimate the rates referring to other drug abuse appear in Table 5. The policy parameters PDRjh will be treated similarly to the employment and crime policy parameters.

Gearing[25], pp. 190-191 reports that S-10% demonstrate problems of chronic alcohol abuse. In Fig. 16, the diagram indicates that the percentage of alcohol abuse tends to increase with time from 5% at six months to 11% at 30 months. (8) Rates range from 5 to 33% for amphetamines, and from 13 to 39% for barbiturates.

So far, attention has been given only to the way to deal with the damages of drug abuse included in the model. The method described above is an alternative to the one used in the model in Section 4, including the estimated monetary values of the damages in the objective function. As already observed, the approach in Section 4 and that used in this Section have advantages and disad- vantages, and also can be used simultaneously.

DeLong[2], p. 226: A Santa Clara County Program found that S-10% used amphetamines, and 20% abused barbiturates before admission into the methadone program.

Sells[9], Vol. 2, p. 21 shows that of a total of 18,366 patients, 13% used barbiturates daily or weekly (another 8.8% used them less than weekly) 2 months prior to admission, 8% used amphetamines daily or weekly (another 5.8% used them less than weekly) 2 months prior to admission.

Se&b], Vol. 2, p. 179, Table 7-1, shows approximately the same percentage of use for two sub-samples. In Year 1 (N = 3, 131) 13.1% used barbiturates daily or weekly (another 8.1% less than weekly) 2 months prior to admission, and in Year 2 (m = 8, 249) 13.6% used barbiturates daily or weekly (another 8.7% less than weekly) 2 months prior to admission. For amphetamines, in Year 1, 8.4% used them daily or weekly (another 5.6% less than weekly), and in Year 2, 8.2% used them daily or weekly (another 5.8% less than weekly) for 2 months prior to admission.

Notes to Table 5 (1) Alcohol abuse rates range from 5 to 20%.

DeLong121, P. 226: A studv of the Santa Clara Countv Program &wed that about 20% overused alcohol before being admitted to the methadone program.

Bourne[24], p. 11 cites a report by Wilmontb and Goldstein stateing that the incidence of alcoholism is no higher than in the general population.

National Commision on Marihuana and Drug Abuse[32], p, 142 estimates that about 5% of the total population and about 15% of the middle-aged population are chronic alcohol users. In the same report, on p. 190, a national survey by Cahalan and As- sociates indicated that 31% of the sample had some problem with drinking within the last three years (psychological dependence, health problems, etc.). The same book, on p. 196 also cites a 1968 report from the office of the Chief Medical Examiner in New York City which indicates that at least 20% of the heroin de- pendent population demonstrate heavy drinking problems. (2) Authors estimate that the rate among those accepted is equivalent to the rate in the general addict population not in treatment programs. (3) Authors estimate that the rate should be approximately the same as that of the general addict population, since they are not abstaining from heroin. (4) Authors estimate that there may be a slight rise in alcoholism, particularly for the older addicts, since there seems to be a general trend to substitute alcohol for heroin, except in a therapeutic community (see footnote 7). (5) Brecher[lO], p. 87 cites study by O’Donnell, “Narcotic Ad- dicts in Kentucky”, which shows that of the years spent out of institutions and free of narcotics, more than half the time was spent either in alcohol or barbituates. Additionally, O’Donnell’s study showed that out of a sample of 47 male addicts, 16 (34%) were non-alcoholics. (6) Authors estimate that the alcoholism rate would probably decrease, since most therapeutic communities do not tolerate much drug use, although the older addicts may show a smaller decrease in alcoholism. (7) Alcohol abuse rates range from 5 to 34%.

DeLong[Z], p. 226: The Santa Clara County Program showed

J. Langrad[36] reports that 33% of 422 heroin addicts used amphetamines and 34% used barbiturates more than six times. This classification of use more or less than six times was based on a study concluding that the majority of persons who had tried heroin six or more times became addicted to it. Since this does not seem valid for amphetamines or barbiturates, actual abuse of these substances should be lower than these usage rates indicate.

Carl Chambers[37], p. 47-48 reports that in a sample of 100 narcotics abusers, 54% also used barbiturate sedatives and 35% were addicted to them in 1966. In 1957, only 39% were abusing them and 18% were addicted. Rate use in model for both is 14%. (9) Authors estimate that abuse rate should be similar to that of general addict population before being accepted for treatment, since it is likely that they will revert to former behavior pattern. (10) Authors estimate that the rate will be similar to that of rate in general addict population, since they are not abstaining from heroin. (11) Authors estimate that rate might increase slightly, since these are “cure” programs (see footnote 12). (12) Brecher[lO], p. 87 cites study by O’Donnell, “Narcotic Addicts in Kentucky” which shows that of the years spent out of institutions and free of narcotics, more than half the time was spent either in alcohol or barbiturates. (13) Authors estimate that use of amphetamines and barbiturates would likely decrease, since most therapeutic communities do not tolerate drug use. (14) Abuse rates range from 4 to 16% for barbiturates and 4 to 15% for amphetamines.

Gearing[2S], pp. 180-191 reports that about 4-10% continue to abuse amphetamines or barbiturates. Figure 16 also seems to indicate that drug abuse increases with time up to two years, then decreases.

Bourne[24], p. 11 states that with non-opiate drugs, primarily amphetamines and barbiturates, incidence of positive urines runs about S-10% and changes very little over time.

284 HECTOR C~RREA and JUDITH SANGL

C. Chambers and W. Taylor[38], p. 123, Table I: Of 119 patients. 11.5% were detected abusing barbiturates, and 14.4%. amphetamines.

I’. Wcppner, R. Slephcns asd H. Conrad(391. p. 14. Table 3: With 11 I patients in methadone programs, 4 were addicted to sedatives, 3 abused fairly often, 4 abused one or a few times, 7 abused but unable to remember how often. with a total of 18 (16.2%). With stimulants, none were addicted. 3 abused fairly often. 2 abused once or a few times and 4 others abused but could not remember how often, with a total of 9 (8%).

DeLong[2], p. 226 cites a study of the Santa Clara County Program showing that 5 and 10% used amphetamines hefore and after admission, and that while 20% abused barbiturates before admission, only 6% abused them after admission.

Next, the constraints imposed by the monetary resources available to the program will be considered.

The total cost in program i for addicts age j is equal to

Cl ii ( 1 - GR:i) I’, t CziiGR;i Yai (19)

where Chii is cost h = I by dropout and h = 2 by graduate age j in program i.

It is assumed below that

for all i and j.

c,ij = OSCZij (20)

In addition, for civil commitment (i = 2) and maximum methadone maintenance (i = 7), costs arc weighted for the proportion of in-patient and out-patient expenses and first-year induction costs and later year costs.

With the assumption in (20), formula (19) reduces to

O.S( 1 + GRi,)Cl, Y,i = C&Y, (21)

where CSiJ cost per addict age j beginning treatment i. The cost constraint takes the form

where R are the total resources available. The cost information used for the model has the

following limitations, due to lack of data: (a) only operating expenses are considered. leaving out capital expenses, and (b) no attempt was made to consider the discounted values of future costs. This means that the costs of programs that take a longer time with ex- penditures distributed over the entire period are un- derestimated.

Since most reports give ranges of values for the costs, the minimum and maximum estimates presented in the literature appear in Table 6.

Final obvious constraints are that the number of addicts age j accepted in all programs-assuming that they cannot enroll in more than one program-is less than or equal to the total number in the population, i.e.

c Y,, - T, j = I, 2 (23)

and that the values that the Y,, can take are non-negative.

5.3 Numerical example As observed before, the object of a model like the one

described in 5.1 is not to replace the directors of agencies

for drug control, nor is it to provide them with the solutions for the problems they face. Models for decision-making simply constitute a framework for a systematic analysis of the consequences of alternative policies. Using the models, the decisionmakers can simulate the consequences of their policies, and in this way they are better able to select the policy having the simulated consequences that they prefer.

With a model like the one described in Section 5.1, the consequences of an infinite number of policies could be simulated. Below, due to limitations of space. only a few examples will be presented.

Of the two sets of costs presented in Table 6. only the minimum one will be used. In more realistic studies, analyses using both should be made.

In Table 7, the effects of changes in the financial resources available for the control of drug abuse are presented under conditions in which none of the other constraints is binding. According to these results, the most cost-effective treatment is minimum methadone maintenance for addicts 30yr of age or older. With increasing resources, the use of minimum methadone for addicts younger than 30 yr should tend to increase. Once all these addicts receive treatment, it is useful to transfer addicts 30yr of age or older to maximum methadone maintenance.

Table 6. Cost of treatment per addict accepted (dollars per patient per year*)

Treatments i j Minimum Maximum

Detoxification < 30”’ I 1 220 660 Civil commitment < 30”’ 2 I 1310 6144 Therapeutic community < 30”) 3 1 1725 5750

Outpatient abstinence < 30(4’ 4 1 900 1500 Antagonist treatment < 30”’ 5 I 2100 3500 Min. methadone maint. < 30c6’ 6 I 425 510 Max. methadone maint. < 30”’ 7 I 1750 Heroin maintenance < 30”’ 8 I 500 1000 Prescription heroin < 3019’ 9 I 500 600

Detoxification 2 30”’ 1 2 240 720 Civil commitment 2 30’*’ 2 2 1392 6528 Therapeutic community 2 30”’ 3 2 1650 5500 Outpatient abstinence 2 30”’ 4 2 915 162.5 Antagonist treatment 2 30’“’ 5 2 2100 3500 Min. methadone maint. 2 30’h’ 6 2 460 525 IMax. methadone maint. > 30”’ 7 2 1800 Heroin maintenance 2 30”’ 8 2 500 1000 Prescription heroin 2 30@’ 92 500 600

(I) Costs range from $400 10 $1200 for inpatient hospital detoxification during 4 to I2 days at $100 per day.

Del.ong(21, p. 182: 5-10 day period necessary for detoxification.

McGlothlin[4], p. 43: average stay of I2 days. (2) Costs range from $4000 (inpatient) and S85O (outpatient) to 512,000 (inpatient) and 54800 (outpatient).

DcLong[21, p. 190 states that cost estimates for all three of the large programs are $10.000-12,000 per yr per addict for inpatient costs.

McGlothlin[4], p. 112: Costs in New York program for inpatients are $9250 and for outpatients $1750; in California program costs for inpatients arc 134008 and for outpatients $850; in Federal program. inpatient costs are 512,000 and outpatient costs are $4800. The low cost estimate will use California program figures. weighting them with a 2.5% inpatient and a 75% outpatient ratio. yielding $1638. The high cost estimate will use the Federal program figures. weighting them with a 40% inpatient and 60% outpatient ratio, yielding $7680.


(3) Costs range from $3000 to $10,000 per yr per addict. DeLong[Z], p. 629, Table 5: For short-term therapeutic com-

munities, costs are $20.38 per day or $7439 per year. For the family (longterm) program, costs are $16.86 per day or $6154 per yr per patient. (4) Authors estimate range from S1500-2500 per yr per addict, if addict goes for services 3 days a week approx. 50 weeks a year at about $10 per day, or $1500 per yr. If visits are 5 times a week, then it is $2500 per yr. (5) Costs range from $3000 to $5000.

DeLong[Z], p. 235 estimates that costs are $300&5000 per addict per yr. (6) Costs range from $500 to $600 for both first year and other year expenses.

DeLong[2], p. 203: $500 per addict per yr is the minimum for a methadone program with standardized doses, outpatient induction, inexpensive urinalyses and no ancillary services.

Brecher[lOl, p. 165: The programs that lack broad ancillary services are cheaper than the comprehensive methadone programs, operating on about $500-600 per patient per yr, including first-year patients. (7) Costs range from $2000 to $2500 first year induction and 61OW-1500 after first year.

DeLong[2], p. 203: Approx. 62OOO per patient year, with individual doses, inpatient induction, frequent urinalyses and ancillary services. and $1000 per patient per year once stabilized.

The estimates used in the model are $2500 for the first year and $1500 per yr thereafter with a weight of 50% each, which probably overestimates cost. (8) Ref. 1. pp. 104-106: Heroin dispensaries would cost about $500 to $1000 per addict per yr. (9) Costs range from SSOU to $600 per addict per yr.

May(301, p. 358 lists staffing details for Lambeth Hospital Clinic, implying a cost of $500 per addict per yr.

Next, a study will be made of the changes that will occur in the solutions obtained with 100, 250 and 300 M dollars of resources when the constraints for employment, crime and alcohol abuse for addicts under 30 yr of age are made increasingly more restrictive. These constraints are modified by changes in the policy parameters PER, for employment, PCR, for crime and PDRll for alcohol abuse.

The results obtained with the modification of the employment constraint are presented in Table 8. The results in this table differ from those in Table 7 in two respects. First, a shift can be observed from the treatment of addicts 30 yr of age and older to that of less than 30 yr of age. Second, it appears that heroin maintenance for

Table 7. Effect of changes in resources available on treatments selected, number of addicts treated and number abstaining

Resources Min. meth. Min. meth. Max. meth. Total Number IO6 dollars Run < 30 (6) t 30 (15) 2 30 (16) treated abstaining

100 001 40,471 180,WO 0 220,471 144.200 I50 002 158,120 180,000 0 338.120 210,000 200 003 275,600 180,000 0 455,600 275,900 250 004 393,410 180,000 0 573.410 341,800 300 00s 420,ooO 151,119 28,881 600,ooO 359,200 350 oaf, 420,000 113,806 66,194 600,000 362,300 400 007 420,ooO 76,493 103,507 600,000 365,500 450 008 420,000 39,179 140,821 6CQ,OCG 368,700 500 009 420,000 1866 178,134 600,000 371,800

Source: Explained in text.

Table 8. Effects of changes in employment policy parameters for addicts < 30 on treatments selected, number of addicts treated and number abstaining

Treatments Policy

parameters No. Min. meth. Wax. meth. Her. main. Min. meth. Max. meth. No. No. PER, employed < 30 (6.1) < 30 (7.1) < 30 (8.1) 2 30 (6,2) z 30 (7.2) treated abstaining

0.25 150,000~ 0.30 171,000 0.35 192,000 0.40 213,OOOi

0.45 234,000+ 0.50 255,000 0.55 276,000 0.60 297,000$

0.50 255,oOOt 420,000 0 0.55 276,000 3037 5942 0.60 297,000$ - -

40,471 82,353

164,710 -

393,410 411,760

0 -

0 0 0 -

0 0 0 -

Resources $100 x lo6 0 18,000 0 141,300 0 65.217 - -

Resources $250 x 10” 0 180,ooo 0 163,040

420,000 86,957 - -

Resources $300 X lo6 0 151,119

411,021 180,000 - -

0 220,471 0 223,653 0 229,927 - -

0 0 0 -

28,881 600,000 0 ~,~

- -

573,410 341,800 574,800 340,600 506,957 58,700

- -

144,200 141,500 136,300

-

359,200 127,OW

-

tconstraint not binding: same values as in Table 7. SNo feasible solution.


addicts less than 30yr of age is more employment- reducing the shift of addicts from heroin abuse to alcohol effective than minimum methadone maintenance. abuse. It also appears that treatment in therapeutic

The results in Table 9 dealing with the effects of more communities is the most effective method when the restrictive policies with respect to crime are very similar target is to reduce alcohol abuse. to those in Table 8 referring to employment policies. In the case of Table 9, a shift is observed toward treatment of addicts less than 30yr old, first with minimum 6. MULTIPERIOD AND DYNAMIC MODELS

methadone maintenance, and, at the lower levels of 6.1 Introduction accepted crime, with heroin maintenance. Due to limitations of space, no attempt will be made

Table 10 indicates that there is little possibility of here to extend the models presented in Sections 3 and 4

Table 9. Effects on changes in crime policy parameter for addicts < 30 on treatments selected, number of addicts treated and number abstaining

Treatments Policy No. crimes

parameter committed Min. meth. Max. meth. Her. main. Min. meth. Max. meth. No. No. PC& <30 < 30 (6,l) < 30 (7,l) < 30 (8,l) 2 30 (6,2) 2 30 (7,2) treated abstaining

0.75 315,0001 40,471 0.70 294,000 54,545 0.65 273,000 109,090 0.60 252,000 163,640 0.55 23 1,000 218,180 0.50 210,000 120,000 0.45 189,0006 -

0.40 168,000? 0.35 147,000 0.30 126,000 0.25 105,000 0.20 84,000 0.15 63,000 0.10 42,000$

0.40 168,ooot 0.35 147,000 0.30 126,000 0.25 105,000 0.20 84,000 0.15 63,000 0.10 42,0001:

393,410 390,700 293,020 195,350 97,670

0 -

420,000 370,630 276,990 183,340 89,694

0 -

Resources $100 x lo6 0 0 180,000 0 0 167,000 0 0 116,600 0 0 66,206 0 0 15,810 0 98,000 0

- - -

Resources $250 x lo6 0 0 180,000 0 29,302 150,660 0 126,980 134,730 0 224,650 118,810 0 322,330 102,880 0 420,000 86,957

- - -

Resources $300 x lo6 0 0 151,119

28,000 21,360 180,000 22,380 120,630 180,000 16,760 219,900 180,000 11,146 319,160 180,000

0 420,000 174,630 - - -

-

28,881 0 0 0 0

5370 -

220,47 1 144,200 221,545 143,300 225,690 139,800 229,846 136,300 233,990 132,900 218,000 67,200

- -

573,410 341,800 570,662 320,000 554,730 255,000 538,810 189,600 522,880 124,100 506,957 58,700

- -

600,000 359,200 600,000 346,900 600,000 290,900 600,000 234,900 600,000 178,800 600,000 122,000

- -

Konstraints not binding: same values as in Table 7. SNo feasible solution.

Table 10. Effect of changes in alcohol abuse policy parameters for addicts < 30 on treatments selected, number of addicts treated and number abstaining

Treatments Policy

parameters No. abuse Ther. comm Min. meth. Min. meth. Max. meth. No. No. PDR~I alcohol < 30 (3,l) < 30 (6,l) z 30 (6,2) 2 30 (7,2) treated abstaining

0.25 150,000t 0 0.20 120,000 8947 0.15 90,000$ -

0.25 150,0001 0 0.20 120,000 86,978 0.15 90.000$ -

0.25 0.20 0.15

150,000t 120,000 90,ooot

0 112,990

-

Resources $100 x lo6 40,471 180,000

4154 180,000 0 220,471 144,200 0 193,101 125,000

- - -

Resources $250 x IO6 393,410 180,000 40,383 180,000

- -

Resources $300 x lo6 420,000 151,119

52,459 180,000 - -

0 573,410 341,800 0 307,361 155,900

- - -

28,881 600,000 359,200 0 359,449 166,100

- - -

Wonstraint not binding: same values as in Table 7. *No feasible solution.


to consider several periods. A brief analysis shows that no conceptual or mathematical problems exist.

The notation would become more cumbersome, because time subscripts would have to be added. Also, explicit attention would have to be paid to the time interdependence of the variables. The first two of these interdependences that come to mind are implicit in the models already presented: the fact that the population of one age group in one period determines the older population in future periods, and the influence of the time that treatments take. In addition, two elements not considered in the previous models could be included. The first is the effect of renewal of the population of addicts. However, here an important limitation would be the lack of a theory explaining the attitudes of the population towards drug addiction, and the reasons why persons become addicted, including the process of contagion from addicts to non-addicts. The second important element is the non-existence of an upper limit for the potential supply of drugs, and the influence of price on their actual supply.

In the present section a simple model that adapts the standard economic theory of demand, supply and price will be used to study the policies that drug control agencies could follow.

6.2 The simplest dynamic model In this model, as in the simplest economic model of

price determination, it will be assumed that

and

D,=-mdpt+b, (23)

S,+, = mspt + b, - rL, (24)

where D, demand for the drug; S, gross supply of the drug; p, price; m, b, r, parameters.

The meanings of all the subscripts should be clear. Gross supply is defined as the total supply, including the amount L lost for the market due to law enforcement.

Special attention is paid to the determination of L. To begin, it will be assumed that

Dt=St-L, (25)

i.e. that the net supply of the drug is completely sold in the market.

In Section 4 it was observed that one objective frequently used is the maximization of the net benefits of a drug control program. To specify these benefits it will be assumed that the crime reduction due to a program that eliminates the amount L, of the drug from the market is proportional to the value of that drug, i.e. yp,L, where y is a known constant. The benefits derived from the employment of addicts could also be added. However, if these benefits are assumed to be proportional to L,, its inclusion only adds to the complexity of the notation.

With the observations above, the simplest objective that could be assumed is the maximization of

Z = rp,Lt - k,Lz (26)

where k is the cost of confiscating one unit of the drug. One of the problems of drug control is that the higher

the price of the drug, the more sophisticated the methods of avoiding the controls become, and, as a consequence, the higher the value k. On the basis of this observation, it

will be assumed here.that

k, = apt t c. (27)

To solve this problem, the following relationship is first obtained from the equality between demand and net supply:

md (28)

Next, from the first order conditions of the maximization of 2 with respect to L*, one obtains

L, =-!T!E-_ (? _ a) mdpt.

The second order conditions are satisfied if y < a. With eqns (28) and (29), the following finite difference

equation for L, is obtained

(30)

This equation specifies the time path of Lt. In particular, from the analysis of the coefficient of LtM1 it follows that Lt will tend to oscillate around its equilibrium values. These oscillations will be damped if

-(2_r)md <-m,. (31)

This relationship shows that Lt will not converge to its equilibrium value if r > 2.

The dynamic model in eqns (23)-(31) is included here because, despite its extreme degree of sirilpliication, it has all the elements of an optimum control model presented in a much more intuitive way than is usual. All these models have an objective function, constraints in the form of difference and/or differential equations with time as an independent variable, and, as could be expected from the nature of the constraints, a solution in the form of a difference or differential equation. When this functional equation is solved, the values that the control variable should have for every value of t are specified. The example in eqns (23)-(31) clearly shows that the controls that must be specified are unknown functions of t, i.e. the values of Lt, and how using the standard procedures of differential calculus treating the Lt as usual independent variables, a solution is obtained in the form of a function L(t).

The model in eqns (23)-(31) can be extended to a standard form of an optimal control model by replacing the objective function in (26) with

2 = i (ypYPt - k,)L, f=O

(32)

where T is the total number of periods considered. No attempt will be made here to analyze this new version of the dynamic model.

From an intuitive point of view, it does not seem acceptable to assume that the objective of an agency working on drug control should be to maximize the net benefits of its programs. A more acceptable objective is that suggested in Hannan, namely, the minimization of the total damages of drug addiction. With this, the ob-


jective of the optimal control model would be to minimize

T 2 = c (~$0, + k&l.

t=*

It is interesting to observe that it is possible to obtain the conditions that the function Lt has to satisfy when the objective function has either of the forms in eqns (26), (32) or (33), using only methods of constrained optimization of the usual differential calculus. The results are not presented here because the presentation of the formulas without statistically estimated parameters is of little practical value.

Whatever the method of solution used in the models with objective (32) or (33), their actual use is fimited, because Computational methods for non-models are not particularly efficient. However, it should be noticed that models that can be solved with linear programming methods can easily be derived as modifications of those presented above.

7. PROBABILITY MODELS

7.1 fnbroduction In Section 3 it was observed that both descriptive and

decision-making models for drug abuse programs can be constructed. It is useful to retail this classi~cation at this point, because it seems that only descriptive proba~~stic models for drug abuse have been constructed so far. The available example of these models will be described briefly in Section 3.2. In this section some observations with respect to the application of probabilistic decision models to the control of drug abuse will be made.

The deterministic models for decision-making that have been considered are models in which it is assumed that all the parameters are known with certainty. However, it is more reasonable to assume that this is not the case, and that only part of the information on the probability distribution function of the parameters is available. This assumption leads to probabilistic models.

No attempt wili be made here to describe the probabilistic models in detail. It is enough to say that, at least in principle, for each deterministic model of decision-making a probabilistic counterpart can be defined. Mathematical methods to deal with many of these models are available.

The main problem for the use of the probabilistic models is the amount of data required to estimate the characteristics of the probability distribution function of the parameters. For this reason, they will have no immediate use in planning the control of drug abuse without more systematic efforts in the collection of data.

7.2 A ~~ochas~~c model of heroin epidemics The model prepared by Hunt will be described brieAy

here. He observes that: (1) The contagion of heroin addiction can be con-

sidered a branching process; i.e. a process with the following mechanism: an addict (the 0th generation) is capable of producing k = 0, 1,2, . . . successors (addicts initiated by him) to form the first generation; each successor in turn produces k = 0, 1,2,. . . offspring, which constitutes the second generation, and so on.

(2) From observed evidence, the probability of a heroin addict of any generation having X successors follows quite closely a geometric distribution:

PI-(X = k) = (o.4)k(o.6). 04

These two observations form a sufficient basis for applying to heroin epidemics the conclusions from the theory of the geometric distribution and the stochastic branching process. The principal conclusions obtained are:

(1) The average number of successors of a heroin addict is, according to (34),

g = 0.67.

(2) Assure that an epidemic begins with a single individu~, the average number of addicts in the nth generation-to be denoted by G,,-is given by

G,, = G,” = 0.67”. (35)

This result shows that, on the average, a heroin epidemic tends to die out quite rapidly. Hunt estimates that this will happen in a period of roughly 6 yr.

8. CONCLUSIONS

We hope this paper will contribute to an understanding of the valuable assistance that mathematical models can provide to managers of drug control programs.

In conclusion, it should be observed that the results presented here should be inte~reted and used with great care, for three reasons: (a) the data used cannot be considered reliable; (b) the models used as examples are extremely simplified; and (c) not enough simulation runs have been conducted.

Problems (b) and (c) can easily be solved. It might be time for some drug control agency to develop, maintain and frequently run an applied mathematical model for decision-making.

Problem (a) requires more time and money for its solution. However, the lack of reliable data is not an obstacle for the use of mathematical models, since decisions have to be made and are being made on the basis of the available data. However, the results obtained from models when unreliable data are used should be interpreted with special care, and the false sense of security that a model provides should be avoided.

REFERENCES

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2. J. DeLong, Treatment and rehabilitation. In Dealing with Drug Abuse, pp. 173-254. Drug Abuse Survey Project, Praeger Publishers, New York (1972).

3. R. g Marks, The heroin problem: policy aiternatives in dealing with heroin use. .E Drag Issues 4, 69-91 (Winter 1974)._

4. W. McGlothlin, V. Tabbush, C. Chambers and K. Jamison, Alternative Approaches of Opiate Addiction Control: Costs, Benefits and Potential. Bureau of Narcotics and Dangerous Drugs Report SCID-TR-7, U.S. Govt. Printing Office, Washington, D.C. (1972).

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