Bulletin on Narcotics

Volume LIII, Nos. 1 and 2, 2001

Dynamic drug policy: Understanding and controlling drug epidemics

<<   contents   >>

A mover-stayer type model for epidemics of problematic drug use

C. ROSSI*
Department of Mathematics, University of Rome “Tor Vergata”, Italy

Abstract
Introduction: what only mathematical models can reveal
An operational model of problematic drug use epidemics
The crucial parameters to be estimated externally and the scenario parameters
Some scenario analyses
Further developments
References

ABSTRACT

A modified version of a model for the human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS) epidemic, proposed at the beginning of the 1990s and recently generalized, is presented as a representation of a problematic drug use epidemic. The model can be used both to estimate relevant epidemic macro-parameters and to carry out scenario analyses. The model is of the “mover-stayer” type and allows for heterogeneous risk behaviour among the susceptible population. Such models treat the “susceptibles” as subdivided into two groups: the “stayers”, that is, the individuals who are considered not to be at risk of infection, and the “movers”, who are at risk. Because of the interactions between infectious individuals (pushers) and the susceptibles, some of the latter may pass into the drug user “compartments” and begin a drug user “career”. The model described here comprises two different stages of hidden drug use. The first (light use) stage, which can be more strictly defined, is the initial (or non-problematic) stage of drug use, following which light drug users can either stop using drugs or pass to hard drug use (or death). Other relevant stages taken into account are the therapy stage, the recidivist use stage (either light or hard use) and the temporary non-use stage (the latter are the visible stages). A simpler version of the model is studied using a Markov hybrid approximation and some “what if” scenario analyses are obtained by simulation. A more complex and realistic model is also outlined for possible further development.

Introduction: what only mathematical models can reveal

The drug problem and its consequences for society represent a complicated field for research. Policy makers and researchers are seeking answers to a number of questions concerning drug use, its consequences and related costs. They are examining the extent of drug use, the consequences of drug use, which policies are effective, the costs of such policies and the cost of drug use to society. The con sequences comprise, among many others, the adverse effects of infectious diseases and the costs to society of drug-related criminality. It is thus important to understand and measure drug use and how it responds to drug control interventions [1]. The present article will contribute to that objective by introducing a simple compartmental epidemic model of drug use that incorporates various effects on initiation into new use or relapse into recidivist use of drugs.

Compartmental models represent a powerful mathematical tool that is well established in modelling the spread of diseases in a population. They provide a framework in which numbers of people in different “compartments” (each one homogeneous with respect to some specified characteristic) and the relationships between such compartments, modelling the dynamics of the population, can be described in mathematical terms.

Dynamic compartmental modelling of epidemic processes, producing either operational or transmission models, occurs through the usual representation of the dynamics involved by means of a system of stochastic or deterministic differential (or difference) equations; this is the case for both the operational and the transmission models. The main difference between the two kinds of model lies in the fact that transmission models take into account the dynamic processes at the microlevel, modelling the interactions between individuals belonging to the different subgroups involved in the epidemic, whereas operational models work on macro-variables or indicators suitable for use in estimating the size of the phenomena or monitoring the impact of various interventions, modelled by suitable scenario parameters. Many models of the two types have been developed to study the human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS) epidemic and they can also be used, with some modifications, to represent problematic drug use epidemics.

Some examples of transmission models are given in Dietz [2], Hadeler [3], Kretzschmar and Dietz [4] in relation to the HIV/AIDS epidemic and in Billard and Dayananda [5] and in Behrens [1] for problematic drug use epidemics. Those models are complex because the researchers introduce into the modelling process a detailed formalization of the interactions existing, or assumed to exist, between a large number of subjects involved in the epidemic process. This is the case of the analysis of HIV transmission across risk groups, such as drug users and heterosexuals, when some specific hypothesis is formulated as regards the contact pattern [4, 6] or when the time at which the contact occurs is assumed somehow to affect the probability of transmission of the infection [4]. As already mentioned, such models can be helpful when specific contact or transmission patterns need to be thoroughly analysed, but they may be extremely cumbersome when the epidemic as a whole is being examined. Detailed model structures contain all sorts of parameters regulating every single interaction in the simulation process and seldom are enough data available for reliable estimation of all those many parameters.

A more efficient way of obtaining a picture of a simulated epidemic is by using a “simple” operational model. In contrast to transmission models, simple models do not attempt to include all the possible group or individual interactions into the modelling structure, but summarize the dynamics of the epidemic in terms of certain non-linear interactions and group the infected individuals in chains of “compartments”. Most of the parameters controlling the dynamics in such systems are derived from epidemiological studies, external to the model, and their values are based simply on specialized studies or on the literature; only a limited number of “internal” parameters are usually left to be estimated by fitting the existing data or to be used for scenario analyses. Typically, the set of internal parameters includes some form of control of the transmission and of the size of the core group; other internal parameters may have different origins and interpretations, depending on the design of each individual model. The ability of simple models to describe an epidemic correctly is theoretically much more limited when compared with that of complex transmission models. In general, however, simple models turn out to be much more tractable, both because of the limited number of parameters required for their functioning and because of the quality of their output. In fact, while the large number of parameters in the transmission models often results in their unidentifiability, parameter estimates in simple models are usually reliable enough for the models to be used successfully in complex scenario analyses and under other difficult conditions.

Most of the simple models have a stochastic basis, although their output results in some deterministic forecast curves. Such models, defined as “hybrid” models in Bailey [7-9], are the result of a deterministic approximation of stochastic models, using mean times of stay in compartments with conditional Poisson arrivals and maximum likelihood (ML) or numerical estimates of internal para meters. Their use, however theoretically disputable, is more and more frequent among applied researchers because of their technical simplicity and the clarity and immediate possibility of use and interpretation of their results: their output consists of deterministic curves of epidemic indicators (incidence, prevalence, dynamics of the core group and so on), while the parameter ML estimation procedure is derived directly from the distributional hypotheses of the original stochastic model. One such model for the HIV/AIDS epidemic was proposed at the beginning of the 1990s and has recently been generalized. It makes it possible to obtain scenario analyses with ease [10, 11]. That model, when used in conjunction with suitable back-calculation methods, can reduce uncertainties in the estimates of incidence curves [12] and has also been used for indirectly estimating the prevalence of injecting drug use in Italy [13]. A modified version of the model is presented below for use in relation to epidemics of problematic drug use. (The model will be presented from a mathematical point of view elsewhere [14].)

An operational model of problematic drug use epidemics

The graphs provided in figure I describe the main features of the proposed model. The model is of the “mover-stayer” type [10, 15, 16] and allows for heterogeneous risk behaviour among the susceptible population. Such models treat the susceptible population as subdivided into two groups: the “stayers”, that is the individuals who are considered not to be at risk of infection (such models are suit able for scenario analyses in order to assess the impact of various proportions of vaccinated persons on the probability of extinction of a given epidemic) and the “movers”, who are at risk. Because of the interactions between infectious individuals (for the present study these are taken to be problematic drug users who are also pushers)** and the susceptibles, or as a result of the pressure of the black market on the susceptibles, some of the latter may pass into the drug user “compartments” and begin a drug user “career”. As in the model proposed by Behrens and colleagues [1], the model described here comprises two different stages of hidden drug use. The first (light drug use) stage, which can be more strictly defined, is the initial (or non-problematic) stage of drug use, following which light drug users can either stop using drugs or pass to hard drug use (or death). The other arrows in the diagram show all the other possible transitions in a drug user career. The curves connecting the drug use (infectious) compartments and the susceptible (or temporary non-use) compartments indicate the possible interactions that may produce transitions from susceptibles (or temporary non-use) to infectious (non-linear terms in the equations); the other possible transitions from susceptibles (or temporary non-use) to infectious are induced by the pressure of the black market and are assumed to be represented in linear terms in the equations.

Fig I (Click on the figure for a larger view)

In order to formulate the corresponding equations (either deterministic or stochastic), some further hypotheses must be explored and the known and unknown parameters described. A first approximation may be through a Markov model, possibly a marked Markov process. In such a case the length of stay in each compartment is assumed to be distributed exponentially and the results are useful in providing a first qualitative insight into the epidemic process. A more realistic approximation is by using semi-Markov processes. In such a case the length of stay in each compartment may be assumed to be distributed differently with respect to an exponential variable, possibly depending also on covariates describing the individual experience of drug use. Such a study is much more complex, but suitable mathematical and simulation techniques can be used. In the present article, only the Markov model is considered and used to make some scenario analyses.

On the basis of figure I, it is immediately possible to formulate the equations of the model either in the form of deterministic (continuous or discrete) equations or as stochastic (continuous or discrete) equations. The discrete stochastic equations will be reported elsewhere [14]; only the simulation programme and some scenarios aimed at evaluating the impact of different parameters on the epidemic are considered in the present article. The state variables used in the model (with the exception of S ( t ), which is the proportion of stayers at time t , represent the incidence per one million inhabitants; the unit time for the simulation runs is taken to equal one week. The equation for the proportion of stayers, S , is derived from the hypothesis [10] that the new entrances in the susceptible compartment are divided into stayers and movers according to constant proportions S o and 1-S o (stationarity), with 0 < S o < 1, even if other hypotheses can be incidentally incorporated into the model as well as possible transitions from the movers to the stayers compartments as a result of prevention campaigns or law enforcement activities. This further development is outlined briefly in the last part of this article.

A qualitative analysis of the epidemic (transient and asymptotic behaviours), in particular as regards the incidence of drug use, that is the transitions from susceptible to drug user, can be conducted by analysing jointly the equation for X and the equation for S . The method is analogous to that used in Rossi [10] and will be considered elsewhere [14].

The crucial parameters to be estimated externally and the scenario parameters

As regards the parameters and the distributions of the length of stay, some are already available from study of the latency period [17]. Some can be derived using therapy data already available in various places. The demographic parameters regulating the dynamics of the susceptible population, µ01 , µ10 and π17 , are assumed to be known and are country-specific. The other µ parameters can be estimated externally using the information from mortality studies among drug users, which is available for most countries of the European Union [18]. The parameters µ23 and µ34 (natural history parameters) can be estimated on the basis of data available from study of the latency period [19, 20]. The parameters µ45 , µ46 , µ54 , µ56 , µ61 and µ65 (therapy parameters) can be obtained, at least as regards their order of magnitude, from therapy data available in most countries; the values of all these parameters for Italy (order of magnitude) are given in table 1. All the other parameters, µ12 , µ26 , v12 , v13 , v15 , v26 , v36 and v56 , can be used as scenario parameters, as well as the parameter “initial proportion of stayers”, S o .

In the present article it is assumed (following Billard and Dayananda [5]) that most pushers are hard drug users (basic v-scenarios). It has been suggested elsewhere that most pushers are soft drug users [21], so some simulation runs will be devoted to obtaining scenarios based on this latter hypothesis (alternative v-scenarios) by modifying the order of magnitude of the v parameters reported in table 1 (10-5 instead of 10-6 and vice versa).

Table 1

All the µ and π parameters represent transition rates per person of the origin compartment per week, the n parameters are rates per week per pair. Using the simulation procedure described below, some impact analyses can be conducted to evaluate the influence of the scenario parameters on the course of the epidemic. It is also possible to make some further scenario analyses to evaluate the impact of increased efficacy of therapy services by taking the scenario parameters as fixed and using the therapy parameters as variable. Similarly, the natural history parameters can be used to make further “what if” scenario analyses (the results of such analyses will be presented elsewhere).

Some scenario analyses

The simulation procedure, used to obtain scenario analyses, is written in S-plus for personal computers. All the parameters can be modified at the beginning of each run. The standard output comprises graphs of the prevalence curves in each compartment and of the incidence curves of major interest. It is possible to choose the total simulation time, which is measured in weeks.

Some scenario analyses have been conducted to study the impact of S o and some other parameters on the qualitative behaviour of the epidemic. The results are reported in graphic form below. As expected, the effect of S o is higher with respect to the other parameters [11] considered for the simulation runs reported here.

Basic v-scenarios

Some scenarios have been obtained based on various hypotheses concerning the parameters. The results are reported in graphic form and a summary table (table 2) describes the macro-characteristics of the different epidemics using the location and the size of the peaks for the incidence and prevalence curves, the value of the cumulative curve of deaths at the end of the period taken into account and the location of the maximum of the proportion of stayers, which represents the time of the beginning of the endemic phase (saturation effect).

Scenario 1

Scenario 1 is obtained using the parameters listed in table 1, with S o = 0.98 (this means that the size of the population at risk is taken to be about 20,000 per one million inhabitants), µ12 = 10-5 (this parameter measures the strength of the black market: its value means that the black market can be assumed to induce about one person of the risk group per month to try drugs at the beginning of the epidemic) and µ26 = 0.004 (this is the parameter regulating spontaneous exits from the non-addicted user population: its value means that the proportion of non-addicts who stop using drugs is about half those who proceed to addictive behavour). Various graphs representing the curves of interest are shown in figures II-VII.

 


Figure II. Scenario 1: Incidence curve from susceptibles to light users
Figure II

 

Figure III. Scenario 1: Prevalence curve of light users
Figure III

 

Figure IV. Scenario 1: Incidence curve from light users to heavy users
Figure IV

 

Figure V. Scenario 1: Prevalence curve of hard users
Figure V

 

Figure VI. Scenario 1: Prevalence curve of clients of therapy services
Figure VI


Figure VII. Scenario 1: Proportion of stayers
Figure VII

Scenario 2

This scenario is obtained using the parameters listed in table 1, with S o = 0.99 (here the size of the population at risk is taken to be about 10,000 per one million inhabitants), µ12 = 10-5 (as in scenario 1) and µ26 = 0.004 (as in scenario 1). Graphs representing the curves of interest are shown in figures VIII-XIII.


Figure VIII. Scenario 2: Incidence curve from susceptibles to light users

Figure VIII
 

Figure IX. Scenario 2: Prevalence curve of light users
Figure IX

 

Figure X. Scenario 2: Incidence curve from light users to hard users
Figure X

 

Figure XI. Scenario 2: Prevalence curve of hard users
Figure XI

 

Figure XII. Scenario 2: Prevalence curve of clients of therapy services
Figure XII

 

Figure XIII. Scenario 2: Proportion of stayers
Figure XIII

Scenario 3

Scenario 3 is obtained using the parameters listed in table 1, with S o = 0.98 (as in scenario 1), µ12 = 10-6 (this parameter measures the strength of the black market: its value means that the black market can be assumed to induce about one person of the risk group per 10 months to try drugs at the beginning of the epidemic) and µ26 = 0.004 (as in scenario 2). Graphs representing the curves of interest are shown in figures XIV-XIX.

 

Figure XIV. Scenario 3: Incidence curve from users
Figure XIV
 

Figure XV. Scenario 3: Prevalence curve of light susceptibles to light users
Figure XV

 

Figure XVI. Scenario 3: Incidence curve from light users
Figure XVI

 

Figure XVII. Scenario 3: Prevalence curve of clients of therapy services
Figure XVII

 

Figure XVIII. Scenario 3: Prevalence curve of clients of therapy services
Figure XVIII

 

Figure XIX. Scenario 3: Proportion of stayers
Figure XIX

Scenario 4

This scenario is obtained using the parameters listed in table 1, with S o = 0.98 (as in scenario 3), µ12 = 10-5 (as in scenarios 1 and 2) and µ26 = 0.0004 (this is the parameter regulating spontaneous exits from the non-addicted user population: its value means that the proportion of non-addicts who stop using drugs is about 1/20 of those who proceed to addictive behaviour). Graphs representing the curves of interest are shown in figures XX-XXV.

Figure XX. Scenario 4: Incidence curve from susceptibles to light users
Figure XX
Figure XXI. Scenario 4: Prevalence curve of light users
Figure XXI
Figure XXII. Scenario 4: Incidence curve from light users to hard users
Figure XXII
Figure XXIII. Scenario 4: Prevalence curve of hard users
Figure XXIII
Figure XXIV. Scenario 4: Prevalence curve of clients of therapy services
Figure XXIV
Figure XXV. Scenario 4: Proportion of stayers
Figure XXV

Scenario 5

The present scenario is obtained using the parameters listed in table 1, but modifying the order of magnitude of the n parameters (10-5 instead of 10-6 and vice versa). This means that it is assumed that there is higher possibility that a non-addicted individual can induce a susceptible to try drugs than an addicted one can. The values S o = 0.98, µ12 = 10-5 and µ26 = 0.004 are the same as in scenario 1. Graphs representing the curves of interest are shown in figures XXVI-XXXI.

Figure XXVI. Scenario 5: Incidence curve from susceptibles to light users
Figure XXVI
Figure XXVII. Scenario 5: Prevalence curve of light users
Figure XXVII
Figure XXVIII. Scenario 5: Incidence curve from light users to hard users
Figure XXVIII
Figure XXIX. Scenario 5: Prevalence curve of hard users
Figure XXIX
Figure XXX. Scenario 5: Prevalence curve of clients of therapy services
Figure XXX
Figure XXXI. Scenario 5: Proportion of stayers
Figure XXXI

Scenario 6

The present scenario is obtained using the parameters listed in table 1, but modifying the order of magnitude of the v parameters (10-5 instead of 10-6 and vice versa), as in scenario 5, and with S o = 0.99, µ12 = 10-5 and µ26 = 0.004 (as in scenario 2). Graphs representing the curves of interest are shown in figures XXXII-XXXVII.

Figure XXXII. Scenario 6: Incidence curve from susceptibles to light users
Figure XXXII
Figure XXXIII. Scenario 6: Prevalence curve of light users
Figure XXXIII
Figure XXXIV. Scenario 6: Incidence curve from from light users to hard users
Figure XXXIV
Figure XXXV. Scenario 6: Prevalence curve of hard users
Figure XXXV
Figure XXXVI. Scenario 6: Prevalence curve of clients of therapy services
Figure XXXVI
Figure XXXVII. Scenario 6: Proportion of stayers
Figure XXXVII

An analysis of the graphs shows clearly that the parameter with the highest impact on the course of the epidemic is S o , which is a measure of the size of the group of susceptibles who are at risk of infection (the core group). The bigger the core group, the faster the evolution of the epidemic and the higher the prevalence and incidence curves. The influence of the parameter that measures the pressure of the black market, µ12 , appears to be less important. As regards the comparison between the basic hypothesis for the n parameters, that is, that pushers are mainly hard users, and the alternative hypothesis, that pushers are mainly light users, on the basis of the results of the simulations, it appears that the alternative scenarios show faster evolutions of epidemics. Table 2 summarizes the results in terms of the macro-indicators (location and size of the peaks for the incidence and prevalence curves, value of the cumulative curve of deaths at the end of the period taken into account and the location of the maximum of the proportion of stayers) described above. All the values related to size represent rates per one million inhabitants.

Table 2 (Click on table for a larger view)

It can be observed that the initial proportion of stayers, S o is the parameter with the highest impact: there are higher differences in the macro-parameters between scenario 1 and scenario 2 than between scenarios 1 and 3 or between scenarios 1 and 4.

Higher values of S 0 correspond to larger and faster-growing epidemics, whereas the parameter related to the pressure of the market is not as important, at least in the range of values examined here. A minor effect also arises from parameter µ26 , which models transitions from light use to non-use (spontaneous cessation). As mentioned above, alternative scenarios correspond essentially to faster-growing epidemics, other things beings equal.

Further analyses will be reported elsewhere [14].

Further developments

The use of suitable markers (marked processes) might make it possible to incorporate further descriptions of each individual involved in an epidemic; for in stance, markers may take into account the number of incarcerations, the number of non-fatal overdoses or the number of failed therapy interventions. Possible repeated therapy interventions should be incorporated into the model in order to obtain a more realistic picture of the career of the problematic drug user; there is general agreement that the time spent in the therapy compartment for the first therapy episode is distributed differently with respect to times related to the following episodes. The complexity of such analysis is evident and the interpretation of the results may be quite uncertain and unreliable owing to lack of information about the statistical distribution of the length of stay in the various compartments. It is therefore preferable to wait until the therapy data sets comprise reliable and complete information that will make possible external estimation of the parameters of interest.

In order to obtain more realistic models, the transition parameters should not be taken as constant, but should be represented as functions taking into account the history of drug use for a given individual, represented by statistical variables taken as known (covariates), and the history of policy interventions (availability of services, law enforcement activities and so on), represented by other covariates (time-dependent) and, possibly, by latent variables. This would result in a realistic but at the same time unreliable and intractable transmission model of very limited use. Some years ago, Kaplan [22] published a paper entitled “Can bad models suggest good policies?” in which he clearly explained how simple models, taking into account only the main peculiarities of the phenomenon of interest and neglecting minor effects, which may simply mask the relevant behaviours, are much more useful in order to assess the main consequences of policy interventions than complex models. The title itself is a good indication of the usefulness of simple mathematical models in addressing complex policy issues. Finally, if it were to be necessary to evaluate the impact of some kind of prevention campaign or intervention relating to problematic drug use among susceptibles, using the mover-stayer model proposed above, that issue could be dealt with easily by introducing the possibility that a mover become a stayer during the period of the campaign (or soon after). This can be simply modelled by assuming that during a short period a given proportion of movers become stayers and modifying the equations of the model to incorporate that fact. Analysis of the new equations would then make it possible to state that the efficacy of a primary prevention intervention of that length is higher at the beginning of the epidemic and decreases in the following period [14]. The effect of a secondary prevention intervention can also be taken into account in a similar manner by modifying the equations of the model. Analysis of the new equations would make it possible to state that the efficacy of a secondary prevention intervention of the given length is lower at the beginning of the epidemic and increases in the following period. The model may thus always be used to forecast the effect of such types of intervention and to determine the best combination of the two [14]. Similarly, the effect of law enforcement, which may influence movers, pushing a proportion of them to become stayers as a result of information about the adverse consequences of using illegal drugs [1], can be modelled assuming some parameters depending either on the number of addicts assisted by the health-care services or the number of incarcerated addicts or both.

Further generalizations might concern a more realistic approach to modelling the length of stay in the various compartments, taking into account the heterogeneity of individual behaviours. Most of these issues will be addressed in future contributions.

References

  1. D. A. Behrens and others, “A dynamic model of drug initiation: implications for treatment and drug control”, Mathematical Biosciences, vol. 159, No. 1 (1999), pp. 1-20.
  2. K. Dietz, “On the transmission dynamics of HIV”, Mathematical Biosciences, vol. 90, 1988, pp. 397-414.
  3. K. P. Hadeler, “Modeling AIDS in structured populations”, in Proceedings of I.S.I. 47th Session, Paris, 1989, pp. 83-99.
  4. M. Kretzschmar and K. Dietz, “The effect of pair formation and variable infectivity on the spread of an infection without recovery”, Mathematical Biosciences , vol. 148, No. 1 (1998), pp. 83-113.
  5. L. Billard and P.W.A. Dayananda, “Drug addiction—pushers generated from addicts”, Biometrical Journal, vol. 35, 1993, pp. 227-244.
  6. J. A. Jacquez and others, “Modeling and analyzing HIV transmission: the effect of contact patterns”, Mathematical Biosciences, vol. 92, 1988, pp. 119-199.
  7. N. T. J. Bailey, “The use of operational modelling of HIV/AIDS in a system approach to public health decision making”, Mathematical Biosciences, vol. 107, 1991, pp. 413-430.
  8. N. T. J. Bailey, “An improved hybrid HIV/AIDS model geared to specific public health data and decision making”, Mathematical Biosciences, vol. 117, 1993, pp. 221-237.
  9. N. T. J. Bailey, “HIV/AIDS core-group dynamic and public health action”, in Modeling the AIDS epidemic, E. H. Kaplan and M. L. Brandeau, eds. (New York, Raven Press, 1994), pp. 29-52.
  10. C. Rossi, “A stochastic mover-stayer model for HIV epidemics”, Mathematical Biosciences , vol. 107, 1991, pp. 521-545.
  11. C. Rossi and G. Schinaia, “The mover-stayer model for the HIV/AIDS epidemic in action”, Interfaces, vol. 28, No. 3 (1998), pp. 127-143.
  12. L. Ravà and others, “Estimating the size of the HIV/AIDS epidemic: complementary use of empirical Bayesian back calculation and the mover-stayer model for gathering the largest amount of information”, Simulation, vol. 71, No. 4 (1998), pp. 213-227.
  13. C. Rossi, “Estimating the prevalence of injecting drug users on the basis of Markov models of the HIV/AIDS epidemic: applications to Italian data”, Health Care Management Science, vol. 2, 1999, pp. 173-179.
  14. C. Rossi, “Operational models for problematic drug use epidemics: the mover-stayer approach to heterogeneity”, forthcoming.
  15. T. M. Morgan, C. S. Aneshensel and V. A. Clark, “Parameter estimation for mover-stayer models: analysing depression over time”, Sociological Methods and Research, vol. 11, 1983, pp. 345-366.
  16. P. S. Albert, “A mover-stayer model for longitudinal marker data”, Biometrics, vol. 55, No. 4 (1999), pp. 1252-1257.
  17. European Monitoring Centre for Drugs and Drug Addiction, “Pilot project to estimate time trends and incidence of problem drug use in the European Union”, final report, Lisbon, 1999.
  18. European Monitoring Centre for Drugs and Drug Addiction, “Coordination of implementation, follow-up and analysis of cohort studies on mortality among drug users in European Union Member States”, final report, Lisbon, 1999.
  19. C. Rossi, “Monitoring drug control strategies: hidden phenomena, observable events, observable times”, International Journal of Drug Policy, vol. 10, No. 1 (1999), pp. 131-144.
  20. L. Ravà and others, “The use of back-calculation for investigating temporal-gegraphic spread of problem drug use: nationwide and regional analysis in Italy”, forthcoming.
  21. L. G. Hunt and C. D. Chambers, The Heroin Epidemics (New York, Spectrum Publications, 1976).
  22. E. Kaplan, “Can bad models suggest good policies? Sexual mixing and the AIDS epidemic”, Journal of Sex Research, vol. 26, 1989, pp. 301-314.

FOOTNOTES

*The present article was funded in part by the European Monitoring Centre on Drugs and Drug Addiction in the framework of the European Network to Develop Policy Relevant Models and Socio-Economic Analyses of Drug Use, Consequences and Interventions. The simulation procedure used to obtain scenario analyses was developed by Alessandro Ghizzoni.

**From surveys conducted among military conscripts, reported in the annual report on the state of the drug problem in Italy for the year 1999, published by the National Focal Point, it appears that, in terms of the reasons for drug use, the two most mentioned factors were curiosity (more than 40 per cent) and peer group pressure (more than 30 per cent).

<<   contents   >>