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

Structural instability and linear allocation control in generalized models of substance use disorder

Substance use disorder (SUD) is a complex disease involving nontrivial biological, psychological, environmental, and social factors. While many mathematical studies have proposed compartmental models for SUD, almost all of these exclusively model new cases as the result of an infectious process, neglecting any SUD that was primarily developed in social isolation. While these decisions were likely made to facilitate mathematical analysis, isolated SUD development is critical for the most common substances of abuse today, including opioid use disorder developed through prescription use and alcoholism developed primarily due to genetic factors or stress, depression, and other psychological factors. In this paper we will demonstrate that even a simple infectious disease model is structurally unstable with respect to a linear perturbation in the infection term - precisely the sort of term necessary to model SUD development in isolation. This implies that models of SUD which exclusively treat problematic substance use as an infectious disease will have misleading dynamics whenever a non-trivial rate of isolated SUD development exists in actuality. As we will show, linearly perturbed SUD models do not have a use disorder-free equilibrium. To investigate management strategies, we implement optimal control techniques with the goal of minimizing the number of SUD cases over time.

A mathematical model with nonlinear relapse: conditions for a forward-backward bifurcation

We constructed a Susceptible-Addicted-Reformed model and explored the dynamics of nonlinear relapse in the Reformed population. The transition from susceptible considered at-risk is modeled using a strictly decreasing general function, mimicking an influential factor that reduces the flow into the addicted class. The basic reproductive number is computed, which determines the local asymptotically stability of the addicted-free equilibrium. Conditions for a forward-backward bifurcation were established using the basic reproductive number and other threshold quantities. A stochastic version of the model is presented, and some numerical examples are shown. Results showed that the influence of the temporarily reformed individuals is highly sensitive to the initial addicted population.

Methodological approaches for the prediction of opioid use-related epidemics in the United States: a narrative review and cross-disciplinary call to action

The opioid crisis in the United States has been defined by waves of drug- and locality-specific Opioid use-Related Epidemics (OREs) of overdose and bloodborne infections, among a range of health harms. The ability to identify localities at risk of such OREs, and better yet, to predict which ones will experience them, holds the potential to mitigate further morbidity and mortality. This narrative review was conducted to identify and describe quantitative approaches aimed at the "risk assessment," "detection" or "prediction" of OREs in the United States. We implemented a PubMed search composed of the: (1) objective (eg, prediction), (2) epidemiologic outcome (eg, outbreak), (3) underlying cause (ie, opioid use), (4) health outcome (eg, overdose, HIV), (5) location (ie, US). In total, 46 studies were included, and the following information extracted: discipline, objective, health outcome, drug/substance type, geographic region/unit of analysis, and data sources. Studies identified relied on clinical, epidemiological, behavioral and drug markets surveillance and applied a range of methods including statistical regression, geospatial analyses, dynamic modeling, phylogenetic analyses and machine learning. Studies for the prediction of overdose mortality at national/state/county and zip code level are rapidly emerging. Geospatial methods are increasingly used to identify hotspots of opioid use and overdose. In the context of infectious disease OREs, routine genetic sequencing of patient samples to identify growing transmission clusters via phylogenetic methods could increase early detection capacity. A coordinated implementation of multiple, complementary approaches would increase our ability to successfully anticipate outbreak risk and respond preemptively. We present a multi-disciplinary framework for the prediction of OREs in the US and reflect on challenges research teams will face in implementing such strategies along with good practices.

Opioid agonist treatment scale-up and the initiation of injection drug use: A dynamic modeling analysis

BACKGROUND

Injection drug use (IDU) is associated with multiple health harms. The vast majority of IDU initiation events (in which injection-naïve persons first adopt IDU) are assisted by a person who injects drugs (PWID), and as such, IDU could be considered as a dynamic behavioral transmission process. Data suggest that opioid agonist treatment (OAT) enrollment is associated with a reduced likelihood of assisting with IDU initiation. We assessed the association between recent OAT enrollment and assisting IDU initiation across several North American settings and used dynamic modeling to project the potential population-level impact of OAT scale-up within the PWID population on IDU initiation.

METHODS AND FINDINGS

We employed data from a prospective multicohort study of PWID in 3 settings (Vancouver, Canada [n = 1,737]; San Diego, United States [n = 346]; and Tijuana, Mexico [n = 532]) from 2014 to 2017. Site-specific modified Poisson regression models were constructed to assess the association between recent (past 6 month) OAT enrollment and history of ever having assisted an IDU initiation with recently assisting IDU initiation. Findings were then pooled using linear mixed-effects techniques. A dynamic transmission model of IDU among the general population was developed, stratified by known factors associated with assisting IDU initiation and relevant drug use behaviors. The model was parameterized to a generic North American setting (approximately 1% PWID) and used to estimate the impact of increasing OAT coverage among PWID from baseline (approximately 21%) to 40%, 50%, and 60% on annual IDU initiation incidence and corresponding PWID population size across a decade. From Vancouver, San Diego, and Tijuana, respectively, 4.5%, 5.2%, and 4.3% of participants reported recently assisting an IDU initiation, and 49.4%, 19.7%, and 2.1% reported recent enrollment in OAT. Recent OAT enrollment was significantly associated with a 45% lower likelihood of providing recent IDU initiation assistance among PWID (relative risk [RR] 0.55 [95% CI 0.36-0.84], p = 0.006) compared to those not recently on OAT. Our dynamic model predicts a baseline mean of 1,067 (2.5%-97.5% interval [95% I 490-2,082]) annual IDU initiations per 1,000,000 individuals, of which 886 (95% I 406-1,750) are assisted by PWID. Based on our observed statistical associations, our dynamic model predicts that increasing OAT coverage from approximately 21% to 40%, 50%, or 60% among PWID could reduce annual IDU initiations by 11.5% (95% I 2.4-21.7), 17.3% (95% I 5.6-29.4), and 22.8% (95% I 8.1-36.8) and reduce the PWID population size by 5.4% (95% I 0.1-12.0), 8.2% (95% I 2.2-16.9), and 10.9% (95% I 3.2-21.8) relative to baseline, respectively, in a decade. Less impact occurs when the protective effect of OAT is diminished, when a greater proportion of IDU initiations are unassisted by PWID, and when average IDU career length is longer. The study's main limitations are uncertainty in the causal pathway between OAT enrollment and assisting with IDU initiation and the use of a simplified model of IDU initiation.

CONCLUSIONS

In addition to its known benefits on preventing HIV, hepatitis C virus (HCV), and overdose among PWID, our modeling suggests that OAT scale-up may also reduce the number of IDU initiations and PWID population size.

System dynamics applications to injury and violence prevention: a systematic review

Purpose of review

System dynamics (SD) is an approach to solving problems in the context of dynamic complexity. The purpose of this review was to summarize SD applications in injury prevention and highlight opportunities for SD to contribute to injury prevention research and practice.

Recent findings

While SD has been increasingly used to study public health problems over the last few decades, uptake in the injury field has been slow. We identified 18 studies, mostly conducted in the last 10 years. Applications covered a range of topics (e.g., road traffic injury; overdose; violence), employed different types of SD tools (i.e., qualitative and quantitative), and served a variety of research and practice purposes (e.g., deepen understanding of a problem, policy analysis).

Summary

Given the many ways that SD can add value and complement traditional research and practice approaches (e.g., through novel stakeholder engagement and policy analysis tools), increased investment in SD-related capacity building and opportunities that support SD use are warranted.

Modelling the trends of inpatient and outpatient rehabilitation for methamphetamine in the Western Cape province of South Africa

BACKGROUND

Dependence on methamphetamine remains one of the major health and social problem in the Western Cape province of South Africa. We consider a mathematical model that takes into account two forms of rehabilitation, namely; inpatient and outpatient. We examine the trends of these two types of rehabilitation. We also seek to investigate the global dynamics of the developed methamphetamine epidemic model.

METHODS

The model is designed by likening the initiation process to an infection that spreads in a community through interactions between methamphetamine users and non-users. We make use of Lyapunov functions obtained from a suitable combination of common quadratic and Volterra-type functions to establish the global stability of the methamphetamine-persistent steady state. The least squares curve fit routine (lsqcurvefit) in Matlab with optimization is used to estimate the parameter values.

RESULTS

The model analysis shows that the model has two equilibria, the methamphetamine free equilibrium and the methamphetamine persistent equilibrium, that are both globally stable when the threshold R(a) < 1 and R(a) > 1, respectively. Upon fitting the model to data on drug users under rehabilitation, parameter values that give the best fit were obtained. The projections carried out the long term trends of these forms of rehabilitation.

CONCLUSION

The results suggest that inpatient rehabilitation programs have an increased potential of enhancing the chances of recovery for methamphetamine addicts.

Development of a brief substance use sensation seeking scale: validation and prediction of injection-related behaviors

Sensation seeking, a personality trait, has been shown to predict engagement in high-risk behaviors. However, little is known regarding the impact of sensation seeking on substance use among street youth. We therefore sought to modify a sensation seeking scale (SSS) for use among this population. Street youth from the Vancouver-based At-Risk Youth Study (n = 226) completed the modified SSS. Exploratory and confirmatory factor analysis (EFA/CFA) were undertaken to establish the scale's dimensionality and internal validity. The association between SSS score and injection-related behaviors was tested using generalized estimating equation analysis. EFA results indicated scale unidimensionality. The comparative fit index (CFI) suggested acceptable fit (CFI = 0.914). In multivariate analysis, sensation seeking was independently associated with injection drug use, crystal methamphetamine use, polysubstance use, and binge drug use (all p < 0.05). Our findings provide preliminary support for the use of the modified SSS among street youth.

Should anyone be riding to glory on the now-descending limb of the crack-cocaine epidemic curve in the United States?

BACKGROUND

Many pre-clinical and clinical researchers do not appreciate the recent decline in United States (US) population-level incidence of crack-cocaine smoking. At present, no more than about 200 young people start using crack-cocaine each day. Ten years ago, the corresponding estimated daily rate was 1000. This short communication looks into these trends, surrounding evidence on this important public health topic, and checks whether duration-reducing treatment interventions might be responsible, versus selected alternatives.

METHODS

Via analyses of standardized computer-assisted self-interview data from the US National Surveys on Drug Use and Health (NSDUH, 2002-2011; n>500,000), we evaluated change in incidence estimates, perceived difficulty to acquire crack, risk of using cocaine, treatment entries, and persistence once crack use has started.

RESULTS

We draw attention to a marked overall decline in year-specific incidence rates for crack-cocaine smoking from 2002 to 2011, especially 2007-2011. There is some variation in estimates of difficulty to acquire crack (p<0.001) and observed risk of using cocaine among 'at risk' susceptibles (p<0.001), but no appreciable shifts in duration of crack smoking among active users (p>0.05) or in proportion of crack users receiving treatment (p>0.05).

CONCLUSIONS

Changing epidemiology of crack-cocaine smoking may rest largely on reductions in newly incident use with no major direct effects due to US cocaine treatment, incarceration, or interdiction. Concurrently, we see quite modest declines in survey-based estimates of cocaine-attributed perceived risk and cocaine availability. As such, we posit that no specific US agency should claim it is 'riding to glory' on the descending limb of this epidemic curve.

A system dynamics approach to the study of Colombian coca cultivation and the counter-intuitive consequence of law enforcement

A large-scale expansion of the Colombian coca cultivation is one of the most revealing signs of a structural change in the illegal cocaine market in the Andean region. From being a modest and domestic production, in the space of five years Colombian coca cultivation supplied a competitive market, capable of substituting almost completely the foreign sources of supply. The purpose of this work is to explore the role and potential of system dynamics (SD) as a modeling methodology to better understand the consequences of drug policy. As a case study, this work tests the hypothesis that the outbreak of Colombian coca cultivations is a consequence of the take down of large cartels, leading to the surge of small drug-trafficking firms called "cartelitos." Using an SD model, and elements from the economic theory of the criminal firm, our work shows how the formation of these small firms might significantly contribute to the configuring of a more competitive domestic coca industry (and hence to a more efficient crime industry). We conclude that SD seems an appropriate dynamic modeling-based approach to address policy issues regarding drug markets. The methodology takes into account the dynamic nature of drug markets and their multi-dimensional responses to policy interventions.

Is harm reduction profitable? An analytical framework for corporate social responsibility based on an epidemic model of addictive consumption

This article aims to help resolve the apparent paradox of producers of addictive goods who claim to be socially responsible while marketing a product clearly identified as harmful. It advances that reputation effects are crucial in this issue and that determining whether harm reduction practices are costly or profitable for the producers can help to assess the sincerity of their discourse. An analytical framework based on an epidemic model of addictive consumption that includes a deterrent effect of heavy use on initiation is developed. This framework enables us to establish a clear distinction between a simple responsible discourse and genuine harm reduction practices and, among harm reduction practices, between use reduction practices and micro harm reduction practices. Using simulations based on tobacco sales in France from 1950 to 2008, we explore the impact of three corresponding types of actions: communication on damage, restraining selling practices and development of safer products on total sales and on the social cost. We notably find that restraining selling practices toward light users, that is, preventing light users from escalating to heavy use, can be profitable for the producer, especially at early stages of the epidemic, but that such practices also contribute to increase the social cost. These results suggest that the existence of a deterrent effect of heavy use on the initiation of the consumption of an addictive good can shed new light on important issues, such as the motivations for corporate social responsibility and the definition of responsible actions in the particular case of harm reduction.

Types of drinkers and drinking settings: an application of a mathematical model

AIMS

US college drinking data and a simple population model of alcohol consumption are used to explore the impact of social and contextual parameters on the distribution of light, moderate and heavy drinkers. Light drinkers become moderate drinkers under social influence, moderate drinkers may change environments and become heavy drinkers. We estimate the drinking reproduction number, R(d) , the average number of individual transitions from light to moderate drinking that result from the introduction of a moderate drinker in a population of light drinkers.

DESIGN AND SETTINGS

Ways of assessing and ranking progression of drinking risks and data-driven definitions of high- and low-risk drinking environments are introduced. Uncertainty and sensitivity analyses, via a novel statistical approach, are conducted to assess R(d) variability and to analyze the role of context on drinking dynamics.

FINDINGS

Our estimates show R(d) well above the critical value of 1. R(d) estimates correlate positively with the proportion of time spent by moderate drinkers in high-risk drinking environments. R(d) is most sensitive to variations in local social mixing contact rates within low-risk environments. The parameterized model with college data suggests that high residence times of moderate drinkers in low-risk environments maintain heavy drinking.

CONCLUSIONS

With regard to alcohol consumption in US college students, drinking places, the connectivity (traffic) between drinking venues and the strength of socialization in local environments are important determinants in transitions between light, moderate and heavy drinking as well as in long-term prediction of the drinking dynamics.

Initiation to the first drug use among substance-dependent persons in Iran

In the present study, we report the initiation to the first drug in a representative sample of drug dependents in Iran. This is a cross-sectional study with a sample size of 7,743. The participants were characterized as drug dependents according to the Diagnostic and Statistical Manual-Fourth Edition and were sampled from the medical centers, prisons, and streets of the capitals of 29 provinces in the Islamic Republic of Iran. The mean age at first drug use in the sample was 21.1 (6.5), which differed for various first drugs of abuse. Opium was the first drug of use among 46.4% of the sample, and cannabis was the first drug only among 23.5%. The most frequent place of initiation was at a friend's house (n = 2,476, 33.2%). The same-gender friend out of school (the highest rate) was the most frequent person who suggested the first use. A total of 43.5% of the participants started drug use with friends and 18.8% at a friend's party. Policymakers may benefit from considering the information provided here in designing programs with the purpose of prevention.

Impact of Relative Residence Times in Highly Distinct Environments on the Distribution of Heavy Drinkers

Alcohol consumption is a function of social dynamics, environmental contexts, individuals' preferences and family history. Empirical surveys have focused primarily on identification of risk factors for high-level drinking but have done little to clarify the underlying mechanisms at work. Also, there have been few attempts to apply nonlinear dynamics to the study of these mechanisms and processes at the population level. A simple framework where drinking is modeled as a socially contagious process in low- and high-risk connected environments is introduced. Individuals are classified as light, moderate (assumed mobile), and heavy drinkers. Moderate drinkers provide the link between both environments, that is, they are assumed to be the only individuals drinking in both settings. The focus here is on the effect of moderate drinkers, measured by the proportion of their time spent in "low-" versus "high-" risk drinking environments, on the distribution of drinkers.A simple model within our contact framework predicts that if the relative residence times of moderate drinkers is distributed randomly between low- and high-risk environments then the proportion of heavy drinkers is likely to be higher than expected. However, the full story even in a highly simplified setting is not so simple because "strong" local social mixing tends to increase high-risk drinking on its own. High levels of social interaction between light and moderate drinkers in low-risk environments can diminish the importance of the distribution of relative drinking times on the prevalence of heavy drinking.

Heroin users in Australia: population trends

The aim of this paper is to identify certain important population trends among heroin users in Australia for the period 1971 - 97, such as: population growth, initiation, i.e. the number who were initiated to heroin in a given year, and quitting, i.e. the number that quit using heroin. For this purpose, we summarize and extract relevant characteristics from data from National Drug Strategy Household Survey (NDSHS 1998) conducted in Australia in 1998. We devise a systematic procedure to estimate historical trends from questions concerning past events. It is observed from our findings that the size of the heroin user population in Australia is in a sharp increase, especially from the early 1980s onwards. The general trend obtained for the period 1971 - 97 is strikingly similar to that obtained by Hall et al. (2000) for the dependent heroin user population in Australia, even though their study was based on different datasets and a different methodology. In our reconstruction of the time history we also detect a levelling-off prior to 1990. Initiation is also observed to be on a sharp increase. The latter trend is accompanied by a similar trend of quitting, perhaps indicating a relatively short heroin use career. A sharp decrease in both initiation and quitting is observed after 1990. In conclusion, in the case of the trend in the population of heroin users a high rate of growth has been identified that is consistent with the existing literature. In the process, we demonstrated that even a static survey such as NDSHS 1998 can, sometimes, be used to extract historical (dynamic) trends of certain important variables.

Initiation into methamphetamine use for young gay and bisexual men

Research over the past 10 years has suggested that methamphetamine use has become a significant problem and is associated with risky sexual behaviors among gay and bisexual men. In order to better understand initiation into methamphetamine use among gay and bisexual men, qualitative analyses were performed on a sample of young gay and bisexual men (ages 18-29) in New York City. Participants were recruited as part of a larger study which used time-space sampling to enroll club-going young adults who indicated recent club drug (ecstasy, ketamine, GHB, methamphetamine, cocaine, and/or LSD) use. The data for this paper are derived from the qualitative interviews of 54 gay and bisexual male methamphetamine users. At initiation (1) methamphetamine was used in a social, non-sexual setting for a majority of the participants; (2) participants expressed limited knowledge of methamphetamine; and (3) many participants used cocaine as a basis for comparison when describing various effects of the drug. The understanding that at initiation methamphetamine was not solely used as a sexual enhancement for members of this community may enable health workers to more accurately target potential users when putting forth intervention efforts. Future research should aim to gain a better understanding into the role that methamphetamine plays in non-sexual contexts, particularly among gay and bisexual men who may not be part of the club "scene." The relationship between attitudes towards methamphetamine and other drugs, particularly cocaine, among gay and bisexual men should be explored.

Dynamic compartmental model of trends in Australian drug use

A five-state compartment model of trends in illicit drug use in Australia is parameterized using data from multiple sources. The model reproduces historical prevalence and supports what-if analyses under the assumption that past trajectories of drug escalation and desistance persist. For fixed initiation, the system has a unique stable equilibrium. The chief qualitative finding is that even though some users escalate rapidly, regular injection drug use still adjusts to changes in incidence with considerable inertia and delay. This has important policy implications, e.g., concerning the timing of reductions in drug-related social cost generated by interventions that reduce the social cost per injection user versus those that cut drug initiation.

Risk of becoming cocaine dependent: epidemiological estimates for the United States, 2000-2001

In this paper, we present new estimates for the risk of becoming cocaine dependent within 24 months after first use of the drug, and study subgroup variation in this risk. The study estimates are based on the National Household Survey on Drug Abuse conducted during 2000-2001, with a representative sample of US residents aged 12 years and older (n=114 241). A total of 1081 respondents were found to have used cocaine for the first time within 24 months prior to assessment. Between 5 and 6% of these recent-onset users had become cocaine dependent since onset of use. Less [corrected] risk of recent cocaine dependence soon after onset of cocaine use was found for female subjects, young adults aged 21-25 years, and non-Hispanic Black/African-Americans. Use of crack-cocaine and taking cocaine by injection were associated with having become cocaine dependent soon after onset of use. These epidemiologic findings help to quantify the continuing public health burden associated with new onsets of cocaine use in the 21st century.

Evaluating explanations of the Australian 'heroin shortage'

AIMS

In this paper we outline and evaluate competing explanations for a heroin shortage that occurred in Australia during 2001 with an abrupt onset at the beginning of 2001.

METHODS

We evaluated each of the explanations offered for the shortage against evidence from a variety of sources: government reports, police and drug law enforcement documents and briefings, key informant (KI) interviews, indicator data and research data.

RESULTS

No similar shortage occurred at the same time in other markets (e.g. Vancouver, Canada or Hong Kong) whose heroin originated in the same countries as Australia's. The shortage was due most probably to a combination of factors that operated synergistically and sequentially. The heroin market had grown rapidly in the late 1990s, perhaps helped by a decline in drug law enforcement (DLE) in Australia in the early 1990s that facilitated high-level heroin suppliers in Asia to establish large-scale importation heroin networks into Australia. This led to an increase in the availability of heroin, increasingly visible street-based drug markets, increased purity and decreased price of heroin around the country. The Australian heroin market was well established by the late 1990s, but it had a low profit margin with high heroin purity, and a lower price than ever before. The surge in heroin problems led to increased funding of the Australian Federal Police and Customs as part of the National Illicit Drug Strategy in 1998-99, with the result that a number of key individuals and large seizures occurred during 1999-2000, probably increasing the risks of large-scale importation. The combination of low profits and increased success of law enforcement may have reduced the dependability of key suppliers of heroin to Australia at a time when seized heroin was becoming more difficult to replace because of reduced supplies in the Golden Triangle. These factors may have reduced the attractiveness of Australia as a destination for heroin trafficking.

CONCLUSIONS

The Australian heroin shortage in 2001 was due probably to a combination of factors that included increased effectiveness of law enforcement efforts to disrupt networks bringing large shipments of heroin from traditional source countries, and decreased capacity or willingness of major traffickers to continue large scale shipments to Australia.

Markov chain modeling of initiation and demand: the case of the U.S. cocaine epidemic

Everingham and Rydell's Markov chain model of cocaine demand is modified and updated in light of recent data. Key insights continue to hold, e.g., that the proportion of cocaine demand stemming from heavy vs. light users changed dramatically over the 1980s. New insights emerge, e.g., pertaining to the average duration of a career of heavy use (about 12 years) and the negative relationship between levels of heavy use and epidemic "infectivity" or the number of new initiates per current user per year. This illustrates how simple modeling can yield insights directly relevant to managing complex drug control policy questions.

Short and long period optimization of drug doses in the treatment of AIDS

Numerical optimization techniques are useful in solving problems of computing the best inputs for systems described by mathematical models and when the objectives can be stated in a quantitative form. This work concerns the problem of optimizing the drug doses in the treatment of AIDS in terms of achieving a balance between the therapeutic response and the side effects. A mathematical model describing the dynamics of HIV viruses and CD4 cells is used to compute the short term optimal drug doses in the treatments of patients with AIDS by a direct method of optimization using a cost function of Bolza type. The model parameters were fitted to actual published clinical data. In order to simplify the numerical procedures, the control law is expressed as a series and the sub-optimal control is obtained by truncating the higher terms. When the patient reaches a clinically satisfactory state, the LQR - Linear Quadratic Regulator technique is used to determine the long period maintenance doses for the drugs. The doses computed using the LQR technique tend to be smaller than equivalent constant-dose therapy in terms of increase in the counts of CD4+T cells and reduction of the density of free viruses.

Using simulation for AIDS policy modeling: benefits for HIV/AIDS prevention policy makers in Vienna, Austria

In spite of advanced therapies and the success of additional prevention programs, the HIV/AIDS epidemic still remains a challenge. Our paper refers academics, health care managers, and policy makers to the relevance of AIDS policy simulators in better decision-making. By highlighting the types of decisions AIDS policy models can support, we demonstrate the strategic role of AIDS policy simulators for the efficient and effective planning of scarce resources to fight the epidemic. For each type of decision, we then review exemplary AIDS policy simulators that have helped policy makers make better decisions. Finally, we present the benefits of an AIDS policy simulator for HIV/AIDS prevention policy makers in Vienna, Austria.