Analysis of a heroin epidemic model with saturated treatment function

A mathematical model is developed that examines how heroin addiction spreads in society. The model is formulated to take into account the treatment of heroin users by incorporating a realistic functional form that "saturates" representing the limited availability of treatment. Bifurcation analysis reveals that the model has an intrinsic backward bifurcation whenever the saturation parameter is larger than a fixed threshold. We are particularly interested in studying the model's global stability. In the absence of backward bifurcations, Lyapunov functions can often be found and used to prove global stability. However, in the presence of backward bifurcations, such Lyapunov functions may not exist or may be difficult to construct. We make use of the geometric approach to global stability to derive a condition that ensures that the system is globally asymptotically stable. Numerical simulations are also presented to give a more complete representation of the model dynamics. Sensitivity analysis performed by Latin hypercube sampling (LHS) suggests that the effective contact rate in the population, the relapse rate of heroin users undergoing treatment, and the extent of saturation of heroin users are mechanisms fuelling heroin epidemic proliferation

Survival analysis and probability density function of switching heroin model

We study a switching heroin epidemic model in this paper, in which the switching of supply of heroin occurs due to the flowering period and fruiting period of opium poppy plants. Precisely, we give three equations to represent the dynamics of the susceptible, the dynamics of the untreated drug addicts and the dynamics of the drug addicts under treatment, respectively, within a local population, and the coefficients of each equation are functions of Markov chains taking values in a finite state space. The first concern is to prove the existence and uniqueness of a global positive solution to the switching model. Then, the survival dynamics including the extinction and persistence of the untreated drug addicts under some moderate conditions are derived. The corresponding numerical simulations reveal that the densities of sample paths depend on regime switching, and larger intensities of the white noises yield earlier times for extinction of the untreated drug addicts. Especially, when the switching model degenerates to the constant model, we show the existence of the positive equilibrium point under moderate conditions, and we give the expression of the probability density function around the positive equilibrium point

Stability analysis of drinking epidemic models and investigation of optimal treatment strategy

In this research we investigate a class of drinking epidemic models, namely the SPARS type models. The basic reproduction number is derived, and the system dynamical behaviours are investigated for both drinking free equilibrium and drinking persistent equilibrium. The purpose is to determine the long term optimal treatment method and the optimal short period vaccination strategy for controlling the population of the periodic drinkers and alcoholics

A Quantitative Study on the Impact of COVID-19 on Drug Treatment Courts

The purpose of this research study was to investigate how the COVID-19 pandemic impacted drug treatment courts (DTCs) in Pennsylvania. DTCs were created as an alternative to the traditional court system by allowing those convicted of a drug offense to receive treatment. Just as these DTCs were dealing with the opioid crisis, a new obstacle occurred in the form of the COVID-19 pandemic, forcing DTCs to adapt to the quickly evolving and changing mandates and policies implemented by the federal and local governments. The impact of the COVID-19 pandemic on these essential DTCs was examined using a therapeutic jurisprudence perspective. A quantitative methodology was utilized to compare graduation rates before and during the pandemic. The data for this research were obtained from publicly archivable data on DTCs in Pennsylvania. The study sample comprised 3,782 participants admitted into a Pennsylvania DTC between 2017 and 2021. The study utilized a chi-square test of independence to determine if COVID-19 impacted drug treatment court graduation rates. The chi-square test findings showed statistically significant differences between the DTC graduation rates of the opiate group, but no statistically significant difference for the remaining targeted drugs of cocaine, heroin, methamphetamine, and benzodiazepines. This showed the opiate group had a higher graduation rate during COVID-19 than before the pandemic. Odds testing was completed, and the findings showed that every drug type had higher odds of graduating during the pandemic than before, except for the drug type benzodiazepine

Backward bifurcation and reinfection in mathematical models of tuberculosis

Mathematical models are widely used for understanding the transmission mechanisms and control of infectious diseases. Numerous infectious diseases such as those caused by bacterial and viral infections do not confer life long immunity after recovering from the first episode. Consequently, they are characterized by partial or complete loss of immunity and subsequent reinfection. This thesis explores the epidemiological implications of loss of immunity using simple and complex mathematical models. First, a simple basic model mimicking transmission mechanisms of tuberculosis (TB) is proposed with the aim of correcting problems that are often repeated by mathematical modellers when determining underlying bifurcation structures. Specifically, the model makes transparent the problems that may arise if one aggregates all the bifurcation parameters when computing backward bifurcation thresholds and structures. The backward bifurcation phenomenon is an important concept for public health and disease management. This is because backward bifurcation signals that disease will not be eliminated even when the basic reproduction number R0 is decreased below unity; rather, for the disease to be eliminated, R0 has to be reduced below another critical threshold. I provide conditions to find the threshold correctly. Secondly, the simple basic TB model is extended to incorporate epidemiological and biological aspects pertinent to TB transmission such as recurrent TB, which is defined as a second episode of TB following successful recovery from a previous episode. I study the conditions for backward bifurcation in this extended model that features recurrent TB. Mathematical techniques based on the center manifold approach, are used to derive an exact backward bifurcation threshold. Furthermore, both analytical and numerical findings reveal that recurrent TB is capable of inducing a new and rare hysteresis effect where TB will persist when the basic reproduction number is below unity even though there is no backward bifurcation. Moreover, when the reinfection pathway among latently infected individuals is switched off, leaving only recurrent TB, the model analysis indicates that recurrent TB can independently induce a backward bifurcation. However, this will only occur if recurrent TB transmission exceeds a certain threshold. Although this threshold seems to be relatively high when realistic parameters are used, it falls within the recent range estimated in the relevant literature. The second TB model is extended by dividing the latent compartment into two: fast (early latent) and slow (late latent) latent compartments, to enhance realism. Individuals in both early and late compartments are subjected to treatment. The proposed TB model is used to investigate how heterogeneity in host susceptibility influences the effectiveness of treatment. It is found that making the assumption that individuals treated with preventive therapy and recovered individuals (previously treated for active TB) acquire equal levels of protection after initial infection, and are therefore reinfected at the same rate, may obscure dynamics that are imperative when designing intervention strategies. Comparison of reinfection rates between cohorts treated with preventive therapy and recovered individuals who were previously treated from active TB provides important epidemiological insights. That is, the reinfection parameter accounting for the relative rate of reinfection of the cohort treated with preventive therapy is the one that plays the key role in generating qualitative changes in TB dynamics. In contrast, the parameter accounting for the risk of reinfection among recovered individuals (previously treated for active TB) does not play a significant role. The study shows that preventive treatment during early latency is always beneficial regardless of the level of susceptibility to reinfection. And if patients have greater immunity following treatment for late latent infection, then treatment is again beneficial. However, if susceptibility increases following treatment for late latent infection, the effect of treatment depends on the epidemiological setting: (a) for (very) low burden settings, the effect on reactivation predominates and burden declines; (b) for high burden settings, the effect on reinfection predominates and burden increases. This is mostly observed between the two reinfection thresholds, RT2 and RT1, respectively associated with individuals being treated with preventive therapy and individuals with untreated late latent TB infection. Finally, a mathematical model that examines how heroin addiction spreads in society is formulated. The model has many commonalities with the TB model. The global stability properties of the proposed model are analysed using both the Lyapunov direct method and the geometric approach by Li and Muldowney. It is shown that even for a four dimensional model, the use of two well known nonlinear stability techniques becomes nontrivial. When all the parameters of the model are accounted for, it is difficult if not impossible, to design a Lyapunov function. Here I apply the geometric approach to establish a global condition that accounts for all model parameters. If the condition is satisfied, then heroin persistence within the community is globally stable. However, if the global condition is not satisfied heroin users can oscillate periodically in number. Numerical simulations are also presented to give a more complete representation of the model dynamics. Sensitivity analysis performed by Latin hypercube sampling (LHS) suggests that the effective contact rate in the population, the relapse rate of heroin users undergoing treatment, and the extent of saturation of heroin users, are the key mechanisms fuelling heroin epidemic proliferation. However, in the long term, relapse of heroin users undergoing treatment back to a heroin using career, has the most significant impact

Three Essays on Health Economics and Policy Evaluation

This dissertation consists of three essays on the U.S. Health care policy. Each paragraph below refers to the three abstracts for the three chapters in this dissertation, respectively. I provide quantitative evidence on how much Prescription Drug Monitoring Programs (PDMPs) affects the retail opioid prescribing behaviors. Using the American Community Survey (ACS), I retrieve county-level high dimensional panel data set from 2010 to 2017. I employ three separate identification strategies: difference-in-difference, double selection post-LASSO, and spatial difference-in-difference. I compare how the retail opioid prescribing behaviors of counties, that are mandatory for prescribers to check the PDMP before prescribing controlled substances (must-access PDMPs), vary from the counties where such a PDMP check is voluntary. I find must-access PDMP reduces about seven retail opioid prescriptions dispensed per 100 persons per year in each county. But, when I compare must-access PDMPs counties with bordering counties without such law, I find a reduction of three retail opioid prescriptions dispensed per 100 persons per year suggesting the possibility of spillovers of retail opioid prescribing behaviors. As of 2019, all U.S. states, except Missouri, have enacted voluntary Prescription Drug Monitoring Programs (PDMPs). In response to the relatively low uptake of voluntary access, several states have strengthened their PDPMs by requiring providers to access information regarding prescription drug use under certain circumstances. These “must-access” PDPMs require states to view a patient\u27s prescription history to facilitate the detection of suspicious prescription and utilization behaviors. This paper develops causal evidence of the effectiveness of “must-access PDPM laws in reducing prescription opioid overdose death rates relative to voluntary PDMP states. I find that PDMPs are ineffective in reducing prescription opioid overdose deaths overall, but the effects are heterogeneous across states with “must-access” PDMP states. I find that marijuana and naloxone access laws, poverty level, income, and education confound the impact of must-access PDMPs on prescription opioid overdose deaths. The optional provision of Medicaid expansion, through the Affordable Care Act (ACA), has triggered a national debate among diverse stakeholders regarding the impacts of Medicaid coverage on various dimensions of public health, costs, and benefits. Randomized experiments like the Rand Health Insurance Experiment and the Oregon Health Insurance Experiment have generated some credible estimates of the average treatment effects of insurance access. However, identical policy interventions can have heterogeneous effects on different subpopulations. This paper uses data from the Oregon Health Insurance Experiment to estimate the heterogeneous treatment effects of access to Medicaid on health care utilization, preventive care utilization, financial strain, and self-reported physical and mental health. I detect heterogeneous treatment effects using a cluster-robust generalized random forest, a causal machine learning approach. I find that the impact of Medicaid is more pronounced among relatively older non-elderly and poorer households, consistent with standard adverse selection theory. Furthermore, I implement the “efficient policy learning, another machine learning strategy, to identify policy changes that prioritize providing Medicaid coverage to the subgroups that are likely to benefit the most. On average, the proposed reforms would improve the average probability of outpatient visits, preventive care use, overall health outcomes, having a personal doctor and clinic, and happiness by a range of 2% to 9% over a random assignment baseline. These findings help design Medicaid Section 1115 waiver

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

CITATION: Mushanyu, J., Nyabadza, F. & Stewart, A. G. R. 2015. Modelling the trends of inpatient and outpatient rehabilitation for methamphetamine in the Western Cape province of South Africa. BMC Research Notes, 8:797, doi:10.1186/s13104-015-1741-4.The original publication is available at http://bmcresnotes.biomedcentral.comBackground: 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 Ra 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.http://bmcresnotes.biomedcentral.com/articles/10.1186/s13104-015-1741-4Publisher's versio

Opioid and stimulant use among a sample of corrections-involved drug users : seeking an understanding of high-risk drug decisions within a system of constraint.

In the United States, high-risk drug use remains a significant social problem. Opioids and stimulants are two drug classes that have contributed to substantial recent increases in drug-related arrests, overdose, and mortality. Kentucky has been particularly devastated by high rates of opioid and stimulant use. Opioid and stimulant effects, while highly rewarding, can result in adverse consequences. Still, some people choose to use these drugs, and choose to continue using even after experiencing adverse consequences, such as incarceration. The aim of this study was to explore high-risk drug use among a sample of corrections-involved adults in Kentucky and to identify endogenous and exogenous factors with the potential to have influenced drug-related decision-making prior and subsequent to incarceration. Attention was paid to understanding concomitant opioid and stimulant use and heroin use. Survey data collected as part of an ongoing corrections-based substance use treatment program outcomes study were examined. The final sample (N=1,563) included adults released into Kentucky counties between 2012-2017. Non-parametric statistical tests and multinomial logistic regression were used to identify factors associated with opioid, stimulant, and concomitant use; binary logistic regression was used to identify factors associated with heroin use. Results indicate that opioid and stimulant use was endemic in this sample, though rates of use subsequent to incarceration were lower than pre-incarceration rates. During the 30-day period prior to incarceration, 29.0% of participants reported concomitant use, 28.5% reported opioid use, and 18.0% reported stimulant use. During the one-year post-release period, 11.9% of participants reported concomitant use, 12.5% reported opioid use, and 8.3% reported stimulant use. During this post-release period, 10.7% reported heroin use. Concomitant and heroin use positively correlated with many factors with the potential to adversely influence cognition and constrain choice. Similar relationships between many of these factors and outcomes involving other drug or no drug use were not observed. Behavioral economics, a molar view of choice and behavior, was used to conceptualize how factors in the lives of participants had the potential to influence and constrain decision-making in respect to high-risk drugs. Findings are discussed in light of how they may inform future research, policy, and practice

Understanding Mediators of the Relationship Between HIV-Related Stigma and Medication Adherence Among People Living with HIV (HIV)

Although accumulated research demonstrates that HIV-related stigma is associated with medication adherence difficulties for PLWH, this literature is limited by inconsistencies in the conceptualization and assessment of both stigma and adherence. Additionally, few empirical research studies provide tests of the psychosocial mechanisms that may account for the stigma-adherence relationship. The present study sought to clarify these relationships using more refined measures, which assessed multiple domains of stigma (i.e., internalized, anticipated, and enacted stigmas) and multiple domains of adherence (i.e., global and intentional nonadherence). It was further hypothesized that two mediational pathways, depressive symptoms and disclosure concerns, may mediate the association between HIV-related stigmas and adherence. A clinic-based sample of 205 people living with HIV (PLWH) completed a questionnaire battery using audio computer-assisted self-interviewing (ACASI). Contrary to previous research, there was no association between HIV-related stigmas and adherence. There was also no evidence that disclosure concerns acted as a mediational pathway, but there was evidence that depression mediated the association between internalized and enacted stigmas and intentional nonadherence since diagnosis. This lack of an association between stigma and adherence may be attributable to a different stage of the HIV epidemic, in which simplified medication regimens present fewer obstacles for PLWH. Future research should prospectively explore the interrelationships among HIV-related stigmas, mediators, and adherence outcomes in the current era of the HIV epidemic and with differing populations

Modelling the dynamics of methamphetamine abuse in the Western Cape

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2011.Includes bibliographyENGLISH ABSTRACT: The production and abuse of methamphetamine has increased dramatically in South Africa, especially in the Western Cape province. A typical methamphetamine use cycle consists of concealed use after initiation, addiction, treatment and recovery. The model by Nyabadza and Musekwa in [32], is extended to include a core group, fast and slow progression to addiction. The model is analysed analytically and numerically using mass action incidence function and non-linear incidence function. The analysis of the model with mass action incidence is presented in terms of the methamphetamine epidemic threshold R0. The analysis shows that the drug free equilibrium is locally asymptotically stable when R0 < 1 and drug persistent equilibrium is locally asymptotically stable when R0 > 1. The model also exhibits a backward bifurcation. Sensitivity analysis of the model on R0 is performed. The most sensitive parameters are transmission rate and recruitment rate of individuals into the core group. The non-linear incidence incorporates innovators and behaviour change. Analytically, the model is analysed in the absence of behaviour change. With behaviour change two cases were considered. Firstly without innovators and secondly with innovators. In the absence of innovators the non-linear incidence reduced to standard incidence and similar results to the ones in the first model were obtained. With the presence of innovators there is no drug free equilibrium. Numerically we fit the model to data on the number of patients who enter into treatment centers for rehabilitation. Using the fitted model, we determine the prevalence and incidence of methamphetamine abuse. We investigate the impact of behaviour change, ‘reinfection’ rate as well as uptake rate into treatment on prevalence. Our results suggest that intervention and prevention programs focusing on behaviour change and uptake rate into treatment would reduce the prevalence. Projections are made to determine the possible long term trends of the prevalence of methamphetamine abuse in the Western Cape. We give suggestions related to data that should be collected from a modelling perspective.AFRIKAANSE OPSOMMING: Die vervaardiging en misbruik van metamfetamien het dramaties in Suid-Afrika toegeneem, veral in die Wes-Kaap provinsie. ’n Tipiese metamfetamien gebruiksiklus bestaan uit heimlike gebruik na aanvang, verslawing, behandeling en herstel. Die model deur Nyabadza en Musekwa in [32], is uitgebrei om ’n kerngroep in te sluit, vinnige en stadige verloop tot verslawing. Die model is analities en numeries ontleed deur van massa-aksie insidensie funksie en ’n nie-liniêre insidensie funksie gebruik te maak. Die ontleding van die model met massa-aksie insidensie word voorgestel in terme van die metamfetamien epidemiese drempel R0. Die ontleding toon dat die dwelmvrye ewewig lokaal asimptoties stabiel is as R0 1. Die model beeld ook ’n terugwaartse bifurkasie uit. Sensitiwiteitsontleding van die model ten opsigte van R0 is uitgevoer. Die mees sensitiewe parameters is die oordraagbaarheidskoers en die rekrute koers van individue in die kerngroep in. Die nuwelinge en gedragsverandering word deur die nie-liniêre insidensie opgeneem. Analities, is die model ontleed in die afwesigheid van gedragsverandering. Met gedragsverandering is twee gevalle beskou. Eerstens sonder nuwelinge en tweedens met nuwelinge. In die afwesigheid van nuwelinge is die nie-liniêre insidensie herlei tot standaard insidensie en soortgelyke resultate is verkry, as dié wat in die eerste model verkry is. Met die aanwesigheid van nuwelinge is daar geen dwelmvrye ewewig nie. Numeries pas ons die model aan die data wat betrekking het met die aantal pasiënte wat in rehabilitasie sentra opgeneem word vir behandeling. Deur die gepaste model te gebruik, het ons die voorkoms en insidensie van metamfetamien misbruik bepaal. Ons ondersoek die impak van gedragsverandering, “re-infeksie” koers sowel as die koers van opname in behandeling op voorkoms. Ons resultate toon dat intervensie- en voorkomingsprogramme sal voorkoms verlaag, wat op die gedragsverandering en die koers van opname in behandeling konsentreer. Die model is ook gebruik om die aantal metamfetamien gebruikers te projekteer. Ons maak voorstelle verwant aan die data, wat vanuit ‘n modellerings-oogpunt ingesamel moet word