BackgroundDependence 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.MethodsThe 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.ResultsThe 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 and , 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.ConclusionThe results suggest that inpatient rehabilitation programs have an increased potential of enhancing the chances of recovery for methamphetamine addicts.

Adolescence methamphetamine use is an issue of considerable concern due to its correlation with later delinquency, divorce, unemployment and health problems. Understanding how adolescents initiate methamphetamine abuse is important in developing effective prevention programs. We formulate a mathematical model for the spread of methamphetamine abuse using nonlinear ordinary differential equations. It is assumed that susceptibles are recruited into methamphetamine use through imitation. An epidemic threshold value, [Formula: see text], termed the abuse reproduction number, is proposed and defined herein in the drug-using context. The model is shown to exhibit the phenomenon of backward bifurcation. This means that methamphetamine problems may persist in the population even if [Formula: see text] is less than one. Sensitivity analysis of [Formula: see text] was performed to determine the relative importance of different parameters in methamphetamine abuse initiation. The model is then fitted to data on methamphetamine users less than 20 years old reporting methamphetamine as their primary substance of abuse in the treatment centres of Cape Town and parameter values that give the best fit are chosen. Results show that the proportion of methamphetamine users less than 20 years old reporting methamphetamine as their primary substance of abuse will continue to decrease in Cape Town of South Africa. The results suggest that intervention programs targeted at reducing adolescence methamphetamine abuse, are positively impacting methamphetamine abuse.

A study is made of the minimum number of generators of the n-th direct power of certain finitely generated groups.

The abuse of drugs is now an epidemic globally whose control has been mainly through rehabilitation. The demand for drug abuse rehabilitation has not been matched with the available capacity resulting in limited placement of addicts into rehabilitation. In this paper, we model limited rehabilitation through the Hill function incorporated into a system of nonlinear ordinary differential equations. Not every member of the community is equally likely to embark on drug use, risk structure is included to help differentiate those more likely (high risk) to abuse drugs and those less likely (low risk) to abuse drugs. It is shown that the model has multiple equilibria, and using the centre manifold theory, the model exhibits the phenomenon of backward bifurcation whose implications to rehabilitation are discussed. Sensitivity analysis and numerical simulations are performed. The results show that saturation in rehabilitation will in the long run lead to the escalation of drug abuse. This means that limited access to rehabilitation has negative implications in the fight against drug abuse where rehabilitation is the main form of control. This suggests that increased access to rehabilitation is likely to lower the drug abuse epidemic.

Drug dependence is a 'chronic disease' treatable through rehabilitation. Many drug addicts progress through a series of rehabilitation and relapsing episodes. In this paper, we formulate a mathematical model with n-alternate stages of rehabilitation and relapsing. The dynamics of drug abuse are treated as an infectious disease that spreads through a population. The model analysis shows that the model has two equilibria, the drug free equilibrium and the drug persistent equilibrium, that are both globally stable when the threshold R 0 < 1 and R 0 > 1 respectively. The model is fitted to data on individuals under repeated rehabilitation and parameter values that give the best fit chosen. The projections carried out the long term trends of proportions for repeated rehabilitants. The relative impact for each subgroup is determined to find out which population subgroup is responsible for a disproportionate number of initiations. The results have huge implications to designing policies aligned to rehabilitation processes.