Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2011. / Includes bibliography / ENGLISH 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 en die dwelmblydende ewewig is lokaal asimptoties stabiel 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.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Clinical results have indicated that abuse of multiple drugs/substances has devastating health and social consequences. The combined abuse of alcohol and the highly addictive methamphetamine has worsened the drug epidemic in South Africa, especially in the Western Cape Province. Using non-linear ordinary differential equations, we formulate a deterministic mathematical model for alcohol-methamphetamine coabuse epidemic. We prove that the growth of the co-abuse epidemic is dependent on the threshold parameters of the individual substances of abuse. The substance with the maximum reproduction number dominates the epidemic. We also prove that the equilibria points of the co-abuse sub-models are locally and globally asymptotically stable when the sub-model threshold parameters are less than unity. Using parameters values derived from the sub-model fittings to data, a population estimate of co-users of alcohol and methamphetamine under treatment is estimated with a prevalence of about 1%. Although the results show of a small proportion of co-users of alcohol and methamphetamine in the province, the prevalence curve is indicative of a persistent problem. Numerical simulation results reveal that co-abuse epidemic would persists when both reproduction numbers are greater than one. Results from sensitivity analysis shows that the individual substance transmission rates between users of methamphetamine and/or alcohol and the general susceptible population are the most vital parameters in the co-abuse epidemic. This suggests the need to emphasise on preventive measures through educational campaigns and social programs that ensure minimal recruitment into alcohol or methamphetamine abuse. Model analysis using the time-dependent controls (policies) emphasizes the need to allocate even more resources on educational campaigns against substance abuse and on effective treatment services that minimizes or eliminates rampant cases of relapse into substance abuse. / AFRIKAANSE OPSOMMING: Kliniese resultate toon dat die misbruik van meer as een dwelmmiddel verwoestende gesondheids-en sosiale gevolge het. Die gekombineerde misbruik van alkohol en die hoogsverslawende methamphetamine het die dwelm-epidemie in Suid-Afrika vererger, veral in die Wes-Kaapse provinsie. Deur van nie-lineere gewone diffensiaalvergelykings gebruik te maak, formuleer ons ’n deterministiese wiskundige model vir epidemie van die gesamentlike misbruik van alkohol en methamphetamine. Ons toon aan dat die groei van die sogenaamde mede-misbruik epidemie afhanklik is van die drumpelparameters van die individuele middels wat misbruik word. Die middels met die grootste voortbringende syfer domineer die epidemie. Ons bewys ook dat die ekwilibriumpunte van die mede-misbruik submodelle plaaslik en globaal asimptoties stabiel is wanneer die sub-model drumpelparameters kleiner as een is. Deur die submodelle op werklike data te pas word waardes vir die drumpelparameters afgelei en word daar beraam dat daar ongeveer 1% van die populasie mede-misbruikers van alkohol en methamphetamine onder behandeling is. Alhoewel die data ’n klein persentasie van mede-misbruikers van alkohol en methamphetamine in die provinsie toon, dui die voorkomskurwe op ’n groeiende endemie en voortdurende probleem. Resultate uit numeriese simulasie toon dat die mede-misbruik epidemie sal voortduur indien beide reproduserende syfers groter as een sal wees. Resultate van sensitiwiteitsanalise toon dat die individuele middeloordragkoerse tussen gebruikers van methamphetamine en/of alkohol en die gewone vatbare populasie die mees noodsaaklike parameters in die mede-misbruik epidemie is. Dit stel voor dat daar meer klem gelê moet word op voorkomingsmaatreëls deur opvoedkundige veldtogte en sosiale programme om te verseker dat minder alkohol en/of methamphetamine misbruik sal word. Model-analise wat gebruik maak van tyd-afhanklike kontroles (beleide) lê verder klem op die feit dat selfs meer hulpbronne aan opvoedkundige veldtogte teen dwelmmisbruik toegewy moet word, asook die effektiewe behandeling wat gevalle van terugval in dwelmmisbruik sal minimeer of elimineer.