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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/6817 |
| Date | 03 1900 |
| Creators | Kalula, Asha Saidi |
| Contributors | Nyabadza, Farai, University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 86 p. : ill. |
| Rights | University of Stellenbosch |
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