Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The high sickle cell gene frequency has been hypothesised to be related to the protective advantage against malaria disease among heterozygous individuals. In this thesis, we study the interaction between the dynamics of malaria and sickle cell gene. The main aim is to investigate the impact of malaria treatment on the frequency of sickle cell gene. For this, we develop a mathematical model that describes the interactions between malaria and sickle cell gene under malaria treatment. The model includes both homozygous for the normal gene (AA) and heterozygous for sickle cell gene (AS) and assumes that AS individuals are not treated since they do not show clinical symptoms. We first analyse the model without malaria treatment, using singular perturbation techniques, basing on the fact that epidemiological and demographical dynamics occur on two different time scales (fast and slow dynamics). Our analysis on the fast time scale shows that high sickle cell gene frequency leads to high endemic levels for longer duration of parasitemia among heterozygous individuals. However, if the duration of parasitemia is reduced then high sickle cell gene frequency is associated with low endemic levels. We also note that on the slow time scale, the invasion ability of sickle cell gene is dependent on the malaria epidemiological parameters. The invasion coefficient given as the difference in the weighted death rates of AA and AS individuals is used as a measure to determine the establishment of sickle cell gene in the population. Results show that, the gene may establish itself if the weighted death rate of AA individuals is greater than that of AS individuals otherwise it fails. We note that, high mortality of AA individuals leads to establishment of sickle cell gene in the population. Then we analysed the model with treatment, our results indicate that the frequency of sickle cell gene decreases with an increase in the recovery rate of AA individuals. We thus conclude that eradication of malaria disease will lead to a reduction in sickle cell gene frequency. / AFRIKAANSE OPSOMMING: Daar word veronderstel dat die hoë sekelsel geenfrekwensie onder heterosigotiese individue verwant is aan die beskermende voordeel teen malaria siekte. In hierdie verhandeling ondersoek ons die wisselwerking tussen die dinamika van malaria en die sekelsel geen. Die hoofdoel is om die invloed van malaria behandeling op die frekwensie van die sekelsel geen te ondersoek. Hiervoor het ons ‘n wiskundige model ontwikkel, wat die wisselwerking tussen die dinamika van malaria en die sekelsel geen met malaria behandeling, beskryf. Die model sluit beide homosigotiese vir die normale geen (AA) en heterosigotiese vir die sekelsel geen (AS) in, en neem aan dat AS individue nie behandel is nie omdat hulle nie die eerste kliniese simptome getoon het nie. Ons ontleed eers die model sonder malaria behandeling, deur gebruik te maak van enkelvoudige pertubasie tegnieke, wat gegrond is op die feit dat epidemiologiese en demografiese dinamika plaasvind op twee verskillende tydskale (vinnige en stadige dinamika). Ons ontleding op die vinnige tydskaal dui dat hoë sekelsel geenfrekwensie onder heterosigotiese individue lei tot hoë endemiese vlakke vir ‘n langer duur van parasitemie. Nietemin, as die duur van parasitemie afneem, dan word hoë sekelsel geenfrekwensie verbind met lae endemiese vlakke. Ons neem ook waar dat op die stadige skaal die indringingsvermoë van die sekelsel afhanklik is van malaria se epidemiologiese parameters. Die indringingskoëffisiënt wat bereken word as die verskil van die geweegde sterftekoerse van AA en AS individue, word gebruik as ‘n maatstaf om die vestiging van die sekelsel geen in die bevolking te bepaal. Resultate toon dat die geen homself kan vestig as die geweegde sterftekoers van AA individue groter is as di e van die AS individue, andersins misluk dit. Ons let op dat hoë mortaliteit van AA individue lei tot die vestiging van die sekelsel geen in die bevolking. Daarna het ons die model wat behandeling insluit ge-analiseer en ons resultate toon dat die frekwensie van die sekelsel geen afneem met ‘n toename in die herstelkoers van AA individue. Ons kom dus tot die gevolgtrekking dat die uitwissing van malaria siekte sal lei tot die afname in sekelsel geenfrekwensie.