Thesis (PhD (Physics))--Stellenbosch University, 2009. / ENGLISH ABSTRACT: This dissertation investigates certain key aspects of mathematical modeling of HIV and TB epidemics in South Africa with particular emphasis on data from a single well-studied community. Data collected over a period of 15 years (1994 to 2009) in Masiphumelele, a township near Cape Town, South Africa are used to develop a community-level mathematical model of the local HIV-TB epidemic. The population is divided into six compartments and a system of di®erential equations is derived to describe the spread of the dual epidemic. Our numerical results suggest that increased access to antiretroviral therapy (ART) could decrease not only the HIV prevalence, but also the TB noti¯cation rate. We present a modeling framework for studying the statistical properties of °uctuations in models of any population of a similar size. Viewing the epidemic as a jump process, the method entails an expansion of a master equation in a small parameter; in this case in inverse powers of the square root of the population size. We derive two-time correlation functions to study the correlation between di®erent types of active TB events, and show how a temporal element could be added to the de¯nition of TB clusters, which are currently de¯ned solely by DNA type. We add age structure to the HIV-TB model in order to investigate the demographical impact of HIV-TB epidemics. Our analysis suggests that, contrary to general belief, HIV-positive cases are not making a substantial contribution to the spread of TB in Masiphumelele. We develop an age-structured model of the HIV-TB epidemic at a national level in order to study the potential impact of a proposed universal test and treat program for HIV on dual HIV-TB epidemics. Our simulations show that generalized ART could signi¯cantly reduce the TB noti¯cation rate and the TB-related mortality rate in the short term. The timescale of the impact of ART on HIV prevalence is likely to be longer. We study the potential impact of more conventional control measures against HIV. Guidance for possible future and/or additional interventions emerge naturally from the results. We advocate a reduction in intergenerational sex, based on our ¯nding that 1.5-2.5 standard deviation in the age di®erence between sexual partners is necessary to create and sustain a major HIV epidemic. A simulation framework is developed to help quantify variance in age-structured epidemic models. The expansion technique is generalized to derive a Fokker-Planck equation. Directions for future work, particularly in terms of developing methods to model °uctuations and validate mixing assumptions in epidemiological models, are identi¯ed. / AFRIKAANSE OPSOMMING: Hierdie proefskrif ondersoek aspekte van die wiskundige modelering van HIV en TB epi- demies in Suid Afrika en fokus ook op 'n spesi¯eke gemeenskap. Data wat oor 'n periode van 15 jaar ingesamel is (1994 tot 2009) in Masiphumelele, 'n woonbuurt naby Kaapstad, Suid Afrika word gebruik om 'n wiskundige model te skep wat HIV-TB in die gemeen- skap modeleer. Die populasie word in ses kompartemente verdeel en 'n stel di®erensiaal vergelykings word afgelei om die verspreiding van di¶e epidemies te ondersoek. Ons nu- meriese resultate toon aan dat verhoogde toegang tot antiretrovirale behandeling (ARB) die potensiaal het om HIV prevalensie die TB koers beduidend te laat daal. Ons ontwikkel 'n raamwerk waarmee die statistiese eienskappe van °uktuasies ondersoek kan word in enige populasie van dieselfde grootte. Die metode ontwikkel 'n meester vergelyking vir die on- derliggende geboorte-dood stogastiese proses en brei dit uit in terme van 'n klein parameter; in di¶e geval in inverse magte van die vierkantswortel van die populasie grootte. Die twee-tyd korrelasie funksies word afgelei, en word gebruik om die korrelasie tussen verskillende tipes van TB episodes te bestudeer, asook om te wys hoe 'n tydselement aan die de¯nisie van TB groeperings gegee kan word. Di¶e word tans slegs d.m.v DNA tipe geklassi¯seer. Ouderdom- struktuur word aan die model toegevoeg om die demogra¯ese impak van HIV-TB epidemies te bestudeer. Ons analise toon aan dat, anders as wat algemeen aanvaar word, maak HIV- positiewe gevalle nie 'n groot bydrae tot die verspreiding van TB in Masiphumelele nie. Ons ontwikkel 'n ouderdom-gestruktureerde model van HIV-TB op nasionale vlak en gebruik die model om die potensiÄele impak van 'n universele toets- en behandel strategie op die HIV-TB epidemies te ondersoek. Ons simulasies toon aan dat algemene ARB waarskynlik 'n groot impak op die TB aanmeldings koers asook die TB-verwante mortaliteits koers kan h^e binne 'n relatiewe kort tydperk. Die impak op HIV prevalensie sal eers oor 'n veel langer periode duidelik word. Ons ondersoek ook die moontlikheid van meer konvensionele beheermaa- treels. Ons ontmoedig tussengenerasie seksuale omgang, gegrond op ons bevinding dat 'n standaard afwyking van 1.5-2.5 in die ouderdoms verskil tussen seksuele vennote, nodig is om 'n HIV epidemie van stapel te stuur en te onderhou. Ons ontwikkel 'n simulasie raamwerk om variansie in ouderdomgestruktureerde modelle te benader. Die uitbreidingstegniek word veralgemeen om 'n Fokker-Planck vergelyking af te lei. Ons identi¯seer probleme in die on- twikkeling van metodes om interaksie patrone en °uktuasies te modeleer in epidemiologiese modelle as opgawe vir toekomstige werk.

MENVSC (Geography) / Department of Geography and Geo-Information Sciences / Semi-arid regions of Africa are prone to epidemics of malaria. Epidemic malaria occurs along the geographical margins of endemic regions, when the equilibrium between the human, parasite and mosquito vector populations are occasionally disturbed by changes in one or more meteorological factors and a sharp but temporary increase in disease incidence results. Monthly rainfall and temperature data from the South African Weather Service and malaria incidence data from Department of Health were used to determine the influence of meteorological variables on malaria transmission in Limpopo from 1998-2014. Meteorological influences on malaria transmission were analyzed using time series analysis techniques. Climate suitability for malaria transmission was determined using MARA distribution model. There are three distinct modes of rainfall variability over Limpopo which can be associated with land falling tropical cyclones, cloud bands and intensity of the Botswana upper high. ENSO and ENSO-Modoki explains about 58% of this variability. Malaria epidemics were identified using a standardized index, where cases greater than two standard deviations from the mean are identified as epidemics. Significant positive correlations between meteorological variables and monthly malaria incidence is observed at least one month lag time, except for rainfall which shows positive correlation at three months lag time. Malaria transmission appears to be strongly influenced by minimum temperature and relative humidity (R = 0.52, p<0.001). A SARIMA (2, 1, 2) (1, 0, 0)12 model fitted with only malaria cases has prediction performance of about 53%. Warm SSTs of the SWIO and Benguela Niño region west of Angola are the dominant predictors of malaria epidemics in Limpopo in the absence of La Niña. Warm SSTs over the equatorial Atlantic and Benguela Niño region results in the relaxation of the St. Helena high thus shifting the rainy weather to south-east Africa. La Niña have been linked with increased malaria cases in south-east Africa. During El Niño when rain bearing systems have migrated east of Madagascar ridging of the St. Helena high may produce conducive conditions for malaria transmission. Anomalously warmer and moist winters preceding the malaria transmission season are likely to allow for high mosquito survival and the availability of the breeding sites thus high population in the beginning of the transmission season hence resulting in increased epidemics. / NRF