| Summary: | 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.
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