Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: This thesis is a study of a generalisation, due to R. Harte (see [9]), of Fredholm theory in the context of bounded linear operators on Banach spaces to a theory in a Banach algebra setting. A bounded linear operator T on a Banach space X is Fredholm if it has closed range and the dimension of its kernel as well as the dimension of the quotient space X/T(X) are finite. The index of a Fredholm operator is the integer dim T−1(0)−dimX/T(X). Weyl operators are those Fredholm operators of which the index is zero. Browder operators are Fredholm operators with finite ascent and descent. Harte’s generalisation is motivated by Atkinson’s theorem, according to which a bounded linear operator on a Banach space is Fredholm if and only if its coset is invertible in the Banach algebra L(X) /K(X), where L(X) is the Banach algebra of bounded linear operators on X and K(X) the two-sided ideal of compact linear operators in L(X). By Harte’s definition, an element a of a Banach algebra A is Fredholm relative to a Banach algebra homomorphism T : A ! B if Ta is invertible in B. Furthermore, an element of the form a + b where a is invertible in A and b is in the kernel of T is called Weyl relative to T and if ab = ba as well, the element is called Browder. Harte consequently introduced spectra corresponding to the sets of Fredholm, Weyl and Browder elements, respectively. He obtained several interesting inclusion results of these sets and their spectra as well as some spectral mapping and inclusion results. We also convey a related result due to Harte which was obtained by using the exponential spectrum. We show what H. du T. Mouton and H. Raubenheimer found when they considered two homomorphisms. They also introduced Ruston and almost Ruston elements which led to an interesting result related to work by B. Aupetit. Finally, we introduce the notions of upper and lower semi-regularities – concepts due to V. M¨uller. M¨uller obtained spectral inclusion results for spectra corresponding to upper and lower semi-regularities. We could use them to recover certain spectral mapping and inclusion results obtained earlier in the thesis, and some could even be improved. / AFRIKAANSE OPSOMMING: Hierdie tesis is ‘n studie van ’n veralgemening deur R. Harte (sien [9]) van Fredholm-teorie in die konteks van begrensde lineˆere operatore op Banachruimtes tot ’n teorie in die konteks van Banach-algebras. ’n Begrensde lineˆere operator T op ’n Banach-ruimte X is Fredholm as sy waardeversameling geslote is en die dimensie van sy kern, sowel as di´e van die kwosi¨entruimte X/T(X), eindig is. Die indeks van ’n Fredholm-operator is die heelgetal dim T−1(0) − dimX/T(X). Weyl-operatore is daardie Fredholm-operatore waarvan die indeks gelyk is aan nul. Fredholm-operatore met eindige styging en daling word Browder-operatore genoem. Harte se veralgemening is gemotiveer deur Atkinson se stelling, waarvolgens ’n begrensde lineˆere operator op ’n Banach-ruimte Fredholm is as en slegs as sy neweklas inverteerbaar is in die Banach-algebra L(X) /K(X), waar L(X) die Banach-algebra van begrensde lineˆere operatore op X is en K(X) die twee-sydige ideaal van kompakte lineˆere operatore in L(X) is. Volgens Harte se definisie is ’n element a van ’n Banach-algebra A Fredholm relatief tot ’n Banach-algebrahomomorfisme T : A ! B as Ta inverteerbaar is in B. Verder word ’n Weyl-element relatief tot ’n Banach-algebrahomomorfisme T : A ! B gedefinieer as ’n element met die vorm a + b, waar a inverteerbaar in A is en b in die kern van T is. As ab = ba met a en b soos in die definisie van ’n Weyl-element, dan word die element Browder relatief tot T genoem. Harte het vervolgens spektra gedefinieer in ooreenstemming met die versamelings van Fredholm-, Weylen Browder-elemente, onderskeidelik. Hy het heelparty interessante resultate met betrekking tot insluitings van die verskillende versamelings en hulle spektra verkry, asook ’n paar spektrale-afbeeldingsresultate en spektraleinsluitingsresultate. Ons dra ook ’n verwante resultaat te danke aan Harte oor, wat verkry is deur van die eksponensi¨ele-spektrum gebruik te maak. Ons wys wat H. du T. Mouton en H. Raubenheimer verkry het deur twee homomorfismes gelyktydig te beskou. Hulle het ook Ruston- en byna Rustonelemente gedefinieer, wat tot ’n interessante resultaat, verwant aan werk van B. Aupetit, gelei het. Ten slotte stel ons nog twee begrippe bekend, naamlik ’n onder-semi-regulariteit en ’n bo-semi-regulariteit – konsepte te danke aan V. M¨uller. M¨uller het spektrale-insluitingsresultate verkry vir spektra wat ooreenstem met bo- en onder-semi-regulariteite. Ons kon dit gebruik om sekere spektrale-afbeeldingsresultate en spektrale-insluitingsresultate wat vroe¨er in hierdie tesis verkry is, te herwin, en sommige kon selfs verbeter word.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: In this thesis, we discuss and apply the Monte Carlo and integral transform methods in pricing options. These methods have proved to be very e ective in the valuation of options especially when acceleration techniques are introduced. By rst pricing European call options we have motivated the use of these methods in pricing arithmetic Asian options which have proved to be di cult to price and hedge under the Black􀀀Scholes framework. The arithmetic average of the prices in this framework, is a sum of correlated lognormal distributions whose distribution does not admit a simple analytic expression. However, many approaches have been reported in the academic literature for pricing these options. We provide a hedging strategy by manipulating the results by Geman and Yor [42] for continuous xed strike arithmetic Asian call options. We then derive a double Laplace transform formula for pricing continuous Asian call options following the approach by Fu et al. [39]. By applying the multi-Laguerre and iterated Talbot inversion techniques for Laplace transforms to the resulting pricing formula we obtain the option prices. Finally, we discuss the shortcomings of using the Laplace transform in pricing options. / AFRIKAANSE OPSOMMING: In hierdie tesis bespreek ons Monte Carlo- en integraaltransform metodes om die pryse van nansi ele opsies te bepaal. Hierdie metodes is baie e ektief, veral wanneer versnellingsmetodes ingevoer word. Ons bepaal eers die pryse van Europese opsies as motivering, voordat ons die bostaande metodes gebruik vir prysbepaling van Asiatiese opsies met rekenkundige gemiddeldes, wat baie moeiliker is om te hanteer in die Black􀀀Scholes raamwerk. Die rekenkundige gemiddelde van batepryse in hierdie raamwerk is 'n som van gekorreleerde lognormale distribusies wie se distribusie nie oor 'n eenvoudige analitiese vorm beskik nie. Daar is egter talle benaderings vir die prysbepaling van hierdie opsies in die akademiese literatuur. Ons bied 'n verskansingsstrategie vir Asiatiese opsies in kontinue tyd met 'n vaste trefprys aan deur die resultate van Geman en Yor [42] te manipuleer. Daarna volg ons Fu et al. [39] om 'n dubbele Laplace transform formule vir die pryse af te lei. Deur toepassing van multi-Laguerre en herhaalde Talbotinversie tegnieke vir Laplace transforms op hierdie formule, bepaal ons dan die opsiepryse. Ons sluit af met 'n bespreking van die tekortkominge van die gebruik van die Laplace transform vir prysbepaling.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: In this thesis we consider the essential topics relating to stereo-vision and the correspondence problem in general. The aim is to reconstruct a dense 3D scene from images captured by two spatially related cameras. Our main focus, however, is on speed and real-time implementation on a standard desktop PC. We wish to use the CPU to solve the correspondence problem and to reserve the GPU for model rendering. We discuss three fundamental types of algorithms and evaluate their suitability to this end. We eventually choose to implement a hierarchical version of the dynamic programming algorithm, because of the good balance between accuracy and speed. As we build our system from the ground up we gradually introduce necessary concepts and established geometric principles, common to most stereovision systems, and discuss them as they become relevant. It becomes clear that the greatest weakness of the hierarchical dynamic programming algorithm is scanline inconsistency. We nd that the one-dimensional LULU- lter is computationally inexpensive and e ective at removing outliers when applied across the scanlines. We take advantage of the hierarchical structure of our algorithm and sub-pixel re nement to produce results at video rates (roughly 20 frames per second). A 3D model is also constructed at video rates in an on-line system with only a small delay between obtaining the input images and rendering the model. Not only is the quality of our results highly competitive with those of other state of the art algorithms, but the achievable speed is also considerably faster. / AFRIKAANSE OPSOMMING: In hierdie tesis beskou ons die noodsaaklike onderwerpe wat in die algemeen verband hou met stereovisie en die ooreenstemmingsprobleem. Die mikpunt is om 'n digte 3D toneel te rekonstrueer vanaf beelde wat deur twee ruimtelik-verwante kameras vasgelê is. Ons hoofdoel is egter spoed, en intydse implementering op 'n standaard rekenaar. Ons wil die SVE (CPU) gebruik om die ooreenstemmingsprobleem op te los, en reserveer die GVE (GPU) vir model-beraping. Ons bespreek drie fundamentele tipes algoritmes en evalueer hul geskiktheid vir hierdie doel. Ons kies uiteindelik om 'n hiërargiese weergawe van die dinamiese programmeringsalgoritme te implementeer, as gevolg van die goeie balans tussen akkuraatheid en spoed. Soos wat ons ons stelsel van die grond af opbou, stel ons geleidelik nodige konsepte voor en vestig meetkundige beginsels, algemeen tot meeste stereovisie stelsels, en bespreek dit soos dit toepaslik word. Dit word duidelik dat skandeerlyn-strydigheid die grootste swakheid van die hiërargiese dinamiese programmeringsalgoritme is. Ons vind dat die een-dimensionele LULU- lter goedkoop is in terme van berekeninge, en e ektief aangewend kan word om uitskieters te verwyder as dit dwarsoor skandeerlyne toegepas word. Ons buit die hiërargiese struktuur van ons algoritme uit en kombineer dit met sub-piksel verfyning om resultate te produseer teen video tempo (ongeveer 20 raampies per sekonde). 'n 3D model word ook gekonstrueer teen video tempo in 'n stelsel wat aanlyn loop, met slegs 'n klein vertraging tussen die verkryging van die intree-beelde en die beraping van die model. Die kwaliteit van ons resultate is nie net hoogs mededingend met dié van die heel beste algoritmes nie, maar die verkrygbare spoed is ook beduidend vinniger.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: An integro-differential equation describing the dynamical behaviour of phytoplankton cells is considered in which the effects of cell division and aggregation are incorporated by coupling the coagulation-fragmentation equation with growth, and the McKendrick-von Foerster renewal model of an age-structured population. Under appropriate conditions on the model parameters, the associated initial-boundary value problem is shown to be well posed in a physically relevant Banach space using the theory of strongly continuous semigroups of operators, the theory of perturbation of positive semigroups and the semilinear abstract Cauchy problems theory. In particular, we provide sufficient conditions for honesty of the model. Finally, the results on the effects of the growth-fragmentation-coagulation on the overall evolution of the phytoplankton population are summarised. / AFRIKAANSE OPSOMMING: ’n Integro-differensiaalvergelyking wat die dinamiese ontwikkeling van fitoplanktonselle beskryf, word beskou. Die uitwerking van seldeling en -aggregasie is geïnkorporeer deur die vergelyking van koagulasie en fragmentasie met groeiaan die McKendrick-von Foerster hernuwingsmodel van ’n ouderdomsgestruktureerde populasie te koppel. Die teorie van sterk kontinue semigroepe van operatore, steuringsteorie van positiewe semigroepe en die teorie van semilineêre abstrakte Cauchy probleme word aangewend om, onder gepaste voorwaardes met betrekking tot die model se parameters, te bewys dat die geassosieerde beginwaarde-probleem met randvoorwaardes ‘goed gestel’ is in ’n fisies relevante Banach-ruimte. In die besonder word voldoende voorwaardes vir eerlikheid van die model verskaf. Ten slotte word ’n opsomming van die resultate met betrekking tot die gekombineerde uitwerking van groei-fragmentasie- koagulasie op die gesamentlike ontwikkeling van die fitoplanktonpopulasie verskaf.

Thesis (MSc)--Stellenbosch University, 2011 / ENGLISH ABSTRACT: With billions of current internet users interacting through social networks, the need has arisen to analyze the structure of these networks. Many authors have proposed random graph models for social networks in an attempt to understand and reproduce the dynamics that govern social network development. This thesis proposes a random graph model that generates social networks using a community-based approach, in which users’ affiliations to communities are explicitly modeled and then translated into a social network. Our approach explicitly models the tendency of communities to overlap, and also proposes a method for determining the probability of two users being connected based on their levels of commitment to the communities they both belong to. Previous community-based models do not incorporate community overlap, and assume mutual members of any community are automatically connected. We provide a method for fitting our model to real-world social networks and demonstrate the effectiveness of our approach in reproducing real-world social network characteristics by investigating its fit on two data sets of current online social networks. The results verify that our proposed model is promising: it is the first community-based model that can accurately reproduce a variety of important social network characteristics, namely average separation, clustering, degree distribution, transitivity and network densification, simultaneously. / AFRIKAANSE OPSOMMING: Met biljoene huidige internet-gebruikers wat deesdae met behulp van aanlyn sosiale netwerke kommunikeer, het die analise van hierdie netwerke in die navorsingsgemeenskap toegeneem. Navorsers het al verskeie toevalsgrafiekmodelle vir sosiale netwerke voorgestel in ’n poging om die dinamika van die ontwikkeling van dié netwerke beter te verstaan en te dupliseer. In hierdie tesis word ’n nuwe toevalsgrafiekmodel vir sosiale netwerke voorgestel wat ’n gemeenskapsgebaseerde benadering volg, deurdat gebruikers se verbintenisse aan gemeenskappe eksplisiet gemodelleer word, en dié gemeenskapsmodel dan in ’n sosiale netwerk omskep word. Ons metode modelleer uitdruklik die geneigdheid van gemeenskappe om te oorvleuel, en verskaf ’n metode waardeur die waarskynlikheid van vriendskap tussen twee gebruikers bepaal kan word, op grond van hulle toewyding aan hulle wedersydse gemeenskappe. Vorige modelle inkorporeer nie gemeenskapsoorvleueling nie, en aanvaar ook dat alle lede van dieselfde gemeenskap vriende sal wees. Ons verskaf ’n metode om ons model se parameters te pas op sosiale netwerk datastelle en vertoon die vermoë van ons model om eienskappe van sosiale netwerke te dupliseer. Die resultate van ons model lyk belowend: dit is die eerste gemeenskapsgebaseerde model wat gelyktydig ’n belangrike verskeidenheid van sosiale netwerk eienskappe, naamlik gemiddelde skeidingsafstand, samedromming, graadverdeling, transitiwiteit en netwerksverdigting, akkuraat kan weerspieël.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Phylogenetic analysis is the study of evolutionary relationships among organisms. To this end, phylogenetic trees, or evolutionary trees, are used to depict the evolutionary relationships between organisms as reconstructed from DNA sequence data. The likelihood of a given tree is commonly calculated for many purposes including inferring phylogenies, sampling from the space of likely trees and inferring other parameters governing the evolutionary process. This is done using Felsenstein’s algorithm, a widely implemented dynamic programming approach that reduces the computational complexity from exponential to linear in the number of taxa. However, with the advent of efficient modern sequencing techniques the size of data sets are rapidly increasing beyond current computational capability. Parallel computing has been used successfully to address many similar problems and is currently receiving attention in the realm of phylogenetic analysis. Work has been done using data decomposition, where the likelihood calculation is parallelised over DNA sequence sites. We propose an alternative way of parallelising the likelihood calculation, which we call segmentation, where the tree is broken down into subtrees and the likelihood of each subtree is calculated concurrently over multiple processes. We introduce our proposed system, which aims to drastically increase the size of trees that can be practically used in phylogenetic analysis. Then, we evaluate the system on large phylogenies which are constructed from both real and synthetic data, to show that a larger decrease of run times are obtained when the system is used. / AFRIKAANSE OPSOMMING:Filogenetiese analise is die studie van evolusionêre verwantskappe tussen organismes. Filogenetiese of evolusionêre bome word aangewend om die evolusionêre verwantskappe, soos herwin vanuit DNS-kettings data, tussen organismes uit te beeld. Die aanneemlikheid van ’n gegewe filogenie word oor die algemeen bereken en aangewend vir menigte doeleindes, insluitende die afleiding van filogenetiese bome, om te monster vanuit ’n versameling van sulke moontlike bome en vir die afleiding van ander belangrike parameters in die evolusionêre proses. Dit word vermag met behulp van Felsenstein se algoritme, ’n alombekende benaderingwyse wat gebruik maak van dinamiese programmering om die berekeningskompleksiteit van eksponensieel na lineêr in die aantal taxa, te herlei. Desnieteenstaande, het die koms van moderne, doeltreffender orderingsmetodes groter datastelle tot gevolg wat vinnig besig is om bestaande berekeningsvermoë te oorskry. Parallelle berekeningsmetodes is reeds suksesvol toegepas om vele soortgelyke probleme op te los, met groot belangstelling tans in die sfeer van filogenetiese analise. Werk is al gedoen wat gebruik maak van data dekomposisie, waar die aanneemlikheidsberekening oor die DNS basisse geparallelliseer word. Ons stel ’n alternatiewe metode voor, wat ons segmentasie noem, om die aanneemlikheidsberekening te parallelliseer, deur die filogenetiese boom op te breek in sub-bome, en die aanneemlikheid van elke sub-boom gelyklopend te bereken oor verskeie verwerkingseenhede. Ons stel ’n stelsel voor wat dit ten doel het om ’n drastiese toename in die grootte van die bome wat gebruik kan word in filogenetiese analise, teweeg te bring. Dan, word ons voorgestelde stelsel op groot filogenetiese bome, wat vanaf werklike en sintetiese data gekonstrueer is, evalueer. Dit toon aan dat ’n groter afname in looptyd verkry word wanneer die stelsel in gebruik is.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: 3D virtual environments can be used as therapy tools in patients with autism spectrum disorders (ASDs); however, the development of such tools is time-consuming. A 3D virtual environment development platform for such tools has been developed specifically for the South African context, because of the language and culture sensitivity of these therapy tools. The 3D virtual environment development platform has a Lua scripting interface for specifying logic in the virtual environments. Lua is a textual programming language, and presents a challenge to ASDs therapists’ ability to create therapy tools without engaging an expert programmer. The aim of this research was to investigate the design and implementation of a visual programming environment to support non-expert programmers in scripting within the 3D virtual environment development platform. Various visual program representation techniques, reported in the literature, were examined to determine their appropriateness for adoption in our design. A visual programming language based on the “building-block” approach was considered the most suitable. The research resulted in the development of a visual script editor (VSE), based on an open source framework called the OpenBlocks library. The VSE successfully alleviated the syntax burden that textual programming languages place on non-expert programmers. The fitness of purpose of our VSE was exemplified in a sample 3D virtual environment that was scripted using the VSE. Despite the success, we argue that the applicability of the “building-block” approach is limited to domain-specific programming languages due to the absence of visual expressions for defining user-defined types, and for specifying hierarchy. / AFRIKAANSE OPSOMMING: Geen opsomming

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: In any ecological system, organisms need to interact with each other for their survival. Such interactions form ecological networks which are usually very complex. Nevertheless, they exhibit well de ned patterns; these regularities are often interpreted as products of meaningful ecological processes. As the networks are evolving through time, biological evolution is one of the factors that affects ecological network architecture. In this work, we develop a mathematical model that represents the evolution through time of such ecological interaction networks. The problem is approached by modelling network evolution as a continuous time Markov process, in such a way that the interactions in which a parent species is involved are potentially inherited by its descendant species. This approach allows us to infer ecological parameters and ecological network histories from real-world network data, as well as to simulate ecological networks under our model. While ecologists have long been aware of the in uence of evolutionary processes in shaping ecological networks, we are now able to evaluate the importance of such in uence. / AFRIKAANSE OPSOMMING: In enige ekologiese stelsel benodig organismes wisselwerkings met mekaar ten einde te oorleef. Sulke interaksies vorm ekologiese netwerke wat gewoonlik baie kompleks is maar nogtans goed-gede nieerde patrone vertoon. Hierdie patrone word dikwels geïnterpreteer as die produk van betekenisvolle ekologiese prosesse. Aangesien die netwerke met die verloop van tyd ontwikkel, is biologiese ewolusie een van die faktore wat ekologiese netwerkargitektuur beïnvloed. In hierdie studie ontwikkel ons 'n wiskundige model wat die ewolusie van sulke ekologiese interaksienetwerke voorstel. Die probleem word benader deur netwerkewolusie as 'n kontinue-tyd Markov-proses te modelleer, op so 'n manier dat die interaksies waarin 'n voorouerspesie betrokke is potensieel oorerf kan word deur die afstammelingspesies. Hierdie benadering laat ons toe om ekologiese parameters en ekologiese netwerkgeskiedenisse vanuit regte-wêreld data af te lei, sowel as om ekologiese netwerke onder ons model te simuleer. Alhoewel ekoloë al lank reeds bewus is van die invloed wat ewolusionêre prosesse het op die vorming van ekologiese netwerke, is ons nou in staat om die belangrikheid van hierdie invloed te evalueer.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The annual risk of infection (ARI), a measure of recent transmission, has been described as the most important parameter in tuberculosis (TB) epidemics. Nevertheless, mounting evidence suggests all factors contributing to TB transmission are not yet completely understood. This research was performed to investigate the role various parameters, e.g. overcrowding, period of infectivity, ventilation, and infectivity of source cases, play in TB transmission. An established airborne transmission risk model, the Wells-Riley equation (WRE), was modified to account for scenarios where unknown numbers of infectious individuals may be present. Subsequently, the ARI for three indoor locations conducive to TB transmission were calculated. Two locations (households and minibus taxis) were identified in a social mixing survey conducted within a South African community where TB is endemic as a part of this research. The third location (prison) was identified in an earlier independent study in the same community. The impact various interventions could have in reducing the ARI associated with each location was explored. Poor ventilation, severe overcrowding, extended exposure periods, and high incidence rates contributed to high TB transmission risks in each location. The household-associated ARI was related to the number of resident adults. Current TB control programs will only reduce the ARI if household ventilation levels are improved simultaneously. Similar reductions in the ARI could be achieved by trebling current ventilation levels or by separating child and adult sleeping areas. Neighbouring households can also contribute substantially to the ARI. The minibus taxi-associated ARI for drivers and commuters was considerable but readily reduced by opening windows or keeping the fresh-air fan on. Reducing TB case prevalence through active or passive case-finding would reduce the ARI substantially. The prison-associated ARI was proportional to levels of overcrowding. No single intervention, such as improved ventilation, decreased lock-up time, or improved case-finding, would decrease the ARI substantially, but concurrent implementation of all of them to meet national or international standards would. This research shows TB is not only transmitted in epidemics by highly infectious TB cases, but that any TB case, no matter how infectious, has the potential to infect susceptible people under the right conditions. / AFRIKAANSE OPSOMMING:Die jaarlikse infeksierisiko (ARI) – maatstaf van onlangse siekteoordrag – word as die belangrikste parameter in tuberkulose- (TB-)epidemies bestempel. Nietemin dui toenemende bewyse daarop dat nie álle faktore wat tot TB-oordrag bydra, volledig verstaan word nie. Hierdie navorsing is onderneem om ondersoek in te stel na die rol van verskillende parameters – byvoorbeeld oorbevolking, tydperk van aansteeklikheid, ventilasie en die aansteeklikheid van brongevalle – in TB-verspreiding. Gevestigde model vir die raming van siekteverspreiding deur die lug, die Wells-Riley-vergelyking (WRE), is aangepas vir scenario’s waar onbekende aantal aansteeklike individue moontlik aanwesig is. Daarna is die ARI bereken vir drie ingeslote ruimtes wat TB-oordrag bevorder. Twee van die ruimtes (huishoudings en minibustaxi’s) is ten tyde van die navorsing uitgewys in sosialevermengingsopname in Suid-Afrikaanse gemeenskap waar TB endemies is. Die derde ruimte (gevangenisse) is uitgewys in vroeëre onafhanklike studie in dieselfde gemeenskap. Die navorser het gevolglik bepaal watter moontlike impak verskillende intervensies op die verlaging van die ARI in elke ruimte het. Swak ventilasie, ernstige oorbevolking, verlengde blootstellingstydperke en hoë voorkomsyfers het in elke ruimte tot hoë TB-oordragrisiko bygedra. Die huishoudingsverwante ARI het verband gehou met die aantal volwassenes wat in die huis woon. Huidige TB-beheerprogramme sal slegs die ARI kan verlaag indien huishoudelike ventilasievlakke terselfdertyd verbeter word. Drie keer beter ventilasievlakke of die skeiding van kinders en volwassenes se slaapareas kan soortgelyke verlagings in die ARI teweegbring. Buurhuishoudings kan ook aansienlik tot die ARI bydra. Die minibustaxi-verwante ARI vir bestuurders en pendelaars was beduidend, maar kan betreklik maklik verlaag word deur vensters oop te maak of die varslugwaaier aan te hou. Die vermindering van die voorkoms van TBgevalle deur aktiewe óf passiewe gevalle-opsporing kan die ARI ook beduidend verlaag. Die gevangenisverwante ARI het met vlakke van oorbevolking verband gehou. Geen enkele intervensie soos beter ventilasie, korter toesluittye of beter gevalle-opsporing sal die ARI aansienlik verlaag nie, maar die gelyktydige inwerkingstelling van ál hierdie intervensies in pas met nasionale of internasionale standaarde kan wél. Hierdie navorsing toon dat TB in epidemies nie net deur hoogs aansteeklike TB-gevalle oorgedra word nie, maar dat enige TB-geval, ongeag hoe aansteeklik, die siekte in die regte omstandighede na vatbare mense kan oordra.

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2011. / Imported from http://etd.sun.ac.za / ENGLISH ABSTRACT: In this thesis, we propose to use Levy processes to model the dynamics of asset prices. In the first part, we deal with single asset options and model the log stock prices with a Levy process. We employ pure jump Levy processes of infinite activity, in particular variance gamma and CGMY processes. We fit the log-returns of six stocks to variance gamma and CGMY distributions and check the goodness of fit using statistical tests. It is observed that the variance gamma and the CGMY distributions fit the financial market data much better than the normal distribution. Calibration shows that at given maturity time the two models fit into the option prices very well. In the second part, we investigate the effect of dependence structure to multivariate option pricing. We use the new concept of Levy copula introduced in the literature by Tankov [40]. Levy copulas allow us to separate the dependence structure from the behavior of the marginal components. We consider bivariate variance gamma and bivariate CGMY models. To model the dependence structure between underlying assets we use the Clayton Levy copula. The empirical results on six stocks indicate a strong dependence between two different stock prices. Subsequently, we compute bivariate option prices taking into account the dependence structure. It is observed that option prices are highly sensitive to the dependence structure between underlying assets, and neglecting tail dependence will lead to errors in option pricing. / AFRIKAANSE OPSOMMING: In hierdie proefskrif word Levy prosesse voorgestel om die bewegings van batepryse te modelleer. Levy prosesse besit die vermoe om die risiko van spronge in ag te neem, asook om die implisiete volatiliteite, wat in finansiele opsie pryse voorkom, te reproduseer. Ons gebruik suiwer–sprong Levy prosesse met oneindige aktiwiteit, in besonder die gamma– variansie (Eng. variance gamma) en CGMY–prosesse. Ons pas die log–opbrengste van ses aandele op die gamma–variansie en CGMY distribusies, en kontroleer die resultate met behulp van statistiese pasgehaltetoetse. Die resultate bevestig dat die gamma–variansie en CGMY modelle die finansiele data beter pas as die normaalverdeling. Kalibrasie toon ook aan dat vir ’n gegewe verstryktyd die twee modelle ook die opsiepryse goed pas. Ons ondersoek daarna die gebruik van Levy prosesse vir opsies op meervoudige bates. Ons gebruik die nuwe konsep van Levy copulas, wat deur Tankov[40] ingelei is. Levy copulas laat toe om die onderlinge afhanklikheid tussen bateprysspronge te skei van die randkomponente. Ons bespreek daarna die simulasie van meerveranderlike Levy prosesse met behulp van Levy copulas. Daarna bepaal ons die pryse van opsies op meervoudige bates in multi–dimensionele exponensiele Levy modelle met behulp van Monte Carlo–metodes. Ons beskou die tweeveranderlike gamma-variansie en – CGMY modelle en modelleer die afhanklikheidsstruktuur tussen onderleggende bates met ’n Levy Clayton copula. Daarna bereken ons tweeveranderlike opsiepryse. Kalibrasie toon aan dat hierdie opsiepryse baie sensitief is vir die afhanlikheidsstruktuur, en dat prysbepaling foutief is as die afhanklikheid tussen die sterte van die onderleggende verdelings verontagsaam word.