abstract: Mosquitoes are the greatest killers of mankind, and diseases caused by mosquitoes continue to induce major public health and socio-economic burden in many parts of the world (notably in the tropical sub-regions). This dissertation contributes in providing deeper qualitative insights into the transmission dynamics and control of some mosquito-borne diseases of major public health significance, such as malaria and dengue. The widespread use of chemical insecticides, in the form of long-lasting insecticidal nets (LLINs) and indoor residual spraying, has led to a dramatic decline in malaria burden in endemic areas for the period 2000-2015. This prompted a concerted global effort aiming for malaria eradication by 2040. Unfortunately, the gains recorded are threatened (or not sustainable) due to it Anopheles resistance to all the chemicals embedded in the existing insecticides. This dissertation addresses the all-important question of whether or not malaria eradication can indeed be achieved using insecticides-based control. A novel mathematical model, which incorporates the detailed Anopheles lifecycle and local temperature fluctuations, was designed to address this question. Rigorous analysis of the model, together with numerical simulations using relevant data from endemic areas, show that malaria elimination in meso- and holo-endemic areas is feasible using moderate coverage of moderately-effective and high coverage of highly-effective LLINs, respectively. Biological controls, such as the use of sterile insect technology, have also been advocated as vital for the malaria eradication effort. A new model was developed to determine whether the release of sterile male mosquitoes into the population of wild adult female Anopheles mosquito could lead to a significant reduction (or elimination) of the wild adult female mosquito population. It is shown that the frequent release of a large number of sterile male mosquitoes, over a one year period, could lead to the effective control of the targeted mosquito population. Finally, a new model was designed and used to study the transmission dynamics of dengue serotypes in a population where the Dengvaxia vaccine is used. It is shown that using of the vaccine in dengue-naive populations may induce increased risk of severe disease in these populations. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020

This dissertation investigates part of the strong nuclear force in point form QCD. The quark sector is neglected to focus on gluons and their self-interactions. The structure of gluons is investigated by building up a field theory for massless particles. Single gluon states are defined, and a condition is implemented to make the wave function inner product positive definite. The transformation between gluon and classical gluon fields generates a differentiation inner product, and the kernels allow for transition between momentum and position space. Then, multiparticle gluon states are introduced as symmetric tensor products of gluon Hilbert spaces generated by creation and annihilation operators. In order to assure that the resulting Fock space inner product is positive definite, an annihilator condition is needed and gauge transformations are introduced. The four momentum operator consists of the stress-energy tensor integrated over the forward hyperboloid. The free gluon four momentum operator introduced via the Lagrangian and stress-energy tensor is shown to be equivalent to that generated by gluon irreducible representations when acting on the physical Fock space. Next the vacuum problem is discussed, where the vacuum state is the state that is annihilated by the the four momentum operator and is invariant under Lorentz and color transformations. To find such a state, the vacuum problem is simplified by considering a one degree of freedom model. The Hamiltonian for such a model, the one dimensional energy operator, is solved under a variety of different ansatzes. It is shown that the Hamiltonian has a continuous eigenvalue spectrum, and that the vacuum can be constructed in a way that eliminates the interaction term of the Hamiltonian. This one dimensional vacuum model is adapted to the full problem where it is shown that such a result cannot be replicated.

Radiative transfer theory describes the interaction of radiation with scattering and absorbing media. It has applications in neutron transport, atmospheric physics, heat transfer, molecular imaging, and so on. In steady state, the radiative transfer equation is an integro-differential equation of five independent variables. This high dimensionality and presence of integral term present a serious challenge when one tries to solve the equation numerically. Over the past 50 years, several techniques for solving the radiative transfer equation have been introduced. One among them is to use approximations of RTE. Various approximations of RTE have been proposed in the literature. These include, but are certainly not limited to, the delta-Eddington approximation, the Fokker-Planck approximation, the Boltzmann-Fokker-Planck approximation, the generalized Fokker-Planck approximation, the Fokker-Planck-Eddington approximation and the generalized Fokker-Planck-Eddington approximation. The Fokker-Planck approximation and differential approximation have received particular attention in the literature due to their relatively high accuracy, relatively low computational cost, and flexibility to potential large scale parallel computing. In this thesis we present a well-posed result for the Fokker-Planck equation that may be used to approximate the radiative transfer equation in highly forward-peaked media. Then we study the differential approximation of radiative transfer (RT/DA) equations. Well-posedness of these approximations is studied. A convergent iteration method for the RT/DA equation is presented. Then we turn to a study of RT/DA based inverse problems. The inverse problems are ill-posed and regularization is needed in solving the inverse problems. We present an existence result for solutions of regularized formulations of the inverse problems. Finally examples are included to illustrate numerical results in solving the inverse problems.

The kidneys are organs that play several important roles in the body, including the removal of waste and the regulation of blood pressure. When the kidneys stop functioning correctly, the human body begins to shut down. Because many diseases affect the kidneys, it is important for doctors to be able to evaluate kidney function. We can think of the kidney as a “black box” -- doctors can measure inputs and outputs through blood and urine tests, but rarely know exactly what occurs inside the kidney. Mathematical models that help doctors use those measured inputs and outputs to make predictions are an important method of evaluating kidney function. This thesis focuses on the ways multiple myeloma, a type of plasma cell cancer, affects kidney function. In some patients with multiple myeloma, proteins produced by myeloma cells cause inflammation in the kidney, which causes loss of kidney function and greatly decreases life expectancy. In these chapters, we discuss kidney physiology and describe the process of inflammation caused by myeloma. We introduce the mathematical background for our model, present and analyze a model for kidney function in healthy patients, and then present our model for kidney function in patients with multiple myeloma. Finally, we discuss using the results of patient blood and urine tests as a way to improve our model's prediction potential. The long-term goal of the work in this thesis is to create a tool that physicians can use to more accurately predict the course of disease for multiple myeloma patients with kidney involvement.

The thesis is motivated by recent advances in signal and image processing, a part of electrical and computer engineering. In the first part, we begin with a new approach to the mathematical signal processing as used in the digital processing of images. We prove such results in the 2D case, and we explain their use. A key point we explore is the interplay between the two cases, continuous and discrete. Our discrete algorithms present fast matrix-operations to be applied to images in pixel form. This part of the thesis in turn is based on tools from wavelet analysis, and more generally from the theory of operators in Hilbert space. In the second part, we address encoding and quantization of wavelet coefficients obtained after applying the DWT (mentioned in first part) to 1-D signals.This is the last crucial step in A/D conversion, i.e., analog to digital. By quantization we mean the conversion and encoding of processing-output into bits; bits that in turn are transmitted and fed into a decoder. We isolate and make mathematically precise a particular family of quantizers which are efficient in that they produce error terms of exponential fall-off. We do this with a family of discrete algorithms, each one governed by a quantizer. In Theorems 3.2, 3.5, 3.11, we obtain quite precise a priori estimates. In the last part, we address the compression of a matrix (a 2-D image) obtained by applying the DWT on an image mentioned in the first part. Embedded Zerotree Wavelet algorithm is introduced and implemented.

Includes bibliography. / The governing equations for interactions between surface gravity wavetrains and slowly-varying water currents are derived and the incorporation of Vocoidal water wave theory into this framework is discussed. The emphasis throughout is on the derivation of the general form of the governing equations plus a detailed discussion of the qualitative physical behaviour implied by the equations. Particular solutions are usually given only where they serve to clarify the general method or some physical feature of the analysis. The thesis proper is introduced by a derivation of wave kinematics on still water. A review of the kinematics and dynamics of an inviscid and irrotational fluid follows. The wave and fluid properties are then combined via the definition of wave integral properties. A derivation of the Airy and Stokes O(a2) wave theories is given and used to illustrate a number of points. Water currents (following or opposing the waves) are introduced via their influence on the wave-kinematics. The wave/current dynamics are then presented in two ways: firstly using a wave energy approach and secondly by introducing the wave action concept. Wave action is more convenient because it is a conserved quantity unlike wave energy. The general equations for two dimensional wave/current interactions are derived and discussed. At this point three topics are reconsidered: group velocity, momentum density in wave motion and Lagrangian mean forms of averaging. The general equations for wave/current interaction are shown to be compatible with the Vocoidal water wave theory and applications of the theory to wave/current problems are discussed.

Includes bibliographical references. / The fundamental degrees of freedom in string theory are extended objects. Solving their equations of motion can be difficult unless they are considered in very constrained situations. We investigate the dynamics of gravitational D-brane radiation. Results of others are reviewed which show that in the static case the string prole of Newtonian and relativistic strings are the same. We show that for slow moving strings the relativistic solution agrees with the classical one.

This thesis presents a study of potential calibrators observed by the Karoo Array Telescope (KAT-7). The KAT-7 is an engineering prototype for the coming sensitive array, the MeerKAT, one of the pathfinders for the Square Kilometre Array (SKA). This thesis plays a supporting role in the ongoing commissioning activities of the KAT-7, whose construction started in early 2008, and which has been undergoing engineering and science verifications since late 2010. This thesis has achieved the first steps towards identifying possible flux-density standards for short baseline interferometers such as the KAT-7. The systematic error for flux-density calibration at KAT-7 was estimated relative to 3C123, and it was found to be ~5% of the measured flux density. 18 (~47%) of the 38 sources were identified as good flux density calibrator candidates, for their Modulation Index (MI) and Variability Index (VI) values were less than 0.05 and all the sources in the field had less than 10% of the peak flux density of the calibrator candidate. One source (PKS J0837-1951), which showed low variability and hence appeared to be a good flux-density calibrator candidate (class A), was deemed to be not a good flux-density calibrator candidate because a strong confusing source was found within its primary beam. Eight sources had MI or VI values between 0.05 and 0.09 and may deserve further study as potential flux calibrator candidates. Five sources had MI or VI values greater than 0.09 suggesting significant variability while 17 sources had only one or two observations so their suitability as potential flux calibrator candidates could not be evaluated. In addition, we used another measure of variability, the de-biased modulation index, for our sources comparing the observed modulation indices against Monte Carlo simulations. The result showed that the values of Md are largely as expected and the uncertainties therefore not substantially underestimated.

In this project we study the three fundamental theories that explain the phenomenon of superconductivity: The London theory, the Ginzburg-Landau theory and the BCS theory. We review works by several authors who utilized these theories as the basis for their investigation. In our literature review we study the behavior of single and multivortex states in mesoscopic thin superconducting discs whose dimensions are comparable to the penetration depth λ and the coherence length ξ of a superconductor. We learn about the types of phase transitions that the vortex configurations undergo and the stability of the resulting states. Our aim is to investigate how vortex configurations reorganize after phase transitions and whether their reorganization releases any energy into the system of vortices in the disc. If so, then what is the precise mechanism through which the released energy is transferred into the disc? We aim to answer this question and generalize the results to neutron star interiors in order to explain and predict the behavior of pulsar glitches.

Includes bibliographical references (p. 134-176) / Given that South Africa is an emerging space nation, in a continent of emerging space nations and economies, several technologies need to be developed to progress the space program into a viable and sustainable endeavour. The three main areas of space technology are communications, navigation and remote sensing. Earth science is strongly reliant on the third of these areas for obtaining global scientific data, on a suitable temporal/spatial scale. One of the forms of electro-magnetic remote sensing is microwave radiometry. This dissertation presents a short review of currently available space-faring radiometer technologies and applications, which are then discussed in the context of today's South Africa. For instance Passive Microwave Radiometers (PMR) in the L-Band have huge implications in Soil Moisture (SM) and Sea Salinity (SSS), which in turn affect the global climate, and are being investigated by current and soon to launch missions such as Aquarius, SMOS and SMAP. Multi-frequency radiometers are used to classify various other aspects of Earth's surface-atmosphere system. The structure of this dissertation is to introduce the concepts of radiometry with a review of current and future radiometers from literature (up to November 2011). The user communities, current and possible, are also analysed. There is a discussion of South Africa's history, needs and presence in space, along with possible constraints on a future South African instrument going to space. A technology demonstrator passive microwave radiometer, for SM and Sea Surface Temperature (SST) along with some atmospheric correction channels, is presented. Synergy with data obtained from other instruments, such as an Infra-Red (IR) sounder, is also discussed.