>Magister Scientiae - MSc / HIV/AIDS is nowadays considered as the greatest public health disaster of modern time. Its progression has challenged the global population for decades. Through mathematical modelling, researchers have studied different interventions on the HIV pandemic, such as treatment, education, condom use, etc. Our research focuses on different compartmental models with emphasis on the effect of public health education. From the point of view of statistics, it is well known how the public health educational programs contribute towards the reduction of the spread of HIV/AIDS epidemic. Many models have been studied towards understanding the dynamics of the HIV/AIDS epidemic. The impact of ARV treatment have been observed and analysed by many researchers. Our research studies and investigates a compartmental model of HIV with treatment and education campaign. We study the existence of equilibrium points and their stability. Original contributions of this dissertation are the modifications on the model of Cai et al. [1], which enables us to use optimal control theory to identify optimal roll-out of strategies to control the HIV/AIDS. Furthermore, we introduce randomness into the model and we study the almost sure exponential stability of the disease free equilibrium. The randomness is regarded as environmental perturbations in the system. Another contribution is the global stability analysis on the model of Nyabadza et al. in [3]. The stability thresholds are compared for the HIV/AIDS in the absence of any intervention to assess the possible community benefit of public health educational campaigns. We illustrate the results by way simulation The following papers form the basis of much of the content of this dissertation, [1 ] L. Cai, Xuezhi Li, Mini Ghosh, Boazhu Guo. Stability analysis of an HIV/AIDS epidemic model with treatment, 229 (2009) 313-323. [2 ] C.P. Bhunu, S. Mushayabasa, H. Kojouharov, J.M. Tchuenche. Mathematical Analysis of an HIV/AIDS Model: Impact of Educational Programs and Abstinence in Sub-Saharan Africa. J Math Model Algor 10 (2011),31-55. [3 ] F. Nyabadza, C. Chiyaka, Z. Mukandavire, S.D. Hove-Musekwa. Analysis of an HIV/AIDS model with public-health information campaigns and individual with-drawal. Journal of Biological Systems, 18, 2 (2010) 357-375. Through this dissertation the author has contributed to two manuscripts [4] and [5], which are currently under review towards publication in journals, [4 ] G. Abiodun, S. Maku Vyambwera, N. Marcus, K. Okosun, P. Witbooi. Control and sensitivity of an HIV model with public health education (under submission). [5 ] P.Witbooi, M. Nsuami, S. Maku Vyambwera. Stability of a stochastic model of HIV population dynamics (under submission).

In this paper, we discuss methods for developing a stochastic model which incorporates behavior differences in the predation movements of Anelosimus studiosus (a subsocial spider). Stochastic models for animal movement and, in particular, spider predation movement have been developed previously; however, this paper focuses on the development and implementation of the necessary mathematical and statistical methods required to expand such a model in order to capture a variety of distinct behaviors. A least squares optimization algorithm is used for parameter estimation to fit a single stochastic model to an individual spider during predation resulting in unique parameter values for each spider. Similarities and variations between parameter values across the spiders are analyzed and used to estimate probability distributions for the variable parameter values. An aggregate stochastic model is then created which incorporates the individual dynamics. The comparison between the optimal individual models to the aggregate model indicate the methodology and algorithm developed in this paper are appropriate for simulating a range of individualistic behaviors.

[pt] A filtração profunda de suspensões particuladas ocorre em muitos processos industriais e ambientais, como filtração de água e contaminação do solo. Na indústria petrolífera, a filtração profunda ocorre próximo ao poço injetor durante a injeção de água, causando redução de injetividade. A captura de partículas no meio poroso pode ser causada por diferentes mecanismos físicos (exclusão pelo tamanho, forças elétricas, gravidade (sedimentação), etc.). No caso do mecanismo de exclusão pelo tamanho, quanto maiores forem as partículas e menores forem os poros, mais intensa será a captura. Conseqüentemente, maior será o dano à formação. Entretanto, o modelo tradicional não considera as distribuições de tamanho de partículas e de poros. Assumindo que as partículas são capturadas pelo mecanismo de exclusão pelo tamanho, foram deduzidas as equações básicas para o transporte de suspensões particuladas no meio poroso considerando as distribuições de tamanho de poros e de partículas. Apenas o fluxo de água via poros acessíveis transporta partículas, ou seja, as partículas não podem acessar poros menores do que elas. No presente trabalho, os efeitos da redução do fluxo de partículas e da inacessibilidade devido ao fluxo seletivo de diferentes tamanhos de partículas são incluídos no modelo estocástico para a filtração profunda. As soluções analíticas obtidas mostram um comportamento físico mais realístico do que o previsto pelo modelo tradicional. O modelo de medição (concentrações totais) obtido difere substancialmente do modelo tradicional para a filtração profunda. Vários dados experimentais foram tratados, mostrando boa concordância e validando o modelo proposto. Um sistema de equações estocásticas para modelar a formação do reboco externo foi proposto e soluções analíticas foram obtidas, permitindo tratar a filtração profunda e a formação do reboco externo, utilizando o mesmo formalismo matemático. / [en] Deep bed filtration of water with particles occurs in several industrial and environmental processes like water filtration and soil contamination. In petroleum industry, deep bed filtration occurs near to injection wells during water injection, causing injectivity reduction. It also takes place during well drilling, sand production control, produced water disposal in aquifers, etc. The particle capture in porous media can be caused by different physical mechanisms (size exclusion, electrical forces, bridging, gravity (sedimentation), etc.). In case of size exclusion mechanism, the larger are the particles and the smaller are the pores, the more intensive is the capture and the larger is the formation damage. Nevertheless, the widely used traditional model does not account for particle and pore size distributions. Considering that particles are captured due to size exclusion mechanism, we derived basic equations for transport of particulate suspensions in porous media, accounting for particle and pore radii distributions. Particles are carried by water flowing through the accessible pore space only, i.e. particles cannot access smaller pores. In the current work, the effects of porous space accessibility and particle flux reduction due to selective flow of different size particles are included into the stochastic deep bed filtration model. The particle and pore ensembles for analytical solutions of the derived system show more realistic physics behaviour than that of the traditional model. Averaging of the derived stochastic equations leads to a new deep bed filtration model that significantly differs from the classical deep bed filtration system. Treatment of several experimental data shows good agreement between the laboratory and modelling data and validates the proposed model. The derived stochastic model has been extended to model formation of external filter cake by particles from the injected polydispersed suspension, allowing treating both deep bed filtration and external filter cake formation in the framework of the same system of governing equations.

[pt] A Igualdade de Jarzynski (IJ) é um tipo especial de Teorema de Flutuação, de trabalho, que caracteriza sistemas termodinâmicos microscópicos fora do equilíbrio. A IJ pode ser usada como uma calibração de experimentos e simulações, o que nos permite estudar comportamentos não triviais da dinâmica desses sistemas. Um desses comportamentos é a troca de entropia e informação que o sistema realiza junto a um banho térmico de contato. Neste ensejo, modelamos via uma dinâmica não-Markoviana, i.e., uma dinâmica com memória, que leva a fluxos reversos de informação do reservatório para o sistema. / [en] The Jarzynski Equality (JE) is a special kind of Fluctuation Theorem, of work, which characterizes non-equilibrium small thermodynamics systems. The JE can be used as gauge of experiments and simulations allowing us to study the non-trivial behaviours of these systems dynamics. One of these behaviours is the entropy and information flow the system makes in contact with a thermal bath. In this framework, we modelled through a non-Markovian dynamic, i.e., with a memory effect, leading to reverse flows of information from the reservoir to the system.

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Uncertainties in both loading and structural characteristics can adversely affect the response and reliability of a structure. Parameter uncertainties in structural dynamics can arise due to several sources. These include variations due to intrinsic material property variability, measurement errors, manufacturing and assembly errors, differences in modeling and solution procedures. Problems of structural dynamics with randomly distributed spatial inhomogeneities in elastic, mass, and damping properties, have been receiving wide attention. Several mathematical and computational issues include discretization of random fields, characterization of random eigensolutions, inversion of random matrices, solutions of stochastic boundary-value problems, and description of random matrix products. Difficulties are encountered when one has to include interaction between nonlinear and stochastic system characteristics, or if one is interested in controlling the system response. The study of structural systems including the effects of system nonlinearity in the presence of parameter uncertainties presents serious challenges and difficulties to designers and reliability engineers. In the analysis of large structures, the situation for substructuring frequently arises due to the repetition of identical assemblages (substructures), within a structure, and the general need to reduce the size of the problem, particularly in the case of non-linear inelastic dynamic analysis. A small reduction in the model size can have a large effect on the storage and time requirement. A primary structural dynamic system may be coupled to subsystems such as piping systems in a nuclear reactor or in a chemical plant. Usually subsystem in itself is quite complex and its modeling with finite elements may result in a large number of degrees of freedom. The reduced subsystem model should be of low-order yet capturing the essential dynamics of the subsystem for useful integration with the primary structure. There are two major issues to be studied: one, techniques for analyzing a complex structure into component subsystems, analyzing the individual sub-system dynamics, and from thereon determining the dynamics of the structure after assembling the subsystems. The nonlinearity due to support gap effects such as supports for piping system in nuclear reactors further complicates the problem. The second is the issue of reviewing the methods for reducing the model-order of the component subsystems such that the order of the global dynamics, after assembly, is within some predefined limits. In the reliability analysis of complex engineering structures, a very large number of the system parameters have to be considered as random variables. The parameter uncertainties are modeled as random variables and are assumed to be time independent. Here the problem would be to reduce the number of random variables without sacrificing the accuracy of the reliability analysis. The procedure involves the reduction of the size of the vector of random variables before the calculation of failure probability. The objectives of this thesis are: 1.To use the available model reduction techniques in order to effectively reduce the size of the finite element model, and hence, compare the dynamic responses from such models. 2.Study of propagation of uncertainties in the reduced order/coupled stochastic finite element dynamic models. 3.Addressing the localized nonlinearities due to support gap effects in the built up structures, and also in cases of sudden change in soil behaviour under the footings. The irregularity in soil behaviour due to lateral escape of soil due to failure of quay walls/retaining walls/excavation in neighbouring site, etc. 4.To evolve a procedure for the reduction of size of the vector containing the random variables before the calculation of failure probability. In the reliability analysis of complex engineering structures, a very large number of the system parameters are considered to be random variables. Here the problem would be to reduce the number of random variables without sacrificing the accuracy of the reliability analysis. 5.To analyze the reduced nonlinear stochastic dynamic system (with phase space reduction), and effectively using the network pruning technique for the solution, and also to use filter theory (wavelet theory) for reducing the input earthquake record to save computational time and cost. It is believed that the techniques described provide highly useful insights into the manner structural uncertainties propagate. The cross-sectional area, length, modulus of elasticity and mass density of the structural components are assumed as random variables. Since both the random and design variables are expressed in a discretized parameter space, the stochastic sensitivity function can be modeled in a parallel way. The response of the structures in frequency domain is considered. This thesis is organized into seven chapters. This thesis deals with the reduced order models of the stochastic structural systems under deterministic/random loads. The Chapter 1 consists of a brief introduction to the field of study. In Chapter 2, an extensive literature survey based on the previous works on model order reduction and the response variability of the structural dynamic systems is presented. The discussion on parameter uncertainties, stochastic finite element method, and reliability analysis of structures is covered. The importance of reducing mechanical models for dynamic response variability, the systems with high-dimensional variables and reduction in random variables space, nonlinearity issues are discussed. The next few chapters from Chapter 3 to Chapter 6 are the main contributions in this thesis, on model reduction under various situations for both linear and nonlinear systems. After forming a framework for model reduction, local nonlinearities like support gaps in structural elements are considered. Next, the effect of reduction in number of random variables is tackled. Finally influence of network pruning and decomposition of input signals into low and high frequency parts are investigated. The details are as under. In Chapter 3, the issue of finite element model reduction is looked into. The generalized finite element analysis of the full model of a randomly parametered structure is carried out under a harmonic input. Different well accepted finite element model reduction techniques are used for FE model reduction in the stochastic dynamic system. The structural parameters like, mass density and modulus of elasticity of the structural elements are considered to be non-Gaussian random variables. Since the variables considered here are strictly positive, the probabilistic distribution of the random variables is assumed to be lognormal. The sensitivities in the eigen solutions are compared. The response statistics based on response of models in frequency domain are compared. The dynamic responses of the full FE model, separated into real and imaginary parts, are statistically compared with those from reduced FE models. Monte Carlo simulation is done to validate the analysis results from SFEM. In Chapter 4, the problem of coupling of substructures in a large and complex structure, and FE model reduction, e.g., component mode synthesis (CMS) is studied in the stochastic environment. Here again, the statistics of the response from full model and reduced models are compared. The issues of non-proportional damping, support gap effects and/local nonlinearity are considered in the stochastic sense. Monte Carlo simulation is done to validate the analysis results from SFEM. In Chapter 5, the reduction in size of the vector of random variables in the reliability analysis is attempted. Here, the relative entropy/ K-L divergence/mutual information, between the random variables is considered as a measure for ranking of random variables to study the influence of each random variable on the response/reliability of the structure. The probabilistic distribution of the random variables is considered to be lognormal. The reliability analysis is carried out with the well known Bucher and Bourgund algorithm (1990), along with the probabilistic model reduction of the stochastic structural dynamic systems, within the framework of response surface method. The reduction in number of random variables reduces the computational effort required to construct an approximate closed form expression in response surface approach. In Chapter 6, issues regarding the nonlinearity effects in the reduced stochastic structural dynamic systems (with phase space reduction), along with network pruning are attempted. The network pruning is also adopted for reduction in computational effort. The earthquake accelerogram is decomposed using Fast Mallat Algorithm (Wavelet theory) into smaller number of points and the dynamic analysis of structures is carried out against these reduced points, effectively reducing the computational time and cost. Chapter 7 outlines the contributions made in this thesis, together with a few suggestions made for further research. All the finite element codes were developed using MATLAB5.3. Final pages of the thesis contain the references made in the preparation of this thesis.

This experimental study analyses the dynamics of concentration fluctuations in a passive plume emitted by a point source within the turbulent boundary layer. We aim to extend the popular study of Fackrell and Robins (1982) about concentration fluctuations and fluxes from point sources by including third and fourth moments of concentration. We also further inquire into the influence of source conditions, such as the source size, source elevation and emission velocity, on higher order concentration moments. The data set is completed by a detailed description of the velocity statistics within the TBL, with exhaustive information on both the temporal and spatial structure of the flow. The experimental data-set has been used to test two different modeling ap- proaches: an analytical meandering plume model (in one and in three dimen- sions) and a Lagrangian stochastic micro-mixing model.

Dispersal, especially long-distance dispersal, is an important component in many disciplines within biology. Many species are passively dispersed by wind, not least spore-dispersed organisms. In this thesis I investigated the dispersal capacity of bryophytes by studying the colonization patterns from local scales (100 m) to landscape scales (20 km). The dispersal distances were measured from a known source (up to 600 m away) or inferred from a connectivity measure (1–20 km). I introduced acidic clay to measure the colonization rates over one season of a pioneer moss, Discelium nudum (I–III). I also investigated which vascular plants and bryophytes that had colonized limed mires approximately 20–30 years after the first disturbance (IV). Discelium effectively colonized new disturbed substrates over one season. Most spores were deposited up to 50 meters from a source but the relationship between local colonization rates and connectivity increased with distance up to 20 km (I–III). Also calcicolous wetland bryophyte species were good colonizers over similar distances, while vascular plants in the same environment colonized less frequently. Common bryophytes that produce spores frequently were more effective colonizers, while no effect of spore size was detected (IV). A mechanistic model that take into account meteorological parameters to simulate the trajectories for spores of Discelium nudum fitted rather well to the observed colonization pattern, especially if spore release thresholds in wind variation and humidity were accounted for (III). This thesis conclude that bryophytes in open habitats can disperse effectively across landscapes given that the regional spore source is large enough (i.e. are common in the region and produce spores abundantly). For spore-dispersed organisms in open landscapes I suggest that it is often the colonization phase and not the transport that is the main bottle-neck for maintaining populations across landscapes. / <p>At the time of the doctoral defence the following papesr were unpublished and had  a status as follows: Paper 2: Epubl ahead of print; Paper 3: Manuscript; Paper 4: Manuscript</p>

The aim of this thesis is to estimate the volatility function of continuoustime stochastic models. The estimation of the volatility of the following wellknown international stock market indexes is presented as an application: Dow Jones Industrial Average, Standard and Poor’s 500, NIKKEI 225, CAC 40, DAX 30, FTSE 100 and IBEX 35. This estimation is studied from two different perspectives: a) assuming that the volatility of the stock market indexes displays shortrange dependence (SRD), and b) extending the previous model for processes with longrange dependence (LRD), intermediaterange dependence (IRD) or SRD. Under the efficient market hypothesis (EMH), the compatibility of the Vasicek, the CIR, the Anh and Gao, and the CKLS models with the stock market indexes is being tested. Nonparametric techniques are presented to test the affinity of these parametric volatility functions with the volatility observed from the data. Under the assumption of possible statistical patterns in the volatility process, a new estimation procedure based on the Whittle estimation is proposed. This procedure is theoretically and empirically proven. In addition, its application to the stock market indexes provides interesting results.

A percussion performance by a skilled musician will often extend beyond a written score in terms of expressiveness. This assertion is clearly evident when comparing a human performance with one that has been rendered by some form of automaton that expressly follows a transcription. Although music notation enforces a significant set of constraints, it is the responsibility of the performer to interpret the piece and “bring it to life” in the context of the composition, style, and perhaps with a historical perspective. In this sense, the sheet music serves as a general guideline upon which to build a credible performance that can carry with it a myriad of subtle nuances. Variations in such attributes as timing, dynamics, and timbre all contribute to the quality of the performance that will make it unique within a population of musicians. The ultimate goal of this research is to gain a greater understanding of these subtle nuances, while simultaneously developing a set of stochastic motion models that can similarly approximate minute variations in multiple dimensions on a purpose-built robot. Live or recorded motion data, and algorithmic models will drive an articulated robust multi-axis mechatronic system that can render a unique and audibly pleasing performance that is comparable to its human counterpart using the same percussion instruments. By utilizing a non-invasive and flexible design, the robot can use any type of drum along with different types of striking implements to achieve an acoustic richness that would be hard if not impossible to capture by sampling or sound synthesis. The flow of this thesis will follow the course of this research by introducing high-level topics and providing an overview of related work. Next, a systematic method for gesture acquisition of a set of well-defined percussion scores will be introduced, followed by an analysis that will be used to derive a set of requirements for motion control and its associated electromechanical subsystems. A detailed multidiscipline engineering effort will be described that culminates in a robotic platform design within which the stochastic motion models can be utilized. An analysis will be performed to evaluate the characteristics of the robotic renderings when compared to human reference performances. Finally, this thesis will conclude by highlighting a set of contributions as well as topics that can be pursued in the future to advance percussion robotics. / Graduate / 2019-12-10