>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).

No description available.

In this study we are interested on the following 4-dimensional system of stochastic differential equations.dS=(-βSI+μ(K-S)+αI+ζR)dt-σ_1 SIF_1 (S,E,I,R)dW_1+σ_4 RF_4 (S,E,I,R)dW_4 dE=(βSI-(μ+η)E)dt+σ_1 SIF_1 (S,E,I,R)dW_1-σ_2 EF_2 (S,E,I,R)dW_2 dI=(ηE-(α+γ+μ)I)dt+σ_2 EF_2 (S,E,I,R)dW_2-σ_3 IF_3 (S,E,I,R)dW_3 dR=(γI-(μ+ζ)R)dt+σ_3 IF_3 (S,E,I,R)dW_3-σ_4 RF_4 (S,E,I,R)dW_4 with variance parameters σ_i≥0 and constants α,β,η,γ,μ ζ≥0. This system may be used to model the dynamics of susceptible, exposed, infected and recovering individuals subject to a present virus with state-dependent random transitions. Our main goal is to prove the existence of a bounded, unique, strong (pathwise), global solution to this system, and to discuss asymptotic stochastic and moment stability of the two equilibrium points, namely the disease free and the endemic equilibria. In this model, as suggested by our advisor, diffusion coefficients can be any local Lipschitz continuous functions on bounded domain D={(S,E,I,R)∈R_+^4:00 of maximum carrying capacity and W_i are independent and identical Wiener processes defined on a complete probability space (Ω,F,{F_t }_(t≥0),P). At the end we carry out some simulations to illustrate our results.

The stochastic SEIR(S) model with random total population is given by the system of stochastic differential equations:dS=(-βSI+μ(K-S)+αI+ζR)dt-σ_1 SIF_1 (S,E,I,R)dW_1+σ_4 RF_4 (S,E,I,R)dW_4+σ_5 S(K-N)dW_5\\ dE=(βSI-(μ+η)E)dt+σ_1 SIF_1 (S,E,I,R)dW_1-σ_2 EF_2 (S,E,I,R)dW_2+σ_5 E(K-N)dW_5 \\ dI=(ηE-(α+γ+μ)I)dt+σ_2 EF_2 (S,E,I,R)dW_2-σ_3 IF_3 (S,E,I,R)dW_3+σ_5 I(K-N)dW_5 \\ dR=(γI-(μ+ζ)R)dt+σ_3 IF_3 (S,E,I,R)dW_3-σ_4 RF_4 (S,E,I,R)dW_4+σ_5 R(K-N)dW_5, where σ_i>0 and constants α, β, η, γ, ζ, μ≥0. K represents the maximum carrying capacity for the total population and W_k=(W_k (t))_(t≥0) are independent, standard Wiener processes on a complete probability space (Ω,F,(F_t )_(t≥0),P). The SDE for the total population N=S+E+I+R has the form dN(t)=μ(K-N)dt+σ_5 N(K-N)dW_5 on D_0=(0,K). The goal of our study is to prove the existence of unique, Markovian, continuous time solutions on the 4D prism D={(S,E,I,R)∈R_+^4:0≤S, E,I,R≤K, S+E+I+R≤K}. Then using the method of Lyapunov functions we prove the asymptotic stochastic and moment stability of disease-free and endemic equilibria. Finally, we use numerical simulations to illustrate our results.

The freshwater dinoflagellate species, Ceratium hirundinella (C. hirundinella) possesses unique characteristics, such as a thecal-plate cell covering of cellulose, spines and flagella. Unlike most other algae and cyanobacteria, C. hirundinella cells are relatively large in size (up to 450 μm in length and 50 μm in width). These unique characteristics (e.g. cell covering and flagella) and adaptations (e.g. spines) give the dinoflagellate cells the ability to reduce their sinking rate from the euphotic zone and to migrate easily through the water column. When source water contains high concentrations of C. hirundinella cells, water treatment problems and poor aesthetic water quality can be expected. These water treatment problems may include 1) the disruption of coagulation and flocculation, 2) clogging of sand filters and 3) taste and odour problems when cells penetrate into the final water. In Chapter 9 of this study, a list of operational guidelines (including alert levels) and recommendations to assist managers and operators of plants when C. hirundinella cells are causing water treatment problems. During events of high C. hirundinella concentrations in source water, managers and operators of conventional water treatment plants need strategies to optimize coagulants and unit processes. Thus when source water contains motile nuisance algae, such as C. hirundinella, in moderate or abundant quantities, it is advisable to conduct jar stirring test experiments using both turbidity and total photosynthetic pigment (or chlorophyll-a) analyses as indicators of appropriate coagulant choice and dosages. The aims of this study are summarized as follows:  To optimize coagulants and conventional water treatment processes by implementing relevant algal removal strategies and indicators during jar stirring test experiments,  To investigate the changes in surface charge (known as zeta potential) on C. hirundinella cells before and after adding coagulants as part of the treatment processes,  To investigate the physical and chemical impacts on the morphology of C. hirundinella cells after coagulation, flocculation and sedimentation,  To identify organic compounds that may be responsible for taste and odour problems associated with C. hirundinella,  To investigate the efficiency of pre-chlorination on the removal C. hirundinella cells when dosing various coagulants, and  Give recommendations and operational guidelines relevant for a conventional water treatment plant to improve C. hirundinella removal A combined water treatment system (Phipps and Bird Model), consisting of a six paddle jar test apparatus and six sand filter columns, was used to simulate conventional processes (coagulation, flocculation, sedimentation and rapid sand filtration). Source water samples containing relatively high C. hirundinella concentrations (> 500 cell/mℓ) were collected from Benoni Lake (26º10’50.40’’S; 28º17’50.11’’ E) in plastic containers and stored as a homogenous sample in a 200 litre container under laboratory conditions (± 22 °C). Samples were collected from the source water as well as after sedimentation (from the supernatant or sludge) to determine turbidity, total photosynthetic pigment analyses (chlorophyll) and for phytoplankton analyses. Flocs (containing C. hirundinella cells) were collected from the sludge or sediment for scanning electron microscopy investigations and to perform zeta potential analyses. Concentrated C. hirundinella samples were frozen at -80 °C according to the proposed sampling protocol for organic compound analyses. Results obtained from this study proved that using the relevant indicators to determine the appropriate coagulant dosages during jar stirring tests may generally improve the removal of problem-causing algae, such as C. hirundinella cells. Improved algal removal efficiencies will subsequently ensure final water with good aesthetic quality. The surface charge (zeta potential) on C. hirundinella cells can be used to evaluate the best coagulation conditions within an operating window of -10 mV to +3 mV when dosing various coagulants. Scanning electron microscopy investigations revealed major damaging effects to C. hirundinella cells when dosing high Ca(OH)2 concentrations. However, when dosing lower Ca(OH)2 concentrations, in combination with organic polymer, better C. hirundinella cell removal efficiencies with less damaging effects to cells was observed. This study also indicated that the pre-chlorination, without causing cell lyses, can be applied to render the highly motile cells immobile which will subsequently assist the coagulation unit process. The aesthetic quality (e.g. tastes and odours) of drinking water may be influenced when C. hirundinella cells release organic material into the water as a result of cell lyses. Organic compounds, such as fatty acids and dicarboxylic acids can lead to taste and odour problems which associate with the presence of C. hirundinella. Organic compounds also serve as precursors for the formation of harmful chlorine by-products formed during chlorination. / PhD (Environmental Sciences), North-West University, Potchefstroom Campus, 2015

The freshwater dinoflagellate species, Ceratium hirundinella (C. hirundinella) possesses unique characteristics, such as a thecal-plate cell covering of cellulose, spines and flagella. Unlike most other algae and cyanobacteria, C. hirundinella cells are relatively large in size (up to 450 μm in length and 50 μm in width). These unique characteristics (e.g. cell covering and flagella) and adaptations (e.g. spines) give the dinoflagellate cells the ability to reduce their sinking rate from the euphotic zone and to migrate easily through the water column. When source water contains high concentrations of C. hirundinella cells, water treatment problems and poor aesthetic water quality can be expected. These water treatment problems may include 1) the disruption of coagulation and flocculation, 2) clogging of sand filters and 3) taste and odour problems when cells penetrate into the final water. In Chapter 9 of this study, a list of operational guidelines (including alert levels) and recommendations to assist managers and operators of plants when C. hirundinella cells are causing water treatment problems. During events of high C. hirundinella concentrations in source water, managers and operators of conventional water treatment plants need strategies to optimize coagulants and unit processes. Thus when source water contains motile nuisance algae, such as C. hirundinella, in moderate or abundant quantities, it is advisable to conduct jar stirring test experiments using both turbidity and total photosynthetic pigment (or chlorophyll-a) analyses as indicators of appropriate coagulant choice and dosages. The aims of this study are summarized as follows:  To optimize coagulants and conventional water treatment processes by implementing relevant algal removal strategies and indicators during jar stirring test experiments,  To investigate the changes in surface charge (known as zeta potential) on C. hirundinella cells before and after adding coagulants as part of the treatment processes,  To investigate the physical and chemical impacts on the morphology of C. hirundinella cells after coagulation, flocculation and sedimentation,  To identify organic compounds that may be responsible for taste and odour problems associated with C. hirundinella,  To investigate the efficiency of pre-chlorination on the removal C. hirundinella cells when dosing various coagulants, and  Give recommendations and operational guidelines relevant for a conventional water treatment plant to improve C. hirundinella removal A combined water treatment system (Phipps and Bird Model), consisting of a six paddle jar test apparatus and six sand filter columns, was used to simulate conventional processes (coagulation, flocculation, sedimentation and rapid sand filtration). Source water samples containing relatively high C. hirundinella concentrations (> 500 cell/mℓ) were collected from Benoni Lake (26º10’50.40’’S; 28º17’50.11’’ E) in plastic containers and stored as a homogenous sample in a 200 litre container under laboratory conditions (± 22 °C). Samples were collected from the source water as well as after sedimentation (from the supernatant or sludge) to determine turbidity, total photosynthetic pigment analyses (chlorophyll) and for phytoplankton analyses. Flocs (containing C. hirundinella cells) were collected from the sludge or sediment for scanning electron microscopy investigations and to perform zeta potential analyses. Concentrated C. hirundinella samples were frozen at -80 °C according to the proposed sampling protocol for organic compound analyses. Results obtained from this study proved that using the relevant indicators to determine the appropriate coagulant dosages during jar stirring tests may generally improve the removal of problem-causing algae, such as C. hirundinella cells. Improved algal removal efficiencies will subsequently ensure final water with good aesthetic quality. The surface charge (zeta potential) on C. hirundinella cells can be used to evaluate the best coagulation conditions within an operating window of -10 mV to +3 mV when dosing various coagulants. Scanning electron microscopy investigations revealed major damaging effects to C. hirundinella cells when dosing high Ca(OH)2 concentrations. However, when dosing lower Ca(OH)2 concentrations, in combination with organic polymer, better C. hirundinella cell removal efficiencies with less damaging effects to cells was observed. This study also indicated that the pre-chlorination, without causing cell lyses, can be applied to render the highly motile cells immobile which will subsequently assist the coagulation unit process. The aesthetic quality (e.g. tastes and odours) of drinking water may be influenced when C. hirundinella cells release organic material into the water as a result of cell lyses. Organic compounds, such as fatty acids and dicarboxylic acids can lead to taste and odour problems which associate with the presence of C. hirundinella. Organic compounds also serve as precursors for the formation of harmful chlorine by-products formed during chlorination. / PhD (Environmental Sciences), North-West University, Potchefstroom Campus, 2015

Dans cette thèse nous étudions les théorèmes limites dans l’analyse statistique dessystèmes dynamiques. Le premier chapitre est consacré aux notions des bases des systèmesdynamiques ainsi que la théorie ergodique. Dans le deuxième chapitre nous introduisonsun cadre fonctionnel abstrait pour lequel la version quenched du théorème de la limitecentrale (TLC) en dimension 1 pour les systèmes dynamiques uniformément dilatantsest satisfaite sous une condition de validité nécessaire et suffisante. Le troisième chapitreest consacré au principe d’invariance presque sûr (PIPS) pour les application aléatoiresdilatantes par morceaux. Nous présentons certaines hypothèses sous lesquelles le (PIPS)est vérifié en utilisant la méthode d’approximation des martingales de Cuny et Merlèvede.Nous étudions aussi le théorème de Sprindzuk et ses conséquences. Nous établissons dansle chapitre quatre la décroissance des corrélations pour les systèmes dynamiques aléatoiresuniformément dilatants par la méthode de couplage en dimension 1. Nous terminons cetravail par une présentation des concepts de base de la théorie des mesures et probabilitéset une présentation de l’espace des fonctions à variation bornée. / In this thesis we study the limit theorems in the statistical analysis of dynamicalsystems. The first chapter is devoted to the basic notions in dynamical systems as well asthe ergodic theory. In the second chapter we introduce an abstract functional frameworkunder which the quenched version of the central limit theorem (CLT) in dimension 1for uniformly expanding dynamic systems is satisfied under a necessary and sufficientcondition validity. The third chapter is devoted to the almost sure invariance principle(ASIP) for random piecewise expanding maps. We present some hypotheses under whichthe (ASIP) is verified using the method of approximation of the martingales of Cuny andMerlèvede. We also study the Sprindzuk theorem and its consequences. In chapter four,we define the decay of correlations for the random dynamical systems uniformly expandingby the coupling method in dimension 1. We finish this work with a presentation of thebasic concepts of the theory of measures and probabilities and a presentation of the spaceof functions with bounded variation.

Le travail présenté dans ce mémoire concerne une méthode de conception sûre de systèmes(COTS). Un COTS est un composant matériel ou logiciel générique qui est naturellement conçu pour être réutilisable et cela se traduit par une forme de flexibilité dans la mise en oeuvre de sa fonctionnalité : en clair, une même fonction peut être réalisée par un ensemble (potentiellement infini) de scénarios différents, tous réalisables par le COTS. La complexité grandissante des fonctions implémentées fait que ces situations sont très difficiles à anticiper d’une part, et encore plus difficiles à éviter par un codage correct. Réaliser manuellement une fonction composite correcte sur un système de taille industrielle, s’avère être très coûteuse. Elle nécessite une connaissance approfondie du comportement des COTS assemblés. Or cette connaissance est souvent manquante, vu qu’il s’agit de composants acquis, ou développés par un tiers, et dont la documentation porte sur la description de leur fonction et non sur sa mise en IJuvre. Par ailleurs, il arrive souvent que la correction manuelle d’une faute engendre une ou plusieurs autres fautes, provoquant un cercle vicieux difficile à maîtriser. En plus, le fait de modifier le code d’un composant diminue l’avantage lié à sa réutilisation. C’est dans ce contexte que nous proposons l’utilisation de la technique de synthèse du contrôleur discret (SCD) pour générer automatiquement du code de contrôle commande correct par construction. Cette technique produit des composants, nommés contrôleurs, qui agissent en contraignant le comportement d’un (ou d’un assemblage de) COTS afin de garantir si possible la satisfaction d’une exigence fonctionnelle. La méthode que nous proposons possède plusieurs étapes de conception. La première étape concerne la formalisation des COTS et des propriété de sûreté et de vivacité (P) en modèles automate à états et/ou en logique temporelle. L’étape suivante concerne la vérification formelle du modèle d’un(des) COTS pour l’ensemble des propriétés (P). Cette étape découvrir les états de violation des propriétés (P) appelés états d’erreur. La troisième étape concerne la correction automatique des erreurs détectées en utilisant la technique SCD. Dans cette étape génère on génère un composant correcteur qui sera assemblé au(x) COTS original(aux) pour que leur comportement général respecte les propriétés souhaitées. L’étape suivante concerne la vérification du système contrôlé pour un ensemble de propriétés de vivacité pour assurer la passivité du contrôleur et la vivacité du système. En fin, une étape de simulation est proposée pour observer le comportement du système pour quelque scénarios intéressent par rapport à son implémentation finale. / This PhD dissertation contributes to the safe design of COTS-based control-command embedded systems. Due to design constraints bounding delays, costs and engineering resources, component re-usability has become a key issue in embedded design. Our proposal is a design method which ensures correction of COTS-based designs. This method uses in synergy a number of design techniques and tools. It starts from modeling of the COTS components which are stored in a generic COTS library, and ends with a design of the global control-command system, verified to be free of errors and ready to be implemented over a hardware chip such as an ASIC or an FPGA "Field Programmable Gate Array". The designer starts by modeling the temporal and logical local preconditions and postconditions of each COTS component, then the global pre/post conditions of the assembly which are not necessary a simple combination of local properties. He models also a list of properties that must be satisfied by the assembly. Any violation of these properties is defined as a design error. Then, by using the model checking approach the model of the assembly is verified against the predefined local and global properties. Some design errors can be corrected automatically through the Discrete Controller Synthesis method (DCS), others however must be manually corrected. After the correction step, the controlled control-command system is verified. Finally a global simulation step is proposed in order to perform a system-level verification beyond the capabilities of available formal tools. We apply the method on two different systems, one concerns transferring data from senders to receivers through FIFO unit, the other is controlcommand system of a train passengers’ access.

對樣本資料之機率密度函數f(x)的無母數估計方法，一直是統計推論領域的研究重點之一，而且在通訊理論與圖形辨別上有非常重要的地位。傳統的文獻對密度函數的估計方法大部分著重於獨立樣本的情形。對於時間數列的相關樣本（例如：經濟指標或加權股票指數資料）比較少提到。本文針對具有弱相關性的穩定時間數列樣本，嘗試提出一個核密度估計的方法並探討其性質。 / For a sample data, the nonparametric estimation of a probability density f(x) is always one point of research problem in statistical inference and plays an important role in communication theory and pattern recognition. Traditionally, the literature dealing with density estimation when the observations are independent is extensive. Time series sample with weak dependence, (for example, an economic indicator or a stock market index data), less in this aspect of discussion. Our main purpose is concerned with the estimation of the probability density function f(x) of a stationary time series sample and discusses some properties of this kernel density.

Cette Thèse est composée de deux parties portant respectivement sur la discrétisation des équations différentielles stochastiques et sur le théorème de la limite centrale presque sûre pour les martingales.<br /><br />La première Partie est composée de trois chapitres: Le premier chapitre introduit le cadre de l'étude et présente les résultats obtenus. Le deuxième chapitre est consacré à l'étude d'une nouvelle méthode d'accélération de convergence, appelée méthode de Romberg statistique, pour le calcul d'espérances de fonctions ou de fonctionnelles d'une diffusion.<br />Ce chapitre est la version augmentée d'un article à paraître dans la revue Annals of Applied Probability.<br /><br />Le troisième chapitre traite de l'application de cette méthode à l'approximation de densité par des méthodes de noyaux.<br />Ce chapitre est basé sur un travail en collaboration avec Arturo Kohatsu-Higa.<br /><br />La deuxième partie de la thèse est composée de deux chapitres: le premier chapitre présente la littérature récente concernant le théorème de la limite centrale presque sûre et ses extensions. Le deuxième chapitre, basé sur un travail en collaboration avec Faouzi Chaâbane, étend divers résultats de type TLCPS à des martingales quasi-continues à gauche.