abstract: Infectious diseases are a leading cause of death worldwide. With the development of drugs, vaccines and antibiotics, it was believed that for the first time in human history diseases would no longer be a major cause of mortality. Newly emerging diseases, re-emerging diseases and the emergence of microorganisms resistant to existing treatment have forced us to re-evaluate our optimistic perspective. In this study, a simple mathematical framework for super-infection is considered in order to explore the transmission dynamics of drug-resistance. Through its theoretical analysis, we identify the conditions necessary for the coexistence between sensitive strains and drug-resistant strains. Farther, in order to investigate the effectiveness of control measures, the model is extended so as to include vaccination and treatment. The impact that these preventive and control measures may have on its disease dynamics is evaluated. Theoretical results being confirmed via numerical simulations. Our theoretical results on two-strain drug-resistance models are applied in the context of Malaria, antimalarial drugs, and the administration of a possible partially effective vaccine. The objective is to develop a monitoring epidemiological framework that help evaluate the impact of antimalarial drugs and partially-effective vaccine in reducing the disease burden at the population level. Optimal control theory is applied in the context of this framework in order to assess the impact of time dependent cost-effective treatment efforts. It is shown that cost-effective combinations of treatment efforts depend on the population size, cost of implementing treatment controls, and the parameters of the model. We use these results to identify optimal control strategies for several scenarios. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2011

Abordaremos a teoria do transporte otimo demonstrando o teorema da dualidade de Kantorovich para uma classe ampla de funções custo. Tal resultado desempenha um papel de suma importância na teoria do transporte otimo. Uma ferramenta importante utilizada e o teorema da dualidade de Fenchel-Rockafellar, aqui enunciado e demonstrado em bastante generalidade. Demonstramos tamb em o teorema da dualidade de Kantorovich-Rubinstein, que trata do caso particular da função custo distância. / We analyze the optimal transport theory proving the Kantorovich duality theorem for a wide class of cost functions. Such result plays an extremely important role in the optimal transport theory. An important tool used here is the Fenchel-Rockafellar duality theorem, which we state and prove in a general case. We also prove the Kantorovich-Rubinstein duality theorem, which deals with the particular case of cost function given by the distance.

In the history of cancer treatment, the immunotherapy is considered to be the most promising treatment approach. The idea behind this breakthrough is to stimulate the patient’s own immune system to recognize the cancer cells and destroy them. In this therapy, the antibodies are known to be powerful medications to activate the immune system in different ways. They circulate throughout the body until they discover a substance that body recognize as alien i.e. antigen and bind to them. Similarly, cancer cells often have molecules on their surface known as tumor-associated antigens. The researchers can design many clones of the antibody that only target a certain antigen type such as one found on tumors or cancer cells. Then, these are used as an effective drug for treating cancer. Thus, the antigen specific antibody interactions play a vital role in cancer immunotherapy. In this study, we propose a dynamic model to represent the population of antigens and antibodies in cancer patients; in particular we focus on the antigen-specific-antibody interactions to elicit an immune response that leads to the death of cancer cells. We formulate a terminal control problem where the schedule and doses of these antibodies are considered as control variables. The objective functional has been formulated as a measure of antigen population at the end of the treatment period. Pontryagin minimum principle (PMP) has been used to obtain the optimal control policies. For illustration, a series of numerical results is presented showing the effectiveness of immune therapy for cancer treatment corresponding to the different scenarios, choices of parameters and treatment periods. The results indicate that the control doses are followed by the emergence of antigen population. This approach would be potentially applicable to determine and prescribe the optimal doses and schedules for cancer patients.

>Magister Scientiae - MSc / Tuberculosis (TB) is currently one of the major public health challenges in South Africa, and in many countries. Mycobacterium tuberculosis is among the leading causes of morbidity and mortality. It is known that tuberculosis is a curable infectious disease. In the case of incomplete treatment, however, the remains of Mycobacterium tuberculosis in the human system often results in the bacterium developing resistance to antibiotics. This leads to relapse and treatment against the resistant bacterium is extremely expensive and difficult. The aim of this work is to present and analyse mathematical models of the population dynamics of tuberculosis for the purpose of studying the effects of efficient treatment versus incomplete treatment. We analyse the spread, asymptotic behavior and possible eradication of the disease, versus persistence of tuberculosis. In particular, we consider inflow of infectives into the population, and we study the effects of screening. A sub-model will be studied to analyse the transmission dynamics of TB in an isolated population. The full model will take care of the inflow of susceptibles as well as inflow of TB infectives into the population. This dissertation enriches the existing literature with contributions in the form of optimal control and stochastic perturbation. We also show how stochastic perturbation can improve the stability of an equilibrium point. Our methods include Lyapunov functions, optimal control and stochastic differential equations. In the stability analysis of the DFE we show how backward bifurcation appears. Various phenomena are illustrated by way of simulations.

Abordaremos a teoria do transporte otimo demonstrando o teorema da dualidade de Kantorovich para uma classe ampla de funções custo. Tal resultado desempenha um papel de suma importância na teoria do transporte otimo. Uma ferramenta importante utilizada e o teorema da dualidade de Fenchel-Rockafellar, aqui enunciado e demonstrado em bastante generalidade. Demonstramos tamb em o teorema da dualidade de Kantorovich-Rubinstein, que trata do caso particular da função custo distância. / We analyze the optimal transport theory proving the Kantorovich duality theorem for a wide class of cost functions. Such result plays an extremely important role in the optimal transport theory. An important tool used here is the Fenchel-Rockafellar duality theorem, which we state and prove in a general case. We also prove the Kantorovich-Rubinstein duality theorem, which deals with the particular case of cost function given by the distance.

Pakistan has been riddled with energy shortage crisis. Long hours of load shedding have caused major economic setbacks in urban areas and rural areas do not even make the cut. Some rural parts, which are connected to the grid, suffer major load shedding and so economic growth is minimal. Most energy is directed towards industrial demand; hence the domestic demand suffers and causes long hours of load shedding. To aid this supply-demand gap, microgrids can be helpful in relieving some of the domestic load on the grid. A microgrid may be more economical only as a support for the main grid in an area, depending on its configuration. Since microgrids are generally composed of renewable energy sources like wind or solar or a combination of both, the supply from just these sources may result in high intermittency. To allow uniform supply, a backup energy source or energy storage is included with the renewable sources. Sizing a microgrid for the targeted region is critical. Some major sizing factors include the availability of renewable resource, load profile of the region, land availability, grid availability, etc. For this thesis, a region near Gharo, a town in Thatta District in Sindh, Pakistan, is selected to deploy the microgrid with a wind farm and battery energy storage system. The microgrid is connected to the main feeder, which supplies grid electricity to a small town of 30 small homes, a school and a small hospital. Hourly wind speed data and an annual load profile is used to calculate the most economic size of the microgrid, depending on the energy dispatch philosophy. To find the most economical solution, this thesis incorporates a stochastic technique, known as the Particle Swarm Optimization (PSO), which is a powerful intelligence evolution algorithm for solving optimization problems. Over the years, PSO has gained popularity due to its simple structure and high performance in solving linear or non-linear objective functions with any number of constraints. In this case, the objective function to be minimized is the net present cost of the microgrid, which comprises of annual capital cost, annual operation and maintenance cost, annual replacement cost of all equipment involved and the annual net cost of buying/selling electricity from/to the grid, respectively.

De nos jours, de nouvelles technologies sont développées pour produire efficacement des produits dérivés de matières premières autresque le pétrole, comme par exemple la biomasse. En effet, la biomasse et plus spécifiquement la biomasse non alimentaire possède un fort potentielcomme substitut aux ressources fossiles pour des raisons environnementales, économiques et politiques. Dans ce contexte, l’étude des bioraffineries offre de nouvelles opportunités pour le Process System Engineering et plus particulièrement pour des activités de recherche quivisent la conception de systèmes constitués d’entités interconnectés. En effet, le verrou principal se concentre sur la modélisation et l’optimisation multi-échelle de la bioraffinerie qui permet l’intégration de plusieurs échelles spatiales allant de l’échelle moléculaire à celle de l’unité de production. Ces différentes échelles sont essentielles pour décrire correctement le système puisqu’elles interagissent en permanence. La forte dilution des courants est le meilleur exemple pour illustrer ces interactions. En effet, la présence d’eau induit de nombreux problèmes thermodynamiques (azéotropes, etc.) à l’échelle moléculaire, ce qui impacte fortement la topologie du procédé notamment sur les étapes de séparation, de purification et detraitement des purges (pour limiter les pertes en produits). Ainsi, la performance de la séquence d’opérations unitaires de l’étape de purification dépend entièrement de la concentration en eau. De plus dans la conception de bioraffinerie, il est fréquent de coupler fermentation et séparation afin d’améliorer les performances de la fermentation et de limiter la présence d’eau dans l’étapede purification. Par ailleurs, la grande quantité d’eau à chauffer ou refroidir entraine la nécessité de réaliser l’intégration énergétique du réseaud’échangeurs du procédé afin de minimiser le coût les dépenses énergétiques. L’objectif de ce travail est alors de proposer une méthodologie générique et les outils associés afin de lever certains verrous de la modélisation et l’optimisation multi-échelle de la bioraffinerie. Basée sur une approche par superstructure, la finalité de la méthodologie est d’évaluer les performances des alternatives étudiées en termes technico-économiques, environnementaux et d’efficacité énergétique en vue de son optimisation multi-objectifs pour trouver la voie de traitement optimale pour le(s) bioproduit(s) d’intérêt. Le cas d’application retenu se focalise sur la production de biobutanol à partir du système Acétone-Butanol-Ethanolet d’une biomasse d’origine forestière. La première étape de la méthodologie proposée concerne la création de la superstructure de la bioraffineriebasée sur une décomposition de cette dernière en 5 étapes principales : le prétraitement, la fermentation, la séparation, la purification et letraitement des purges. Ensuite, la seconde étape consiste à modéliser chaque alternative de procédé. Cette modélisation utilise un modèlethermodynamique à coefficients d’activité afin de décrire le comportement fortement non-idéal des molécules du milieu. De plus, l’intégration du traitement des purges et de l’intégration énergétique durant cette étape permet d’améliorer le procédé. Enfin, la dernière étape s’intéresse à l’optimisation multiobjectif qui se focalise sur différents aspects : maximisation de la production, minimisation des coûts, du prix minimal de vente des bioproduits, des pertes en produits et de l’impact environnemental. Cette dernière étape inclut également des études de sensibilité sur les différents paramètres de la méthodologie : opératoires, économiques, environnementaux... A l’issu de l’optimisation, un compromis seratrouvé afin d’obtenir une bioraffinerie durable. / Nowadays, to replace chemical products derived from petrol, new technologies are developed to produce products derived from others feedstock than crude oil like biomass. Indeed, biomass and especially nonfood biomass has a high potential as substitute due to its environmental, economic and political interests. Inthis context, the study of biorefineries offers new opportunities in the Process System Engineering and especially in research activities which aim to design systems with interlinked compounds. Indeed, the main hurdle focuses on the modeling and the multiscale optimization of thebiorefinery that allows integratingseveral spatial scales from the molecular scale to the plant scale. These scales are essential to describe accurately the system because they interact. The large dilution of flows is the best example to show these interactions. Indeed, water induces many thermodynamic problems (azeotropes, etc.) at the moleculescale, that impact on the process design and mainly on the separation, the purification and the treatment of purges (to limit losses of products). In consequence, the sequence of unit operations of the purification step depends of the water concentration. Furthermore, in the design of the biorefinery, the fermentation and theseparation are usually combined in order to improve performances of the fermentation and limit the water concentration in the purification step. Moreover, the large amount of water that needs to be heated or cooled induces the need of the energy integration of the heat exchangers network to minimize energy consumption. The aim of this work is to propose a generic methodology with connected tools in order to overcome some hurdles caused by the modeling and the multiscaleoptimization of the biorefinery. Based on the superstructure approach, the purpose of the methodology is to estimate performances of considered alternatives in the technical, economic, environmental and energy efficient aspects in preparation for the multiobjective optimization which finds the optimal process for the productionof the interesting bioproduct. This work focuses especially on the production of biobutanol through the Acetone-Butanol-Ethanol system from forest biomass. The methodology begins with the creation of the superstructure of the biorefinery composed by 5 major steps: the pretreatment, the fermentation, the separation, the purification and the treatment of purges. Next, the methodology consists in modeling each alternative of process. It integrates a thermodynamic model with activity coefficients in order to describe accurately the greatly nonideal behavior of molecules. Moreover, the treatment of purges and the energy integration are integratedat this step in order to improve the process. Finally, the last step interests to the multiobjective optimization which focuses on different aspects: the maximization of production and the minimization of the costs, the minimal selling price of bioproducts, the losses of bioproducts and the environmental impact. This step includes also sensitivity analysis on different parameters of the methodology: operating, economic, environmental… After the optimization, a compromise is made in order to obtain sustainable biorefinery.

In Chapter 1 of this thesis we examine two key empirical questions in public economics by exploiting the 2009 Polish tax reforms. First, we estimate the degree of substitution between employment and self-employment tax bases – on the extensive margin. In particular, we quantify the impact of changes in the differential in rates of taxation between the two tax bases on the propensity of taxpayers to declare any positive level of employment or self-employment income. Second, we contribute to the literature on elasticities of taxable income on the intensive margin – the responsiveness of taxable income to changes in marginal tax rates – by providing estimates which are more likely to be robust to changes in year-to-year income dynamics than previous estimates. To identify these effects, we exploit variation in marginal and total tax rates around the 2009 reforms which occurs independently of an individual’s position in the income distribution as a result of joint reporting with a spouse. At the same time, to obtain the extensive-margin responses, we exploit the uniqueness of the 2009 Polish tax reforms, which left the tax schedule un-changed for some types of self-employment while changing the tax schedule for the employed. The baseline estimates of the intensive-margin elasticities are around 0.2-0.3 for the employed and around 0.5-0.7 for the self-employed. The estimates jointly make possible the decomposition of the deadweight losses of the tax reform into intensive and extensive-margin responses, with the contribution of the extensive margin found to be around 7% of the total. In Chapter 2, we examine the optimal non-linear taxation in an environment in which individuals have the option of engaging in either employment or self-employment activity. We build on the estimates from Chapter 1 to calibrate an extension of the classic Mirrleesian model which allows for extensive-margin transitions between employment and self-employment. The results help rationalise the preferential tax treatment of self-employment income versus employment income given in certain tax systems. They also illustrate the ways in which the possibility of extensive-margin transitions between tax bases moderate the incentive to give such preferential treatment. Based on the parameterisation here, the presence of the extensive-margin ap- pears to have a limited impact on the optimal marginal and total tax rates faced by the employed and self-employed. This, together with the earlier decomposition of deadweight losses in Chapter 1 by types of response, points towards a limited role of the extensive margin as a consideration for optimal-tax design, at least as far as the employment and self-employment tax bases are concerned. Chapter 3 turns to a fundamental question in health economics: how do health states change over the life-cycle, and how does the risk of adverse health-shocks change over the life-cycle? Most economic models of individuals’ behaviour over the life-cycle, to the extent to which they incorporate a measure of health risk, assume a simplified unidimensional measure of health. We contribute to this literature by estimating a flexible dynamic factor model of health and health risk over an individual’s life using the rich health data from the Health and Retirement Study (HRS). We find that the many potentially collinear health indicators found in the HRS can broadly be summarized into four underlying factors. Three of these correspond to what may be termed subjective health measures, such as self-reported mobility, while a fourth corresponds to objective measures, including the number of overnight hospital stays, doctor visits and medical spending. The persistence of these underlying factors and the variance of their shocks are estimated as parameters of a vector auto-regressive process. We obtain results about the deterministic evolution of the health factors with age, the level of risk relating to each health measure, as well as heterogeneity by level of education. These are intended as building blocks of an ongoing project concerning the optimal design of disability insurance, given the health risks faced by individuals.

This thesis is of the three article format. All three articles contribute to the literature on optimal fiscal policy with exogeneous government expenditures and distortionary taxation following Lucas and Stokey (1983) and Aiyagari et al. (2002) (AMSS). The first article extends the framework of AMSS by modelling agents ex ante heterogeneous in deterministic labour productivity trends in an infinite-horizon production economy with incomplete markets. The government does not use transfers. When the productivities of different agents grow at different rates, there is a conflict over the timing of tax collection. This is explored in a two-period model. The infinite-horizon model with two agents (‘lowskilled’ and ‘high-skilled’) is used to quantitatively analyse the impact of productivity trends observed in recent decades on the optimal policy. The impact is significant. The model can contribute to explaining the increase in government debt in many advanced economies in recent decades. The optimal policy strongly depends on Pareto weights but welfare of the agents does not. Political economics implications are discussed. The second article analyses the impact of heterogeneous productivity trends on the optimal policy when the social planner can use transfers. There is now conflict over the timing and the level of taxation, and it is explored in a two-period model. The optimal policy is studied in the same environment as in the first article. For most Pareto weights, the change in the tax rate is less pronounced than in the model without transfers, but still greater than the expected change due to shocks. The optimal policy and the welfare of the agents strongly depend on Pareto weights. Policy implications are discussed. The optimal policy in the horizon of decades is significantly affected by even a modest heterogeneity in the growth rates of the agents. Solution methods common to all three articles are discussed. In the third article the closed economy model of AMSS is extended into an open economy setting with two countries. The government of each country finances its exogeneous stochastic expenditures by distortionary labour taxation, and issues one-period bonds. The Ramsey planner chooses policy for both countries, and a no-arbitrage condition on the return of bonds of the two countries restricts her choices. The optimal policy is quantitatively studied in a calibrated model with ex-ante identical countries and equal Pareto weights, and three settings are compared in terms of policy and welfare: autarky (closed economy), partial union (international borrowing allowed), and full union (transfers between governments allowed).

La détection de la peau constitue une phase primordiale de prétraitement dans plusieurs applications telles que la vidéo surveillance et le filtrage d'Internet. Toutefois, c'est une tâche difficile à accomplir étant donné la diversité des couleurs de la peau et la variété des conditions de prise de vue. Dans l'objectif de surmonter ces dernières contraintes, nos travaux de thèse consistent à définir un modèle robuste de la distribution de la peau capable de différencier les pixels de peau de ceux de non-peau dans des images variées. Notre modélisation est fondée sur le concept des modèles graphiques probabilistes connus par leur intuitivité et efficacité pour la représentation d'une distribution jointe de probabilités sur un ensemble de variables aléatoires, plus particulièrement les arbres indexant des probabilités. En vue de définir le modèle de l'arbre idéal indexant la distribution de la peau, nous avons proposé trois approches différentes : le modèle d'arbre de dépendances à b probabilité peau et non peau, le modèle de mélange des arbres et celui de leur combinaison. Le modèle d'arbre de dépendances à bi-probabilité peau et non peau proposé, exploite les propriétés d'interclasse et d'intra classe entre les deux classes peau et non peau ainsi que les interactions entre un pixel et ses voisins que nous traduisons par un arbre de dépendance optimal. L'arbre élaboré est un arbre idéal unique indexant conjointement les distributions de probabilités peau et non peau. Le modèle de mélange des arbres est proposé pour remédier à la multiplicité des arbres de dépendances optimaux possibles sur un graphe. L'entité du mélange proposée concerne aussi bien les structures des arbres considérés que les probabilités portées par ces dernières. Ainsi, l'arbre idéal indexant probabilité peau est l'arbre résultant du mélange portant la probabilité du mélange. Quant au modèle de combinaison des arbres élaboré, il constitue une approche alternative au mélange proposé visant l'exploitation des différent informations emmagasinées dans les différents arbres de dépendances optimaux possibles. Un fondement théorique est présenté dans cette thèse pour déterminer la meilleure approche à adopter, le mélange des arbres ou la combinaison des arbres, et ce en fonction des arbres de dépendances optimaL considérés. Les expérimentations réalisées sur la base Compaq montrent l'efficacité et la faisabilité de nos approches. En outre, des études comparatives entre n< modèles de peau et l'existant prouvent qu'en termes de qualité et de quantité des résultats obtenus, les modèles proposés permettent de discriminer les pixels de peau et ceux de non peau dans des images couleurs variées. / Skin detection or segmentation is considered as an important preliminary process in a number of existing systems ranging over face detection, filtering Internet images, and diverse human interaction areas. Nevertheless, there are two skin segmentation challenges: the pattern variability and the scene complexity. This thesis is devoted to define a new approach for modeling the skin probability distribution. ln the aim of dealing with the skin detection problem, we investigate the models of probability trees to approximate skin and non-skin probabilities. These models can represent a joint distribution in an intuitive and efficient way. Hence, we have proposed three main approaches to seek a perfect tree model estimating the skin probability distribution: (1) the model of dependency tree that approximates the skin and the non skin probability distribution together, (2) the mixture of trees' model, and (3) the combination of trees' model. The first proposed model is based on the optimal spanning tree principle combined to an appropriate relevant criterion that we have defined. The contribution takes into account both the interclass and the intra class between skin and non skin classes, and the interactions between a given pixel and its neighbors. The rationale behind proposing the second model is that in sorne cases the approximation of true class probability given by an optimal spanning tree (OST) is not unique and might be chosen randomly, while this model will take the advantages of the useful information represented on each OST. The mixture of trees' model consists in mixing the structures of the OSTs and their probabilities with the aim of seeking a perfect spanning tree. This latter emphasizes the dependencies' degrees of data, and approximates effectively the true probability distribution. Finally, the third model is defined to deal with a particular kind of multiple OSTs. This model is a parallel combination of different classifiers based on the OSTs. A mathematical theory, proving and specifying the appropriate approach to be used (mixture of trees or combination of trees) depending on the considered OSTs' kind, is presented in this thesis. In addition to experimental results, on the Compaq database, showing the effectiveness and the high reliability of our three approaches.