Cette thèse est consacrée à l'étude de décompositions trajectorielles de processus de Lévy spectralement positifs et des relations de dualité pour des processus de ramification, motivée par l'utilisation de ces derniers comme modèles probabilistes d'une dynamique épidémiologique. Nous modélisons l'arbre de transmission d'une maladie comme un arbre de ramification, où les individus évoluent indépendamment les uns des autres, ont des durées de vie i.i.d. (périodes d'infectiosité) et donnent naissance (infections secondaires) à un taux constant durant leur vie. Le processus d'incidence dans ce modèle est un processus de Crump-Mode-Jagers (CMJ) et le but principal des deux premiers chapitres est d'en caractériser la loi conjointement avec l'arbre de transmission partiellement observé, inferé à partir des données de séquences. Dans le Chapitre I, nous obtenons une description en termes de fonctions génératrices de la loi du nombre d'individus infectieux, conditionnellement à l'arbre de transmission reliant les individus actuellement infectés. Une version plus élégante de cette caractérisation est donnée dans le Chapitre II, en passant par un résultat général d'invariance par retournement du temps pour une classe de processus de ramification. Finallement, dans le Chapitre III nous nous intéressons à la loi d'un processus de ramification (sous)critique vu depuis son temps d'extinction. Nous obtenons un résultat de dualité qui implique en particulier l'invariance par retournement du temps depuis leur temps d'extinction des processus CMJ (sous)critiques et de l'excursion hors de 0 de la diffusion de Feller critique (le processus de largeur de l'arbre aléatoire de continuum). / This dissertation is devoted to the study of some pathwise decompositions of spectrally positive Lévy processes, and duality relationships for certain (possibly non-Markovian) branching processes, driven by the use of the latter as probabilistic models of epidemiological dynamics. More precisely, we model the transmission tree of a disease as a splitting tree, i.e. individuals evolve independently from one another, have i.i.d. lifetimes (periods of infectiousness) that are not necessarily exponential, and give birth (secondary infections) at a constant rate during their lifetime. The incidence of the disease under this model is a Crump-Mode-Jagers process (CMJ); the overarching goal of the two first chapters is to characterize the law of this incidence process through time, jointly with the partially observed (inferred from sequence data) transmission tree. In Chapter I we obtain a description, in terms of probability generating functions, of the conditional likelihood of the number of infectious individuals at multiple times, given the transmission network linking individuals that are currently infected. In the second chapter, a more elegant version of this characterization is given, passing by a general result of invariance under time reversal for a class of branching processes. Finally, in Chapter III we are interested in the law of the (sub)critical branching process seen from its extinction time. We obtain a duality result that implies in particular the invariance under time reversal from their extinction time of the (sub)critical CMJ processes and the excursion away from 0 of the critical Feller diffusion (the width process of the continuum random tree).

<p>U radu su prvi put za proveru teorija nastajanja i gumolike elastičnosti polimernih mreža primenjeni modelni poliuretanski sistemi čiji su čvorovi veoma stabilni izocijanurat(heksahidro-1,3,5-triazin- 2,4,6-trion) prstenovi. Rad obuhvata: (1) Sintetezu nekoliko serija homogenih poli(uretan-izocijanuratnih) mreža, na bazi 2.4-TDI i poli(oksipropilen)diola, dvoetapnim postupkom u masi, tj. ciklotrimerizacijom teleheličnih diizocijanata uz katalizator Polycat 41. (2) Sintezu manje savr&scaron;enih mreža sa nekom definisanom koncentracijom visećih lanaca koja je postignuta uvodenjem monoola dietilenglikol- monometiletra u toku pripreme prepolimera. (3) Odredivanje povoljnih reakcionih uslove svih etapa sinteze uz primenu i modifikaciju postojećih analitičkih postupaka za karakterizaciju svojstava reakcionih komponenti i praćenja reakcija umrežavanja. (4) Procenu mogućih bočnih reakcija u toku nastajanja mreža jer one mogu značajno uticati na raspodelu različitih fragmenata u mrežama praćenjem koncentracija reakcionih proizvoda kod modelnih reakcija u rastvoru uz isti diol i isti katalizator i odgovarajuće mono-funkcionalne izocijanatne komponente. (5) Karakterizaciju svojstava i strukturnih parametara modelnih mreža i to primenom postupaka ekstrakcije i bubrenja, diferencijalne skanirajuće kalorimetrije, fotoelastičnih merenja, dinamičko-mehaničke spektroskopije i merenja napon-istezanje. Izborom nominalne Mn diola od 400 do 4000 postigne gustina umreženja 0.1x10^-4mol/cm³ do 10x10^-4mol/cm³ sa rasponom temperatura staklastog prelaza od -60&deg;C do +30&deg;C. Za procenu strukturnih parametara mreža primenjena je teorija granajućih procesa sa kaskadnom zamenom za slučaj nastajanja mreža ciklotrimerizacijom i dvoetapni postupak. U zavisnosti od konverzije NCO grupa u mrežama izračunate su masa po monomernoj jedinici, maseni udeo, broj po monomernoj jedinici, koncentracija po jedinici zapremine, Mn, Mw, Mz i Mw/Mn kako za elastično aktivne, osnovne i viseće lance tako i zamolekule sola. Ključni ekspehmentalni podaci koji su kori&scaron;ćeni za proveru teorija nastajanja bio sadržaj gela dobijen vi&scaron;estrukim ekstrakcijama, a za teorije gumolike elastičnosti ravnotežni modul dobijen iz fotoelastičnih merenja. Provera teorija gumolike elastičnosti se zasnivala se u nalaženje vrednosti pred- faktora 3 (tzv. faktora pamćenja) u jednačini koja daje vezu ravnotežnog modula elastičnosti i koncentracije elastično aktivnih lanaca mreže koji ukazuje na način fluktuacije cvorova mreže. Fotoelastična merenja su omogućila da se preko njih parameter pred-faktor 3 proceni pri malim deformacijama. Potvrden je koncept stvarnih mreža sa delimično potisnutim fluktuacijama čvorova uz aditivni doprinos zapletenosti lanaca modulu elastičnosti prema Langley-Graesley konceptu zarobljenih prepletaja.</p> / <p>In this thesis polymer network formation theories and rubber elasticity of polymer networks on model polyurethane systems having as a junction very stable isocyanurate(hexahidro-1,3,5-tгiazin-2,4,6- trion) rings are for the first time investigated. The work contains: (1) Synthesis of homogenious series of several poly(urethane-isocyanurate) networks based on poly(oxypropylene)diol and 2.4-TDI by two stages procedures in bulk i.e. cyclotrimerization of telechelic diisoicyanates with Polycat 41 as catalyst. (2) Synthesis of less perfect networks with some concentration of dangling chains obtained by introduction of monofunctional component diethyleneglycolmonomethylether during prepolymer formation. (3) Determination of optimal reaction conditions for both stages of the synthesis. (4) Estimation of possible side reaction during network formation because of theirs important influence on nework fragment distribution. (5) Caracterization of networks structure and properties by swelling and multiply extraction, differential scanning calorimetry, photoelastical measurements, dynamic-mechanical spectroscopy and stress-strain measurements. By using the different Mn of diols (400 - 4000) the network density from<br />0.1x10^-4mol/cm³ to 10x10^-4mol/cm³ and glass transition temperatures from -60&deg;C to +30&deg;C was achieved. For network structure determinations the theory of branching processes with cascade substitution was used. Mass per monomer unit, mass fraction, number per monomer unit, concentration per unit volume, Mn, Mw, Mz and Mw/Mn for elastically active network chains, backbone chains, dangling chains and sol molecules In the dependence on NCO groups conversion are calculated. Crucial experimental data for examination of network formation theories was the gel content and for rubber<br />elasticity theories the equilibrium modulus received from photoelastical measurement. The estimation of rubber elasticity theories was based on the analysis of the front factor 3 in the equation which gives the relation between the equilibrium modulus and concentration of elastically active network chains. It is estimated the additive contribution of chain entanglements to the equilibrium modulus, especially in the case of high networks density according to Langley-Graesley theory of trapped entanglements.</p>

In this thesis, we use multitype Galton-Watson branching processes in random environments as individual-based models for the evolution of structured populations with both demographic stochasticity and environmental stochasticity, and investigate the phenotype allocation problem. We explore a variational characterization for the stochastic evolution of a structured population modeled by a multitype Galton-Watson branching process. When the population under consideration is large and the time scale is fast, we deduce the continuous approximation for multitype Markov branching processes in random environments. Many problems in evolutionary biology involve the allocation of some limited resource among several investments. It is often of interest to know whether, and how, allocation strategies can be optimized for the evolution of a structured population with randomness. In our work, the investments represent different types of offspring, or alternative strategies for allocations to offspring. As payoffs we consider the long-term growth rate, the expected number of descendants with some future discount factor, the extinction probability of the lineage, or the expected survival time. Two different kinds of population randomness are considered: demographic stochasticity and environmental stochasticity. In chapter 2, we solve the allocation problem w.r.t. the above payoff functions in three stochastic population models depending on different kinds of population randomness. Evolution is often understood as an optimization problem, and there is a long tradition to look at evolutionary models from a variational perspective. In chapter 3, we deduce a variational characterization for the stochastic evolution of a structured population modeled by a multitype Galton-Watson branching process. In particular, the so-called retrospective process plays an important role in the description of the equilibrium state used in the variational characterization. We define the retrospective process associated with a multitype Galton-Watson branching process and identify it with the mutation process describing the type evolution along typical lineages of the multitype Galton-Watson branching process. Continuous approximation of branching processes is of both practical and theoretical interest. However, to our knowledge, there is no literature on approximation of multitype branching processes in random environments. In chapter 4, we firstly construct a multitype Markov branching process in a random environment. When conditioned on the random environment, we deduce the Kolmogorov equations and the mean matrix for the conditioned branching process. Then we introduce a parallel mutation-selection Markov branching process in a random environment and analyze its instability property. Finally, we deduce a weak convergence result for a sequence of the parallel Markov branching processes in random environments and give examples for applications.

[pt] Este trabalho modela a evolução temporal de uma epidemia com uma abordagem estocástica. O número de novas infecções por infectado é modelado como uma variável aleatória discreta, chamada aqui de contágio. Logo, a evolução temporal da doença é um processo estocástico. Mais especificamente, a propagação é dada pelo modelo de Bienaymé-Galton-Watson, um tipo de processo de ramificação de parâmetro discreto. Neste processo, para um determinado instante, o número de membros infectados, ou seja, a geração de membros infectados é uma variável aleatória. Na primeira parte da dissertação, dado que o modelo probabilístico do contágio é conhecido, quatro metodologias utilizadas para obter as funções de massa das gerações do processo estocástico são comparadas. As metodologias são: funções geradoras de probabilidade com e sem identidades polinomiais, cadeia de Markov e simulações de Monte Carlo. A primeira e terceira metodologias fornecem expressões analíticas relacionando a variável aleatória de contágio com a variável aleatória do tamanho de uma geração. Essas expressões analíticas são utilizadas na segunda parte desta dissertação, na qual o problema clássico de inferência paramétrica bayesiana é estudado. Com a ajuda do teorema de Bayes, parâmetros da variável aleatória de contágio são inferidos a partir de realizações do processo de ramificação. As expressões analíticas obtidas na primeira parte do trabalho são usadas para construir funções de verossimilhança apropriadas. Para resolver o problema inverso, duas maneiras diferentes de se usar dados provindos do processo de Bienaymé-Galton-Watson são desenvolvidas e comparadas: quando dados são realizações de uma única geração do processo de ramificação ou quando os dados são uma única realização do processo de ramificação observada ao longo de uma quantidade de gerações. O critério abordado neste trabalho para encerrar o processo de atualização na inferência paramétrica usa a distância de L2-Wasserstein, que é uma métrica baseada no transporte ótimo de massa. Todas as rotinas numéricas e simbólicas desenvolvidas neste trabalho são escritas em MATLAB. / [en] This work models an epidemic s spreading over time with a stochastic approach. The number of infections per infector is modeled as a discrete random variable, named here as contagion. Therefore, the evolution of the disease over time is a stochastic process. More specifically, this propagation is modeled as the Bienaymé-Galton-Watson process, one kind of branching process with discrete parameter. In this process, for a given time, the number of infected members, i.e. a generation of infected members, is a random variable. In the first part of this dissertation, given that the mass function of the contagion s random variable is known, four methodologies to find the mass function of the generations of the stochastic process are compared. The methodologies are: probability generating functions with and without polynomial identities, Markov chain and Monte Carlo simulations. The first and the third methodologies provide analytical expressions relating the contagion random variable and the generation s size random variable. These analytical expressions are used in the second part of this dissertation, where a classical inverse problem of bayesian parametric inference is studied. With the help of Bayes rule, parameters of the contagion random variable are inferred from realizations of the stochastic process. The analytical expressions obtained in the first part of the work are used to build appropriate likelihood functions. In order to solve the inverse problem, two different ways of using data from the Bienaymé-Galton-Watson process are developed and compared: when data are realizations of a single generation of the branching process and when data is just one realization of the branching process observed over a certain number of generations. The criteria used in this work to stop the update process in the bayesian parametric inference uses the L2-Wasserstein distance, which is a metric based on optimal mass transference. All numerical and symbolical routines developed to this work are written in MATLAB.

Marine populations worldwide are in decline due to anthropogenic effects. Spatial management via marine reserves may be an effective conservation method for many species, but the requisite theory is still underdeveloped. Integrodifference equation (IDE) models can be used to determine the critical domain size required for persistence and provide a modelling framework suitable for many marine populations. Here, we develop a novel spatially implicit approximation for the proportion of individuals lost outside the reserve areas which consistently outperforms the most common approximation. We examine how results using this approximation compare to the existing IDE results on the critical domain size for populations in a single reserve, in a network of reserves, in the presence of alongshore currents, and in structured populations. We find that the approximation consistently provides results which are in close agreement with those of an IDE model with the advantage of being simpler to convey to a biological audience while providing insights into the significance of certain model components. We also design a stochastic individual based model (IBM) to explore the probability of extinction for a population within a reserve area. We use our spatially implicit approximation to estimate the proportion of individuals which disperse outside the reserve area. We then use this approximation to obtain results on extinction using two different approaches, which we can compare to the baseline IBM; the first approach is based on the Central Limit Theorem and provides efficient simulation results, and the second modifies a simple Galton-Watson branching process to include loss outside the reserve area. We find that this spatially implicit approximation is also effective in obtaining results similar to those produced by the IBM in the presence of both demographic and environmental variability. Overall, this provides a set of complimentary methods for predicting the reserve area required to sustain a population in the presence of strong fishing pressure in the surrounding waters.

Cette thèse étudie le Q-processus de certains processus de branchement (superprocessus inhomogènes) ou de recombinaison (processus de Lambda-Fleming-Viot) via une approche généalogique. Dans le premier cas, le Q-processus est défini comme le processus conditionné à la non-extinction, dans le second cas comme le processus conditionné à la non-absorption. Des constructions trajectorielles des Q-processus sont proposées dans les deux cas. Une nouvelle relation entre superprocessus homogènes et processus de Lambda-Fleming-Viot est établie. Enfin, une étude du Q-processus est menée dans le cadre général des processus régénératifs / This work is concerned with the definition and study of the Q-process of some branching processes (inhomogeneous superprocesses) or recombination processes (Lambda-Fleming-Viot process). In the first case, the Q-process is defined as the process conditioned on non-extinction, whereas in the second case, it is defined as the process conditioned on non-absorbtion. A pathwise construction of the Q-process is given in both cases. A link between a class of homogeneous superprocesses and Lambda-Fleming-Viot processes is provided. Last, a study of the Q-process in the more general framework of regenerative processes is performed

Cette thèse est consacrée à l'étude du comportement asymptotique de grands graphes et arbres aléatoires. Le premier modèle étudié est un modèle de graphe aléatoire inhomogène introduit par Bo Söderberg. Un chapitre de ce manuscrit est consacré à l'étude asymptotique de la taille des composantes connexes à proximité de la fenêtre critique, en le reliant à la longueur des excursions d'un mouvement brownien avec dérive parabolique, étendant les résultats obtenus par Aldous. Le chapitre suivant est consacré à un processus de graphes aléatoires proposé par Itai Benjamini, défini ainsi : les arêtes sont ajoutées indépendamment, à taux fixe. Lorsqu'un sommet atteint le degré k, toutes les arêtes adjacentes à ce sommet sont immédiatement supprimées. Ce processus n'est pas croissant, ce qui empêche d'utiliser directement certaines approches usuelles. L'utilisation de limites locales permet de montrer la présence (resp. l'absence) d'une composante géante à certaines étapes dans le cas k>=5 (resp. k<=3). Dans le cas k=4, ces résultats permettent de caractériser la présence d'une composante géante en fonction du caractère surcritique ou non d'un processus de branchement associé. Dans le dernier chapitre est étudiée la hauteur d'un arbre de Lyndon associé à un mot de Lyndon choisi uniformément parmi les mots de Lyndon de longueur n, prouvant que cette hauteur est approximativement c ln n, avec c=5,092... la solution d'un problème d'optimisation. Afin d'obtenir ce résultat, nous couplons d'abord l'arbre de Lyndon à un arbre de Yule, que nous étudions ensuite à l'aide de techniques provenant des théories des marches branchantes et des grandes déviations. / This thesis is dedicated to the study of the asymptotic behavior of some large random graphs and trees. First is studied a random graph model introduced by Bo Söderberg in 2002. One chapter of this manuscript is devoted to the study of the asymptotic behavior of the size of the connected components near the critical window, linking it to the lengths of excursion of a Brownian motion with parabolic drift. The next chapter talks about a random graph process suggested by Itai Benjamini, defined as follows: edges are independently added at a fixe rate. Whenever a vertex reaches degree k, all adjacent edges are removed. This process is non-increasing, preventing the use of some commonly used methods. By using local limits, in the spirit of the PWIT, we were able to prove the presence (resp. absence) of a giant component at some stages of the process when k>=5 (resp. k<=3). In the case k=4, these results allows to link the presence (resp. absence) of a giant component to the supercriticality (resp. criticality or subcriticality) of an associated branching process. In the last chapter, the height of random Lyndon tree is studied, and is proven to be approximately c ln n, in which c=5.092... the solution of an optimization problem. To obtain this result, we couple the Lyndon tree with a Yule tree, then studied with the help of branching walks and large deviations

La presente tesis doctoral presenta varios resultados fundamentales para la ampliación del conocimiento actual de procesos importantes en la generación de los circuitos neuronales, como son la migración y la ramificación de células neurales. En primer lugar, se ha determinado la expresión de la semaforina transmembranal 4F en cerebro de ratón en desarrollo y adulto. Así, se ha visto que se expresa en diversas áreas del cerebro, y se ha encontrado expresión de esta proteína en precursores neuronales y en neuronas maduras, principalmente en dendritas, y en células del linaje oligodendroglial. Para profundizar más en este aspecto se llevaron a cabo varios marcajes dobles de Sema4F con proteínas marcadoras de oligodendrocitos, observándose marca en el nervio óptico y otras regiones cerebrales, incluídas la materia blanca y vías de migración de oligodendrocitos. La localización de esta semaforina en el nervio óptico a edades embrionarias y su expresión en células precursoras de oligodendrocitos (OPCs), comprobada in vitro, nos llevó a sugerir que Sema4F funciona controlando la migración de OPCs. Una serie de experimentos con explantes de nervio óptico tratados con medio control o medio condicionado 4F nos permitió determinar que Sema4F actúa inhibiendo la migración de OPCs, sin afectar a su proliferación. Además, Sema4F induce la diferenciación de OPCs a oligodendrocitos maduros. Todos estos datos sugieren un posible papel de Sema4F en procesos de remielinización. Los efectos negativos de Sema4F sobre la migración de OPCs deben cursar con cambios en el citoesqueleto celular. La kinasa de adhesión focal (FAK) es un importante mediador de señales extracelulares (como factores tróficos, interacción de integrinas con proteínas de matriz extracelular, etc…) y el interior de las células. Actúa sobre el citoesqueleto de actina y de tubulina, influyendo en la generación de filopodios, lamelipodios y fibras de estrés. Tiene un papel crucial en migración, de modo que dedicimos estudiar si Sema4F ejerce sus efectos en OPCs a través FAK. Hemos visto que Sema4F es capaz de inducir la fosforilación en varios residuos tirosina de FAK en pocos minutos, y que ambas proteínas por separado ejercen efectos opuestos en la migración de oligodendrocitos. La vía de señalización de 4F, de la que se desconoce incluso el receptor, podría cursar mediante la modulación del estado de activación de FAK, aunque faltan experimentos definitivos. FAK presenta varias isoformas específicas del sistema nervioso central, originadas mediante procesos de splicing alternativo. En la presente tesis hemos determinado con gran especificidad la forma mayoritaria expresada en varias áreas cerebrales y en el desarrollo embrionario o el adulto, tanto en neuronas como en células de la glía. FAK responde a neurotrofinas y participa en procesos de ramificación neuronal, si bien su efecto final es controvertido. Otra proteína que responde a neurotrofinas, y actúa promoviendo la ramificación axonal, es la kinasa dependiente de cdc-42 activada 1 (Ack1). En esta tesis hemos determinado que ambas proteínas interaccionan en cerebro específicamente, de manera independiente de la isoforma de FAK presente. Mediante el uso de inhibidores hemos visto que la activación de FAK es necesaria para la fosforilación de Ack1 y viceversa. FAK es la responsable de la atracción ejercida por netrina-1, y hemos determinado que la ausencia de Ack1 elimina el efecto de esta molécula de señalización. Con técnicas de Espetrometría de Masas hemos identificado algunos posibles interactores de ambas proteínas. Además, hemos observado cambios en el estado de fosforilación de varios residuos de FAK y Ack1 en función del estado de desarrollo (ratones P5 Vs. Adultos) y del estado general de activación del cerebro (ratones inyectados con la droga epileptogénica PTZ Vs. Control). / This thesis presents several results related to important processes regarding neural circuit formation, i.e. migration and ramification of Central Nervous System (CNS) cells. First, we have determined the expression of transmembrane semaphorin 4F (Sema4F) in developing and adult mice brain. Expression of this protein is high in neuronal and oligodendrocyte precursor cells (OPCs), and in different areas including optic nerve (ON) and different migratory pathways. In vitro experiments confirmed Sema4F expression in OPCs. We investigated the role of this protein in functions important for OPC physiology, and found that Sema4F inhibits OPC migration from ON explants and induces their differentiation into mature progenitors. Negative effects of Sema4F in migration must involve cytoskeleton changes. Focal adhesion kinase (FAK) is an important integrator of different extracellular signals and modulates cytoskeleton dynamics to control generation of lamellipodia, fillopodia and stress fibers. In the present project we found that Sema4F is able to phosphorylate FAK, and that FAK enhances OPC migration. The exact implications of Sema4F-FAK relationship remain to be elucidated. FAK exists in different spliced isoforms, expressed preferentially in brain. In this project, we characterised the exact isoform expressed in different areas of the brain and by different cell types. Finally, FAK response to neurotrophins is well characterised. FAK also participates in ramification processes, with controversial final effects in neurons. Ack1 is a crucial transducer of neurotrophin-induced ramification. In this thesis we show that both proteins interact specifically in neurons. We have also found that the activation of FAK is necessary for Ack1 phosphorylation upon stimulation, and viceversa. FAK mediates netrin-1 attraction, and here we have determined that knocking-down Ack1 avoids netrin-1 effects in hippocampal explants. By Mass Spectrometry (MS) techniques, we have observed changes in the phosphorylation state of both proteins depending on the developmental stage of the brain (P5 mice) or its activation state (epileptic mice).

This work is devoted to the issue of monitoring of distributed real-time systems. In particular, it focuses on formal aspects of model-based supervision and problems which are related to it. In its first part, we present the basic properties of two well-known formal models used to model distributed systems: networks of timed automata and time Petri nets. We show that the behavior of these models can be represented with so-called branching processes. We also introduce the key conceptual elements of the supervisory system. The second part of the work is dedicated to the issue of constrained unfoldings which enable us to track causal relationships between events in a distributed system. This type of structure can be used to reproduce processes of the system on the basis of a completely unordered set of previously observed events. Moreover, we show that time constraints imposed on a system and observations submitted to the supervisory system can significantly affect a course of events in the system. We also raise the issue of parameters in time constraints. The proposed methods are illustrated with case studies. The third part of the work deals with the issue of unobservable cyclical behaviors in distributed systems. This type of behaviors leads to an infinite number of events in constrained unfoldings. We explain how we can obtain a finite structure that stores information about all observed events in the system, even if this involves processes that are infinite due to such unobservable loops. The fourth and final part of the work is dedicated to implementation issues of the previously described methods.

Les modèles de mutations décrivent le processus d’apparitions rares et aléatoires de mutations au cours de lacroissance d’une population de cellules. Les échantillons obtenus sont constitués de nombres finaux de cellules mutantes,qui peuvent être couplés avec des nombres totaux de cellules ou un nombre moyen de cellules en fin d’expérience.La loi du nombre final de mutantes est une loi à queue lourde : de grands décomptes, appelés “jackpots”,apparaissent fréquemment dans les données.Une construction générale des modèles se décompose en troisniveaux. Le premier niveau est l’apparition de mutations aléatoires au cours d’un processus de croissance de population.En pratique, les divisions cellulaires sont très nombreuses, et la probabilité qu’une de ces divisions conduise à une mutation est faible,ce qui justifie une approximation poissonnienne pour le nombre de mutations survenant pendant un temps d’observation donné.Le second niveau est celui des durées de développement des clones issus de cellules mutantes. Du fait de la croissance exponentielle,la majeure partie des mutations ont lieu à la fin du processus, et les durées de développement sont alors indépendanteset exponentiellement distribuées. Le troisième niveau concerne le nombre decellules qu’un clone issu d’une cellule mutante atteint pendant une durée de développement donnée.La loi de ce nombre dépend principalement de la loi des instants de division des mutantes.Le modèle classique, dit de Luria-Delbrück, suppose que les développements cellulaires des cellules normales aussi bien que mutantess’effectue selon un processus de Yule. On peut dans ce cas calculer expliciter la loi du nombre final de mutantes.Elle dépend de deux paramètres, qui sont le nombre moyen de mutations et le paramètre de fitness (ratio des taux de croissance des deux types de cellules).Le problème statistique consiste à estimer ces deux paramètres au vu d’un échantillon denombres finaux de mutantes. Il peut être résolu par maximisation de la vraisemblance,ou bien par une méthode basée sur la fonction génératrice. Diviser l'estimation du nombre moyen de mutations par le nombre total de cellulespermet alors d'estimer la probabilité d’apparition d’une mutation au cours d’une division cellulaire.L’estimation de cette probabilité est d’une importancecruciale dans plusieurs domaines de la médecine et debiologie: rechute de cancer, résistance aux antibiotiques de Mycobacterium Tuberculosis, etc.La difficulté provient de ce que les hypothèses de modélisation sous lesquelles la distribution du nombre final de mutants est explicitesont irréalistes.Or estimer les paramètres d’un modèle quand la réalité en suit un autre conduit nécessairement à un biais d’estimation.Il est donc nécessaire de disposer de méthodes d’estimation robustes pour lesquelles le biais, en particulier sur la probabilité de mutation,reste le moins sensible possible aux hypothèses de modélisation.Cette thèse contient une étude probabiliste et statistique de modèles de mutations prenant en compte les sources de biais suivantes : durées de vie non exponentielles, morts cellulaires,variabilité du nombre final de cellules, durées de vie non-exponentielles et non-identiquement distribuées, dilution de la population initiale.Des études par simulation des méthodes considérées sont effectuées afin de proposer, selon les caractéristiques du modèle,l’estimation la plus fiable possible. Ces méthodes ont également été appliquées à desjeux de données réelles, afin de comparer les résultats avec les estimations obtenues avec les modèles classiques.Un package R a été implémenté en collaboration avec Rémy Drouilhet et Stéphane Despréaux et est disponible sur le CRAN.Ce package est constitué des différents résultats obtenus au cours de ce travail. Il contient des fonctions dédiées aux modèles de mutations,ainsi qu'à l'estimation des paramètres. Les applications ont été développées pour le Labex TOUCAN (Toulouse Cancer). / Mutation models are probabilistic descriptions of the growth of a population of cells, where mutationsoccur randomly during the process. Data are samples of integers, interpreted as final numbers ofmutant cells. These numbers may be coupled with final numbers of cells (mutant and non mutant) or a mean final number of cells.The frequent appearance in the data of very large mutant counts, usually called “jackpots”, evidencesheavy-tailed probability distributions.Any mutation model can be interpreted as the result of three ingredients. The first ingredient is about the number of mutations occuring with small probabilityamong a large number of cell divisions. Due to the law of small numbers, the number of mutations approximately follows aPoisson distribution. The second ingredient models the developing duration of the clone stemming from each mutation. Due to exponentialgrowth, most mutations occur close to the end of the experiment. Thus the developing time of arandom clone has exponential distribution. The last ingredients represents the number of mutant cells that any clone developing for a given time will produce. Thedistribution of this number depends mainly on the distribution of division times of mutants.One of the most used mutation model is the Luria-Delbrück model.In these model, division times of mutant cells were supposed to be exponentially distributed.Thus a clone develops according to a Yule process and its size at any given time follows a geometric distribution.This approach leads to a family of probability distributions which depend on the expected number of mutations and the relative fitness, which is the ratio between the growth rate of normal cells to that of mutants.The statistic purpose of these models is the estimation of these parameters. The probability for amutant cell to appear upon any given cell division is estimated dividing the mean number of mutations by the mean final number of cells.Given samples of final mutant counts, it is possible to build estimators maximizing the likelihood, or usingprobability generating function.Computing robust estimates is of crucial importance in medical applications, like cancer tumor relapse or multidrug resistance of Mycobacterium Tuberculosis for instance.The problem with classical mutation models, is that they are based on quite unrealistic assumptions: constant final number of cells,no cell deaths, exponential distribution of lifetimes, or time homogeneity. Using a model for estimation, when thedata have been generated by another one, necessarily induces a bias on estimates.Several sources of bias has been partially dealed until now: non-exponential lifetimes, cell deaths, fluctuations of the final count of cells,dependence of the lifetimes, plating efficiency. The time homogeneity remains untreated.This thesis contains probabilistic and statistic study of mutation models taking into account the following bias sources:non-exponential and non-identical lifetimes, cell deaths, fluctuations of the final count of cells, plating efficiency.Simulation studies has been performed in order to propose robust estimation methods, whatever the modeling assumptions.The methods have also been applied to real data sets, to compare the results with the estimates obtained under classical models.An R package based on the different results obtained in this work has been implemented (joint work with Rémy Drouilhetand Stéphane Despréaux) and is available on the CRAN. It includes functions dedicated to the mutation models and parameter estimation.The applications have been developed for the Labex TOUCAN (Toulouse Cancer).