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During my PhD thesis, I mainly worked on three projects involving mathematical and bioinformatics methods in immuno-oncology. 1) Project 1: development of the SmartFACS deconvolution tool. I developed a deconvolution tool called SmartFACS in collaboration with the Mechanics and Applied Mathematics Department (ULB) and the Immunobiology Laboratory (ULB). The role of this computer program is to estimate the cellular composition of a sample from RNA sequencing data. This tool has been developed in the context of immunology, where it is often of interest to identify immune cells present in the blood or in tumors. At the time this project began, no deconvolution tool for mouse data was yet available, despite the fact that much biomedical research is carried out in mouse models. During my thesis, I therefore decided to develop a tool adapted to mouse data, while trying to improve existing tools for human data by detecting a wider range of immune populations. 2) Project 2: research of clinical biomarkers for anti-PD1 immunotherapy treatment. In collaboration with the Immunobiology group (ULB), I worked on anti-PD1 therapy, which is already used clinically for certain cancers. Unfortunately, the success rate of this treatment is still relatively moderate. This is why the aim of my project was to determine predictive markers of response to anti-PD1 in order to better understand the mechanisms of resistance and find avenues of improvement for this treatment, as well as to better identify responder patients. To achieve these objectives, a mouse model was developed in the Immunobiology Laboratory. This model enabled us to reproduce the dichotomous response observed in the clinic. My work focused on analyzing pre-treatment tumor sequencing data obtained using the mouse model, in order to compare the immune infiltrate of responder and non-responder mice. My data analyses, together with experiments conducted by Jelena Gabrilo, revealed two immune populations important for treatmen, Doctorat en Sciences, info:eu-repo/semantics/nonPublished

Deriving the time-dependent expected reward function associated with a continuous-time Markov chain involves the computation of its transient deviation matrix. In this paper we focus on the special case of a finite quasi-birth-and-death (QBD) process, motivated by the desire to compute the expected revenue lost in a MAP/PH/1/C queue. We use two different approaches in this context. The first is based on the solution of a finite system of matrix difference equations; it provides an expression for the blocks of the expected reward vector, the deviation matrix, and the mean first passage time matrix. The second approach, based on some results in the perturbation theory of Markov chains, leads to a recursive method to compute the full deviation matrix of a finite QBD process. We compare the two approaches using some numerical examples., SCOPUS: ar.j, DecretOANoAutActif, info:eu-repo/semantics/published

We consider an infinite fluid queue modulated by a Markovian environment which varies slowly between two different modes of behaviour. We derive the series expansion of the stationary density of its buffer content as a function of the aggregated stationary densities of the buffer contents of the two environmental fluid queues., SCOPUS: cp.p, info:eu-repo/semantics/published

In this paper we analyse Markov-modulated fluid processes over finite time intervals. We study the joint distribution of the level at time θ< ∞ and of the maximum level over [0, θ], as well as the joint distribution of the level at time θ and the minimum level over [0, θ]. We approximate θ by a random variable T with Erlang distribution and so use an approach different from the usual Laplace transform to compute the distributions. We present probabilistic interpretation of the equations and provide a numerical illustration., SCOPUS: ar.j, info:eu-repo/semantics/published

We consider perturbations of positive recurrent Markov modulated fluid models. In addition to the infinitesimal generator of the phases, we also perturb the rate matrix, and analyze the effect of those perturbations on the matrix of first return probabilities to the initial level. Our main contribution is the construction of a substitute for the matrix of first return probabilities, which enables us to analyze the effect of the perturbation under consideration., SCOPUS: ar.j, info:eu-repo/semantics/published

We consider the Poisson equation (I -P)u = g, where P is the transition matrix of a quasi-birth-and-death process with infinitely many levels, g is a given infinite dimensional vector, and u is the unknown. Our main result is to provide the general solution of this equation. To this purpose we use the block tridiagonal and block Toeplitz structure of the matrix P to obtain a set of matrix difference equations, which are solved by constructing suitable resolvent triples., SCOPUS: ar.j, info:eu-repo/semantics/published

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The Erlangian approximation of Markovian fluid queues leads to the problem of computing the matrix exponential of a subgenerator having a block-triangular, block-Toeplitz structure. To this end, we propose some algorithms which exploit the Toeplitz structure and the properties of generators. Such algorithms allow us to compute the exponential of very large matrices, which would otherwise be untreatable with standard methods. We also prove interesting decay properties of the exponential of a generator having a block-triangular, block-Toeplitz structure., info:eu-repo/semantics/published

A Markov process is defined by its transition matrix. A skip-free Markov process is a stochastic system defined by a level that can only change by one unit either upwards or downwards. A regular perturbation is defined as a modification of one or more parameters that is small enough not to change qualitatively the model.This thesis focuses on a category of methods, called matrix analytic methods, that has gained much interest because of good computational properties for the analysis of a large family of stochastic processes. Those methods are used in this work in order i) to analyze the effect of regular perturbations of the transition matrix on the stationary distribution of skip-free Markov processes; ii) to determine transient distributions of skip-free Markov processes by performing regular perturbations.In the class of skip-free Markov processes, we focus in particular on quasi-birth-and-death (QBD) processes and Markov modulated fluid models.We first determine the first order derivative of the stationary distribution - a key vector in Markov models - of a QBD for which we slightly perturb the transition matrix. This leads us to the study of Poisson equations that we analyze for finite and infinite QBDs. The infinite case has to be treated with more caution therefore, we first analyze it using probabilistic arguments based on a decomposition through first passage times to lower levels. Then, we use general algebraic arguments and use the repetitive block structure of the transition matrix to obtain all the solutions of the equation. The solutions of the Poisson equation need a generalized inverse called the deviation matrix. We develop a recursive formula for the computation of this matrix for the finite case and we derive an explicit expression for the elements of this matrix for the infinite case.Then, we analyze the first order derivative of the stationary distribution of a Markov modulated fluid model. This leads to the analysis of the mat, Doctorat en Sciences, info:eu-repo/semantics/nonPublished

We consider Poisson's equation for quasi-birth-and-death processes (QBDs) and we exploit the special transition structure of QBDs to obtain its solutions in two different forms. One is based on a decomposition through first passage times to lower levels, and the other is based on a recursive expression for the deviation matrix. We revisit the link between a solution of Poisson's equation and perturbation analysis and we show that it applies to QBDs. We conclude with the PH/M/1 queue as an illustrative example, and we measure the sensitivity of the expected queue size to the initial value. © 2013 Elsevier B.V. All rights reserved., SCOPUS: ar.j, info:eu-repo/semantics/published

We consider Poisson's equation for quasi-birth-and-death processes (QBDs) and we exploit the special transition structure of QBDs to obtain its solutions in two different forms. One is based on a decomposition through first passage times to lower levels, and the other is based on a recursive expression for the deviation matrix. We revisit the link between a solution of Poisson's equation and perturbation analysis and we show that it applies to QBDs. We conclude with the PH/M/1 queue as an illustrative example, and we measure the sensitivity of the expected queue size to the initial value. © 2013 Elsevier B.V. All rights reserved., SCOPUS: ar.j, info:eu-repo/semantics/published

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info:eu-repo/semantics/published