No description available.

No description available.

No description available.

Chapter 1 gives a brief introduction to modelling epidemics in both a deterministic and stochastic framework. Chapter 2 gives a brief introduction to the single-group Susceptible-Infected-Susceptible (SIS) model and we mention and establish all the necessary background theory we apply to the model with the aim of extending these theories and techniques to our own multi-group SIS model in order to obtain results concerning the short and long-term behaviour of the epidemic. In Chapter 3 we define a k-group SIS model which will allow us to examine the effects of heterogeneity in three different categories - the infectivity of infectious individuals, their mixing behaviour and an individual's susceptibility to the disease. In doing this, we describe the dynamics of the model including how to represent it in a deterministic framework, the quasi-stationary distribution and the time to extinction. We apply a branching process approximation to the model, viable for the early stages of a disease, using well established theory. In Chapter 4 we look specifically at the early stages of the epidemic based on the branching process approximation and produce numerical results on how the probability of disease emergence behaves as the basic reproduction number Ra increases. We contrast 2-group heterogeneous models against a homogeneous model, for epidemics assuming either an exponential or constant infectious period. We then analyse these results with an iterative and inductive proof showing that the emergence probability for a heterogeneous model will always be less than that for a homogeneous model, not just in the limit but at all stages of iterative convergence. Next we provide a proof which shows that for the non-separable general model this ordering exists for any given infectious period. We then go on to look at comparing two heterogeneous models to one another under various sets of parameters and use majorization theory as a tool for doing so. We use orderings referred to as ordinary majorization, p-majorization and pq-majorization to show that there is an inferred ordering of emergence probabilities when comparing multi-group he- terogeneous models to one another. In Chapter 5 we study the long-term behaviour of the stochastic multi-group SIS model. We begin by formulating conditions for the general model under which feasible equilibria exist and conditions where either the disease-free or endemic equilibria are stable. For a 2-group version, we calculate numerically the determi- nistic equilibrium values, stochastic means and quasi-stationary distributions for a range of Ra values. We use an Ornstein-Uhlenbeck process to approximate the quasi-stationary distribution and assess the accuracy of the approximation. We then calculate the expected time to extinction and use a coefficient of variation approximation as a proxy for this and discuss the suitability of such an approxi- mation to the exact results. These analyses are all carried out for models which exhibit heterogeneity in infectivity, mixing or susceptibility.

Cette thèse a pour origine un projet sur l'optimisation de la chimiothérapie, rassemblant trois directeurs: Jean Clairambault, médecin et mathématicien, Michèle Sabbah, biologiste du cancer, et Emmanuel Trélat, mathématicien spécialisé en contrôle optimal. Ainsi, l'essentiel du travail a été motivé par des questions provenant de la biologie ou la thérapie du cancer. Y répondre a nécessité l'utilisation et le développement d'outils empruntés à diverses disciplines mathématiques, parmi lesquelles l'analyse asymptotique d'équations aux dérivées partielles, leur contrôle optimal théorique et numérique. Ces développements ont posé de nouveaux problèmes mathématiques intéressants en eux-mêmes, avec des applications en dynamique adaptative, dynamique des populations, contrôle optimal ou encore analyse numérique. Plus précisément, nous proposons des résultats d'analyse asymptotique pour certaines équations ou systèmes de sélection/mutation et réaction/diffusion non-locaux. Le contrôle Dirichlet des équations monostable et bistable 1D est étudié dans le détail. On considère l'étude numérique et théorique d'un problème de contrôle optimal pour un système représentant des cellules saines et cancéreuses soumises à de la chimiothérapie. Enfin, l'existence d'instabilités de Turing pour un système de Keller-Segel est prouvée. Pour ces équations, nous développons des schémas numériques aux volumes finis qui préservent la positivité, la dissipation de l'énergie, la conservation de la masse et les états stationnaires. / This PhD originates from a joint project on chemotherapy optimisation, bringing together three advisors: Jean Clairambault, medical doctor and mathematician, Michèle Sabbah, cancer biologist, and Emmanuel Trélat, mathematician specialised in optimal control. Most of the work undertaken has thus been motivated by questions from cancer biology or therapy. Answering them has required using and further developing tools from several different mathematical areas, among them the asymptotic analysis for partial differential equations, and theoretical and numerical optimal control. These developments have in turn posed new mathematical problems, interesting in their own right, with applications in the mathematical fields of adaptive dynamics, population dynamics, optimal control or numerical analysis. More precisely, we propose results of asymptotic analysis for some selection/mutation and reaction/diffusion non-local equations or systems. The Dirichlet control towards homogeneous states of 1D monostable and bistable equations is investigated in detail. A numerical and theoretical analysis for an optimal control is performed on a system representing cancer and healthy cells exposed to chemotherapy. Finally, Turing instabilities are shown to be exhibited by some Keller-Segel equations, for which we design finite-volume numerical schemes preserving positivity, energy dissipation, mass conservation and steady states.

Depuis la stagflation observée consécutivement à la forte hausse du prix du pétrole en 1973 et 1979, les chocs pétroliers sont considérés comme l’une des sources de fluctuations potentiellement les plus importantes aux États-Unis comme dans de nombreux pays industrialisés. De nombreux articles ont étudié le rôle des chocs pétroliers dans la fluctuation des principales variables macroéconomiques à savoir, la croissance, le chômage, l’inflation et les salaires. Cependant, ces travaux n’ont pas encore permis d’aboutir à un consensus. Le débat s’est même intensifié au cours de cette dernière décennie, en raison d’une absence de réaction forte de l’économie réelle pendant la période d’augmentation du prix du pétrole entre 2002 et 2007. En effet, la récession qu’aurait dû engendrer une telle hausse des prix ne fut observée qu’au moment de la crise des subprimes en 2008. Plusieurs hypothèses furent avancées pour expliquer la différence entre les crises des années 1970 et 2000. Blanchard & Gali (2009) et Blanchard & Riggi (2013) évoquent, par exemple, la réduction de la quantité de pétrole utilisée dans la production, la plus grande flexibilité des salaires réels et une meilleure crédibilité de la politique monétaire. Hamilton (2009) et Kilian (2008) suggèrent quant à eux de l’expliquer par l’origine différente des deux chocs pétroliers : un choc d’offre pendant les années 70 et un choc de demande pendant les années 2000. L’objectif original de la thèse était de réexaminer l’impact des chocs pétroliers sur l’économie réelle par le canal de la dette. [...]Le développement de ces travaux entamés dans la thèse pourra aboutir à un cadre alternatif de modélisation décisif pour l’intelligence de la macroéconomie. Il devrait permettre une meilleure compréhension de l’évaluation des relations réciproques entre la sphère financière, la réalité des cycles macroéconomiques réels, l’énergie et le climat dans ce qui est sans aucun doute l’enjeu de notre génération : la transition écologique. / Ever since the stagflation that followed the oil price run-ups of 1973 and 1979, oil price shocks have been considered one of the most influential sources of economic fluctuation in the United States and other developed countries. A large body of literature has analyzed oil price shocks as sources of variation for leading macroeconomic variables such as GDP growth, unemployment rate, inflation, and wages. However, scholars have yet to reach a consensus as to the true impact of oil shocks on the macroeconomic environment. Furthermore, the last decade has seen the debate intensify as the results of the relatively (in comparison with the 1970s) muted reaction of the real economy during the 2002-6 oil price run-up. Indeed, the recessionary effect was only observed during the subprime mortgage crisis of 2008-9. Numerous hypotheses have been put forward to explain the difference in impact during the 1970s versus the 2000s. For instance, Blanchard & Gali (2009) and Blanchard & Riggi (2013) evoked the reduction of the quantity of oil used of a unit of production, more flexible real wages, and a better credibility of the monetary policy. Hamilton (2009) and Kilian (2008) pinpointed a difference in the nature of the shock: whereas the oil shocks of the 1970s were driven by supply, that of the 2000s was led by demand. The original aim of this thesis was to reevaluate the impact of the oil shock in the 2000s through the debt channel. First, based on the work of Banchard & Gali, we proposed a new dynamic stochastic general equilibrium model (DSGE), which includes oil as an input of production as well as a consumption good. By relaxing some of the hypotheses of Blanchard & Gali, especially the decoupling of the output elasticity of oil with the cost-share in the production, our work demonstrated that oil is still a fundamental variable of the GDP in the United States. Furthermore, we found that energy efficiency is a key factor that explains the muted macroeconomic impact of an increase in oil prices. A third line of inquiry that may explain the difference between the shocks of the 1970s and the 2000s considers the extra costs implied by a higher price of oil that were absorbed by private debt (which was itself exacerbated by low interest rates set by the Federal Reserve in the 2000s). However, we found that DSGE modeling is unable to replicate the macroeconomic environment that led to the subprime mortgage crisis. In light of these considerations, I reoriented my thesis along the lines of a new angle of research that seeks to represent economic mechanisms differently. Under this new frame-work, private debt is at the core of macroeconomic analysis. It provides an alternative view of the financial crisis that occurred in the 2000s.[...]The conclusions of this thesis demonstrate great potential for providing foundations for new perspectives in macroeconomic modeling. The papers included in the thesis allow, in particular, for a better understanding of situations that most macroeconomic models are not able to cope with, including the over indebtedness crisis. As a result, the framework introduced here may provide an alternative and improved perspective for public policy. Further development of the research presented in this thesis may lead to the improvement of other frameworks in the field of macroeconomics. This would allow for a better understanding of complex interactions between the financial sphere, real business cycles, energy, and climate in what is certainly the biggest challenge of our generation : the ecological shift.