No description available.

No description available.

L’objectif de cette thèse est la modélisation, l’analyse mathématique et la simulation numérique de quelques modèles de transmission du VIH-SIDA dans une population sexuellement active donnée en général et en particulier dans celle du Mali. Nous proposons de modèles basés sur les connaissances actuelles de la transmission du virus du VIH. A cet effet, nous présentons trois modèles : un modèle comportemental en épidémiologie, un modèle qui incorpore le rôle des campagnes de sensibilisation en santé publique et un modèle de co-circulation de deux formes recombinantes du VIH-1. Dans le premier modèle, des résultats d’existence et d’unicité de la solution d’un problème parabolique semi-linéaire décrivant l’évolution d’une population soumise à l’infection du VIH sont présentés. La population est divisée en individus dont le risque comportemental est faible et en individus dont les comportement sont très risqués et qui interagissent entre eux. Une variable continue représentant ce risque comportemental est introduit. Le comportement asymptotique en temps du problème est étudié. Certains résultats numériques concernant la répartition de la population selon la variable représentant le risque comportemental sont présentés dans le cas de l’infection du VIH-SIDA au Mali. Dans les deux derniers modèles, nous obtenons une analyse complète de la stabilité de ces modèles à l’aide des techniques de Lyapunov suivant la valeur du taux de reproduction de base . Nous proposons une méthode alternative au taux de reproduction de base qui permet de confiner l’évolution de la maladie dans des limites fixées. Nous illustrons ces modèles par des simulations numériques. Ces dernières sont faites à partir de nos modèles confrontés aux données du Mali concernant la propagation du VIH-SIDA. / The objective of this thesis is the modeling, mathematical analysis and numerical simulation of a few models of transmission of the HIV-AIDS in a sexually active population given in general and in particular in that of Mali. We propose models based on the epidemiology currently known from the transmission of the HIV virus. Thus, we present three models of the transmission of HIV: a individual behavior and epidemiological model, a model that incorporates the role of public health education program on HIV and a mathematical model for the co-circulating into two circulating recombinants forms of HIV-1. In the first model, Somé results of existence and uniqueness of solution of a semilinear,parabolic problem describing the evolution of a population subjected to a disease are presented. The population is divided into individuals whose behavioral risk is low and in individuals whose behavior are very risky and that interact between them. A continuous variable representing a behavioral risk is introduced. The asymptotic in time of the problem is studied, and the existence of a non zero stationary state is proved. Somé numerical results concerning the distribution of the population according to the variable representing a behavioral risk are presented within the disease of the HIV-AIDS in Mali. In the last two models, we obtain a thorough analysis of the stability of these models using the Lyapunov techniques according to the value of the basic reproduction ratio,R0. We propose an alternative method to the basic reproductive rate R0 which allows to confine the evolution of the disease in the fixed limits. Numerical simulations are done to illustrate the behaviour of the model, using data collected in the literature regarding the spread of HIV in Mali.

Le but de la thèse est d'étudier l'homogénéisation d'équations de convection-diffusion singulières et de problèmes spectraux à poids indéfini. La thèse se compose de deux parties. La première partie contient des résultats qualitatifs et asymptotiques pour les solutions d'équations de type convection-diffusion stationnaires et instationnaires, qui sont définies dans des domaines bornés ou nonbornés. Les problèmes examinés comprennent des études qualitatives pour une équation elliptique avec des termes du premier ordre dans un cylindre semi-infini, l'homogénéisation de modèles de convection-diffusion dans des cylindres minces et une analyse asymptotique d'équations de convection-diffusion instationnaires avec un grand terme du premier ordre, posées dans un domaine borné. La deuxième partie de la thèse porte sur l'homogénéisation de problèmes spectraux à poids indéfini, pouvant changer de signe. On montre que le comportement asymptotique dépend essentiellement de la moyenne du poids, notamment si la moyenne est nulle ou non nulle. On construit alors le développement asymptotique du spectre dans les deux cas.

In the thesis we consider two types of problems. In Paper 1, we study small solutions to a time-independent nonlinear elliptic partial differential equation of Emden-Fowler type in a semi-infnite cylinder. The asymptotic behaviour of these solutions at infnity is determined. First, the equation under the Neumann boundary condition is studied. We show that any solution small enough either vanishes at infnity or tends to a nonzero periodic solution to a nonlinear ordinary differential equation. Thereafter, the same equation under the Dirichlet boundary condition is studied, the non-linear term and right-hand side now being slightly more general than in the Neumann problem. Here, an estimate of the solution in terms of the right-hand side of the equation is given. If the equation is homogeneous, then every solution small enough tends to zero. Moreover, if the cross-section is star-shaped and the nonlinear term in the equation is subject to some additional constraints, then every bounded solution to the homogeneous Dirichlet problem vanishes at infnity. In Paper 2, we study asymptotics as t → ∞ of solutions to a linear, parabolic system of equations with time-dependent coefficients in Ωx(0,∞), where Ω is a bounded domain. On δΩ(0,∞) we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function κ(t). This includes in particular situations when the coefficients may take different values on different parts of Ω and the boundaries between them can move with t but stabilize as t → ∞. The main result is an asymptotic representation of solutions for large t. As a corollary, it is proved that if κєL1(0,∞), then the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.

Made available in DSpace on 2015-05-15T11:46:21Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1068702 bytes, checksum: 4394a0038dff8943184cc1e89156c0df (MD5) Previous issue date: 2014-09-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The Bohnenblust{Hille inequality guarantees the existence of a function C : N ! [1;+1), corresponding to each positive integer m, a constant C(m) with the following property: regardless of the choice of the natural N and the bounded m-linear form U : `N 1 `N 1 ! K, the sequence (U(ei1 ; : : : ; eim))N i1;:::;im=1 belongs to ` 2m m+1 and its 2m m+1-norm is bounded by C(m)kUk, where k k denotes the supremum norm. Apart from the intrinsic mathematical interest, for C(m) does not depend on each natural N, the diversity and relevance of the applications enrich the result further. On the actual scenario, recent explicit estimates for the constants C(m) present optimal asymptotic behaviour and subexponencial growth, in contrast to the exponential growth of the known estimates from the last decades. Valuable informations concerning the optimal constants (the lowest ones with the previous property stated) emerge, once proved that these also enjoy of an optimal asymptotic growth, if it exists. / A Desigualdade de Bohnenblust-Hille assegura a existência de uma função C : N -! [1;+1), que corresponde a cada inteiro positivo m uma constante C(m) 2 [1;+1) com a seguinte propriedade: quaisquer que sejam o natural N e a forma m-linear limitada U : `N 1 ----- `N 1 ! K, a sequência (U(ei1 ; : : : ; eim))N i1;:::;im=1 pertence à 2m m+1 e sua norma 2m m+1 é limitada por C(m) kUk, onde k k denota a norma supremo. Afora o interesse matemático intrínseco, pois C(m) independe de cada natural N, a diversidade e relevância das aplicações enriquecem ainda mais o resultado. No cenário atual, recentes estimativas para as constantes C(m) apresentam comportamento assintótico ótimo e crescimento subexponencial, em contraste ao crescimento exponencial das estimativas conhecidas ao longo das várias décadas anteriores. Informações de valor a respeito das constantes ótimas (as menores possíveis com a propriedade descrita anteriormente) surgem, uma vez provado que essas também usufruirão de crescimento assintótico ótimo, caso este exista.

No description available.

Neste trabalho estudamos problemas de reação-difusão semilineares do tipo \'u IND..t(x, t) = \'DELTA\'u(x, t)+ f (u(x, t)), x \'PERTENCE A\' \'OMEGA\' \'PARTIAL\' U/\'PARTIAL\'V (x, t) = 0, x \'PERTENCE A\' \'PARTIAL\'\' OMEGA\'. Desenvolvemos uma teoria abstrata para a obtenção da continuidade da dinâmica assintótica de (P) sob perturbações singulares do domínio espacial W e aplicamos a uma série de exemplos dos assim chamados domínios finos / In this work we study semilinear reaction-diffusion problems of the type \'u IND.t(x, t) = \'DELTA\'u(x, t)+ f (u(x, t)), x \' PERTENCE A\' \'OMEGA\' \'PARTIAL\'u/\'ARTIAL\' v (x, t) = 0, x \"PERTENCE A\' \'PARTIAL\' \' OMEGA\' We develop a abstract theory to obtain the continuity of the asymptotic dynamics of (P) under singular perturbations of the spatial domain W and we apply that to many examples in thin domains

O objetivo deste trabalho é investigar propriedades qualitativas das equações diferenciais funcionais neutras (EDFNs) e introduzir uma classe geral de equações chamadas EDFNs em medida. Obtemos resultados sobre o comportamento assintótico para uma classe de EDFNs com coeficientes periódicos, onde o período e o retardamento estão racionalmente relacionados. Também, conseguimos mostrar que a dicotomia exponencial do operador solução das equações diferenciais funcionais com retardamento (EDFRs) não autônomas implica na existência de soluções limitadas para EDFRs não homogêneas associadas. Finalmente, através da teoria das equações diferenciais ordinárias generalizas (EDOs generalizadas), obtemos resultados de existência e unicidade, dependência contnua em relação aos dados inicias, das soluções das EDFNs em medida. Os resultados novos apresentados neste trabalho estão contidos nos artigos [31, 43] / The aim of this work is to investigate qualitative properties of neutral functional differential equations (NFDEs) and introduce a general class of equations called measure NFDE . We obtain results on the asymptotic behavior for a class of NFDEs with periodic coefficients, where the period and delay are rationally related. Moreover, we show that the exponential dichotomy of the solution operator of non autonomous retarded functional differential equations (RFDEs) implies the existence of bounded solutions to the associated non homogeneous RFDEs. Finally, using the theory of generalized ordinary differential equations (generalized ODEs), we obtain results of existence and uniqueness, continuous dependence on parameters of the solutions of measure NFDEs. The new results presented in this work are contained in the articles [31,43]

O objetivo deste trabalho é investigar propriedades qualitativas das equações diferenciais funcionais neutras (EDFNs) e introduzir uma classe geral de equações chamadas EDFNs em medida. Obtemos resultados sobre o comportamento assintótico para uma classe de EDFNs com coeficientes periódicos, onde o período e o retardamento estão racionalmente relacionados. Também, conseguimos mostrar que a dicotomia exponencial do operador solução das equações diferenciais funcionais com retardamento (EDFRs) não autônomas implica na existência de soluções limitadas para EDFRs não homogêneas associadas. Finalmente, através da teoria das equações diferenciais ordinárias generalizas (EDOs generalizadas), obtemos resultados de existência e unicidade, dependência contnua em relação aos dados inicias, das soluções das EDFNs em medida. Os resultados novos apresentados neste trabalho estão contidos nos artigos [31, 43] / The aim of this work is to investigate qualitative properties of neutral functional differential equations (NFDEs) and introduce a general class of equations called measure NFDE . We obtain results on the asymptotic behavior for a class of NFDEs with periodic coefficients, where the period and delay are rationally related. Moreover, we show that the exponential dichotomy of the solution operator of non autonomous retarded functional differential equations (RFDEs) implies the existence of bounded solutions to the associated non homogeneous RFDEs. Finally, using the theory of generalized ordinary differential equations (generalized ODEs), we obtain results of existence and uniqueness, continuous dependence on parameters of the solutions of measure NFDEs. The new results presented in this work are contained in the articles [31,43]