Dinâmica de infecções por HTLV-I considerando taxas de lise, morte, proliferação e transmissibilidade da resposta CTL
Carregando...
Arquivos
Data
2018-07-27
Autores
Varalta, Najla
Título da Revista
ISSN da Revista
Título de Volume
Editor
Universidade Estadual Paulista (Unesp)
Resumo
Os avanços no entendimento da dinâmica do sistema imunológico humano utilizando modelos matemáticos baseados na teoria de sistemas dinâmicos estão crescendo continuamente. Modelos não-lineares advindos das clássicas equações de Lotka-Volterra têm sido utilizados para estudo da resposta imunológica na ocasião de infecções. Com isso em mente, uma vez que a compreensão do sistema imunológico humano tem grande destaque, o foco específico desta tese é a análise da resposta imunológica frente a infecções pelo vírus HTLV-I (do inglês "Human T cell lymphotropic virus type I"), utilizando três modelos matemáticos representativos que contemplam esta infecção. Especificamente, estudou-se o uso da função sigmoidal com o parâmetro n genérico e sua implicação nos equilíbrios existentes e estabilidade de cada ponto para descrever a proliferação CTL. Posteriormente, foi proposto um modelo que descreve a interação entre KIR e HLA. Neste modelo, foi possível descrever os equilíbrios e sua respectiva estabilidade em função da taxa de transmissibilidade infecciosa relativa ao desenvolvimento de duas doenças associadas ao HTLV-I: HAM/TSP e ATL. Finalmente, o terceiro modelo, adicionou-se a taxa de produção de células T CD8+ e, com isto, o sistema apresentou apenas dois equilíbrios. O segundo ponto de equilíbrio é escrito em termos de um polinômio de segundo grau, côncavo para cima e com isto, foi definido a região de existência deste ponto bem como critérios para sua estabilidade. Cada modelo matemático apresentado possui três EDOs de segunda ordem acopladas. Utilizou-se métodos analíticos aplicáveis a sistemas não-lineares, bem como métodos numéricos para análise de bifurcações essenciais para um entendimento substancial dos fenômenos envolvidos.
Advances in understanding the dynamics of the human immune system using mathematical models based on the theory of dynamic systems are growing continuously. Nonlinear models derived from the classical Lotka-Volterra equations have been used to study the immune response at the time of infection. With this in mind, since the understanding of the human immune system is prominent, the specific focus of this thesis is the analysis of the immune response to infections by the HTLV-I virus (Human T cell lymphotropic virus type I), using three representative mathematical models that contemplate this infection. Specifically, the use of the sigmoidal function with the generic parameter n and its implication in the existing equilibrium and stability of each point to describe CTL proliferation was studied. Subsequently, a model was proposed that describes the interaction between KIR and HLA. In this model, it was possible to describe the equilibriums and their respective stability as a function of the infectious transmissibility rate related to the development of two HTLV-I associated diseases: HAM/TSP and ATL. Finally, in the third model, the CD8+ T-cell production rate was added and, with this, the system presented only two equilibriums. The second equilibrium point is written in terms of a second-degree polynomial, concave up and with this, the region of existence of this point was defined as well as criteria for its stability. Each mathematical model has three coupled second-order ODEs. Analytical methods were applied to nonlinear systems, as well as numerical methods for analysis of bifurcations essential for a substantial understanding of the involved phenomena.
Advances in understanding the dynamics of the human immune system using mathematical models based on the theory of dynamic systems are growing continuously. Nonlinear models derived from the classical Lotka-Volterra equations have been used to study the immune response at the time of infection. With this in mind, since the understanding of the human immune system is prominent, the specific focus of this thesis is the analysis of the immune response to infections by the HTLV-I virus (Human T cell lymphotropic virus type I), using three representative mathematical models that contemplate this infection. Specifically, the use of the sigmoidal function with the generic parameter n and its implication in the existing equilibrium and stability of each point to describe CTL proliferation was studied. Subsequently, a model was proposed that describes the interaction between KIR and HLA. In this model, it was possible to describe the equilibriums and their respective stability as a function of the infectious transmissibility rate related to the development of two HTLV-I associated diseases: HAM/TSP and ATL. Finally, in the third model, the CD8+ T-cell production rate was added and, with this, the system presented only two equilibriums. The second equilibrium point is written in terms of a second-degree polynomial, concave up and with this, the region of existence of this point was defined as well as criteria for its stability. Each mathematical model has three coupled second-order ODEs. Analytical methods were applied to nonlinear systems, as well as numerical methods for analysis of bifurcations essential for a substantial understanding of the involved phenomena.
Descrição
Palavras-chave
Dinâmica não-linear, Modelagem matemática, Biomatemática, Progressão de infecções, Nonlinear dynamics, Mathematical modeling, Biomathematics, Infection progression