Estudo da dinâmica de evolução do HIV em seres humanos utilizando sistema de equações diferenciais ordinárias
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Data
2019-02-28
Autores
Vicentin, Daniel Chieregato
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Universidade Estadual Paulista (Unesp)
Resumo
O objetivo desta dissertação é abordar aspectos qualitativos de sistemas de equações diferenciais ordinárias e sistemas contínuos suaves por partes aplicados à dinâmica do Vírus da Imunodeficiência Humana (HIV). Neste trabalho, apresentamos um modelo matemático que descreve a dinâmica do HIV no corpo humano e o analisamos através da matriz da próxima geração e teoria de estabilidade, com a finalidade de prever se a doença fica ou não controlada. Posteriormente, estudamos um sistema de equações diferenciais ordinárias usado para modelar a dinâmica do vírus para diferentes tipos de tratamentos. Tal modelo foi explorado qualitativamente de duas maneiras: por um sistema contínuo (pelo método de Korobeinikov) e por um descontínuo (pelas convenções de Filippov). Analisamos o comportamento dinâmico de terapias antirretrovirais, visando a diminuição das concentrações virais no sangue, de acordo com a análise da estabilidade realizada.
The goal of this dissertation is to study qualitative aspects about systems of ordinary differential equations and piecewise smooth systems applied to the dynamic of Human Immunodeficiency Virus (HIV). In this work, we present a mathematical model that describes the dynamic of HIV in the human body and we analyze this model by next-generation matrix and stability theory in order to predict if the disease becomes stable, and thus stop virus transmission. In addition, we studied another system of ordinary differential equations that were proposed to model the HIV dynamics assuming different therapies. We have explored qualitatively the model by two distinct approaches: a continuous system (by Korobeinikov method) and a discontinuous system (by Filippov theory). Due to the stability analysis, it was possible to understand the dynamics of anti-retroviral therapies, which are responsible for decreasing the concentration of detectable HIV in blood.
The goal of this dissertation is to study qualitative aspects about systems of ordinary differential equations and piecewise smooth systems applied to the dynamic of Human Immunodeficiency Virus (HIV). In this work, we present a mathematical model that describes the dynamic of HIV in the human body and we analyze this model by next-generation matrix and stability theory in order to predict if the disease becomes stable, and thus stop virus transmission. In addition, we studied another system of ordinary differential equations that were proposed to model the HIV dynamics assuming different therapies. We have explored qualitatively the model by two distinct approaches: a continuous system (by Korobeinikov method) and a discontinuous system (by Filippov theory). Due to the stability analysis, it was possible to understand the dynamics of anti-retroviral therapies, which are responsible for decreasing the concentration of detectable HIV in blood.
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Palavras-chave
Modelagem Matemática., HIV, Sistemas Suaves por Partes., Mathematical Modelling., Piecewise Smooth Systems.