Teoria qualitativa e estabilidade de Lyapunov para sistemas de equações de ordem fracionária e uma aplicação em um modelo SIR-SI para a dengue

Abstract

This work is a study of the theory of fractional differential equations from the point of view qualitative. To this, we study theorems of existence, uniqueness and continuation of solutions, comparison theory, Lyapunov stability theory to fractional order systems and we apply these results. In recent years, study of fractional-order systems has attracted the attention of many researchers and has generated several publications. The possible applications of this theory are present in several areas of the knowledge such as Viscoelasticity, Electrochemistry, Physics, Epidemiology, among many others that can be modeled by fractional differential equation. We make use of this theory to study a mathematical model to transmission of Dengue Disease, by fractional derivative of Caputo type, whose model is available in the literature. Thus, a fundamental result is the analysis of the behavior asymptotic, that is, what happens to the solutions when the time to infinity, so we will use Lyapunov stability. We use numerical simulations to carried out to validate the model and investigate whether the equilibrium points are globally asymptotically stable, since four values were considered as initial conditions, whose solutions converge to the equilibrium points. In addition, it was compared to a simulation using ordinary differential equations and it was found that the solutions of the fractional order model have less oscillations because they guarantee a damping system, causing the solutions to converge at a lower speed.