Estudio del comportamiento asintótico de un modelo en epidemiología

Abstract

Nowadays, mathematical models are one of the most popular tools to study epidemics due to the fact that they enable to analyze and simulate the incidence, spread, persistence or control of infectious diseases. Within an epidemiological framework, the aim of the present work is to study deterministic SIR models. These models are suitable for the description of infectious diseases with relatively short periods of infection and immune response. Instead of analyzing the behaviour of each single individual, these models are based on the classification of the individuals of the population into three classes (Susceptible, Infectious and Recovered) and they describe the dynamics among these three classes through a system with three differential equations. In particular, in this work we consider two situations: the first and simplest case, in which the population size is constant and is modelled by an autonomous differential system, and the second and more complex one, in which the total number of individuals in the population may change throughout the time and the dynamics among the different classes is modelled by a non-autonomous differential system. In both cases, we study the asymptotic behaviour of the resulting model. For the aforementioned study, it is necessary to review the scientific literature on autonomous and non-autonomous dynamical systems, with the aim to apply the aforesaid results to the considered the SIR models. To that end, we present some results on autonomous dynamical systems and in particular, on the global attractor, which are of interest for their application to the SIR model defined by an autonomous system. Then, we present some of the main results of interest on non-autonomous dynamical systems and more specifically on the pullback attractor, which will be applied to the second SIR model considered, which is defined by a non-autonomous system.