Aplicación de los sistemas diferenciales al estudio de un modelo epidemiológico con reinfección

Abstract

Mathematical modeling, currently, is a very important element in health system, since it allows studying the incidence, spread and persistence of infectious diseases. This study is very useful to combat or control epidemiological diseases with reinfection. The objective of this paper is to study deterministic SIRI models, as well as to compare them with deterministic SIR models. The purpose of this comparison is to find relationships and differences between these models. Both models do not study the behavior of a single individual, but are based on the classification of them in groups. The SIRI models divide the individuals into four classes: Susceptible, Infectious, Recovered and Reinfected. These models are very useful for the description of infectious diseases in which individuals have relatively short periods of infection and have a temporary immune response. The dynamics between these groups is reflected in a system of three differential equations. In this paper we will study the case in which the SIRI model has a population size that remains constant (autonomous differential system), although we will mention other possible situations, among which the size of the population varies over time (non-autonomous differential system). We will also study the SIR model, which is used to analyze diseases without reinfection, in its simplest version, the case in which the population also remains constant (autonomous differential system). For both models we will study the local asymptotic behavior of the resulting system. For this analysis it will be necessary to make a study of the scientific literature about the autonomous dynamic systems, in order to apply them for the SIRI and SIR models.