Propietats globals en models epidemiològics de malalties de transmissió sexual

Abstract

In the study of the dynamics of sexually transmitted diseases, a population is usually subdivided into an active and relatively small core group and a weakly connected and largely inactive remainder, the non-core. In the core group one finds high transmission rates and high disease prevalence. The core group is usually a reservoir for sexually transmitted disease and it has a crucial role in the spread of the disease. The goal of the present work is to study the evolution of the core taking into account different social and epidemic scenarios. In addition to that, a substantial introduction to mathematical epidemiology as well as some basic concepts on stability theory will be presented. In this project we will learn how mathematics could be applied not only to understand the evolution of epidemics, but also to defi ne strategies in order to eradicate them. Although stochastic modeling of epidemics is a current topic of research, we are not going to treat it in these pages. Instead of this, our models are deterministic and are based on diff erential equations. In particular, the well known compartmental models are used within the text, in which variables represent di fferent population groups. This kind of models allows us to identify stationary scenarios and to study their stability. For attracting fixed points, questions such as the basin of attraction are of special interest for the purpose of this work. In this sense, the original part of this work is centered in studying global proprieties of a general model of sexually transmitted diseases. This model has a highly dependence of an arbitrary function that refl ects the uncertainty of social behavior. In fact, this function reproduces the caution degree of people during sexual relations. Although it is expected that an outbreak of a dangerous pathogen will make the people more cautious, the way how it happens is diffi cult to determine. For this reason we keep this functions as general as possible in order to see what can be said in most of possible scenarios. After that, we impose some restrictions to this function and we conclude some other particular results about the spread of the disease depending on some population parameters.