Modelagem de epidemias através de modelos baseados em indivíduos

Abstract

The need to understand the propagation of diseases from the point of view dynamic has motivated the development of the mathematical epidemiology. The mathematical epidemiology considers models that may help in the control of epidemics. Kermack e McKendrick (1927) developed the SIR model, which classifies the individuals in three states: susceptible, infectious and recovered. These three states are related by means of nonlinear differential equations. However, the SIR model is unable to explain some phenomena such as the persistence or eradication of infectious diseases, the main reason for this is that the SIR model considers the spatial distribution and temporally homogeneous of individuals, from the premise that the size of the population is so large as to allow for continuous variables approximation of the various states. One approach is the called Individual Based Model (IBM) proposed by Nepomuceno (2005), that analyzes an individual as a discrete entity, constructed to reproduce the premises involved in the SIR model. In this work the following aspects are investigated: i) Validation of the IBM model compared with the classical SIR model in randomly different situations and creation of a new version, IBM-2, in which were made some changes to the premises of IBM, to get to a model close to the SIR model; ii) Modeling of heterogeneity from the IBM-2 by means Neural Networks to reduce the cost computer in simulations; iii) Modeling the IBM-2, incorporating vaccination in the study of the influence of stochastic fluctuation of the dynamical variables of an epidemics on the time of eradication of such epidemics, where they are shown here that, for small populations, such influence can become predominant; iv) Modeling the spread of an epidemic incorporating the MBI-2 the structure of regular networks that considers local contacts and complex networks from the model of random networks proposed by Erdös e Rényi (1959), in which the contacts between individuals are determined randomly and networks scale-free proposed by Barabási e Albert (1999), , in which some individuals have higher number of contacts that others, following a distribution called the power law. The analysis of propagating of epidemics through networks allows consider different situations of interest in the dynamics of epidemics.