Modelagem epidemiológica com um modelo SIR estocástico utilizando Cadeia de Markov de tempo contínuo

Abstract

In epidemiological modeling, the implementation of mathematical models and their respective computer simulations are advantageous tools for estimating characteristics about the spread of infectious diseases. A well-known and widely used model for this aim is the Susceptible-Infected-Recovered (SIR), proposed by Kermack and McKendrick in 1927, by using a deterministic approach. In this dissertation, the SIR model is presented with a stochastic approach, using Continuous Time Markov Chain. Stochastic modeling allows to insert some randomness into the model's parameters and, thus, analyze situations that are not foreseen by the deterministic model. It brings some advantages such as the access to probability distributions for some properties that are exclusive to the stochastic model. In the present work, an updated bibliographic review is made and the theoretical background of stochastic modeling using Markov Chains is given. An adaptation of the stochastic SIR model using Continuous Time Markov Chain (SIR CTMC) is considered for two epidemiological situations, a measles outbreak in Ilha Grande, Brazil, in 1976, and the COVID-19 pandemic in the state of Rio de Janeiro, Brazil, in 2020. Computer simulations are performed using the Python language, from which a comparison is made between the deterministic and stochastic models. In the simulations, values of the parameters reported in the literature are used. In this way, it has been possible to validate the stochastic model and some of its advantages are in evidence.