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Abstract(s)
As doenças infeciosas podem propagar-se no seio das população e provocar
epidemias com grandes consequências para a saúde pública e para a economia.
A epidemiologia é a ciência que estuda os padrões da ocorrência de doenças
infeciosas em populações humanas assim como os fatores que determinam
esses padrões. Nesse sentido, a epidemiologia matemática (ou computacional)
transcreve as características dos fenómenos biológicos para a linguagem
matemática na forma de modelos que permitem reproduzir computacionalmente
esses fenómenos. Com esses modelos é possível simular a evolução do
contágio por uma doença infeciosa no interior de uma população, prever o
impacto total e testar o efeito de eventuais medidas preventivas ou corretivas.
Um modelo epidemiológico pode ser implementado de forma determinística
ou estocástica. Os modelos estocásticos são em geral mais realistas.
No entanto, como são computacionalmente mais exigentes, eles são normalmente
utilizados apenas em pequenas populações. A adaptação dos modelos
estocásticos a populações de grandes dimensões é pois um tema de grande
pertinência.
Neste projeto procura-se estudar a modelação estocástica com recurso a
cadeias de Markov. Implementam-se modelos estocásticos em dois contexvii
tos diferentes correspondentes a problemas práticos propostos na literatura.
Num primeiro problema considera-se o contágio por uma doença infeciosa
numa população não misturada que corresponde a uma enfermaria hospitalar.
Aplicam-se as técnicas de simulação aleatória designadas por método
de Monte Carlo e avaliam-se os efeitos de diferentes fatores, como a taxa de
mobilidade, de vacinação e de contágio. Avalia-se também a possibilidade de
aplicar as cadeias de Markov no contexto deste problema. Observa-se que
esta abordagem traz vantagens computacionais que resultam da utilização
da matriz de transição.
O segundo contexto, que pressupõe uma população homogénea e uniformemente
misturada, consiste na implementação dos modelos epidemiológicos
SIS e SIR, bem de nidos do ponto de vista determinístico. Faz-se a sua implementa
ção com recurso a cadeias de Markov e comparam-se os resultados com
as soluções determinísticas. Os resultados mostram que é possível obter com
os modelos estocásticos soluções compatíveis com as soluções determinísticas.
Veri ca-se também que a utilização das cadeias de Markov é desa ante
do ponto de vista da investigação. Efetivamente, a construção da matriz
de transição torna-se cada vez mais complexa à medida que o número de
variáveis aleatórias aumenta.
Infectious diseases can spread within the population and cause epidemics with major consequences for public health and for the economy. Epidemiology is the science that studies the patterns of occurrence of infectious diseases in human populations as well as the factors that determine these patterns. In this sense, mathematical (or computational) epidemiology transcribes the characteristics of biological phenomena to the mathematical language in the form of templates that let you play computationally these phenomena. With these models we can simulate the evolution of infectious disease within a population, predict the full impact and test the e ect of any preventive or corrective measures. An epidemiological model can be implemented deterministically or stochastically. Stochastic models are generally more realistic. However, they are computationally more demanding, they are typically only used in small populations. The adjustments to the stochastic models to large populations is therefore a topic of great relevance. This project was looking to study stochastic modeling with resources to Markov chains. We implement stochastic models in two di erent contexts corresponding to practical problems proposed in the literature. In a rst ix problem considered by the spread of an infectious disease in a non-mixed population that corresponds to a hospital ward. We apply the random simulation techniques, called Monte Carlo method, and evaluate the e ects of di erent factors such as the degree of mobility, vaccination and infection. We also adress the possibility of applying the Markov chains in the context of this problem. It is observed that this approach presents some computational advantages resulting from the use of the transition matrix. The second context, that presupposes a homogeneous and mixed population, is the implementation of epidemiological models SIS and SIR, wellde ned from the deterministic point of view. We implemente the models with the use of Markov chains and compare the results with the deterministic solutions. The results show that it is possible to obtain with the stochastic models solutions compatible with the deterministic solutions. It is also veri- ed that the use of Markov chains is challenging the research point of view. Indeed, the construction of the transition matrix becomes increasingly complex as the number of random variables increases.
Infectious diseases can spread within the population and cause epidemics with major consequences for public health and for the economy. Epidemiology is the science that studies the patterns of occurrence of infectious diseases in human populations as well as the factors that determine these patterns. In this sense, mathematical (or computational) epidemiology transcribes the characteristics of biological phenomena to the mathematical language in the form of templates that let you play computationally these phenomena. With these models we can simulate the evolution of infectious disease within a population, predict the full impact and test the e ect of any preventive or corrective measures. An epidemiological model can be implemented deterministically or stochastically. Stochastic models are generally more realistic. However, they are computationally more demanding, they are typically only used in small populations. The adjustments to the stochastic models to large populations is therefore a topic of great relevance. This project was looking to study stochastic modeling with resources to Markov chains. We implement stochastic models in two di erent contexts corresponding to practical problems proposed in the literature. In a rst ix problem considered by the spread of an infectious disease in a non-mixed population that corresponds to a hospital ward. We apply the random simulation techniques, called Monte Carlo method, and evaluate the e ects of di erent factors such as the degree of mobility, vaccination and infection. We also adress the possibility of applying the Markov chains in the context of this problem. It is observed that this approach presents some computational advantages resulting from the use of the transition matrix. The second context, that presupposes a homogeneous and mixed population, is the implementation of epidemiological models SIS and SIR, wellde ned from the deterministic point of view. We implemente the models with the use of Markov chains and compare the results with the deterministic solutions. The results show that it is possible to obtain with the stochastic models solutions compatible with the deterministic solutions. It is also veri- ed that the use of Markov chains is challenging the research point of view. Indeed, the construction of the transition matrix becomes increasingly complex as the number of random variables increases.
Description
Keywords
Epidemias Modelos estocásticos Cadeias de Markov Matriz de transição