This dissertation examines equilibria in epidemics and introduces novel approaches to epidemic modeling, consisting of two separate but closely related chapters. In Chapter 2, we develop an optimizing epidemic model within a dynamic urn-SIR framework, accommodating generic offspring distributions, and derive a trajectory convergence theorem. We study the mean-field equilibrium through numerical simulations. Our findings reveal two key features often overlooked in existing literature: substantial variation in epidemic outcomes despite homogeneous individual behavior, and the potential for resurgence in the number of infections. We demonstrate that the offspring distribution of infections significantly impacts epidemic dynamics, with negative binomial distributions leading to more dispersed outcomes and higher probabilities of minor outbreaks compared to geometric distributions. These results highlight the importance of stochastic modeling in epidemic forecasting and public health policy. Chapter 3 proposes and examines static and dynamic urn-SIR models, a novel approach to epidemic modeling that addresses key limitations of traditional stochastic SIR models. We focus on the critical issue of heterogeneity in individual infectiousness, which is not adequately captured by the geometric offspring distribution inherent in the continuous-time Markov chain SIR models. Our urn-SIR models accommodates generic offspring distributions, including the empirically supported negative binomial distribution. We formally formulate the static and dynamic urn-SIR models. The static model focuses on the end of the epidemic, where primary variables are the epidemic size and the total number of contacts, while the dynamic model captures the dynamic process of the disease progression. The cornerstone of our work is a proven threshold limit theorem, characterizing the asymptotic behavior of the epidemic size as the population approaches infinity. This theorem extends beyond early-stage branching process approximations in the existing literature that considers generic offspring distribution. Moreover, we also show that in the dynamic model, the trajectories of epidemic processes converges in probability to a corresponding deterministic system, allowing comprehensive analysis of entire epidemic courses. Our work bridges crucial gaps in existing literature, providing a more realistic representation of disease spread while maintaining analytical tractability. The findings have significant implications for epidemiology, public health, and related fields, informing more effective strategies for disease control and prevention.

Orientadores: Amir Ordacgi Caldeira, Eduardo Peres Novais de Sá / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-09-01T01:58:28Z (GMT). No. of bitstreams: 1 LopezDelgado_DanielAntonio_D.pdf: 831710 bytes, checksum: 17fbe60b2052b9d8534b963d0e85fe0e (MD5) Previous issue date: 2016 / Resumo: A criação de um computador quântico é um projeto que guia, ao mesmo tempo, avanços tecnológicos e um melhor entendimento das propriedades de sistemas quânticos e da Mecânica Quântica em geral. O teorema do limiar é derivado da teoria quântica de correção de erros e garante que, se o ruido estocástico que afeta os componentes de um computador quântico encontra-se abaixo de um valor limite, podemos operar esse computador quântico confiavelmente. Investigamos como esse teorema é modificado quando consideramos uma memória quântica (a qual usa o código de superfície para corrigir erros) acoplada a um ambiente correlacionado. O limiar de erros nesse caso é relacionado à transição de fase ordem-desordem de um sistema de spin equivalente / Abstract: The design of a quantum computer is a project which drives, at the same time, technological advancement and a better understanding of the properties of quantum systems and of Quantum Mechanics in general. The threshold theorem comes from quantum error correction theory and it guarantees that, if stochastic noise affecting the components of a quantum computer is below some threshold value, we can operate this quantum computer reliably. We investigate how this theorem is modified when we consider a quantum memory (which uses the surface code to correct errors) coupled to a correlated environment. The error threshold in this case is related the order-disorder phase transition of an equivalent spin system / Doutorado / Física / Doutor em Ciências

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