<p dir="ltr">Network models are essential for representing a myriad of real-world problems. Two of the most important categories of networks are centralized and distributed networks. In this thesis, we investigate the efficient resource allocation for one centralized communication network and two distributed epidemic networks.</p><p dir="ltr">In Chapter 2, we study three proposed centralized coded caching schemes with uncoded pre-fetching for scenarios where end users are grouped into classes with different file demand sets. We provide a lower bound for the transmission rate for the system with heterogeneous user profiles. Then the transmission rates of the three schemes are compared with the lower bound to evaluate their gap to optimality, and also compared with each other to show that each scheme can outperform the other two when certain conditions are met. Finally, we propose a cache distribution method that results in a minimal peak rate and a minimal average rate for one of the schemes when the users’ storage is relatively small compared with the size of the library.</p><p dir="ltr">In Chapter 3, we examine a discrete-time networked SIR (susceptible-infected-recovered) epidemic model, where the infection, graph, and recovery parameters may be time-varying. We propose a stochastic framework to estimate the system states from observed testing data and provide an analytic expression for the error of the estimation algorithm. We validate some of our assumptions for the stochastic framework with real COVID-19 testing data. We identify the system parameters with the system states from our estimation algorithm. Employing the estimated system states, we provide a novel distributed eradication strategy that guarantees at least exponential convergence to the set of healthy states. We illustrate the results via simulations over northern Indiana, USA.</p><p dir="ltr">In Chapter 4, we propose a novel discrete-time multi-virus SIR model that captures the spread of competing SIR epidemics over a population network. First, we provide a sufficient condition for the infection level of all the viruses over the networked model to converge to zero in exponential time. Second, we propose an observation model which captures the summation of all the viruses’ infection levels in each node, which represents the individuals who are infected by different viruses but share similar symptoms. We present a sufficient condition for the model to be strongly locally observable. We propose a distributed Luenberger observer for the system state estimation. We demonstrate how to calculate the observer gain for the estimator and prove that the estimation error of our proposed estimator converges to zero asymptotically with the observer gain found. We also propose a distributed feedback controller which guarantees that all viruses are eradicated at an exponential rate. We then show via simulations that the estimation error of the Luenberger observer converges to zero before the viruses die out.</p><p dir="ltr">We conclude in Chapter 5, where we summarize the findings of this thesis and introduce several challenging open research questions that arise from its results. These questions encompass a range of topics, including the design of optimal testing strategies for large populations, the investigation of estimation techniques in the presence of noisy measurement models, the extension of the SIR epidemic model to more complex models like SEIR and SAIR, and the exploration of efficient vaccine allocation schemes.</p>

Made available in DSpace on 2015-03-03T15:15:25Z (GMT). No. of bitstreams: 1 MarceloBS_DISSERT.pdf: 2228867 bytes, checksum: 46ad012b7ecf9d333c9b9a88bbfb0411 (MD5) Previous issue date: 2010-08-12 / Conselho Nacional de Desenvolvimento Cient?fico e Tecnol?gico / The diffusive epidemic process (PED) is a nonequilibrium stochastic model which, exhibits a phase trnasition to an absorbing state. In the model, healthy (A) and sick (B) individuals diffuse on a lattice with diffusion constants DA and DB, respectively. According to a Wilson renormalization calculation, the system presents a first-order phase transition, for the case DA > DB. Several researches performed simulation works for test this is conjecture, but it was not possible to observe this first-order phase transition. The explanation given was that we needed to perform simulation to higher dimensions. In this work had the motivation to investigate the critical behavior of a diffusive epidemic propagation with L?vy interaction(PEDL), in one-dimension. The L?vy distribution has the interaction of diffusion of all sizes taking the one-dimensional system for a higher-dimensional. We try to explain this is controversy that remains unresolved, for the case DA > DB. For this work, we use the Monte Carlo Method with resuscitation. This is method is to add a sick individual in the system when the order parameter (sick density) go to zero. We apply a finite size scalling for estimates the critical point and the exponent critical =, e z, for the case DA > DB / O processo epid?mico difusivo (PED) ? um modelo estoc?stico de n?o equil?brio que se inspira no processo de contato e que exibe uma transi??o de fase para um estado absorvente. No modelo, temos indiv?duos saud?veis (A) e indiv?duos doentes (B) se difundindo numa rede unidimensional com uma difus?o constante DA e DB, respectivamente. De acordo com os c?lculos do grupo de renormaliza??o, o sistema apresentou uma transi??o de fase de primeira ordem, para o caso DA > DB. V?rios pesquisadores realizaram trabalhos de simula??o para testar esta conjectura e n?o conseguiram observar esta transi??o de primeira ordem. A explica??o dada era que precis?vamos realizar simula??o para dimens?es maiores. Por isso, neste trabalho tivemos a motiva??o de investigarmos o comportamento cr?tico de um processo de propaga??o epid?mico difusivo com intera??o de L?vy (PEDL) em uma dimens?o. A distribui??o de L?vy tem intera??o de difus?o de todos os tamanhos levando o sistema unidimensional a um sistema de dimens?es maiores. Com isso, poderemos tentar explicar esta controv?rsia que existe at? hoje, para o caso DA > DB. Para este trabalho utilizamos o M?todo de Monte Carlo com ressuscitamento. Este m?todo consiste em acrescentar um indiv?duo doente no sistema quando o par?metro de ordem (densidade de doente) vai ? zero. Aplicamos a t?cnica de an?lise de escala de tamanho finito para determinarmos com boa precis?o o ponto cr?tico e os expoentes cr?ticos ??/v, v e z, para o caso DA > DB