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Due to an increased amount of applications that can be modeled as large-scale, there has been growing interest in using simple methods for optimization that require low iteration cost as well as limited memory storage. We will be concerned with optimization problems for networked systems with high dimensions, focusing on applications in crowdsourcing and pandemic control. What makes these problems complex is that the objectives relate to aspects of the evolution of the dynamics of the system. We develop first-order optimization methods with low iteration complexity for such applications in this dissertation work. In the first part of this work, we consider the adversarial crowdsourcing problem. We reduce this problem to the robust rank-one matrix completion problem, and we propose a new first-order algorithm with theoretical guarantees. These results are then applied to the problem of classification from crowdsourced data under the assumption that while the majority of the workers are governed by the standard single-coin David-Skene model, some of the workers can deviate arbitrarily from this model. Extensive experimental results show our algorithm outperforms all other state-of-the-art methods in such an adversarial scenario. In the second part of the work, we consider the optimal lockdown problem for pandemic control. As a common strategy of contagious disease containment, lockdowns will inevitably weaken the economy. Here we propose a mathematical framework with first-order methods to achieve pandemic control through an optimal stabilizing non-uniform lockdown, where our goal is to reduce the economic activity as little as possible while decreasing the number of infected individuals at a prescribed rate. We demonstrate the power of this framework by analyzing a model of COVID-19 spread in the 62 counties of New York State. We find that an optimal stabilizing lockdown based on epidemic status in April 2020 would have reduced economic activity more stringently outside of New York City compared to within it, even though the epidemic was much more prevalent in New York City at that point. In the third part of the work, we consider the optimal vaccine allocation issue for pandemic control, where our goal is to send the infections to zero as soon as possible with a fixed number of vaccines. To achieve this, we propose a mathematical framework for classical epidemic models as well as a COVID-19 model. Moreover, we also analyzed the epidemic model used in [Bubar et al., 2021], and compared our method with the strategies in [Bubar et al., 2021]. We found that it is better to offer vaccines to younger people when the basic reproduction number R0 is moderately above one.

archives@tulane.edu / 1 / Harley Hanes

The application of spatial methods to epidemic estimation and prediction problems is a vibrant and active area of research. In many cases, however, well thought out and laboratory supported models for epidemic patterns may be easy to specify but extremely difficult to fit efficiently. While this problem exists in many scientific disciplines, epidemic modeling is particularly prone to this challenge due to the rate at which the problem scope grows as a function of the size of the spatial and temporal domains involved. An additional barrier to widespread use of spatiotemporal epidemic models is the lack of user friendly software packages capable of fitting them. In particular, compartmental epidemic models are easy to understand, but in most cases difficult to fit. This class of epidemic models describes a set of states, or compartments, which captures the disease progression in a population. This dissertation attempts to expand the problem scope to which spatio-temporal compartmental epidemic models are applicable both computationally and practically. In particular, a general family of spatially heterogeneous SEIRS models is developed alongside a software library with the dual goals of high computational performance and ease of use in fitting models in this class. We emphasize the task of model specification, and develop a framework describing the components of epidemic behavior. In addition, we establish methods to estimate and interpret reproductive numbers, which are of fundamental importance to the study of infectious disease. Finally, we demonstrate the application of these techniques both under simulation, and in the context of a diverse set of real diseases, including Ebola Virus Disease, Smallpox, Methicillin-resistant Staphylococcus aureus, and Influenza.

Cheyne-Stokes respiration is a distinct breathing pattern consisting of periods of hyperpnea followed by apneas, with unknown etiology. One in two patients with congestive heart failure suffer from this condition. Researchers hypothesize that key factors in CSR are the fluid shift from the standing to supine position and the differences between genders. A mathematical model of the cardio-respiratory system was constructed using parameter values from real data. Hopf bifurcation analysis was used to determine regions of stable versus oscillatory breathing patterns. In the model, Cheyne-Stokes respiration is more likely to occur while in the supine position and males are more likely to develop Cheyne-Stokes than females. These findings, which are in agreement with clinical experience, suggest that both gender and fluid shift contribute to the pathogenesis of Cheyne-Stokes respiration, and that physical quantities such as blood volumes and neural feedback may be sufficient to explain the observations of CSR. / Department of Mathematics and Statistics

Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (p. [115]-122).

Compartmentalization is a general principle in biological systems which is observable on all size scales, ranging from organelles inside of cells, cells in histology, and up to the level of groups, herds, swarms, meta-populations, and populations. Compartmental models are often used to model such phenomena, but such models can be both highly nonlinear and difficult to work with. Fortunately, there are many significant biological systems that are amenable to linear compartmental models which are often more mathematically accessible. Moreover, the biology and mathematics is often so intertwined in such models that one can be used to better understand the other. Indeed, as we demonstrate in this paper, linear compartmental models of migratory dynamics can be used as an exciting and interactive means of introducing sophisticated mathematics, and conversely, the associated mathematics can be used to demonstrate important biological properties not only of seasonal migrations but also of compartmental models in general. We have found this approach to be of great value in introducing derivatives, integrals, and the fundamental theorem of calculus. Additionally, these models are appropriate as applications in a differential equations course, and they can also be used to illustrate important ideas in probability and statistics, such as the Poisson distribution.

Due to their convenience, computers have become a standard in society and therefore, need the utmost care. It is convenient and useful to model the behavior of digital virus outbreaks that occur, globally or locally. Compartmental models will be used to analyze the mannerisms and behaviors of computer malware. This paper will focus on a computer worm, a type of malware, spread within a business network. A mathematical model is proposed consisting of four compartments labeled as Susceptible, Infectious, Treatment, and Antidotal. We shall show that allocating resources into treating infectious computers leads to a reduced peak of infections across the infection period, while pouring resources into treating susceptible computers decreases the total amount of infections throughout the infection period. This is assuming both methods are receiving resources without loss. This result reveals an interesting notion of balance between protecting computers and removing computers from infections, ultimately depending on the business executives' goals and/or preferences.

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