Mathematical modeling of epidemics and adversarial learning in distributed systems
The COVID-19 epidemic has had a major global impact on humanity and the economy. Analyzing the effect of the COVID-19 pandemic can provide guidance for future pandemics. This dissertation studies three aspects of the epidemics during modern times. In the first part of the thesis, we study different aspects of pandemics along with mathematical models to address these aspects. Epidemics affect small communities in a different way than large urban centers. In chapter 2, we develop a mathematical model for finite-size effects using a stochastic compartmental susceptible–infected–recovered (SIR) model with a martingale formulation. The deterministic part coincides with the classical SIR model and we provide an upper bound for the stochastic part. Through analysis of the stochastic component depending on varying population size, we provide a theoretical explanation of finite size effects. Our theory is supported by numerical simulations of theoretical infinitesimal variance. In chapter 3, we propose a coupled model of policy-making and epidemic dynamics based on the SIR model and an optimization scheme. In chapter 4, we propose a hierarchical non-negative matrix factorization (NMF) scheme to classify the literature on COVID-19. We discover eight major latent topics and 52 granular subtopics in the body of literature, related to vaccines, genetic structure and modeling of the disease and patient studies, as well as related diseases and virology.Modern day machine-learning algorithms often operate in a distributed manner and are known to be vulnerable to adversarial attacks. Developing large-scale distributed methods that are robust to the presence of adversarial or corrupted workers is an important part of making such methods practical for real-world problems. In chapter 5, we propose novel methods that guarantees convergence and identify adversarial workers in highly hostile systems. The algorithm utilizes simple statistics (mode) to guarantee convergence and is capable of identifying the adversarial workers. Additionally, the efficiency of the proposed methods is shown in simulations in the presence of adversaries. The results demonstrate the great capability of such methods to tolerate different levels of adversary rates and to identify the adversarial workers with high accuracy.