Perspectives on adaptive dynamical systems

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems, such as the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges and give perspectives on future research directions, looking to inspire interdisciplinary approaches

Virus spread over networks: Modeling, analysis, and control

The spread of viruses in biological networks, computer networks, and human contact networks can have devastating effects; developing and analyzing mathematical models of these systems can provide insights that lead to long-term societal benefits. Basic virus models have been studied for over three centuries; however, as the world continues to become connected and networked in more complex ways, previous models no longer are sufficient. Therefore virus spread over networks is a newer research topic, which provides a compelling modeling technique to capture real world behavior, and interest from the control field has provided an exciting new outlook on the area. Prior research has focused mainly on network models with static graph structures; however, the systems being modeled typically have dynamic graph structures and have not been validated with real spread data over a network. In this dissertation, we consider virus spread models over networks with dynamic graph structures, and we investigate the behavior of these systems. We perform stability analyses of epidemic processes over time-varying networks, providing sufficient conditions for convergence to the disease free equilibrium (the origin, or healthy state), in both the deterministic and stochastic cases. We also explore the scenario of multiple viruses, in the case of competing viruses, including human awareness, and coupled competing viruses. We analyze the healthy state and the endemic states of these models over static and dynamic graph structures. Various control techniques are also proposed to mitigate virus spread in networks. Illustrative figures and simulations are presented throughout. No previous work has explored identification and validation of network dependent virus spread models, which is considered herein using two datasets: 1) John Snow's fundamental 1854 cholera dataset and 2) a 2009-2012 USDA farm subsidy dataset. We conclude by discussing current work and future research directions