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

Persistent and Pernicious Errors in Algebraic Problem Solving

Students hold many misconceptions as they transition from arithmetic to algebraic thinking, and these misconceptions can hinder their performance and learning in the subject. To identify the errors in Algebra I which are most persistent and pernicious in terms of predicting student difficulty on standardized test items, the present study assessed algebraic misconceptions using an in-depth error analysis on algebra students’ problem solving efforts at different points in the school year. Results indicate that different types of errors become more prominent with different content at different points in the year, and that there are certain types of errors that, when made during different levels of content are indicative of math achievement difficulties. Recommendations for the necessity and timing of intervention on particular errors are discussed

3D Geometric Analysis of Tubular Objects based on Surface Normal Accumulation

This paper proposes a simple and efficient method for the reconstruction and extraction of geometric parameters from 3D tubular objects. Our method constructs an image that accumulates surface normal information, then peaks within this image are located by tracking. Finally, the positions of these are optimized to lie precisely on the tubular shape centerline. This method is very versatile, and is able to process various input data types like full or partial mesh acquired from 3D laser scans, 3D height map or discrete volumetric images. The proposed algorithm is simple to implement, contains few parameters and can be computed in linear time with respect to the number of surface faces. Since the extracted tube centerline is accurate, we are able to decompose the tube into rectilinear parts and torus-like parts. This is done with a new linear time 3D torus detection algorithm, which follows the same principle of a previous work on 2D arc circle recognition. Detailed experiments show the versatility, accuracy and robustness of our new method.Comment: in 18th International Conference on Image Analysis and Processing, Sep 2015, Genova, Italy. 201

Instabilities of dispersion-managed solitons in the normal dispersion regime

Dispersion-managed solitons are reviewed within a Gaussian variational approximation and an integral evolution model. In the normal regime of the dispersion map (when the averaged path dispersion is negative), there are two solitons of different pulse duration and energy at a fixed propagation constant. We show that the short soliton with a larger energy is linearly (exponentially) unstable. The other (long) soliton with a smaller energy is linearly stable but hits a resonance with excitations of the dispersion map. The results are compared with the results from the recent publicationsComment: 20 figures, 20 pages. submitted to Phys. Rev.