SIAM Undergraduate Research Online (SIURO)

Volume 10

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Undergraduate Student Research

On the Immunization of Small Computer Networks
[PDF, 3MB]

Published electronically February 1, 2017

Author: Brennen Fagan (California State University, Fresno)

Sponsor: Doreen De Leon (California State University, Fresno)

Abstract: As complex computer networks have become integral to our life, it has become important to create effective and efficient defenses against computer viruses. To model these viruses and networks, a computer network can be treated as a graph of nodes and edges. For small graphs, we can use centrality measures to identify which nodes to immunize first. We examine an existing immunization scheme based on the computation of a bound for the decay of a virus and compare it to some proposed schemes on random graphs. We observe that an immunization scheme based on the organization of a graph can outperform a greedy scheme. We also observe that the computation of the virus decay bound does not correlate in the expected way with the ability of the virus to infect an entire network.


No oscillations in the Michaelis-Menten approximation of the dual futile cycle under a sequential and distributive mechanism
[PDF, 2MB]

Published electronically March 23, 2017

Authors: Adriana Morales (University of Puerto Rico, Rio Piedras Campus) and Luna Bozeman (Clemson University)

Sponsor: Anne Shiu (Texas A&M University)

Abstract: Protein phosphorylation and dephosphorylation are important intracellular processes. The main object of study in this paper is the dual futile cycle, a network that describes the dual-site phosphorylation/dephosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism. Specifically, we analyze the 2-dimensional Michaelis-Menten (M-M) approximation of this system. It has been previously shown that this system is bistable. We also know, from monotone systems theory, that every solution converges to some steady state. Here, we give a new and simpler proof of this convergence result by using Bendixon's criterion to rule out oscillations. Ultimately, understanding the behavior of this system could help us understand the original dual futile cycle (or MAPK cascades that contain it), which has recently been shown to admit oscillations via Hopf bifurcations.


Detecting Foot-Chases from Police Body-Worn Video
[PDF, 2MB]

Published electronically March 31, 2017

Authors: Rafael Aguayo (University of California, San Diego), Alejandro Camacho (California State University, Fullerton), Piyali Mukherjee (Columbia University), and Qi Yang (University of Southern California)

Sponsor: Hayden Schaeffer (Carnegie Mellon University)

Abstract: Existing methods to record interactions between the public and police officers are unable to capture the entirety of police-public interactions. In order to provide a comprehensive understanding of these interactions, the Los Angeles Police Department (LAPD) intends to utilize Body-Worn Video (BWV) collected from cameras fastened to their officers. BWV provides a novel means to collect fine-grained information about police-public interactions. The purpose of this project is to identify foot-chases from the videos using machine-learning algorithms. Our proposed algorithm uses the Bag-of-Intrinsic-Words algorithm followed by classification via support-vector machines. Our training dataset consists of 100 training videos (20 foot-chase & 80 non-foot-chase), and a test dataset of 60 LAPD videos (4 foot-chase & 56 non-foot-chase). We achieved results of 91.6% testing accuracy.


Cats, Rabbits, Birds, and Viruses, Oh My! Modelling the Conservation Implications of a Complex Virus Release in a Predator-Prey System
[PDF, 2MB]

Published electronically May 5, 2017

Author: Jean-Paul R. Soucy (University of Ottawa)

Sponsor: Robert Smith? (University of Ottawa)   

Abstract: Mathematical models have proven useful in planning conservation efforts for threatened species. In this paper, we develop a model based on the Macquarie Island ecosystem, where native seabirds were threatened by invasive pest species. In particular, European rabbits (Oryctolagus cuniculus) destroyed seabird nesting sites, while feral cats (Felis catus) preferentially hunted rabbits but also consumed seabirds. Management strategies included releasing a rabbit-killing disease and shooting feral cats. We investigate the interactions between species in the ecosystem as well as conservation practices using analytical techniques such as the basic reproductive ratio (R0) and partial rank correlation coefficient analysis. The results of this study reveal that the important factors to disease establishment change depending on the ecological interactions present in the system. Additionally, we show that the interaction between disease and predation sometimes produces surprising outcomes. Overall, the results of the study demonstrate the need for mathematical modelling in the conservation process in order to anticipate the complex responses of an ecosystem to management practices. We conclude with a brief list of considerations for conservation planners dealing with ecologically complex systems in the future.


A Simple Model for Human Immunodeficiency Virus Based on Erlang's Method of Stages
[PDF, 1MB]

Published electronically June 13, 2017

Author: Samuel Swanson (University of Florida)

Sponsor: Maia Martcheva (University of Florida)  

Abstract: The purpose of this paper is to select a model for HIV that uses few parameters while fitting the world prevalence and death data well. Here we consider a set of models based on Erlang's method of stages, including some with and some without social distancing. The use of stages is supported by biological studies which suggest that HIV passes through stages in each individual, although the exact number is not known. This set of models can represent such stages using a successive number of classes. To perform model selection, we compute R0 and use it to estimate initial values of the parameters in this model. We run thousands of iterations of a Nelder-Mead simplex search algorithm to determine the optimal values of parameters for each model and the error associated with each model. These errors are used to compute AICc values and then the AICc values are compared to select the most likely model. The selected model from this experiment contains the social distancing term as well as four infected classes/stages. We then perform identifiability analysis and determine that the “true values" of the parameters for this model are uniquely determinable based on the data points.


Propagation of lead in the human body
[PDF, 1MB]

Published electronically July 26, 2017

Authors: Melissa Morrissey (Brown University) and Jordan Collignon (California State University, Monterey Bay)

Sponsor: Todd Kapitula (Calvin College)

Abstract: Lead is a toxin that has well known side effects including fatigue, muscle pain, impaired kidney function, lower IQs for children, and brittle bones. Lead can be absorbed into the body through paint, air, water, and various other consumer products. Once ingested, blood transports lead throughout the body. The vast majority of lead absorbed in the body accumulates in the bone. Here we explore a three-compartment nonlinear ODE model for lead in blood, cortical bone, and trabecular bone. Thereafter, we compare the ODE results with a PDE model in which it is assumed that lead slowly diffuses through the bone. Numerical solutions of the model ODE equations suggest that in order to have results which are consistent with experimental data, one should assume nonlinear interactions between the blood and cortical bone, but linear interactions between the blood and trabecular bone. On the other hand, we find that the PDE model we use does not provide for a good comparison with the data. We briefly touch upon some possible reasons for this discrepancy, and ways in which the model could be improved.


M3 Challenge Introduction
[PDF, 500KB]

Raining Down on Rising Sea Levels
[PDF, 1MB]

Published electronically August 9, 2017

Authors: Deepak Moparthi, Albert Cao, Andrew Hwang, Joshua Yoon, and Haoyang Yu (Adlai E. Stevenson High School, Lincolnshire, IL)

Sponsor: Paul Kim (Adlai E. Stevenson High School, Lincolnshire, IL)

Summary: The National Park Service (NPS) is committed to preserving the beauty of America in order to provide everyone amazing interactions with nature. For over 100 years, the NPS has maintained these wonders of America; however, as it begins its second century of operation, one of the NPS's greatest concerns is the issue of climate change. Climate change greatly influences actions the NPS takes to protect parks and events such as flooding or other disasters can affect how many people visit the park.

In particular, rising sea levels are one of the imminent problems that the United States is faced with because of its impact on flooding, and it is necessary for the NPS to identify which National Parks are at risk. We were initially tasked with developing a model to classify 5 particular parks as having either high, medium, or low risk of sea level change. For each location, we created a probabilistic model of the sea level height in the next t years. We determined whether a site had high, medium, or low risk levels based on the damage that we would expect to occur based on the change in sea level. We calculated the probability of risk associated with each region in 10, 20, 50, and even 100 years from now. Our findings show that Cape Hatteras and the Padre Island possess the greatest risk of all 5 national parks.

After classifying these parks as high, medium, or low risk based on sea level change alone, we sought to determine a set of additional criteria to build a model that would assign each site a "vulnerability score." The vulnerability score is based off of the likelihood and severity of climate related events occurring. We selected our criteria to be the Heat Index, which consisted of temperature and humidity, hurricane intensity and frequency, and the Air Quality Index. These criteria were then used to construct a model that generated the Vulnerability score by first assigning a subscore for each of the individual criteria and then taking a weighted average of these subscores. We found that Padre Island National Seashore and Acadia National Park are in critical condition, with Padre Island being in a worse condition than Acadia National Park. Furthermore, Olympic National Park and Cape Hatteras National Seashore are still safe but almost in a critical condition, and Kenai Fjords National Park is the safest of all five.

Finally, we created a model to determine how to allocate limited funds to the parks based off of factors including the adjusted vulnerability score we calculated in part two, as well as the number of visitors for each park. We accomplished this by first determining the expected number of visitors for the future based on data from previous years and the vulnerability score. We then used the results of our model to convert our predictions of the number of visitors and the vulnerability scores into indices that would calculate the overall Financial Utility index. We used the financial utility indices for each site to decide the optimal distribution of funds between the 5 parks. Based on our results, we found that the percentage of funds should be allocated as follows from most to least: Acadia National Park (30:48%), Olympic National Park (28:27%), Cape Hatteras (21:49%), Padre Island (10:94%), Kenai Fjords (8:82%).


Change-point Detection Methods for Body-Worn Video
[PDF, 1MB]

Published electronically August 22, 2017

Authors: Stephanie Allen (State University of New York at Geneseo), David Madras (University of Toronto), Ye Ye (UCLA), and Greg Zanotti (DePaul University)

Sponsor: Giang Tran (University of Texas at Austin)

Abstract: Body-worn video (BWV) cameras are increasingly utilized by police departments to provide a record of police-public interactions. However, large-scale BWV deployment produces terabytes of data per week, necessitating the development of effective computational methods to identify salient changes in video. In work carried out at the 2016 RIPS program at IPAM, UCLA, we present a novel two-stage framework for video change-point detection. First, we employ state-of-the-art machine learning methods including convolutional neural networks and support vector machines for scene classification. We then develop and compare change-point detection algorithms utilizing mean squared-error minimization, forecasting methods, hidden Markov models, and maximum likelihood estimation to identify noteworthy changes. We test our framework on detection of vehicle exits and entrances in a BWV data set provided by the Los Angeles Police Department and achieve over 90% recall and nearly 70% precision | demonstrating robustness to rapid scene changes, extreme luminance differences, and frequent camera occlusions.


A Comparison of Measles Vaccination Regimes in a Stochastic, Spatial SIR Model
[PDF, 3MB]

Published electronically August 23, 2017

Authors: Emily Nguyen (Muhlenberg College) and Amanda Reeder (Norfolk State University)

Sponsor: John Fricks (Arizona State University)

Abstract: Measles is a disease that continues to affect millions of people; however, it can now be controlled through vaccination. Using a spatial, stochastic, continuous-time SIR model, we investigate four different vaccination regimes in a \country" of 25 cities. The model was constructed using work by Bjørnstad et al. and May and Anderson as a basis for determining the spatial structure of the country and a set of appropriate parameters [May and Anderson, 1984, Bjørnstad et al., 2002]. We examine the behavior of measles under these vaccination regimes over a period of 20 years with the goal of determining an optimal regime that leads to herd immunity. All data was simulated using an Euler approximation of the continuous-time Markov chain. As vaccination rates increase and begin to induce herd immunity, on average, the same proportion of susceptible people are vaccinated across regimes. However, a closer investigation of the qualitative behavior of these regimes reveals distinct differences among them. Briefly, some regimes maintained a much lower proportion of susceptibles in the population, while others allowed the number of susceptible people in the population to fluctuate significantly. A steady, high vaccination rate across the population eliminated cases and led to herd immunity without such fluctuations.


Fusion in Card Collecting Games: A Probable Outcome
[PDF, 1MB]

Published electronically August 30, 2017

Authors: Emili Moan (North Carolina State University), Lindsay Bradley (Winthrop University), and Zoe Vernon (Washington University in St. Louis)

Sponsor: Kristen Abernathy (Winthrop University)

Abstract: Card Collecting Games (CCGs), as well as many games in other genres, often employ a mechanic referred to as gacha-fuse-evolve where players randomly draw items with different levels of rarity (common, uncommon, and rare) that can be fused and evolved to create stronger items. With the free-to-play model that many online companies use, it is important that CCG developers keep the game easy enough that players want to continue to play but difficult enough that players want to spend money to better their experience. To achieve this, developers need to ensure fusions occur often enough to keep the non-paying players engaged, but seldom enough to entice players to purchase additional fusion opportunities. For this project, we explore the probability of players drawing four different types of fusion (unique fusion, quad-fusion, evolutionary trees, and recipe fusion) in a given time period. We also run a sensitivity analysis to determine which parameters - deck size, number of rare cards, or length of play - are most sensitive. Finally, we create a C++ program to run simulations and verify the results of the unique and quad-fusions probabilities.




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