Applied Math Student Projects

Students in the Applied Math major work on a variety of projects under the direction of one or more faculty members.

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A mathematical model of gang membership in North Lawndale accounting for rehabilitation and fringe crime imprisonment (2015)

Students: Nate Annen, Jacob Clement
Adviser: Dr. Danilo Diedrichs, Dr. Noah Toly

The community of North Lawndale is one of the most dangerous neighborhoods in the Chicago area.  The crime rate is high and the gang life is prevalent.  We use a compartment model consisting of four time-dependent nonlinear differential equations to describe how a gang evolves throughout the population of North Lawndale.  Our research builds upon previous research, but accounting for population dynamics which were not previous considered such as the imprisonment of fringe crime and the rehabilitation of core gang members.  The main findings of this research include conditions under which the core gang member population is notably reduced.  Sensitivity analyses indicate which methods of reducing gang membership and crime rates are the most effective.

Exploratory analysis of a High-Speed Intercity Passenger Rail in the Midwest (2015)

Students: Miriam Agamah, Kyu Lim Lee
Adviser: Dr. Danilo Diedrichs

Many nations have developed extensive networks of high-speed passenger rails, such as France’s TGV rail network and Japan’s SCMaglev levitation train which recently (April 21, 2015) broke the rail speed record at 375 miles per hour.  Currently the only high speed rail in the U.S. is the Acela Express linking Boston to Washington D.C.  This study explores the financial feasibility and profitability of constructing a high speed rail between major cities in the Midwest.  Using census data and the gravity model of trade, we estimate the transportation needs between promising city pairs over the next 50 years.  We then conduct a comprehensive cost/benefit comparison between high speed rail, air and automobile transportation, using multinomial regression to quantify the attractiveness and ultimately estimate the ridership of the new railroad.  We use data from the Acela express to estimate ticket prices, construction and maintenance costs to determine which lanes would generate a profit over the 50 year time horizon.

Clustering trends of overweight obesity in students in Dupage County (2015)

Students: Melissa Gray, Robin Kong
Advisers: Dr. Danilo Diedrichs, Dr. Nate Thom

Overweight and obesity are becoming more prevalent in elementary through high school students in the United States. This research analyzes trends of obesity and overweight among students in DuPage County Illinois to explore possible relationships between obesity and factors such as ethnicity, income, education level, or geographic location. Using data collected from over 200 area schools, hierarchical clustering algorithms show representative clustering of schools according to geographic location and the percentage of overweight or obese students in each school. Principal Component Analysis is used to determine factors contributing most to variability in the data. Ethnic distribution, average income, and average education level were explored within each cluster. These findings can contribute to the efforts to decrease overweight and obesity in students and predict future trends.

Agent-based simulation of crime in society (2015)

Students: Johnny Edman, Richard Ndekezi
Adviser: Dr. Danilo Diedrichs

Mathematical models are becoming increasingly common in the field of quantitative and predictive criminology. Previous modeling efforts have had some success using differential equations to determine the transition between different states using a modified SIR (Susceptible-Infected-Recovered) epidemiology model. However these models group a very diverse population in to a small number of compartments, thereby ignoring the individual behavior patterns and factors that determine crime.  We use an Agent-Based Model (ABM) approach based on a computational random number generator to generate a virtual society of 1000 “agents”, each one having different characteristics including age, income level, educational attainment, ethnicity, gender and hostility based on realistic data-driven probability distributions.  The ABM is embedded into a discrete dynamical model that tracks the evolution of each agent’s characteristics over a ten year time horizon, allowing us to determine which characteristic(s) are the most susceptible to induce crime in society.

 

Quantifying communication effects in disaster response logistics: A multiple network system dynamics model (2014)

Student: Kaile Phelps
Advisers: Dr. Danilo Diedrichs, Dr. Paul Isihara

Complementing the importance of adequate relief supplies and transportation capacity in the first two weeks of post-disaster logistics, efficient communication, information sharing and informed decision-making play a crucial yet often underestimated role in reducing wasted material resources and loss of human life.  A mathematical discrete dynamical system is used to model transportation of different commodities from multiple relief suppliers to disaster sites across a network of limited capacity with variable signal delays, information sharing, prioritization, distribution and redistribution strategies.  Simulations results highlight how communication deficiencies and indiscriminate shipping of resources result in material convergence and shortage of urgent supplies observed in actual emergencies, thus providing a useful quantitative tool for decision-making and training volunteer managers in the importance of a smart response protocol.  

 

The Schedule Effect: can recurrent peak infections be reduced without vaccines, quarantines or school closings? (2013)

Student: Doeke Buursma
Advisers: Dr. Danilo Diedrichs, Dr. Paul Isihara

CalendarUsing a standard SIR (Susceptible-Infected-Recovered) model with seasonal dynamics, we study the "schedule effect", which allows for a significant reduction in recurrent peak infections of endemic diseases in schools by varying the traditional school calendar.  Analysis of the phase plane explains the relationship between the maximum recurring infection peaks and the period of an oscillating transmission function. The response may exhibit period-doubling and chaos induced at certain periods, leading to increased peaks. We show how to take these effects into consideration to design an optimum school schedule.

 

A Mathematical Model for the Growth and Decline of the Church in DuPage County (2013)

Student: Daniela Cuba
Adviser: Dr. Danilo Diedrichs

UEB modelThe project analyzes the population dynamics that affect size of the church in DuPage County, Illinois.  A dynamic model (which we name the UEB model), similar to the SIR model used in epidemiology, divides the population into compartments of Unbelievers, Enthusiasts (who actively bring unbelievers into the church), and passive Believers. In addition to the conversion dynamics, our model also incorporates the long-term demographic fluctuations of DuPage County.  The parameters of the model are determined by fitting church and demographic data obtained from the Census Bureau.  The results of this study are useful to identify the most impactful strategies for churches to increase their membership.

 

Constrained Optimization Model for Quantitative Criminology (2013)

Student: Korey Clement
Adviser: Dr. Danilo Diedrichs

FingerprintThe constrained optimization model for quantitative criminology, first introduced by criminologists Alfred Blumstein and Daniel Nagin, is used to control and minimize the crime rate of a given population. Using the most recent data on crime and punishment available (2009), we use this model to determine the lowest crime rate that can achieved in the United States. The sensitivity analysis of the model's parameters reveals what steps must be carried forth in order to reduce the crime rate to a global minimum.

 

Development of an Application for Indoor Temperature Control Efficiency (2013)

Student: Roland Hesse
Adviser: Dr. Danilo Diedrichs

Temperature ProfileUsing numerical techniques to discretize and solve heat equation (a partial differential equation) in three dimensions, we devise a computerized application that models the heat flow and determines temperature gradients in a building.  The application allows for the geometry of the rooms and insulation properties of the boundaries to be specified, as well as the indoor locations where people are most likely to be found.  We use this system to locate the optimal placements of HVAC (Heating, Ventilation, and Air Conditioning) for overall efficiency in temperature control and reduction of wasted energy and climate-control costs.

 

Computational Composition of Traditional Scottish Music (2013)

Student: Tim Macdonald
Adviser: Dr. Danilo Diedrichs

Scottish MusicTraditional Scottish music has recurring patterns at the harmonic, melodic, and structural levels. Using a combination of automated pattern recognition techniques and domain-specific knowledge, we develop a system that, seeded with a corpus of existing tunes, composes original music in the same style. This is accomplished using a long short term memory network---a type of recurrent neural network.  The corpus used was the complete works of 18th century composer William Marshall, which was transformed into a sequence of integers suitable for inputting into the network.  Post-conversion, the music was used to train the network, and the trained network was used for generating new music.

 

Inventory Models in Disaster Relief (2012)

Student: Nate Veldt
Adviser: Dr. Danilo Diedrichs

CycloneModels for supply chain management can be used to assist humanitarian relief organizations in calculating how to efficiently provide for a population affected by a natural disaster. This project explores in detail a single period probabilistic model for fulfilling a demand while minimizing costs.  The model is implemented in MATLAB and several examples are given of how this model might be used in a specific disaster relief situation. A Monte Carlo simulation is used to generate potential values for demand to then analyze how this model might be used to meet a demand that stretches over multiple periods.

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