Titles and Abstracts

9th Annual UMKC Math and Stat Research Day

 Division of Computing, Analytics & Mathematics,

School of Science and Engineering

University of Missouri - Kansas City

Morning Session: Modeling and Analysis of Infectious Diseases

Chair of the Session: Dr. Noah Rhee 

Agent-Based Simulation of Infectious Disease Spread Using MESA Framework

Sarawat Murtaza Sara smtn8@umkc.edu 

Division of Computing, Analytics & Mathematics,

School of Science and Engineering,

University of Missouri - Kansas City

Infectious disease outbreaks in hospitals can severely impact both patient health and the efficiency of the healthcare system as a whole. Agent-based simulations offer researchers and decision-makers a useful tool for studying the transmission of these outbreaks. In this study, we use the MESA framework to present a simple agent-based simulation of the spread of infectious diseases in a hospital setting. For this, we leverage MESA, an agent-based modeling (ABM) framework in Python. MESA is particularly useful for simulating complicated systems such as those involved in hospital-based disease spreading. In MESA, participating entities, called "agents" in ABM's parlance, are separately modeled and interact to produce system-level behavior. By this, ABMs capture individual agents' behavior in a given environment, impacting the whole patterns of disease transmission. In our (over) simplified agent-based simulation of the classical SIR (susceptible, infected, and recovered) Epidemic modeling, we deploy one type of agent, nurses, and let them move and interact on a fictional hospital floor (a 2D grid space) thus spreading diseases among themselves. In our presentation, we will describe how the architecture and design of MESA's framework support this simulation. We will also show some preliminary results regarding the disease spread in the hospital area and show the visualization of the simulation progress in MESA.

Multi-state modeling of methicillin-resistant Staphylococcus aureus infection in a hospital environment

Kathryn Menta, mentak@mail.umkc.edu 

Division of Computing, Analytics & Mathematics,

School of Science and Engineering,

University of Missouri - Kansas City 

The development of antibiotic-resistant strains of bacteria commonly found in healthcare environments is a major public health concern. One such type of bacteria of focus is Methicillin-resistant Staphylococcus aureus, or MRSA, which can cause staph, bloodstream, and other infections and can be lethal. MRSA is also difficult to eradicate, as it has evolved to be resistant to several types of antibiotics. There is an increased emphasis on enhancing statistical modeling methods to gain perspective and knowledge of how these antibiotic-resistant bacteria strains develop and are transmitted in healthcare settings. We develop parallel multi-state MRSA models for hospital patients and healthcare workers. Using the simulated panel data, we aim to unpack the MRSA transmission dynamics within and between patients and hospital workers.

When Care is Shared: Using Agent-Based Models to Analyze Covid-19 Transmission Between Nursing Homes That Share Staff 

Kiel Corkran,  kcbch@umkc.edu 

Division of Computing, Analytics & Mathematics,

School of Science and Engineering,

University of Missouri - Kansas City

The spread of infectious diseases within nursing home facilities has been a concern for epidemiologists, due to the ongoing Covid-19 pandemic and its deadly effects on this specific population. The need for a more detailed understanding of precise structures that govern disease transmission in a nursing home has become increasingly apparent. Current agent-based models focus on understanding disease spread without considering that caregivers may work in multiple nursing homes. These models don’t give detailed information on the dynamics of Covid-19 when caregivers work in multiple homes. To fill this gap, we developed an agent-based model structure to simulate the Covid-19 outbreak in two different nursing homes under a variety of different scenarios. Using the model simulations and the observed Covid-19 outbreaks, we measure the effects of shared caregivers on Covid-19 transmission in multiple facilities. 

What can Nursing Home Quality Measures tell us about pandemic preparedness? 

Julia Pluta, plutaj@umkc.edu 

Division of Computing, Analytics & Mathematics,

School of Science and Engineering,

University of Missouri - Kansas City

The Center for Medicare and Medicaid Services (CMS) tracks several measures to rate the quality of care at nursing homes throughout the United States. These measures produce rating systems, compiled into three domains: Health Inspection Rating, Staffing Rating, and Quality Measure (QM) Rating, which are used to create an overall rating in the Five-Star Quality Rating (FSQR) system. By combining the weekly reported COVID-19 morbidity and mortality data collected by CMS with the FSQR data, I conducted correlation, principal component, and boxplot analyses to determine whether any of the rating systems and their components predicted or reflected the progression of COVID-19 in each of 113 KC Metro Nursing Homes through the end of the first year of the pandemic. I determined that none of the major quality ratings or the FSQR overall rating had any statistically significant correlation with COVID-19-related outcomes during the first year of the pandemic, which raises two new questions: 1) What is the FSQR measuring and why doesn’t that translate to better COVID-19 outcomes?; and 2) What components of a rating system might be better used to generate a rating that would be more predictive of outcomes for a future pandemic?

Afternoon Session: Theory and Applications of Ordinary Differential Equations, Advanced Numerical Methods

Chair of the Session: Dr. Shuhao Cao 

Using the Stable Manifold Theorem: Connection between Solutions to Linear and Non-linear Systems

Makayla Devening, mtdg84@umsystem.edu  

Division of Computing, Analytics & Mathematics,

School of Science and Engineering,

University of Missouri - Kansas City

In differential equations, mathematicians have figured out ways to solve linear differential equations with relative ease, but non-linear equations have proven to be more difficult. The Stable Manifold Theorem states that the stable manifold of a stable equilibrium point is tangent to the eigenspace associated with the stable eigenvalues of the linearized system at the equilibrium point. This theorem is a tool that helps to predict the behavior of non-linear systems. I will present a sketch of the proof through an example and give a real-world application of the theorem.

Establishing Topological Conjugacy of Local Solutions Using Hartman-Grobman

Theorem, Steven Giangreco, smgrmd@umsystem.edu 

Division of Computing, Analytics & Mathematics,

School of Science and Engineering,

University of Missouri - Kansas City

A real-world example of a system of non-linear differential equations is presented to introduce the practical utility of the Hartman-Grobman theorem. First, pre-requisite concepts such as homeomorphisms, trajectories, and flows are reviewed. Then the Hartman-Grobman theorem is stated and explained. Numerical simulations are conducted to demonstrate the implementations of homeomorphisms to map between topologically conjugate systems within the neighborhood of a given solution. Limitations regarding the applicability and scope of the Hartman-Grobman theorem are discussed.

Implications and Applicability of Hartman-Grobman Theorem for Analyzing Nonlinear Dynamical System

Jodi Donald donaldjo@umkc.edu

Division of Computing, Analytics & Mathematics,

School of Science and Engineering,

University of Missouri - Kansas City

In this presentation, we will learn when it is appropriate to use the Hartman-Grobman Theorem. Once established, it will accurately depict what is happening around a hyperbolic equilibrium point, allowing us to determine a nonlinear system's qualitative behavior. Additionally, the stability of the solutions of the system can be determined. The example of a competing species model will be utilized to demonstrate the Theorem.

Finite Element Methods for Phase Field Models of Two-Phase Ferro-Fluid Flows

Mahdi Gharehbaygloo, mgharehbaygloo@mst.edu   

Department of Mathematics and Statistics,

Missouri University of Science and Technology,

Rolla, Missouri

In this research a modified mathematical model for Ferro-Fluids will be developed. The model is being developed using the existing PDE models for ferrofluids, namely Rosensweig and Shliomis models. This resultant model will be investigated for the existence of the solution (convergence), uniqueness of the solution, and stability.

Then, DEAL.ii software will be used to numerically solve this model by applying Finite Element Methods. Due to the existence of different phases, a phase-field approach will be imposed for both the development and simulation of the model. The results are expected to be studied in contrast with the available results from lab experiments.

Decoupled finite element method for a phase field model of two-phase ferrofluid flows

Youxin Yuan yyuan@mst.edu 

Department of Mathematics and Statistics,

Missouri University of Science and Technology

Ferrofluid is a liquid that is attracted to the poles of a magnet and usually does not retain magnetization in the absence of an externally applied magnetic field. In this talk, the decoupled finite element method for the two-dimensional ferrohydrodynamics model and the numerical validation of the method, which consists of the Navier-Stokes equation, the Cahn-Hilliard equation, and the magnetic field equation, will be discussed.

Understanding the Concept of Lyapunov Stability and Applications of Lyapunov Functions

Cole Flackmiller, cjfhmr@umsystem.edu  

Division of Computing, Analytics & Mathematics,

School of Science and Engineering,

University of Missouri - Kansas City

Linear stability analysis is a powerful tool in dynamical systems theory to analyze the behavior of local solutions near a hyperbolic equilibrium solution. Nonetheless, it cannot be applied to systems with nonhyperbolic equilibrium solutions. In this talk, I introduce the concept of Lyapunov stability and the method of Lyapunov function to study the global behavior of solutions. The method of Lyapunov function has important applications in various fields, including control engineering, robotics, and economics. I will provide a few examples to demonstrate its applications to stabilize the behavior of complex systems and predict the long-term behavior of nonlinear systems.

On the fundamental theorem of existence and uniqueness

Dustin Fluderer dwfw4f@umsystem.edu 

Division of Computing, Analytics & Mathematics,

School of Science and Engineering,

University of Missouri - Kansas City 

The fundamental theorem of existence and uniqueness (FTUE) is a central theorem in differential equations that provides conditions for the existence and uniqueness of solutions to initial value problems. In this talk, I will explain how FTUE can be proved and discuss the application of this theorem. An important application of the FTUE is in the analysis of control systems. Control systems are designed to regulate the behavior of complex systems, such as robots, aircraft, and chemical processes. The FTUE provides a way to ensure that the control algorithms are well-defined and have unique solutions. This is important for designing stable control systems that can reliably regulate the behavior of the complex system.