A comparison of the performance of CUDA versus OpenMP for Jacobi, Gauss-Seidel, and S.O.R. iterative methods for Laplace's Equation with Dirichlet boundary conditions is presented. Both the number of cores and the grid size were varied for the OpenMP program, while the grid size was varied for the CUDA program. CUDA outperforms the 8-core OpenMP program with the Jacobi and Gauss-Seidel schemes for all grid sizes, and is competitive with S.O.R for all grid sizes examined.

In this research we consider stochastic models of Glioblastoma Multiforme brain tumors. We first look at a model by K. Swanson et al., which describes the dynamics as random diffusion plus deterministic logistic growth. We introduce a stochastic component in the logistic growth in the form of a random growth rate defined by a Poisson process. We show that this stochastic logistic growth model leads to a more accurate evaluation of the tumor growth compared its deterministic counterpart. We also discuss future plans to incorporate individual patient geometry, extend the model to three dimensions and to incorporate effects of different treatments into our model, in collaboration with a local hospital.

This project discusses simulation results of star formation by Active Galactic Nuclei (AGN) jets using the WENO method. A typical AGN jet with velocity uj=0.3c, density ρj=10^(-2) H/cm3, and temperature Tj=10^(7) K was injected into a 425 light years square region. The jet passes through a stationary inhomogeneous ambient background of temperature Ta=5x10^4 K and density ρa= 2 H/cm^3 to test if AGN jets, by creating bow shocks propagating through the interstellar medium and molecular clouds, can form stars in the densest regions. According to the star formation criteria for gravitational collapse of Cen and Ostriker, the resulting simulations indicate the presence of star formation via AGN jets (1992). The parameters are tuned to match Centaurs A to identify star formation in this galaxy. The simulations will also be run in three dimensions in the future and for longer time intervals to gain a better understanding of the star formation process via AGN jets.

In 2010, two gamma-ray /x-ray bubbles were detected in the center of the Milky Way Galaxy. These bubbles extend symmetrically ≈ 30, 000 light years above and below the Galactic Center, with a width of ≈ 27, 000 light years. These bubbles emit gamma-rays at energies between 1 and 100 giga-electronvolts, have approximately uniform surface brightness, and are expanding at ≈ 30, 000 km/s. We believe that these Fermi Bubbles are the result of an astrophysical jet pulse that occurred millions of years ago. Utilizing high-performance computing and Euler’s Gas Dynamics Equations, we hope to find a realistic simulation that will tell us more about the age of these Fermi Bubbles and better understand the mechanism that powers the bubbles.

The increased number of novel pathogens that potentially threaten the human population has motivated the development of mathematical and computational modeling approaches for forecasting epidemic impact and understanding key environmental characteristics that influence the spread of diseases. Yet, in the case that substantial uncertainty surrounds the transmission process during a rapidly developing infectious disease outbreak, complex mechanistic models may be too difficult to be calibrated quick enough for policy makers to make informed decisions. Simple phenomenological models that rely on a small number of parameters can provide an initial platform for assessing the epidemic trajectory, estimating the reproduction number and quantifying the disease burden from the early epidemic phase.

Chapter 1 provides background information and motivation for infectious disease forecasting and outlines the rest of the thesis.

In chapter 2, logistic patch models are used to assess and forecast the 2013-2015 West Africa Zaire ebolavirus epidemic. In particular, this chapter is concerned with comparing and contrasting the effects that spatial heterogeneity has on the forecasting performance of the cumulative infected case counts reported during the epidemic.

In chapter 3, two simple phenomenological models inspired from population biology are used to assess the Research and Policy for Infectious Disease Dynamics (RAPIDD) Ebola Challenge; a simulated epidemic that generated 4 infectious disease scenarios. Because of the nature of the synthetically generated data, model predictions are compared to exact epidemiological quantities used in the simulation.

In chapter 4, these models are applied to the 1904 Plague epidemic that occurred in Bombay. This chapter provides evidence that these simple models may be applicable to infectious diseases no matter the disease transmission mechanism.

Chapter 5, uses the patch models from chapter 2 to explore how migration in the 1904 Plague epidemic changes the final epidemic size.

The final chapter is an interdisciplinary project concerning within-host dynamics of cereal yellow dwarf virus-RPV, a plant pathogen from a virus group that infects over 150 grass species. Motivated by environmental nutrient enrichment due to anthropological activities, mathematical models are employed to investigate the relevance of resource competition to pathogen and host dynamics.

In recent decades, marine ecologists have conducted extensive field work and experiments to understand the interactions between bacteria and bacteriophage (phage) in marine environments. This dissertation provides a detailed rigorous framework for gaining deeper insight into these interactions. Specific features of the dissertation include the design of a new deterministic Lotka-Volterra model with n + 1 bacteria, n
+ 1 phage, with explicit nutrient, where the jth phage strain infects the first j bacterial strains, a perfectly nested infection network (NIN). This system is subject to trade-off conditions on the life-history traits of both bacteria and phage given in an earlier study Jover et al. (2013). Sufficient conditions are provided to show that a bacteria-phage community of arbitrary size with NIN can arise through the succession of permanent subcommunities, by the successive addition of one new population. Using uniform persistence theory, this entire community is shown to be permanent (uniformly persistent), meaning that all populations ultimately survive.

It is shown that a modified version of the original NIN Lotka-Volterra model with implicit nutrient considered by Jover et al. (2013) is permanent. A new one-to-one infection network (OIN) is also considered where each bacterium is infected by only one phage, and that phage infects only that bacterium. This model does not use the trade-offs on phage infection range, and bacterium resistance to phage. The OIN model is shown to be permanent, and using Lyapunov function theory, coupled with LaSalle’s Invariance Principle, the unique coexistence equilibrium associated with the NIN is globally asymptotically stable provided that the inter- and intra-specific bacterial competition coefficients are equal across all bacteria.

Finally, the OIN model is extended to a “Kill the Winner” (KtW) Lotka-Volterra model

of marine communities consisting of bacteria, phage, and zooplankton. The zooplankton

acts as a super bacteriophage, which infects all bacteria. This model is shown to be permanent.

Chapter 1 introduces some key elements of important topics such as; quantum mechanics,

representation theory of the Lorentz and Poincare groups, and a review of some basic rela- ´

tivistic wave equations that will play an important role in the work to follow. In Chapter 2,

a complex covariant form of the classical Maxwell’s equations in a moving medium or at

rest is introduced. In addition, a compact, Lorentz invariant, form of the energy-momentum

tensor is derived. In chapter 3, the concept of photon helicity is critically analyzed and its

connection with the Pauli-Lubanski vector from the viewpoint of the complex electromag- ´

netic field, E+ iH. To this end, a complex covariant form of Maxwell’s equations is used.

Chapter 4 analyzes basic relativistic wave equations for the classical fields, such as Dirac’s

equation, Weyl’s two-component equation for massless neutrinos and the Proca, Maxwell

and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubanski vector and the Casimir ´

operators of the Poincare group. A connection between the spin of a particle/field and ´

consistency of the corresponding overdetermined system is emphasized in the massless

case. Chapter 5 focuses on the so-called generalized quantum harmonic oscillator, which

is a Schrodinger equation with a time-varying quadratic Hamiltonian operator. The time ¨

evolution of exact wave functions of the generalized harmonic oscillators is determined

in terms of the solutions of certain Ermakov and Riccati-type systems. In addition, it is

shown that the classical Arnold transform is naturally connected with Ehrenfest’s theorem

for generalized harmonic oscillators. In Chapter 6, as an example of the usefulness of the

methods introduced in Chapter 5 a model for the quantization of an electromagnetic field

in a variable media is analyzed. The concept of quantization of an electromagnetic field

in factorizable media is discussed via the Caldirola-Kanai Hamiltonian. A single mode

of radiation for this model is used to find time-dependent photon amplitudes in relation

to Fock states. A multi-parameter family of the squeezed states, photon statistics, and the

uncertainty relation, are explicitly given in terms of the Ermakov-type system.

Cancer is a major health problem in the world today and is expected to become an even larger one in the future. Although cancer therapy has improved for many cancers in the last several decades, there is much room for further improvement. Mathematical modeling has the advantage of being able to test many theoretical therapies without having to perform clinical trials and experiments. Mathematical oncology will continue to be an important tool in the future regarding cancer therapies and management.

This dissertation is structured as a growing tumor. Chapters 2 and 3 consider spheroid models. These models are adept at describing 'early-time' tumors, before the tumor needs to co-opt blood vessels to continue sustained growth. I consider two partial differential equation (PDE) models for spheroid growth of glioblastoma. I compare these models to in vitro experimental data for glioblastoma tumor cell lines and other proposed models. Further, I investigate the conditions under which traveling wave solutions exist and confirm numerically.

As a tumor grows, it can no longer be approximated by a spheroid, and it becomes necessary to use in vivo data and more sophisticated modeling to model the growth and diffusion. In Chapter 4, I explore experimental data and computational models for describing growth and diffusion of glioblastoma in murine brains. I discuss not only how the data was obtained, but how the 3D brain geometry is created from Magnetic Resonance (MR) images. A 3D finite-difference code is used to model tumor growth using a basic reaction-diffusion equation. I formulate and test hypotheses as to why there are large differences between the final tumor sizes between the mice.

Once a tumor has reached a detectable size, it is diagnosed, and treatment begins. Chapter 5 considers modeling the treatment of prostate cancer. I consider a joint model with hormonal therapy as well as immunotherapy. I consider a timing study to determine whether changing the vaccine timing has any effect on the outcome of the patient. In addition, I perform basic analysis on the six-dimensional ordinary differential equation (ODE). I also consider the limiting case, and perform a full global analysis.

Swarms of animals, fish, birds, locusts etc. are a common occurrence but their coherence and method of organization poses a major question for mathematics and biology.The Vicsek and the Attraction-Repulsion are two models that have been proposed to explain the emergence of collective motion. A major issue for the Vicsek Model is that its particles are not attracted to each other, leaving the swarm with alignment in velocity but without spatial coherence. Restricting the particles to a bounded domain generates global spatial coherence of swarms while maintaining velocity alignment. While individual particles are specularly reflected at the boundary, the swarm as a whole is not. As a result, new dynamical swarming solutions are found.

The Attraction-Repulsion Model set with a long-range attraction and short-range repulsion interaction potential typically stabilizes to a well-studied flock steady state solution. The particles for a flock remain spatially coherent but have no spatial bound and explore all space. A bounded domain with specularly reflecting walls traps the particles within a specific region. A fundamental refraction law for a swarm impacting on a planar boundary is derived. The swarm reflection varies from specular for a swarm dominated by

kinetic energy to inelastic for a swarm dominated by potential energy. Inelastic collisions lead to alignment with the wall and to damped pulsating oscillations of the swarm. The fundamental refraction law provides a one-dimensional iterative map that allows for a prediction and analysis of the trajectory of the center of mass of a flock in a channel and a square domain.

The extension of the wall collisions to a scattering experiment is conducted by setting two identical flocks to collide. The two particle dynamics is studied analytically and shows a transition from scattering: diverging flocks to bound states in the form of oscillations or parallel motions. Numerical studies of collisions of flocks show the same transition where the bound states become either a single translating flock or a rotating (mill).

Factory production is stochastic in nature with time varying input and output processes that are non-stationary stochastic processes. Hence, the principle quantities of interest are random variables. Typical modeling of such behavior involves numerical simulation and statistical analysis. A deterministic closure model leading to a second order model for the product density and product speed has previously been proposed. The resulting partial differential equations (PDE) are compared to discrete event simulations (DES) that simulate factory production as a time dependent M/M/1 queuing system. Three fundamental scenarios for the time dependent influx are studied: An instant step up/down of the mean arrival rate; an exponential step up/down of the mean arrival rate; and periodic variation of the mean arrival rate. It is shown that the second order model, in general, yields significant improvement over current first order models. Specifically, the agreement between the DES and the PDE for the step up and for periodic forcing that is not too rapid is very good. Adding diffusion to the PDE further improves the agreement. The analysis also points to fundamental open issues regarding the deterministic modeling of low signal-to-noise ratio for some stochastic processes and the possibility of resonance in deterministic models that is not present in the original stochastic process.