An impulsive differential equation model for Marek's disease

Many dynamical processes are subject to abrupt changes in state. Often these perturbations can be periodic and of short duration relative to the evolving process. These types of phenomena are described well by what are referred to as impulsive differential equations, systems of differential equations coupled with discrete mappings in state space. In this thesis we employ impulsive differential equations to model disease transmission within an industrial livestock barn. In particular we focus on the poultry industry and a viral disease of poultry called Marek's disease. This system lends itself well to impulsive differential equations. Entire cohorts of poultry are introduced and removed from a barn concurrently. Additionally, Marek's disease is transmitted indirectly and the viral particles can survive outside the host for weeks. Therefore, depopulating, cleaning, and restocking of the barn are integral factors in modelling disease transmission and can be completely captured by the impulsive component of the model. Our model allows us to investigate how modern broiler farm practices can make disease elimination difficult or impossible to achieve. It also enables us to investigate factors that may contribute to virulence evolution. Our model suggests that by decrease the cohort duration or by decreasing the flock density, Marek's disease can be eliminated from a barn with no increase in cleaning effort. Unfortunately our model also suggests that these practices will lead to disease evolution towards greater virulence. Additionally, our model suggests that if intensive cleaning between cohorts does not rid the barn of disease, it may drive evolution and cause the disease to become more virulent.

Electron acoustic soliton energy of the Kadomtsev-Petviashvili equation in the Earth’s magnetotail region at critical ion density

The nonlinear properties of small amplitude electron-acoustic solitary waves (EAWs) in a homogeneous system of unmagnetized collisionless plasma consisted of a cold electron fluid and isothermal ions with two different temperatures obeying Boltzmann type distributions have been investigated. A reductive perturbation method was employed to obtain the Kadomstev-Petviashvili (KP) equation. At the critical ion density, the KP equation is not appropriate for describing the system. Hence, a new set of stretched coordinates
is considered to derive the modified KP equation. Moreover, the solitary solution, soliton energy and the associated electric field at the critical ion density were computed. The present investigation can be of relevance to the electrostatic solitary structures observed in various space plasma environments, such as Earth’s magnetotail region.

Symmetries and exact solutions of KP equation with an arbitrary nonlinear term

The generalized KP (GKP) equations with an arbitrary nonlinear term model and characterize many nonlinear physical phenomena. The symmetries of GKP equation with an arbitrary nonlinear term are obtained. The condition that must satisfy for existence the symmetries group of GKP is derived and also the obtained symmetries are classified according to different forms of the nonlinear term. The resulting similarity reductions are studied by performing the bifurcation and the phase portrait of GKP and also the corresponding solitary wave solutions of GKP
equation are constructed.

A Simultaneous Equation Approach to Estimating HIV Prevalence with Non-Ignorable Missing Responses

Estimates of HIV prevalence are important for policy in order to establish the health status of a country's population and to evaluate the effectiveness of population-based interventions and campaigns. However, participation rates in testing for surveillance conducted as part of household surveys, on which many of these estimates are based, can be low. HIV positive individuals may be less likely to participate because they fear disclosure, in which case estimates obtained using conventional approaches to deal with missing data, such as imputation-based methods, will be biased. We develop a Heckman-type simultaneous equation approach which accounts for non-ignorable selection, but unlike previous implementations, allows for spatial dependence and does not impose a homogeneous selection process on all respondents. In addition, our framework addresses the issue of separation, where for instance some factors are severely unbalanced and highly predictive of the response, which would ordinarily prevent model convergence. Estimation is carried out within a penalized likelihood framework where smoothing is achieved using a parametrization of the smoothing criterion which makes estimation more stable and efficient. We provide the software for straightforward implementation of the proposed approach, and apply our methodology to estimating national and sub-national HIV prevalence in Swaziland, Zimbabwe and Zambia.

Automated Equation Formulation for Causal Loop Diagrams

The annotation of Business Dynamics models with parameters and equations, to simulate the system under study and further evaluate its simulation output, typically involves a lot of manual work. In this paper we present an approach for automated equation formulation of a given Causal Loop Diagram (CLD) and a set of associated time series with the help of neural network evolution (NEvo). NEvo enables the automated retrieval of surrogate equations for each quantity in the given CLD, hence it produces a fully annotated CLD that can be used for later simulations to predict future KPI development. In the end of the paper, we provide a detailed evaluation of NEvo on a business use-case to demonstrate its single step prediction capabilities.

Combined Conduction-Convection-Radiation Heat Transfer of Slip Flow inside a Micro-channel Filled with a Cellular Porous Material

Combined conduction–convection–radiation heat transfer is investigated numerically in a micro-channel filled with a saturated cellular porous medium, with the channel walls held at a constant heat flux. Invoking the velocity slip and temperature jump, the thermal behaviour of the porous–fluid system are studied by considering hydrodynamically fully developed flow and applying the Darcy–Brinkman flow model. One energy equation model based on the local thermal equilibrium condition is adopted to evaluate the temperature field within the porous medium. Combined conduction and radiation heat transfer is treated as an effective conduction process with a temperature-dependent effective thermal conductivity. Results are reported in terms of the average Nusselt number and dimensionless temperature distribution, as a function of velocity slip coefficient, temperature jump coefficient, porous medium shape parameter and radiation parameters. Results show that increasing the radiation parameter (Tr)(Tr) and the temperature jump coefficient flattens the dimensionless temperature profile. The Nusselt numbers are more sensitive to the variation in the temperature jump coefficient rather than to the velocity slip coefficient. Such that for high porous medium shape parameter, the Nusselt number is found to be independent of velocity slip. Furthermore, it is found that as the temperature jump coefficient increases, the Nusselt number decrease. In addition, for high temperature jump coefficients, the Nusselt number is found to be insensitive to the radiation parameters and porous medium shape parameter. It is also concluded that compared with the conventional macro-channels, wherein using a porous material enhances the rate of heat transfer (up to about 40 % compared to the clear channel), insertion of a porous material inside a micro-channel in slip regime does not effectively enhance the rate of heat transfer that is about 2 %.

Stochastic Modeling of Reversible Biochemical Reaction-Diffusion Systems and High-Resolution Shock-Capturing Methods for Fluid Interfaces

Thesis (Ph.D.)--University of Washington, 2016-08

The Helmholtzian Equation: Stabilizing Mechanisms for Quadratic Nonlinear Fields

Hexagonal Resonant Triad patterns are shown to exist as stable solutions of a particular type of nonlinear field where no cubic field nonlinearity is present. The zero ‘dc’ Fourier mode is shown to stabilize these patterns produced by a pure quadratic field nonlinearity. Closed form solutions and stability results are obtained near the critical point, complimented by numerical studies far from the critical point. These results are obtained using a neural field based on the Helmholtzian operator. Constraints on structure and parameters for a general pure quadratic neural field which supports hexagonal patterns are obtained.

The role of nuclear physics in supernovae and the evolution of neutron stars, Neutrino Opacities, Equation of State, Transport Coefficients, and Dark Matter Production

Thesis (Ph.D.)--University of Washington, 2016-08

Algebro-Geometric Aspects of the Classical Yang-Baxter Equation

In this thesis we consider algebro-geometric aspects of the Classical Yang-Baxter Equation and the Generalised Classical Yang-Baxter Equation. In chapter one we present a method to construct solutions of the Generalised Classical Yang-Baxter Equation starting with certain sheaves of Lie algebras on algebraic curves. Furthermore we discuss a criterion to check unitarity of such solutions. In chapter two we consider the special class of solutions coming from sheaves of traceless endomorphisms of simple vector bundles on the nodal cubic curve. These solutions are quasi-trigonometric and we describe how they fit into the classification scheme of such solutions. Moreover, we describe a concrete formula for these solutions. In the third and final chapter we show that any unitary, rational solution of the Classical Yang-Baxter Equation can be obtained via the method of chapter one applied to a sheaf of Lie algebras on the cuspidal cubic curve.

Dispersive estimates for the Schrödinger equation

In this document we explore the issue of $L^1\to L^\infty$ estimates for the solution operator of the linear Schr\"{o}dinger equation, \begin{align*} iu_t-\Delta u+Vu&=0 &u(x,0)=f(x)\in \mathcal S(\R^n). \end{align*} We focus particularly on the five and seven dimensional cases. We prove that the solution operator precomposed with projection onto the absolutely continuous spectrum of $H=-\Delta+V$ satisfies the following estimate $\|e^{itH} P_{ac}(H)\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{n}{2}}$ under certain conditions on the potential $V$. Specifically, we prove the dispersive estimate is satisfied with optimal assumptions on smoothness, that is $V\in C^{\frac{n-3}{2}}(\R^n)$ for $n=5,7$ assuming that zero is regular, $|V(x)|\lesssim \langle x\rangle^{-\beta}$ and $|\nabla^j V(x)|\lesssim \langle x\rangle^{-\alpha}$, $1\leq j\leq \frac{n-3}{2}$ for some $\beta>\frac{3n+5}{2}$ and $\alpha>3,8$ in dimensions five and seven respectively. We also show that for the five dimensional result one only needs that $|V(x)|\lesssim \langle x\rangle^{-4-}$ in addition to the assumptions on the derivative and regularity of the potential. This more than cuts in half the required decay rate in the first chapter. Finally we consider a problem involving the non-linear Schr\"{o}dinger equation. In particular, we consider the following equation that arises in fiber optic communication systems, \begin{align*} iu_t+d(t) u_{xx}+|u|^2 u=0. \end{align*} We can reduce this to a non-linear, non-local eigenvalue equation that describes the so-called dispersion management solitons. We prove that the dispersion management solitons decay exponentially in $x$ and in the Fourier transform of $x$.

A study of electrochemical transport and diffuse charge dynamics using a Langevin equation

This thesis aims to develop new numerical and computational tools to study electrochemical transport and diffuse charge dynamics at small scales. Previous efforts at modeling electrokinetic phenomena at scales where the noncontinuum effects become significant have included continuum models based on the Poisson-Nernst-Planck equations and atomic simulations using molecular dynamics algorithms. Neither of them is easy to use or conducive to electrokinetic transport modeling in strong confinement or over long time scales. This work introduces a new approach based on a Langevin equation for diffuse charge dynamics in nanofluidic devices, which incorporates features from both continuum and atomistic methods. The model is then extended to include steric effects resulting from finite ion size, and applied to the phenomenon of double layer charging in a symmetric binary electrolyte between parallel-plate blocking electrodes, between which a voltage is applied. Finally, the results of this approach are compared to those of the continuum model based on the Poisson-Nernst-Planck equations.

Fully Anisotropic Solution of the Three Dimensional Boltzmann Transport Equation

The development of accurate modeling techniques for nanoscale thermal transport is an active area of research. Modern day nanoscale devices have length scales of tens of nanometers and are prone to overheating, which reduces device performance and lifetime. Therefore, accurate temperature profiles are needed to predict the reliability of nanoscale devices. The majority of models that appear in the literature obtain temperature profiles through the solution of the Boltzmann transport equation (BTE). These models often make simplifying assumptions about the nature of the quantized energy carriers (phonons). Additionally, most previous work has focused on simulation of planar two dimensional structures. This thesis presents a method which captures the full anisotropy of the Brillouin zone within a three dimensional solution to the BTE. The anisotropy of the Brillouin zone is captured by solving the BTE for all vibrational modes allowed by the Born Von-Karman boundary conditions.