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Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 127-131).

Thesis (Ph. D.)--University of Chicago, Committee of Neurobiology, December 1999. / Includes bibliographical references. Also available on the Internet.

Combination antiviral drug therapy improves the survival rates of patients chronically infected with hepatitis B virus by controlling viral replication and enhancing immune responses. Some of these drugs have side effects that make them unsuitable for long-term administration. To address the trade-off between the positive and negative effects of the combination therapy, we investigated an optimal control problem for a delay differential equation model of immune responses to hepatitis virus B infection. Our optimal control problem investigates the interplay between virological and immunomodulatory effects of therapy, the control of viremia and the administration of the minimal dosage over a short period of time. Our numerical results show that the high drug levels that induce immune modulation rather than suppression of virological factors are essential for the clearance of hepatitis B virus.

Stochastic delay differential equations (SDDEs) are systems of differential equations with a time lag in a noisy or random environment. There are many nonlinear SDDEs where a linear growth condition is not satisfied, for example, the stochastic delay Lotka-Volterra model of food chain discussed by Xuerong Mao and Martina John Rassias in 2005. Much research has been done using discrete delay where the dynamics of a process at time t depend on the state of the process in the past after a single fixed time lag \tau. We are researching processes with continuously distributed delay which depend on weighted averages of past states over the entire time lag interval [t-\tau,t].By using martingale concepts, we prove sufficient conditions for the existence of a unique solution, ultimate boundedness, and non-extinction of one-dimensional nonlinear SDDE with continuously distributed delay. We give generalized Khasminskii-type conditions which along with local Lipschitz conditions are sufficient to guarantee the existence of a unique global solution of certain n-dimensional nonlinear SDDEs with continuously distributed delay. Further, we give conditions under which Euler-Maruyama numerical approximations of such nonlinear SDDEs converge in probability to their exact solutions.We give some examples of one-dimensional and 2-dimensional stochastic differential equations with continuously distributed delay which satisfy the sufficient conditions of our theorems. Moreover, we simulate their solutions and analyze the error of approximation using MATLAB to implement the Euler-Maruyama algorithm.

A generalized low order model (LOM) for the fluctuating lift coefficient caused by vortex shedding from a tandem cylinder pair is proposed to expand upon models from previous authors. This model could provide a reduced computational time method for collecting qualitative and quantitive data from a tandem shedding pair. A delay coupled system with sufficient bifurcation characteristics is developed to account for the different flow regimes (extended-body, reattachment, and co-shedding) which occur as cylinder spacing is varied. Coefficient and parameter fitting is performed to fit experimental data. Finally, results and physical interpretations of the interactions in the model are discussed. It was found that many aspects of the flow at varying L/D ratios could be modeled by the LOM, including vortex suppression in the forward cylinder at the critical spacing, and amplitude growth in the rear cylinder in the co-shedding regime.

Delay differential equations (DDEs) are a class of differential equations that have received considerable recent attention and been shown to model many real life problems, traditionally formulated as systems of ordinary differential equations (ODEs), more naturally and more accurately. Ideally a DDE modeling package should provide facilities for approximating the solution, performing a sensitivity analysis and estimating unknown parameters. In this thesis we propose new techniques for efficient simulation, accurate sensitivity analysis and reliable parameter estimation of DDEs. We propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a general linear method (GLM) and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. We identify a precise process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. We introduce an equation governing the dynamics of sensitivities for the most general system of parametric DDEs. Then, having a similar view as the simulation (DDEs as discontinuous ODEs), we introduce a formula for finding the size of jumps that appear at discontinuity points when the sensitivity equations are integrated. This leads to an algorithm which can compute sensitivities for various kind of parameters very accurately. We also develop an algorithm for reliable parameter identification of DDEs. We propose a method for adding extra constraints to the optimization problem, changing a possibly non-smooth optimization to a smooth problem. These constraints are effectively handled using information from the simulator and the sensitivity analyzer. Finally, we discuss the structure of our evolving modeling package DDEM. We present a process that has been used for incorporating existing codes to reduce the implementation time. We discuss the object-oriented paradigm as a way of having a manageable design with reusable and customizable components. The package is programmed in C++ and provides a user-friendly calling sequences. The numerical results are very encouraging and show the effectiveness of the techniques.

Delay differential equations (DDEs) are a class of differential equations that have received considerable recent attention and been shown to model many real life problems, traditionally formulated as systems of ordinary differential equations (ODEs), more naturally and more accurately. Ideally a DDE modeling package should provide facilities for approximating the solution, performing a sensitivity analysis and estimating unknown parameters. In this thesis we propose new techniques for efficient simulation, accurate sensitivity analysis and reliable parameter estimation of DDEs. We propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a general linear method (GLM) and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. We identify a precise process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. We introduce an equation governing the dynamics of sensitivities for the most general system of parametric DDEs. Then, having a similar view as the simulation (DDEs as discontinuous ODEs), we introduce a formula for finding the size of jumps that appear at discontinuity points when the sensitivity equations are integrated. This leads to an algorithm which can compute sensitivities for various kind of parameters very accurately. We also develop an algorithm for reliable parameter identification of DDEs. We propose a method for adding extra constraints to the optimization problem, changing a possibly non-smooth optimization to a smooth problem. These constraints are effectively handled using information from the simulator and the sensitivity analyzer. Finally, we discuss the structure of our evolving modeling package DDEM. We present a process that has been used for incorporating existing codes to reduce the implementation time. We discuss the object-oriented paradigm as a way of having a manageable design with reusable and customizable components. The package is programmed in C++ and provides a user-friendly calling sequences. The numerical results are very encouraging and show the effectiveness of the techniques.

We investigate the linear stability and perform the bifurcation analysis for Hopfield neural networks with a general distribution of delays, where the neurons are identical. We start by analyzing the scalar model and show what kind of information can be gained with only minimal information about the exact distribution of delays. We determine a mean delay and distribution independent stability region. We then illustrate a way of improving on this conservative result by approximating the true region of stability when the actual distribution is not known, but some moments or cumulants of the distribution are. We compare the approximate stability regions with the stability regions in the case of the uniform and gamma distributions. We show that, in general, the approximations improve as more moments or cumulants are used, and that the approximations using cumulants give better results than the ones using moments. Further, we extend these results to a network of n identical neurons, where we examine the stability of a symmetrical equilibrium point via the analysis of the characteristic equation both when the connection matrix is symmetric and when it is not. Finally, for the scalar model, we show under what conditions a Hopf bifurcation occurs and we use the centre manifold technique to determine the criticality of the bifurcation. When the kernel represents the gamma distribution with p=1 and p=2, we transform the delay differential equation into a system of ordinary differential equations and we compare the centre manifold computation to the one we obtain in the ordinary differential case.

We investigate the linear stability and perform the bifurcation analysis for Hopfield neural networks with a general distribution of delays, where the neurons are identical. We start by analyzing the scalar model and show what kind of information can be gained with only minimal information about the exact distribution of delays. We determine a mean delay and distribution independent stability region. We then illustrate a way of improving on this conservative result by approximating the true region of stability when the actual distribution is not known, but some moments or cumulants of the distribution are. We compare the approximate stability regions with the stability regions in the case of the uniform and gamma distributions. We show that, in general, the approximations improve as more moments or cumulants are used, and that the approximations using cumulants give better results than the ones using moments. Further, we extend these results to a network of n identical neurons, where we examine the stability of a symmetrical equilibrium point via the analysis of the characteristic equation both when the connection matrix is symmetric and when it is not. Finally, for the scalar model, we show under what conditions a Hopf bifurcation occurs and we use the centre manifold technique to determine the criticality of the bifurcation. When the kernel represents the gamma distribution with p=1 and p=2, we transform the delay differential equation into a system of ordinary differential equations and we compare the centre manifold computation to the one we obtain in the ordinary differential case.