This thesis is based on the use of mathematical theories, modelling, and simulations to study the transmission dynamics of HIV/AIDS in the presence of PrEP (pre-exposure prophylaxis) in the MSM (men who have sex with men) population in the United States. A new deterministic model for HIV/AIDS that incorporates PrEP is designed and used to assess the population-level impact of the use of PrEP on the transmission dynamics within an MSM population. Conditions for the effective control (or elimination) and persistence of HIV/AIDS in the MSM population are determined by rigorously analyzing this model. Uncertainty and sensitivity analysis is carried out to determine the effect of the uncertainties in the parameter values on the response variable (the associated reproduction number) and to identify the top-five parameters that have the most effect on the disease transmission dynamics. Numerical simulations show that HIV burden decreases with increasing PrEP coverage. / October 2015

The main goal of this dissertation was to study the bifurcation structure underlying families of low dimensional dynamical systems that model cellular excitability. One of the main contributions of this work is a mathematical characterization of profiles of electrophysiological activity in excitable cells of the same identified type, and across cell types, as a function of the relative levels of expression of ion channels coded by specific genes. In doing so, a generic formulation for transmembrane transport was derived from first principles in two different ways, expanding previous work by other researchers. The relationship between the expression of specific membrane proteins mediating transmembrane transport and the electrophysiological profile of excitable cells is well reproduced by electrodiffusion models of membrane potential involving as few as 2 state variables and as little as 2 transmembrane currents. Different forms of the generic electrodiffusion model presented here can be used to study the geometry underlying different forms of excitability in cardiocytes, neurons, and other excitable cells, and to simulate different patterns of response to constant, time-dependent, and (stochastic) time- and voltage-dependent stimuli. In all cases, an initial analysis performed on a deterministic, autonoumous version of the system of interest is presented to develop basic intuition that can be used to guide analyses of non-autonomous or stochastic versions of the model. Modifications of the biophysical models presented here can be used to study complex physiological systems involving single cells with specific membrane proteins, possibly linking different levels of biological organization and spatio-temporal scales.

The author examines nonlinear aeroelastic responses of air vehicle systems. Herein, the governing equations for a cantilevered configuration are developed and the methods of analysis are explored. Based on the developed nonlinear bending-bending-torsion equations, internal resonance, which is possible in future air vehicles, and the possible cause of limit cycle oscillations of aircraft wings with stores are investigated. The nonlinear equations have three types of nonlinearities caused by wing flexibility, store geometry and aerodynamic stall, and retain up to third-order nonlinear terms. The internal resonance conditions are examined by the Method of Multiple Scales and demonstrated by time simulations. The effect of velocity change for various physical parameters and stiffness ratio is investigated through bifurcation diagrams derived from Poinar´e maps. The dominant factor causing limit cycle oscillations is the stiffness ratio between in-plane and out-of-plane motion.

An analog controller has been analyzed and built for an electrostatic micro-cantilever beam. The closed loop MEMS device can be used as both actuator and sensor. As an actuator it will have the advantage of large stable travel range up to 90% of the gap. As a sensor the beam is to be driven into chaotic motion which is very sensitive changes in the system parameters. Two versions of the controller have been analyzed and implemented, one for the actuator and one for the sensor. For the actuator, preliminary experiments show good matching with the model. As for the sensor, the dynamic behavior have been studied and the best operating regions have been determined.

A multifunctional central pattern generator (CPG) can produce bursting polyrhythms that determine locomotive activity in an animal: for example, swimming and crawling in a leech. Each rhythm corresponds to a specific attractor of the CPG. We employ a Hodgkin-Huxley type model of a bursting leech heart interneuron, and connect three such neurons by fast inhibitory synapses to form a ring. This network motif exhibits multistable co-existing bursting rhythms. The problem of determining rhythmic outcomes is reduced to an analysis of fixed points of Poincare mappings and their attractor basins, in a phase plane defined by the interneurons' phase differences along bursting orbits. Using computer assisted analysis, we examine stability, bifurcations of attractors, and transformations of their basins in the phase plane. These structures determine the global bursting rhythms emitted by the CPG. By varying the coupling synaptic strength, we examine the dynamics and patterns produced by inhibitory networks.

Dans ce mémoire, nous étudions le problème centre-foyer sur un système polynomial. Nous développons ainsi deux mécanismes permettant de conclure qu’un point singulier monodromique dans ce système non-linéaire polynomial est un centre. Le premier mécanisme est la méthode de Darboux. Cette méthode utilise des courbes algébriques invariantes dans la construction d’une intégrale première. La deuxième méthode analyse la réversibilité algébrique ou analytique du système. Un système possédant une singularité monodromique et étant algébriquement ou analytiquement réversible à ce point sera nécessairement un centre. Comme application, dans le dernier chapitre, nous considérons le modèle de Gauss généralisé avec récolte de proies. / In this thesis, we study the center-focus problem in a polynomial system. We describe two mechanisms to conclude that a monodromic singular point in this polynomial system is a center. The first one is the method of Darboux. In this method, one uses invariant algebraic curves to build a first integral. The second method is the algebraic (and analytic) reversibility. A monodromic singularity, which is algebraically or analytically reversible at the singular point, is necessarily a center. As an application, in the last chapter, we consider the generalized Gause model with prey harvesting and a generalized Holling response function of type III.

The thesis is based on the use of mathematical modeling and analysis to gain insightinto the transmission dynamics of malaria in a community. A new deterministic model for assessing the role of age-structure on the disease dynamics is designed. The model undergoes backward bifurcation, a dynamic phenomenon characterized by the co-existence of a stable disease-free and an endemic equilibrium of the model when the associated reproduction number is less than unity. It is shown that adding age-structure to the basic model for malaria transmission does not alter its essential qualitative dynamics. The study is extended to incorporate the use of anti-malaria drugs. Numerical simulations of the extended model suggest that for the case when treatment does not cause drug resistance (and the reproduction number of each of the two strains exceed unity), the model undergoes competitive exclusion. The impact of various effectiveness levels of the treatment strategy is assessed.

<p>To understand the underlying mechanisms of cardiac arrhythmias, computational models are used to study heart rhythm dynamics. The parameters of these models carry inherent uncertainty. Therefore, to interpret the results of these models, uncertainty quantification (UQ) and sensitivity analysis (SA) are important. Polynomial chaos (PC) is a computationally efficient method for UQ and SA in which a model output Y, dependent on some independent uncertain parameters represented by a random vector &xi;, is approximated as a spectral expansion in multidimensional orthogonal polynomials in &xi;. The expansion can then be used to characterize the uncertainty in Y.</p><p>PC methods were applied to UQ and SA of the dynamics of a two-dimensional return-map model of cardiac action potential duration (APD) restitution in a paced single cell. Uncertainty was considered in four parameters of the model: three time constants and the pacing stimulus strength. The basic cycle length (BCL) (the period between stimuli) was treated as the control parameter. Model dynamics was characterized with bifurcation analysis, which determines the APD and stability of fixed points of the model at a range of BCLs, and the BCLs at which bifurcations occur. These quantities can be plotted in a bifurcation diagram, which summarizes the dynamics of the model. PC UQ and SA were performed for these quantities. UQ results were summarized in a novel probabilistic bifurcation diagram that visualizes the APD and stability of fixed points as uncertain quantities.</p><p>Classical PC methods assume that model outputs exist and reasonably smooth over the full domain of &xi;. Because models of heart rhythm often exhibit bifurcations and discontinuities, their outputs may not obey the existence and smoothness assumptions on the full domain, but only on some subdomains which may be irregularly shaped. On these subdomains, the random variables representing the parameters may no longer be independent. PC methods therefore must be modified for analysis of these discontinuous quantities. The Rosenblatt transformation maps the variables on the subdomain onto a rectangular domain; the transformed variables are independent and uniformly distributed. A new numerical estimation of the Rosenblatt transformation was developed that improves accuracy and computational efficiency compared to existing kernel density estimation methods. PC representations of the outputs in the transformed variables were then constructed. Coefficients of the PC expansions were estimated using Bayesian inference methods. For discontinuous model outputs, SA was performed using a sampling-based variance-reduction method, with the PC estimation used as an efficient proxy for the full model.</p><p>To evaluate the accuracy of the PC methods, PC UQ and SA results were compared to large-sample Monte Carlo UQ and SA results. PC UQ and SA of the fixed point APDs, and of the probability that a stable fixed point existed at each BCL, was very close to MC UQ results for those quantities. However, PC UQ and SA of the bifurcation BCLs was less accurate compared to MC results.</p><p>The computational time required for PC and Monte Carlo methods was also compared. PC analysis (including Rosenblatt transformation and Bayesian inference) required less than 10 total hours of computational time, of which approximately 30 minutes was devoted to model evaluations, compared to approximately 65 hours required for Monte Carlo sampling of the model outputs at 1 &times; 10<super>6</super> &xi; points.</p><p>PC methods provide a useful framework for efficient UQ and SA of the bifurcation diagram of a model of cardiac APD dynamics. Model outputs with bifurcations and discontinuities can be analyzed using modified PC methods. The methods applied and developed in this study may be extended to other models of heart rhythm dynamics. These methods have potential for use for uncertainty and sensitivity analysis in many applications of these models, including simulation studies of heart rate variability, cardiac pathologies, and interventions.</p> / Dissertation

Dans cette thèse, nous étudions un système à temps discret de dimension N, dont les paramètres varient périodiquement. Le système de dimension N est construit à partir de n sous-systèmes de dimension un couplés symétriquement. Dans un premier temps, nous donnons les propriétés générales du système de dimension N. Dans un second temps, nous étudions le cas particulier où le sous-système de dimension un est défini à l'aide d'une transformation logistique. Nous nous intéressons plus particulièrement à la structure des bifurcations lorsque N=1 ou 2. Des zones échangeurs centrées sur des points cuspidaux sont obtenues dans le cas de courbes de bifurcation de type fold (nœud-col). Ensuite, nous nous intéressons au comportement de circuits de type Chua couplés lorsqu'un paramètre varie lui aussi périodiquement, la période étant celle d'une des variables d'état interne au système. A partir de l'étude des bifurcations du système, la non existence de cycles d'ordre impair et la coexistence de plusieurs attracteurs est mise en évidence. D'autre part, on peut mettre en évidence la coexistence de différents attracteurs pour lesquels les états de synchronisation sont distincts. Le cas continu est comparé avec le cas discret. Des phénomènes tout à fait similaires sont obtenus. Il est important de noter que l'étude d'un système à temps discret est plus facile et plus rapide que celle d'un système à temps continu. L'étude du premier système permet donc d'avoir des informations sur ce qui peut se produire dans le cas continu. Pour terminer, nous analysons le comportement d'un autre système couplé à temps continu, basé lui aussi sur le circuit de Chua, mais pour lequel la commutation qui contrôle la variation du paramètre s'effectue différemment du premier système. Ce type de commutation génère une augmentation du nombre d'attracteurs.

The thesis is based on the use of mathematical modeling and analysis to gain insightinto the transmission dynamics of malaria in a community. A new deterministic model for assessing the role of age-structure on the disease dynamics is designed. The model undergoes backward bifurcation, a dynamic phenomenon characterized by the co-existence of a stable disease-free and an endemic equilibrium of the model when the associated reproduction number is less than unity. It is shown that adding age-structure to the basic model for malaria transmission does not alter its essential qualitative dynamics. The study is extended to incorporate the use of anti-malaria drugs. Numerical simulations of the extended model suggest that for the case when treatment does not cause drug resistance (and the reproduction number of each of the two strains exceed unity), the model undergoes competitive exclusion. The impact of various effectiveness levels of the treatment strategy is assessed.