Applied Mathematics Seminar 2009/10


Date: April 28, 2010
Speaker: Lin Wang
Title: Threshold Dynamics in Disease Models with Latency and Relapse
Abstract: In this talk, I will present a  general mathematical  model for a disease with an exposed (latent)  period and relapse. Such a model is appropriate for tuberculosis, including bovine tuberculosis in cattle and wildlife, and for herpes. For this  model with a general probability of remaining in the exposed class, the basic reproduction number is identified and its threshold property is discussed.


Date: April 13, 2010
Speaker: Marina Chugunova
Title: On the speed of the propagation of the thin film interface
Abstract: The equation $u_t + [u^n (u_{xxx} + \alpha^2 u_x - \sin(x))]_x = 0$ with periodic boundary conditions is a model of the evolution of a thin liquid film on the outer surface of a horizontal cylinder in the presence of gravity field. We use energy-entropy methods to study different properties of generalized weak solutions of this equation. For example: finite speed of the compact support propagation for $n \in (1,3)$ is proved by application of local energy-entropy estimates.

Joint work with A. Burchard, M. Pugh, B. Stephens, and R. Taranets

Date: March 23, 2010
Speaker: Zaheerabbas Patwa
Title: Beneficial mutations in lytic viruses: fixation probabilities and adaptation rates
Abstract: Lytic viruses are obligate parasites whose adaptation rates have attracted considerable scientific interest, as they are a key model organism in experimental evolution. Adaptation experiments using these viruses are characterized by growth, mutation, and periodic sampling (population bottlenecks) that ultimately influence the fate of a rare beneficial mutation. We first develop a life-history model for lytic viruses. Using this life-history model, we proceed to study the fixation probabilities and adaptation rates of rare beneficial mutations under constant and growing host-cell densities. The talk will be non-technical for the most part, so it should be accessible to the general audience.

Date: March 2, 2010
Speaker: Sven-Joachim Kimmerle
Title: Macroscopic diffusion models for precipitation in crystalline GaAs - Modeling, analysis and simulation
Abstract: From a thermodynamically consistent model for precipitation in gallium arsenide crystals including mechanics we derive different mathematical models to describe the size evolution of arsenic-rich liquid droplets in crystalline gallium arsenide. These models generalise the well-known Mullins-Sekerka model for Ostwald ripening.
In particular we consider a model, which leads to a quasilinear parabolic PDE for the chemical potential, which is coupled to an elliptic boundary value problem for the displacement and to ODEs for the free boundaries. Due to different scales for typical distances between droplets and typical radii of liquid droplets we can reduce this problem by homogenisation methods to a large system of ODEs coupled by a mean field, so-called mean field models.
The mean field models capture the main properties of our system and are better adapted for numerics and stability analysis than the original problem. Numerical simulations allow to decide in which case which one of the models might be appropriate to the experimental situation.


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Date: December 2, 2009
Speaker: Motassem Araydah
Title: Existence of a Positive Solution to a Nonlinear System of PDEs in a Domain with a Triple-phase Boundary, application to hydrogen fuel cells
Abstract: We consider a system of nonlinear PDEs describing the reaction-diffusion dynamics near a triple-phase boundary in the catalyst layer of hydrogen fuel cells. The system involves bulk diffusion and surface reaction-diffusion processes and is an approximation of a model with a thin layer. The coupling of surface and bulk diffusion involves a nonlinear equation (adsorption-desorption process) and a singular boundary condition. Using certain a priori estimates, variational methods techniques and the fixed point theorem, we prove the existence of a positive bounded weak solution. Moreover, we prove the validity of the model by computing the solution numerically and comparing it with the solution obtained with a thin layer model.

Date: November 23, 2009
Speaker: Xiaoqiang Zhao
Title: Spreading speeds and traveling waves for monostable biological systems
Abstract: In this talk, I will first give a brief review on asymptotic speeds of spread (in short, spreading speeds) and traveling waves for biological evolution systems with monostable nonlinearities. Then I will present the mathematical theory of spreading speeds and traveling waves for monotone semiflows. Finally I will discuss its applications to some deterministic models on biological invasions and disease spread.

Date: November 18, 2009
Speaker: Michel Tchuenche
Title: Optimal Control and Sensitivity Analysis of an Influenza Model with Treatment and Vaccination
Abstract: We formulate and analyze the dynamics of an influenza pandemic model with vaccination and treatment using two preventive scenarios: increase and decrease in vaccine uptake. We focus primarily on controlling the disease with a possible minimal cost and side effects using Control theory which is therefore applied via the Pontryagin's maximum principle, and it is observed that full treatment effort should be given while increasing vaccination at the onset of the outbreak. Next, sensitivity analysis and simulations are carried out in order to determine the relative importance of different factors responsible for disease transmission and prevalence. The most sensitive parameter of the various reproductive numbers apart from the death rate is the inflow rate, while the proportion of new recruits and the vaccine efficacy are the most sensitive parameters for the endemic equilibrium point.

Date: November 4, 2009
Speaker: Allison Kolpas
Title: Effects of Demographic Stochasticity on Population Persistence in Advective Media
Abstract: Many populations live and disperse in advective media. A fundamental question, known as the ``drift paradox" in stream ecology, is how a closed population can survive when it is constantly being transported downstream by the flow.  Recent population-level models have focused on the role of diffusive movement in balancing the effects of advection, predicting critical conditions for persistence.  Here, we formulate an individual-based stochastic model to quantify the effects of demographic stochasticity on persistence.
When there is no correlation in the movement of inter-patch residents, stochasticity smooths the persistence-extinction boundary, while when individuals disperse in ``packets" from one patch to another and the flow field is memoryless on the timescale of packet transport, the probability of persistence is greatly enhanced. 

Date: October 28, 2009
Speaker: Stefan Soltuz
Title: Non-negative Matrix Factorisation for the Cocktail Party Problem
Abstract: In many applications, the negative components contradict physical realities. For example, the pixels in a grayscale image have nonnegative intensities, so an image with negative intensities cannot be reasonably interpreted. The Power Spectrum of two mixed sources is also provided by a non-negative matrix. To address this problem, a recent approach called nonnegative matrix factorization (NMF) was proposed to search for a representative basis with only nonnegative vectors. (NMF) attempts to decompose a non-negative data matrix into a product of two nonnegative matrices. The convergence and the numerical performance of a new iteration for  (NMF) will be shown.


Date: October 21, 2009
Speaker: Marc Ethier
Title: Analysis of Singular Zones in Multidimensional Discrete Data
Abstract: The exploration and visualization of scalar data fields, such as those obtained e.g. in medical image processing, are commonly based on the study of their isosurfaces for given isovalues. Due to computational constraints, it is often necessary to limit the study to isovalues where "interesting" behaviour occurs. Critical isosurfaces fall in this category as they tend to indicate topology changes in the domain.
In this talk, we propose a homological method inspired by the Conley index theory for the detection and classification of singularities in discrete multidimensional scalar fields on cubical grids. Having grouped critical cubes into components, the Conley index theory can then be applied to analyse their criticality using suitably chosen isolating neighbourhoods. We validate and illustrate our method with computational results on 2D and 3D data fields.


Date: October 16, 2009
Speaker: Minh Van Nguyen
Title: Asymptotic Behavior of Individual Orbits of Discrete Systems
Abstract: In this talk I will speak about individual versions of the Katznelson-Tzafriri Theorem on the asymptotic behavior of orbits of power-bounded operators and related stability results. The obtained results based on elementary proofs extend previous results on the subject. We will further the techniques of spectral theory of sequences to study the stability of discrete systems

Date: September 30, 2009
Speaker: Olivier Rousseau
Title: Geometrical Modeling of the Heart
Abstract: Cardiovascular diseases have been the highest cause of death in North America and in Europe for decades. For this reason, a lot of research is made to understand the heart's physiology.
One way to better understand the heart is via theoretical modeling of physiological mechanisms and Finite Element Methods are often used to solve the Partial Differential Equations (PDEs) involved. Finite Element Methods require a mesh of the domain (the heart muscle).
Most of the time, simulations are made on idealized geometries and the community lacks realistic 3D models of the heart's geometry. In this talk, it is seen how a precise and realistic 3D geometrical model of the heart can be built from CT images. Its construction required the use, and adaptation of several PDE methods for image processing.
This talk will give an overview of the building process as well as theoretical and numerical insights on PDE methods in image processing.


Date: September 23, 2009
Speaker: Robert Smith?
Title: Can the viral reservoir of latently infected CD4+ T cells be eradicated with antiretroviral HIV drugs?
Abstract: The majority of cells infected with the human immunodeficiency virus are activated CD4+ T cells, which can be treated with antiretoviral drugs. However, an obstacle to eradication is the presence of viral reservoirs, such as latently infected CD4+ T cells. Such cells may be less susceptible to antiretroviral drugs and may persist at low levels during treatment. We introduce a model of impulsive differential equations that describe T cell and drug interactions. We make the extreme assumption that latently infected cells are unaffected by drugs, in order to answer the research question: Can the viral reservoir of latently infected cells be eradicated using current antiretroviral therapy? We analyse the model in both the presence and absence of drugs, showing that, if the frequency of drug taking is sufficiently high, then the number of uninfected CD4+ T cells approaches the number of T cells in the uninfected immune system. In particular, this implies that the latent reservoir will be eliminated. It follows that, with sufficient application of drugs, latently infected cells cannot sustain a viral reservoir on their own.

Date: September 9, 2009  
Speaker: Christina Cobbold  
Title: A quantitative genetics approach to model the evolution of insect development.
Abstract: Insects such as the Mountain Pine Beetle develop through a sequence of life stages at rates directly dependent on temperature. Together with seasonal temperature swings this can serve to synchronise developmental timing. Climate change threatens to destroy this synchrony. We couple an existing model for the insect life-cycle with quantitative genetics theory to predict how developmental traits evolve. Using the method of steepest descents and numerical simulations we demonstrate that species are likely to be slow to converge on the evolutionary stable strategy and this strategy is at the threshold for maintaining the synchrony of developmental milestones in a fixed environment. Thus the species is optimally positioned to respond to short term survival challenges via developmental plasticity.