Math Biology Seminar - Spring 2009

Time: Fridays 3:30 PM
Room: PSA 102
(unless otherwise specified)

Date

Speaker

Topic/Abstract

 Organizational meeting

Derek Moulton
University of Arizona

Title: Mathematical Modeling with Soap-Films

Abstract: Soap bubbles have been described as theplaythings of children and mathematicians alike. Studied byscientists such as Newton, da Vinci, and Plateau, “play” with soap bubbles andfilms has led to a surprising array of discoveries in both mathematics andphysics. In this talk, I present several soap bubble and soap film systems,the physical systems which motivated their study, and the mathematicalmodels we have used to explore them.

Pavlo Cherepanov
University of New Mexico

Title: Shock Layer Formation in Numerical Solutions of1-D Bhatnagar-Gross-Krook Model and Its Comparison toNumerical Solutions of 1-D Navier-Stokes Equations.

Abstract: Continuum Mechanics and Kinetic Theory aretwo mathematical theories with fundamentally different approaches to thesame physical phenomenon. Continuum Mechanics together withThermodynamics treat a substance (a gas or a fluid) as a continuous medium anddescribes the evolution of its macro characteristics via application of theConservation Laws to small packets of the substance. Kinetic Theory attempts todescribe the evolution of the macro parameters by treating a substance as afamily of colliding objects. The number of objects must be large enough so astatistical approach can be taken. In this work we introduce a numerical scheme to solve1-D Bhatnagar-Gross-Krook (BGK) model equations and examine the formation of astationary viscous shock. Obtained results are compared to a stationary numericalsolution of 1-D Navier-Stokes equation with a similar set of shockforming conditions. This is joint work with Jens Lorenz.

Omayra Ortega

Title: Evaluation of Rotavirus Models withCo-Infection and Vaccination

Abstract:Rotavirus diarrhea causes a disproportionate amountof the world's childhood mortality. Approximately 611,000children die each year due to complications of rotavirus infections.In this study we formulate a model of the spread of rotavirus diarrheabased on a continuous time ordinary differential equations model of two viralstrains of influenza. We expand this model to include the case ofco-infection. We further expand the original model to explore the effects ofvaccination. Computer simulations show that the spread of the disease is highly sensitiveto the levels of cross-immunity between strains, and the level ofvaccination in the population. Threshold vaccination and cross-immunity rates foreradication of the disease are found analytically.

Qing  Nie 

University of California, Irvine

Title: Systems Biology of Cell Signaling 

Abstract: The proper growth, development, and survival of an organism require extensive and accurate communication among the cells of the organism. Hence, cells sense and react to a wide variety of stimuli, which convey information such as nutrients, harmful insults, and the state of neighboring cells.
Using a systems biology approach that integrates modeling and experimentation, <>we study two cell signaling  systems: 1) robust sensing and signal transduction during mating of yeast cells, and 2) proliferative control of cell lineages in mammalian olfactory epithelium.

Christopher Kribs

University of Texas at Arlington (UTA)

Title: Sharpness of saturation in harvesting and predation

Abstract: Harvesting and predation occur via contact processes in which the rate at which the managed (prey) population can be found depends upon the population size, usually saturating at high densities. Many models incorporate saturation in this process without considering the effects of the particular function used to describe it. We show that the sharpness with which this saturation occurs has an important effect upon the resulting population dynamics, with bistability (sometimes involving a stable equilibrium and a stable limit cycle) occurring for saturation that is any sharper than the commonly used Michaelis-Menten (Holling type II) functional response. This sharpness threshold occurs across a wide range of model types, from simple harvesting to both density-dependent and ratio-dependent predation.

Haiyan Wang 

Arizona State University 

Title: Traveling waves for nonmonotone discrete-time integrodifference models

Abstract: Discrete-time integrodifference models arise naturally in population biology as models for organisms with discrete nonoverlapping generations and well-defined growth. Spreading speeds of integrodifference describe the speed at which the geographic range of the population expands. It has been shown that the spreading speed can be characterized as the slowest speed of non-constant traveling wave solutions of integrodifference models for monotone growth functions. Nonmonotone growth may arise some phenomena such as overcompensation. We will revisit some related recent results on traveling waves for nonmonotone models and present a result which can cover a large class of the models with overcompensation.

Kalle Parvinen

The University of Turku in Finland

Title: Evolution of altruism in a metapopulation 

Abstract: Consider a population, where individuals can contribute to a common resourcepool with a cost of increased personal mortality rate. All individuals in thepopulation get an equal share of the common resource pool, which increasestheir fecundity. Although such altruistic contribution is beneficial on thepopulation level, altruism will not evolve in large well-mixed population. Thereason for this is that a rare mutant does not gain anything by contributing,but gets an increased mortality rate, and thus such a population is vulnerableto the invasion of free-riders. However, in a metapopulation model with smalllocal population sizes [1-3], a mutant is not rare locally, although it is rareglobally. Therefore altruistic acts benefit also the kin of the mutant, andaltruism can evolve. In this presentation I will illustrate how variousparameters, such as the catastrophe rate and dispersal affect the adaptivedynamics of altruism.  

Paul Hasler

Georgia Tech 

SPRING BREAK 

Zhilan Feng 

 Purdue University 

Title: Transmission dynamics of an influenza model with vaccination and antiviral treatment

Abstract: Vaccination and antiviral treatment are two important prevention and control measures for the spread of influenza. However, the benefit of antiviral use can be compromised if drug-resistant strains arise. In this talk, a mathematical model will be prsented which is used to explore the impact of vaccination and antiviral treatment on the transmission dynamics of influenza. The model includes both drug-sensitive and -resistant strains. Analytical results of the model show that the two reproduction numbers, R_{SC} andl R_{RC}, associated with the sensitive and resistant strains respectively, provide threshold conditions that determine the competitive outcomes of the two strains. These threshold conditions can be used to gain important insights into the effect of vaccination and treatment on the prevention and control of influenza. The findings imply that in some cases, higher levels of treatment may lead to an increase of epidemic size, and the extend to which this occurs depends on other factors such as the vaccination level and resistance development.

Henry Huang

Department of Biomathematics,
School of Medicine
University of California, Los Angeles

Title: Mathematical Analysis of Tracer Kinetics: An Evolutionary History 

Abstract: Mathematical analysis has been an integral part of tracer methods since theearly use of tracers for biological investigations. As the type of tracers and the device for their measurements advance, themathematical analysis method used has evolved. Over the years, the type of tracers has gone from dyes to stable isotopes toradioisotopes, which also went from beta, gamma to positron emitters. The measuring instruments likewise have gone fromspectrophotometer to mass spectrograph, scintillation detectors (beta and gamma), gamma camera, andemission tomographs. The biological information obtained has expanded from blood volume, blood perfusion, totransport and biochemical reaction rates, and its use has gone from basic research to clinical applications. A few majormathematical analysis methods in this evolutionary process will be reviewed to illustrate the changes. Some on-goingdevelopments will also be discussed to show what lie ahead.  

Emily Stone

University of Montana 

Title: Detecting Spill-over: A dynamical systems modeling approach toglutamatergic synaptic signaling  

Abstract: The connectivity of neurons in the hippocampus depends in part on whether neurotransmitter from one release site can leak out and activate receptors in another synapse or extrasynaptic patch. The existence of such ``spill-over" is under debate in the neuroscience community, since direct measurements of neurotranmitter in such detail cannot, as of yet, be made. Experimental evidence of spillover is thus indirect, and should be sifted through as many different filters as possible. In this talk I will present the contributions of dynamical systems modeling to this effort.  

Peter Hinow

University of Minnesota

Title: Analysis of a model for transfer phenomena in biological populations

Abstract: We study the problem of transfer in a population structured by a continuum variable corresponding to the quantity being transferred. The transfer of the quantity occurs between individuals according to specified rules. The model is of Boltzmann type with kernel corresponding to the transfer process. We prove that the transfer process preserves total mass of the transferred quantity and the solutions of the simple model converge weakly to Radon measures. We generalize the model by introducing proliferation of individuals and production and diffusion of the transferable quantity. It is shown that the generalized model admits a globally asymptotically stable steady state, provided that transfer is sufficiently small. We discuss an application of our model to cancer cell populations, in which individual cells exchange the surface protein P-glycoprotein, an important factor in acquired multidrug resistance against cancer chemotherapy.

This is joint work with Pierre Magal (University of Le Havre, France) and Glenn Webb (Vanderbilt University).

Saber Elaydi

Trinity University 

Title: Towards a theory of periodic discrete dynamical systems/difference equations and population biology

Abstract: Recent questions in theoretical ecology on attenuance and the global stability of periodically forced discrete population models led the author and his collaborators to develop the theory of periodic difference equations. In this talk we will introduce the notion of skew-product dynamical systems in its most simplified form and indicate how it would give insights into topics of current interest. In particular, we will indicate how the recent conjectures by Henson and Cushing on the Beverton-Holt model can be settled.

Reene Moore

University of Pennsylvania School of Medicine

Title: Estimating Random Effects from Longitudinal Data with and without Limits of Detection 

Abstract: Estimating random effects corresponding to subject-specific characteristics is often of interest in many epidemiological and biomedical studies. The complexity of estimating these true mean levels varies depending on the outcome of interest. In this talk, I will discuss the estimation of two random effects from longitudinal data. The first outcome is mean HIV RNA levels of subjects from a cohort study. A challenge in estimating true mean HIV RNA levels is that some values fall below a limit of detection associated with the measuring instrument (assay). A widely accepted predictor of subject-specific random effects is the empirical Bayes estimate, which has the well-known potential disadvantage of overshrinking toward the population mean. Alternative “constrained Bayes” predictors maintain favorable properties while reducing this shrinkage and in our work we present this methodology adapted to handle non-detectable values. The second outcome we will explore in this talk is the estimation of adiposity change in children. A challenge in studying childhood obesity is the lack of consensus on the most appropriate primary outcome. Change in weight is the standard primary outcome for studies in adults, but in children changes in height also must be taken into account. In this talk, I will discuss different approaches taken in childhood obesity studies and the extent to which existing data are most supportive of a particular approach. 

Tanya Kostova (NSF) 

Title: On some problems related to two dynamical models in mathematical biology  

Abstract: The goal of my talk is to present two comparatively simple mathematical models and some problems related to them. The first model is a discrete – time epidemic network model. The goal of the modeling is to establish the dependence of the epidemic reproductive number on the epidemic rates and on the architecture of the underlying network. Further research is needed to connect the value of the Perron root of the nonnegative matrix representing the network and the network connectivity. The second model is the Fitz-Hugh-Nagumo model. I will describe a series of bifurcation events observed as a part of the rich dynamics of this model and will point to a numerically observed phenomenon that does not seem to have a known theoretical explanation. If time permits, I will discuss other models and problems of possible common interest.  

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