The Second International Conference on Mathematical Modeling and Analysis of Populations in Biological Systems

October 9-11, 2009

Huntsville, Alabama

Titles and Abstracts

Plenary Speakers

Louis Gross

Integrating into the Future

Dr. Louis Gross

Departments of Ecology and Evolutionary Biology and Mathematics, University of Tennessee

NIMBioS and the Math/Biology Interface One indicator of the potential for mathematical approaches to enhance research across the biological sciences is the increased funding in this area by agencies such as the NSF. The advent of the National Institute for Mathematical and Biological Synthesis, as a second major NSF-funded center at this interface (MBI being the other), provides evidence that the field is not only deserving of enhanced support, but additionally that there are advantages to multiple approaches to foster the growth of these interdisciplinary interactions. NIMBioS provides multiple routes for increased research and educational connections between these fields and to foster connections as well to other areas such as computation and social science. I will describe the opportunities that NIMBioS provides, give examples of the research and educational initiatives already underway, and provide some personal thoughts about future directions in both the development of general theory and the application to important practical issues.

Hans Heesterbeek

The Construction and Use of Next Generation Matrices for Compartmental Epidemic Systems

Dr. Hans Heesterbeek

Faculty of Veterinary Medicine, University of Utrecht

The basic reproduction number $R_{0}$ is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of $R_{0}$ where finitely many different categories of individuals are recognised. I clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. I present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be NGM with large domain and the NGM with small domain. The three matrices together reflect the range of possibilities encountered in the literature for the characterisation of $R_{0}$. I show how they are connected and how their construction follows from the basic model ingredie! nts, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number $R_{0}$. Although I present formal recipes based on linear algebra, the construction of the NGM by way of direct epidemiological reasoning is strongly encouraged, using the clear interpretation of the elements of the NGM and of the model ingredients. I present a selection of examples as a practical guide to the methods. Finally, I will show several applications of next-generation matrices for epidemiological systems, notably the possible insights that can be gained from sensitivity analysis of $R_{0}$ using the NGM. The largest part of this lecture is based on recent joint work with Odo Diekmann (Utrecht) and Mick Roberts (Auckland): The construction of next-genenation matrices for compartmental epidemic systems (Diekmann, Heesterbeek & Roberts, submitted). The latter part is based on the paper: Elasticity analysis in epidemiology: an application to tick-borne infections (Matser, Hartemink, Heesterbeek, Galvani & Davis; Ecology Letters, 2009, in press)

Alun Lloyd

Modeling Genetic Strategies for Controlling Mosquito Borne Diseases

Dr. Alun Lloyd

Department of Mathematics, North Carolina State University

Mosquito borne infections, most notably malaria and dengue, kill over a million people every year. Traditional control measures (such as insecticides) against these infections in developing countries have had mixed success. A novel avenue of attack involves the production and release of mosquitoes that have been manipulated or genetically engineered to be less able, or even unable, to transmit infection. The manipulated mosquitoes will, however, be less fit than native (wildtype) mosquitoes, and so would not be expected to spread in the wild. Selfish genesones that are able to "bend" the laws of Mendelian inheritance, getting transmitted to a higher fraction of offspring than would be expectedhave been suggested as a way of overcoming this problem, driving the desired trait into wild populations. Mathematical modeling is playing an important role in several largescale projects that are currently under way to assess the feasibility of these genetic techniques. In this talk I shall discuss the biology of some of the approches and the accompanying modeling work, illustrating how a number of different models are being used as the projects move along the path from labbased studies to possible field deployment.

Horst Thieme

Population Persistence without a Compact Attractor

Dr. Horst R. Thieme

School of Mathematical and Statistical Sciences, Arizona State University

(Joint work with Hal L. Smith)

The theory of persistence is designed to provide an answer to such questions as which species, in a mathematical model of interacting species, will survive over the long term. In a mathematical model of an epidemic, will the disease drive a host population to extinction or will the host persist? Can a disease remain endemic in a population? Persistence theory can give a mathematically rigorous answer to these questions: it establishes an positive long-term lower bound for the component of a dynamical system such as population size or disease prevalence; if persistence is uniform, this lower bound does not depend on the initial state of the system.

Persistence theory conveniently uses the language of dynamical systems, notably semiflows on metric spaces. A powerful but also restricting assumption is the existence of a compact attractor of points. This assumption excludes, among other things, the consideration of growing populations. This talk explores how much it can be relaxed. The Lotka-Volterra predator-prey system shows that some features of a compact attractor must be retained for uniform persistence. Applications are presented to the spread of infectious diseases in growing populations and to dividing cells in a chemostat with age-dependent resource uptake and division rates.

Jianhong Wu

Migratory Birds and Avian Influenza Outbreak: Models and Satellite Surveillance Data for Spatiotemporal Dynamics of H5N1 Spread

Dr. Jianhong Wu

Department of Mathematics and Statistics, Centre for Disease Modeling, York University

Virulent outbreaks of Highly Pathogenic Avian Influenza since 2005 have raised the question about the roles of migratory and wild birds in this disease's transmission dynamics. Despite increased monitoring, the role of wild waterfowl as the primary source of the highly pathogenic H5N1 has not been clearly established, and the consequence of outbreaks of HPAI among species of wild birds for the local and non-local ecology where migratory species are established has not been quantified. Understanding the entangled dynamics of migration and the disease dynamics is key to planning of prevention and control strategies for humans, migratory birds and the poultry industry. This talk will introduce the various factors involved in the spatial spread of H5N1 in Asia and present the results of a few dynamical models of seasonal migration linking the local dynamics during migratory stopovers to the larger-scale migratory routes. The effect of repeated epizootic at specific migratory stopovers for Bar-headed geese (Anser indicus) will be discussed as an illustration of the ecological impact of H5N1 outbreaks. Issues relevant to the co-existence and interaction of low and high pathogenic strains will be addressed, and some challenging problems in the theory of monotone periodic processes and nonlinear dynamical systems described by delay differential equations with periodic coefficients will be presented.

(This talk is based on projects in collaboration with Lydia Bourouiba, Venkaka Duvvuri, Stephen Gourley, Rongsong Liu and Sasha Alexandra Teslya.)

Invited Talks

Competitive Exclusion in a Discrete-Time, Stage-Structured Population Model

Dr. Azmy Ackleh

Department of Mathematics, University of Louisiana at Lafayette

We develop and analyze a discrete-time stage-structured population model that describes the competition of two similar species. We show that if one of the species has invasion reproductive number greater than one and the other has invasion reproduction number less than one, then competitive exclusion occurs and the winner species is the one with the larger invasion reproductive number.

Stationary Periodic and Homoclinic Solutions for 1-D Nonlocal Reaction-diffusion Equations

Dr. Shangbing Ai

Department of Mathematical Sciences, University of Alabama in Huntsville

Spatially periodic patterns for 1-D nonlocal reaction-diffusion equations arise from various biological models. The problem reduces to study periodic and homoclinic solutions of differential equations with perturbations containing convolution terms. We consider the case that the system is time-reversible. Assuming the unperturbed system has a family of periodic orbits surrounded by a homoclinic orbit, we establish the persistence of these solutions for the perturbed system.

Estimation of Cell Proliferation Dynamics Using CFSE Data

Dr. H. T. Banks

Center for Research in Scientific Computation, Center for Quantitative Science in Biomedicine, North Carolina State University

Advances in fluorescent labeling of cells as measured by flow cytometry have stimulated recent illuminating quantitative studies of proliferating populations of cells. We discuss our recent efforts on a new class of mathematical models based on fluorescence intensity as a structure variable to describe the evolution in time of proliferating cells labeled by carboxyfluorescein succinimidyl ester (CFSE). Early models and several extensions/modifications are discussed. Suggestions for improvements are presented and analyzed with respect to statistical significance for better agreement between model solutions and experimental data. These investigations reveal that the new decay/label loss and time dependent effective proliferation and death rates which we introduce do indeed provide improved fits of the model to data as well as new understanding of the data itself. Statistical models for the observed variability/noise in the data are discussed with implications for uncertainty quantification. The resulting new cell dynamics models should prove useful in proliferation assay tracking and modeling, with applications in numerous areas of disease progression (such as cancer, HIV and other viruses, etc.) well as in microbiology.

Fragmentation and Aggregation of Bacterial Emboli

Dr. David Bortz

Department of Applied Mathematics, University of Colorado

Klebsiella pneumoniae is one of the most common causes of intravascular catheter infections, potentially leading to life-threatening bacteremia. These bloodstream infections dramatically increase the mortality of illnesses and often serve as an engine for sepsis. Our current model for the dynamics of the size-structured population of aggregates in a hydrodynamic system is based on the Smoluchowski coagulation equations.

In this talk, I will discuss the progress of several investigations into properties of our model equations. In particular, I will focus on: a) accurate characterization of the fractal properties for the aggregates, b) a differential geometry approach to fragmentation modeling, and (time permitting) c) self-similar solutions to the equations.

Effect of Cross-Immunity between High and Low Pathogenic Strains of Avian Influenza in Wild Birds at Seasonal Migration Stopovers

Dr. Lydia Bourouiba

Department of Mathematics and Statistics, York University

Many species of wild birds are identified to be highly susceptible to the highly pathogenic strain of H5N1 despite being known as natural reservoirs of low pathogenic avian influenza viruses. Understanding the disease dynamics of avian in uenza in wild birds at various stopovers of their seasonal migration is important for the evaluation of both the role of wild birds in the spread of H5N1, and the ecological impact of H5N1 outbreaks on these species.

Recent experimental studies identified a temporary cross-immunity between low pathogenic and the high pathogenic H5N1 strains in certain species of wild birds. The data focused on species of birds which are more susceptible to these strains. In this talk, I will discuss the impact of this cross-immunity observed at the individual bird level on the population as a whole. The effect of the seasonal prevalence of the low pathogenic strains on the change of the highly pathogenic strain disease dynamics will be discussed in the context of previous epidemics observed in bird populations.

This talk is based on projects in collaboration with Sasha Alexandra Teslya and Jianhong Wu.

A Mathematical Feasibility for the Use of Aptamers in Chemotherapy and Imaging

Dr. Khalid Boushaba

Department of Mathematics, Iowa State University

A challenge for drug design is to create molecules with optimal function that also partition efficiently into the appropriate in vivo compartment(s). This is particularly true in cancer treatments because cancer cells upregulate their expression of multidrug resistant transporters, which necessitates a higher concentration of extracellular drug to promote sufficiently high intracellular concentrations for cell killing. Pharmacokinetics can be improved by ancillary molecules, such as cyclodextrins, that increase the effective concentrations of hydrophobic drugs in the blood by providing a hydrophobic binding pocket. However, the extent to which the extracellular concentration of drug can be increased is limited. A second approach, different from the "push" mechanism just discussed, is a "pull" mechanism by which the effective intracellular concentrations of a drug is increased by a molecule with an affinity for the drug that is located inside the cell. Here we propose and give a proof in principle that intracellular RNA aptamers might perform this function. The mathematical model considers the following: Suppose I denotes a drug (inhibitor) which must be distributed spatially throughout a cell, but tends to remain outside the cell due the transport properties of the cell membrane. Suppose that E, a deleterious enzyme that binds to I, is expressed by the cell and remains in the cell. Here we evaluate the use of an intracellular aptamer with affinity for the inhibitor (I) to increase the efficiency of inhibitor transport across the cell membrane. We show that this outcome will occur if (1) the aptamer neither binds too tightly nor too weakly to the inhibitor than the enzyme and (2) the aptamer is much more diffusible in the cell ytoplasm than the enzyme. We illustrate these possibilities with numerical aimulations. The ability of the aptamer to increase the intracellular concentration of aptamer ligand (inhibitor in the above case) could also be put to use for imaging the cell. Thus, we propose and show by simulation that an intracellular aptamer can be enlisted for an integrated approach to increasing inhibitor effectiveness and imaging aptamer-expressing cells.

Backward Bifurcations in a Simple Disease Transmission Model

Dr. Fred Brauer

Department of Mathematics, University of British Columbia

We describe a simple disease transmission model with demographics, imperfect vaccination, and recovery with temporary immunity. We derive a necessary condition for the existence of a backward bifurcation which can not be satisfied if the immunity is permanent. For sufficiently rapid disease dynamics, this condition is also sufficient.

Sensitivity Analysis of Cancer Models with Proliferating and Quiescent Cells

Dr. John A. Burns

Interdisciplinary Center for Applied Mathematics, Virginia Tech

In this presentation we discuss a model of tumor growth that includes quiescence cells and the immune system response to a cycle-phase-specific drug. Tumor cells can be divided into proliferating or cycling cells and non-proliferating or quiescent cells. A cell is considered "cancerous" when it has lost its ability to regulate cell growth and division leading to a rapid uncontrolled growth of malignant cells. The model considers three populations of cancer cells and the immune system. The three populations considered in this talk are the quiescence cells, the tumor cells during interphase and the tumor cells during mitosis. Delay differential equations are used to model the system to take into account the phases of the cell cycle. We then focus on a particular method (the Sensitivity Equation Method) for computing the model sensitivities and use these sensitivities to help predict the long term behavior of the model with and without drug treatment.

Mathematical Modeling of Malaria Epidemiology and Control

Dr. Nakul Chitnis

Malaria Modeling Team, Swiss Tropical Institute, Liverpool School of Tropical Medicine

Malaria interventions are usually prioritized using efficacy estimates from intervention trials, without considering the context of existing intervention packages or long term dynamics. We use numerical simulation of mathematical models of malaria in humans and mosquitoes to provide robust quantitative predictions of effectiveness of different strategies in reducing transmission, morbidity and mortality. We link individual-based stochastic simulation models for malaria in humans with a deterministic model for mosquito infection and survival, incorporating variations in host exposure to infectious bites, naturally acquired immunity to infection and disease, effects of co-infection, and variations in human infectiousness. We can reasonably well reproduce malariological patterns in endemic areas, including non-monotonic relationships between parasite prevalence and disease incidence with host age and force of infection; and provide quantitative relationships between malaria morbidity and mortality, and increasing coverage of vector control interventions, intermittent preventive treatment in infants, and different vaccines.

The Fundamental Bifurcation Theorem and Darwinian Matrix Models

Dr. J. M. Cushing

Department of Mathematics, University of Arizona

Stability of a Delay Equation Arising from a Juvenile-Adult Model

Dr. Keng Deng

Department of Mathematics, University of Louisiana at Lafayette

We consider a delay equation that has been formulated from a juvenile-adult population model. We give conditions on the vital rates to ensure local stability of the positive equilibrium. We also show that under certain conditions the trivial equilibrium is asymptotically stable. We then make numerical simulations to describe the rich dynamical behavior of the model.

VIVO and VITRO HSV_1 Infections, Latency-Reactivation by Systems Theory Approach

Dr. Youssef Dib

Department of Mathematics and Physics, University of Louisiana at Monroe

A nonlinear methematical model for HSV1 viral infections will be produced from its background. Differential cell are the host of this virus. Once infected, this differential cell would survive as long as it host this virus. It is assumed that both HSV1\'s DNA and Nuclear DNA in the differential cell depend on Thyroid Hormone liganded with its receptore. Numerical simulation proving the biological relevence will be shown. In addition, future research direction for this model will be discussed.

Optimal Control for a Tick Disease Model Using Hybrid ODE Systems

Dr. Wandi Ding

Department of Mathematical Sciences, Middle Tennessee State University

We are considering an optimal control problem for a type of hybrid system involving ordinary differential equations and a discrete time feature. One state variable has dynamics in only one season of the year and has a jump condition to obtain the initial condition for that corresponding season in the next year. The other state variable has continuous dynamics. Given a general objective functional, existence, necessary conditions and uniqueness for an optimal control are established. We apply our approach to a tick-transmitted disease model with age structure in which the tick dynamics changes seasonally while hosts have continuous dynamics. The goal is to maximize disease-free ticks, minimize infected ticks through an optimal control strategy of treatment with acaricide. Numerical examples are given to illustrate the results.

Tracking Influenza Using Particle Learning Algorithms

Dr. Vanja Dukic

Department of Health Studies, University of Chicago

In this talk we introduce a novel approach, based on the particle learning (PL) methodology, for classic epidemics models from the family of the susceptible-exposed-infected-recovered (SEIR) models. The proposed approach is particularly well-suited to on-line learning and surveillance of infectious diseases. As compared to the widely used MCMC (O'Neil and Roberts 1999, Elderd et al. 2006, Leman et al. 2009) and perfect sampling (Fearnhead and Meliglokou 2004) based methods, the PL method, which is based on the clever use of sufficient statistics as added states, is more robust, easier to implement, and readily generalizable to problems with more complex dynamics.

Mathematical Modeling of the Effects of HER2 Over-Expression in Breast Cancer

Dr. Amina Eladdadi

Department of Mathematics, The College of Saint Rose

(Joint work with David Isaacson, Department of Mathematical Sciences, Rensselaer Polytechnic Institute)

Members of the type I receptor tyrosine kinase (RTK) family, which consists of the epidermal growth factor receptor (EGFR), HER2 (ErbB2), HER3 (ErbB3) and HER4 (ErbB4) play a crucial role in growth and differentiation of both normal and malignant mammary epithelial cells. The carcinogenic effects of HER2 protein overex-pression on cell growth and cell proliferation have been observed in a variety of experimental systems. These observations suggest that HER2 overexpression provides tumor cells with a growth advantage leading to a more aggressive phenotype. Although these effects have been attributed to high levels of HER2-expression, there have been no quantitative linkages between HER2 expression levels and the proliferation rate of HER2-overexpressing cells. To investigate the effects of HER2 receptor overexpression on cell proliferation, we have developed a mathematical model that describes the proliferative behavior of HER2-overexpressing cells as a function of the HER2 expression level. The proliferation model formulates the cell proliferation rate as a function of the cell surface HER2 and EGFR receptor numbers and ligand concentration. The model enables us to simulate the proliferative behavior of the HER2-overexpressing cells with various HER2 and EGFR expression levels at various ligand concentrations. Numerical simulations of the model give good agreement with the experimental data in which an increase in HER2 receptors leads to increased cell proliferation.

Modeling the Evolutionary Implications of Influenza Medication Strategies

Dr. Zhilan Feng

Department of Mathematics, Purdue University

(Joint work with J. Glasser, R. Liu, Z. Qiu, D. Xu, and Y. Yang)

Medication and treatment are important measures for prevention and control of influenza. However, the benefit of antiviral use can be compromised if drug-resistant strains arise. Consequently, not only the epidemic size may increase with a higher level of treatment but also the viruses may become more resistant to the antiviral drugs. We use a mathematical model to explore the impact of antiviral treatment on the transmission dynamics of influenza. The model includes both drug-sensitive and -resistant strains. Analytical and numerical results of the model show that the conventional quantity for the control reproduction number is not appropriate to use for gaining insights into the disease dynamics. We derive a new reproduction number by considering multiple generations of infection, and demonstrate that this new reproduction number provides a more reasonable measure for evaluating control programs as well as evolutionary implications of influenza medication strategies.

Optimal Control of Tick-Borne Disease

Dr. Holly Gaff

Virginia Modeling, Analysis and Simulation Center, Old Dominion University

Human monocytic ehrlichiosis (Ehrlichia chaffeensis ), or HME, is a ticktransmitted, ricksettisal disease with growing impact in the United States. Risk of a tick-borne disease such as HME to humans can be estimated using the prevalence of that disease in the tick population. A deterministic model for HME is explored to investigate the underlying dynamics of prevalence in tick populations, particularly when spatial considerations are allowed. Optimal control is applied to this model to identify how limited resources can best be used to reduce the risk of tick-borne diseases to humans.

Mathematical Recipe for HIV Elimination in Resource-Poor Settings

Dr. Abba Gumel

Department of Mathematics, University of Manitoba

I will present a model for the transmission dynamics of HIV/AIDS in a population, and show how such a model (and its analyses) could provide a cost-effective roadmap for the effective control and/or elimination of HIV in a resource-poor setting, such as Nigeria.

The Largest Basic Reproduction Number in Multi-Type Epidemic Models

Dr. Karl Hadeler

Department of Mathematical and Statistical Sciences, Arizona State University

The basic reproduction number for a multi-group epidemic model depends on the distribution of types. Determining the worst case amounts to maximizing the spectral radius ρ(XA) where A is a given non-negative matrix and X is a variable non-negative diagonal matrix with trace equal to one. Lower bounds for the maximum can be obtained and improved without computing eigenvalues. Upper bounds can be computed using the max eigenvalue of the matrix A. (Joint work with Ludwig Elsner)

Socially-induced Ovulation Synchrony in a Seabird Colony: a Discrete-Time Model

Dr. James L. Hayward

Department of Biology, Andrews University

(Joint work with Shandelle M. Henson, and J. M. Cushing)

Spontaneous oscillator synchrony has been documented in a wide variety of electrical, mechanical, chemical, and biological systems, including the menstrual cycles of women and estrous cycles of Norway rats. In temperate regions, many colonial birds breed seasonally in a time window set by photoperiod; some studies have suggested that heightened social stimulation in denser colonies can lead to a tightened annual reproductive pulse. It has been unknown, however, whether the analogue of menstrual synchrony occurs in birds, that is, whether avian ovulation cycles can synchronize on a daily timescale within the annual breeding pulse. We present data on every-other-day egg-laying synchrony in a breeding colony of glaucous-winged gulls (Larus glaucescens) and show that the level of synchrony declined with decreasing colony density. We also discuss a discrete-time mathematical model based on the hypothesis that preovulatory luteinizing hormone surges synchronize through social stimulation.

The selective Advantage of Ovulation Synchrony in Colonial Seabirds: a Darwinian Dynamics Model

Dr. Shandelle M. Henson

Department of Mathematics, Andrews University

(Joint work with J. M. Cushing and James L. Hayward)

The existence of socially-induced ovulation synchrony in colonial seabirds begs the question of selective advantage. We pose a discrete-time population model for colonial seabirds, incorporating a social stimulation parameter that can induce ovulation synchrony during the breeding season. Using the birth rate as a bifurcation parameter, we prove the existence of a transcritical bifurcation of positive periodic solutions, and show that the bifurcation is supercritical in the absence of social stimulation. In the presence of social stimulation, the bifurcation can become subcritical, and the branch of positive solutions bends back to the right, lying above the branch for which there is no social stimulation. If the population model is coupled to a dynamic model for an evolving trait related to social stimulation, the resulting Darwinian dynamics model predicts that the system will evolve to a state for which ovulation synchrony exists.

A New Look at Stochastic Resonance Enhancement of Mammalian Auditory Information Processing

Dr. Dawei Hong

Center for computational and integrative biology, Department of computer science, Rutgers University

Dynamics of an SIS Type of Reaction-Diffusion Epidemic Model

Dr. Wenzhang Huang

Department of Mathematical Sciences, University of Alabama in Huntsville

Recently an SIS epidemic reaction-diffusion model with Neumann (or no-flux) boundary condition have been proposed and studied by several authors to understand the dynamics of disease transmission in a spatially heterogeneous environment in which the individuals are subject to a random movement. Many important and interesting properties have been obtained: such as the role of diffusion coefcients in defning the reproductive number; the globally stability of disease-free equilibrium; the existence of positive endemic steady; and the asymptotical profiles of the endemic steady states as one of diffusion coefcients is sufficiently small (or large). In this research we will study a modified SIS diffusion model with the Dirichlet boundary condition. Results on the dynamics of disease transmission and problems on the model will be presented.

Using Models to Help Control the H1N1 Pandemic

Dr. Mac Hyman

Department of Mathematics, Tulane University

We must use all of the tools available to advance epidemic models, from qualitative insight to quantitative predictions, to devise effective strategies to minimize the impact and spread of infectious diseases such as the current H1N1 flu pandemic. I will review the lessons learned from mathematical modeling previous epidemics, with the goal of identifying ways that our mathematical models can be used to help improve the effectiveness of public health interventions measures. In particular, I will describe how mathematical models can estimate the benefits and the costs of projected interventions and project the requirements that an epidemic will place on the health care system.

Dynamics of an Age-Structured Population with Allee Effects and Harvesting

Dr. Sophia Jang

Department of Mathematics and Statistics, Texas Tech University

In this talk we introduce a discrete-time, age-structured single population model with Allee effects and harvesting. It is assumed that survival probabilities from one age class to the next are constants and fertility rate is a function of weighted total population size. Global extinction is certain if the maximal growth rate of the population is less than one. The model can have multiple attractors and the asymptotic dynamics of the population depends on its initial distribution if the maximal growth rate is larger than one. An Allee threshold depending on the components of the unstable interior equilibrium is derived when only the last age class can reproduce. The population becomes extinct if its initial population distribution is below the threshold. Harvesting on any particular age class can decrease the magnitude of the possible stable interior equilibrium and increase the magnitude of the unstable interior equilibrium simultaneously.

On the Existence of Error Threshold of the Quasispecies Model

Dr. Tanya Kostova

National Science Foundation

The quasispecies theory was introduced approximately 30 years ago by Eigen and Schuster. It became very popular within the virology community when experimental evidence showed that viruses have so high mutation rates that the viral populations consist of numerous diverse genotypes. It has been widely accepted that the quasispecies model predicts that the fittest genotype (with the highest replication rate) loses dominance when the mutation rate becomes sufficiently high. These conclusions have been largely based on computer simulations and have led to the definition of the so called error threshold. I show that it is easy to construct counter examples where the fittest genotype remains dominant independently of the value of the mutation rate and therefore the error threshold does not exist.

Multistability and Oscillations in Feedback Loops

Dr. Maria Leite

Department of Mathematics, University of Oklahoma

(Joint work with Yunjiao Wang, University of Manchester)

Feedback loops are found to be important network structures in biological systems. Recently, the dynamical role of feedback loops have received extensive attention. In this talk we discuss some of the interesting dynamical features of those loops such as multistability and oscillations.

Optimal Control of Treatments in a Cholera Model

Dr. Suzanne Lenhart

Department of Mathematics, University of Tennessee

While cholera has been a recognized disease for two centuries, there is no strategy for its effective control. We formulate a mathematical model to include essential components such as a hyperinfectious, short-lived bacterial state, a separate class for mild human infections, and waning disease immunity. A new result quantifies contributions to the basic reproductive number from multiple infectious classes. Using optimal control theory, parameter sensitivity analysis, and numerical simulations, a cost-effective balance of multiple intervention methods is compared for two endemic populations. Results provide a framework for designing cost-effective strategies for diseases with multiple intervention methods.

Spreading Speeds and Traveling Wave Solutions in Partially Monotone Systems

Dr. Bingtuan Li

Department of Mathematics, University of Louisville

Investigating the spreading speeds and traveling waves for spatial multiple species models has been fascinating and challenging. Most of the existing results on spread of species assume that the system is monotone throughout the region of biological interest. In this talk, we will present mathematical results on spatial spread of partially monotone models in the form of reaction-diffusion equations and in the form of integro-difference equations. By a partially monotone model we mean that the model is monotone near an unstable equilibrium from which the spatial transition moves away. In such a model species interact with each other to promote growth and migration in a cooperative manner in one region while they may behave differently in other regions. A partially monotone model may generate complicated dynamics including chaos. We will show results on the so-called "linear determinacy" that equates spreading speed in the full nonlinear model with spread rate in the system linearized about the leading edge of the invasion. We will then show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We will finally discuss the applications of the general mathematical results to some specific ecological models in which the predator-prey interaction can be incorporated. (Joint work with Hans F. Weinberger, University of Minnesota)

Impact of Mosquito Transgenes on Malaria Transmission

Dr. Jia Li

Department of Mathematical Sciences, University of Alabama in Huntsville

We formulate continuous-time models for interactive wild and transgenic mosquitoes. With fundamental analysis of their dynamics, we introduce the transgenic mosquitoes into a simple compartmental malaria transmission mode l. We study the dynamics of the simple malaria model and the model with the transgenic mosquitoes, and investiga te the impact of transgenic mosquitoes on the malaria transmission.

Delay Dependent Conditions for Global Stability of an Intravenous Glucose Tolerance Test Model

Dr. Jiaxu Li

Department of Mathematics, University of Louisville

Diabetes mellitus has become an epidemic disease in the sense of life style. Detecting the onset of diabetes is one of the fundamental steps in treatment of diabetes including determining the insulin sensitivity and glucose effectiveness. An effective method for this end is the intravenous glucose tolerance test (IVGTT). Several mathematical models have been proposed and some are widely used in clinics. The most recent model proposed by P. Palumbo, S. Panunzi and A. De Gaetano (2007) demonstrates reasonable profiles with their experimental data. To analytically ensure the global stability of the basal equilibrium, several attempts have been made. The existing results are either delay independent conditions or the convergence is for a type of specific solutions. In this talk, we study the global stability and obtain delay dependent conditions to ensure the global and asymptotic stable equilibrium. An easy-to-check condition that is an estimate of the upper bound of time delay is given.

An Age-structured Two-Strain Epidemic Model with Super-Infection

Dr. Xue-Zhi Li

Department of Mathematics, Xinyang Normal University

This article focuses on the study of an age-structured two-strain model with super-infection. The explicit expression of basic reproduction numbers and the invasion reproduction numbers corresponding to strain one and strain two are obtained. It is showed that the infection-free steady state is globally stable if the basic reproductive number R_0 is below one. Existence of strain one and strain two exclusive equilibria is established. Conditions for local stability or instability of the exclusive equilibria of the strain one and strain two are established. Existence of coexistence equilibrium is also obtained under the condition that both invasion reproduction numbers are larger than one. Keywords: age-structured; two-strain epidemic model; super-infection; basic reproduction number; invasion reproduction number, the exclusive equilibrium, the coexistence equilibrium, stability.

A Mathematical Approach to Study the Impact of Predators on the Vegetation Succession

Dr. Rongsong Liu

Department of Mathematics, University of Wyoming

In order to study the role of predator of on the vegetation succession, we use a system of ordinary differential equations to model the interaction among two plant species, herbivores, and predators. The toxin-determined functional response is applied to describe the interactions between plant species and herbivores and Holling Type II functional response is used to model the interactions between herbivores and predators. In order to study how the predators impact the succession of vegetation, we derive the invasion condition. Numerical simulations are conducted to reinforce of analytical results.

Tracking Prey or Tracking Prey's Resource?

Dr. Yuan Lou

Department of Mathematics, Ohio State University

We consider a continuous environment with an arbitrary distribution of resources, randomly diffusing prey that consumes the resources, and predators that consume the prey. Our model introduces a class of movement rules in which the direction of predators\' movement is determined (i) randomly, (ii) by prey density, and/or (iii) by the density of the prey's resource. We find that, for some resource distributions, predators that track the gradient of the prey's resource may have an advantage compared to predators that track the gradient of prey directly.

Permanence of the Water Hyacinth in Northeast Louisiana

Dr. Mariette Maroun

Department of Mathematics and Physics, University of Louisiana at Monroe

The invasive specie of Water Hyacinth was introduced into the United States of America in late 1800. This aquatic plant consists of three different stages. It take over upon introduction into fresh water bodies because of its sexual and as-sexual reproductive system. From early 1970's till now, scientists have been trying to control it in different manners from chemical to biological. A model will be provided to show that long term control of this plant depends only on its survival.

Avian Influenza: Modeling, Analysis, and Data Fitting

Dr. Maia Martcheva

Department of Mathematics, University of Florida

Low Pathogenic Avian Influenza (LPAI) virus, which circulates in wild bird populations in mostly benign form, is suspected to have mutated into a highly pathogenic (HPAI) strain after transmission to the domestic birds. HPAI has recently garnered worldwide attention because of the ``spillover" infection of this strain from domestic birds to humans - primarily those in poultry industry - causing significant human fatality and thus creating potentially favorable conditions for another flu pandemic. We use an ordinary differential equation model to describe this complex dynamics of the HPAI virus, which epidemiologically links a number of species in a multi-species community. We include the wild bird population as a periodic source feeding infection to the coupled domestic bird-human system. We also account for mutation between the low and high pathogenic strains. We fit our model to the actual number of human avian influenza cases obtained from WHO, and estimate the relevant reproduction numbers and invasion reproduction numbers. We conclude that low pathogenic avian influenza is maintained in the domestic bird population through "spill over" from wild birds, while high pathogenic avian influenza is endemic in the domestic bird population.

A NSFD Scheme for a Model of Respiratory Virus Transmission

Dr. Ronald Mickens

Department of Physics, Clark Atlanta University

We construct a nonstandard finite difference numerical integration scheme for an SIRS model of respiratory virus transmission. Our work extends that done by A. J. Arenas et al. (Computers and Mathematics with Applications, Vol. 56 (2008), 670-678) by using the system's exact conservation law to place constraints on the discretization. The scheme satisfies a positivity condition for all time step-sizes. We note that neither of the latter two conditions holds for the scheme derived by Arenas et al.

Stability of Choice in the Honey Bee Nest-Site Selection Process

Dr. Andrew Nevai

Department of Mathematics, University of Central Florida

A pair of compartment models for the honey bee nest-site selection process is introduced. The first model represents a swarm of bees deciding whether a site is viable, and the second characterizes its ability to select between two viable sites. The one-site assessment process has two equilibrium states: a disinterested equilibrium (DE) in which the bees show no interest in the site and an interested equilibrium (IE) in which bees show interest. In analogy with epidemic models, basic and absolute recruitment numbers (R0 and B0) are defined as measures of the swarm\'s sensitivity to dancing by a single bee. If R0 is less than one then the DE is locally stable, and if B0 is less than one then it is globally stable. If R0 is greater than one then the DE is unstable and the IE is stable under realistic conditions. In addition, there exists a critical site quality threshold Q* above which the site can attract some interest (at equilibrium) and below which it can! not. There also exists a a second critical site quality threshold Q** above which the site can attract a quorum (at equilibrium) and below which it cannot. The two-site discrimination process, which examines a swarm's ability to simultaneously consider two sites differing in both site quality and discovery time, has a stable DE if and only if both sites\' individual basic recruitment numbers are less than one. Numerical experiments are performed to study the influences of site quality on quorum time and the outcome of competition between a lower quality site discovered first and a higher quality site discovered second.

Stochastic Energy Budgets, Integrate-and-Fire Models, and Population Dynamics

Dr. Roger Nisbet

Department of Ecology, Evolution and Marine Biology, University of California, Santa Barbara

In many adult animals, energy-rich material is accumulated as "reserves" that are eventually converted to reproductive material and released in a pulse. Similarly, achieving reproductive maturity in juveniles typically involves sustained energy allocation, with the onset of reproduction occurring after some threshold level is achieved. These observations have motivated study of a stochastic bioenergetic model involving integration of a varying energy input and firing on achieving some threshold, similar to the well studied integrate-and-fire (I-F) neuron model. The model dynamics in a periodically variable environment give insight on the synchronization of reproduction in many organisms, and I shall show an example involving spawning corals. I-F dynamics are also central to stage-structured population models with time varying maturation delays. I shall discuss an application to zooplankton populations.

Growing, Slowing, Growing: A Branching Process Model of How Cells Survive Telomere Shortening

Dr. Peter Olofsson

Department of Mathematics, Trinity University

Telomeres are specialized structures found at the ends of chromosomes. During DNA replication, telomeres shorten and once a critical length is reached, the cell stops dividing and becomes senescent. A cell population that experiences telomere shortening exhibits initial exponential growth, and once senescent cells start dominating the population, growth slows down and the population size levels off resulting in a typical sigmoid-shaped growth curve. However, experimental data on the yeast Saccharomyces cerevisiae indicate that some populations regain exponential growth after slowing down. The explanation for this phenomenon is that some cells develop ways to maintain short telomeres and become "survivors." We suggest a bracnhing process model that takes into account random variation in individual cell cycle times, telomere shortening, finite lifespan of mother cells, and the possibility of survivorship. We identify and estimate crucial parameters such as cell cycle mean and variance, and the probability of an individual cell becoming a survivor, and compare our model to experimental data.

Evolutionary Changes in Competitive Outcomes

Dr. Rosalyn Rael

Department of Ecology and Evolutionary Biology, University of Michigan

Evolution is a natural process that can occur on time scales commensurate with ecological dynamics and result in changes in expected outcomes of interactions such as competition. Evolutionary game theory is a modeling technique that combines both ecological dynamics and evolution to form "Darwinian dynamics." With this method, natural selection is viewed as a game, where traits are strategies that affect payoff in the form of species\' fitness. I will give a brief introduction to Darwinian dynamics and describe applications of this modeling approach to two-species competition. Using evolutionary game theory, we show how evolution can lead to the coexistence of species or other outcomes not expected in the absence of evolution. I will discuss the conditions necessary to see such changes, and show that these results compare well with data from classic flour beetle experiments.

Formation of Spatial Patterns in Stage-Structured Populations with Density Dependent Dispersal

Dr. Suzanne Robertson

Mathematical Biosciences Institute, Ohio State University

Spatial segregation among life cycle stages has been observed in many stage-structured species, both in homogeneous and heterogeneous environments. We investigate density dependent dispersal of life cycle stages as a mechanism responsible for this separation by using stage-structured, integrodifference equation (IDE) models that incorporate density dependent dispersal kernels. After investigating mechanisms that can lead to spatial patterns in two dimensional Juvenile-Adult IDE models, we construct spatial models to describe the population dynamics of the flour beetle species T. castaneum, T. confusum and T. brevicornis and use them to assess density dependent dispersal mechanisms that are able to explain spatial patterns that have been observed in these species.

Asymmetric Division of Activated Latently Infected Cells May Explain the Divergent Decay Kinetics of the HIV Latent Reservoir

Dr. Libin Rong

Theoretical Biology and Biophysics, Los Alamos National Laboratory

Most HIV-infected patients when treated with combination therapy achieve viral loads that are below the current limit of detection of standard assays after a few months. Despite this, virus eradication from the host has not been achieved. Latent, replication-competent HIV-1 can generally be identified in resting memory CD4+ T cells in patients with "undetectable" viral loads. Turnover of these cells is extremely slow but virus can be released from the latent reservoir quickly upon cessation of therapy. In addition, a number of patients experience transient episodes of viremia, or HIV-1 blips, even with suppression of the viral load to below the limit of detection for many years. The mechanisms underlying the slow decay of the latent reservoir and the occurrence of intermittent viral blips have not been fully elucidated. In this study, we address these two issues by developing a mathematical model that explores a hypothesis about latently infect! ed cell activation. We propose that asymmetric division of latently infected cells upon sporadic antigen encounter may both replenish the latent reservoir and generate intermittent viral blips. Interestingly, we show that occasional replenishment of the latent reservoir induced by reactivation of latently infected cells may reconcile the differences between the divergent estimates of the half-life of the latent reservoir in the literature.

Modelling the Transmission Dynamics and Control of Hepatitis B Virus in China

Dr. Shigui Ruan

Department of Mathematics, University of Miami

Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus (HBV) and is a major global health problem. HBV is the most common serious viral infection and a leading cause of death in mainland China. Around 130 million people in China are carriers of HBV, almost a third of the people infected with HBV worldwide and about 10% of the general population in the country; among them 30 million are chronically infected. Every year, 300,000 people die from HBV-related diseases in China, accounting for 40 - 50% of HBV-related deaths worldwide. Despite an effective vaccination program for newborn babies since the 1990s, which has reduced chronic HBV infection in children, the incidence of hepatitis B is still increasing in China. We propose a mathematical model to understand the transmission dynamics and prevalence of HBV infection in China. Based on the data reported by the Ministry of Health of China, the model provides an approximate estimate of the basic reproduction number R0 =2.406. This indicates that hepatitis B is endemic in China and is approaching its equilibrium with the current immunization programme and control measures. Although China made a great progress in increasing coverage among infants with hepatitis B vaccine, it has a long and hard battle to fight in order to significantly reduce the incidence and eventually eradicate the virus. Keywords: Hepatitis B virus, mathematical modeling, transmission dynamics, basic reproduction number, disease endemic equilibrium.

*Research was partially supported by the State Scholarship Fund of China Scholarship Council (Z.L.), NSFC grant #10825104 and the China MOE Research Grant (W.Z.), and NSF grant DMS-0715772 (S.R.)

Global Stability in a Multi-Species Periodic Leslie-Gower Model

Dr. Robert Sacker

Department of Mathematics, University of Southern California

The $d$-species Leslie-Gower competition model is studied in which all the parameters are $p$-periodic. It is shown that whenever the coupling is small, there is a positive $p$-periodic state that is exponentially asymptotically stable and globally attracts all initial states having positive coordinates.

Global-Stability Problems for Coupled Systems Arising in Population Dynamics

Mr. Zhisheng Shuai

Department of Mathematical and Statistical Sciences, University of Alberta

(Joint work with Michael Li)

We study the global-stability problem of equilibria for coupled systems of differential equations arising in population dynamics. Using results from graph theory, we develop a systematic approach to construct global Lyapunov functions/functionals for coupled systems from individual Lyapunov functions/functionals for vertex systems. We apply our general approach to several coupled systems in ecology and epidemiology, for example, single species model with dispersal, predator-prey model with dispersal, and multi-group epidemic model with time delays.

Modeling the Evolution of Repetitive Sequence in DNA

Dr. Suzanne Sindi

Center for Computational Molecular Biology, Brown University

There are families of nearly identical sequences within the genomes of human, fly, worm and every non-microbial genome that has been determined. Such sequences were originally hypothesized to be "junk DNA", but biologists continue to find many functions these sequences perform. Several features of repetitive DNA follow power law distributions, a natural question is how such distributions have emerged over time from individual duplication events. I will describe mathematical models that demonstrate how power law and generalized Pareto Law distributions can emerge naturally from random duplication and deletion in a genome.

Lyapunov Exponents and Persistence

Dr. Hal Smith

Department of Mathematics and Statistics, Arizona State University

(Joint work with P. Salceanu)

We will show that Lyapunov exponents can be employed in establishing persistence of discrete and continuous-time finite dimensional dynamical systems.

Transmission of Avian Influenza between Two Patches

Dr. Baojun Song

Department of Mathematical Sciences, Montclair State University

Patch models are constructed and analyzed to study the role of a migratory bird population in the transmission of the highly pathogenic H5N1 strain of avian influenza.

Our discrete models consider a migratory bird population and two local bird populations. The local bird populations live in their own patches and the migratory birds migrate back and forth between patches seasonally. The models are tested by using the prevalence of avian influenza in Mallard Duck by season in United States and Canada from 1974 to 1986. Both our analytic results and simulations predict a pattern of seasonal oscillation of the prevalence of avian influenza in Mallard Duck A variety types of demographic growth modes are discussed. The models for most nonlinear reproductions or nonlinear survive functions undergo double period-double bifurcations.

Dynamics of Killer T Cells and Immunodominance in the Influenza Infection

Dr. Abdessamad Tridane

Department of Applied Sciences and Mathematics, Arizona State University at the Polytechnic Campus

Antigen-specific killer T cells ( CD8+ cells ) play an important role in virus clearance. The aim of this talk is to introduce and analyze mathematical models of the dynamics of killer T cells and the differential expansion of antigen-specific CD8+ cell, called immunodominance, in the influenza infection. Understanding qualitative impact of killer T cells is very important for the design of T-cell-based vaccines that promote early virus clearance. The systematical analysis of these model systems show that the behaviors of the models are similar for high killer T cells density generating reasonable dynamics. Our models try to shed some light on possible explanations of the some aspect immunodominance in influenza infection by studying the effect of the epitope of the antigen presented on the surface of the infected cells and the effect of Interferon-γ

Traveling Wave Solutions of Delayed Reaction-Diffusion Equations

Dr. Haiyan Wang

Division of Mathematical and Natural Sciences, Arizona State University

Traveling wave phenomena in reaction-diffusion equations arise from many biological problems. Combining upper and lower solutions, monotone iterations and fixed point theorems, we study the existence and asymptotic behavior of traveling wave solutions for nonmonotone reaction-diffusion equations with nonlocal delay. Our results extend and improve some related results in the literature.

A Formula for the Time-Dependent Transmission Rate in an SIR Model from Data

Dr. Howie Weiss

Department of Mathematics, Georgia Tech

The transmissibility of many infectious diseases varies significantly in time, but has been thought impossible to measure directly. Based on solving an inverse problem for SIR-type systems, we devise a mathematical algorithm to recover the time-dependent transmission rate from infection data. We apply our algorithm to historic UK measles data and observe that for most cities the main spectral peak of the transmission rate has a two-year period. Our construction clearly illustrates the danger of overfitting an epidemic transmission model with a variable transmission rate function.

Models for the Spread of Hantavirus between Reservoir and Spillover Species

Dr. Curtis Wesley

Department of Mathematics, Louisiana State University at Shreveport

The modeling of interspecies transmission has the potential to provide more accurate predictions of disease persistence and emergence dynamics. We describe various models which are motivated by recent work on hantavirus in rodent communities in Paraguay. Each model is a system of ordinary differential equations (ODE) which are developed for modeling the spread of hantavirus between a reservoir and a spillover species. The basic reproduction number is calculated for each model, with global stability results given for some models. Numerical simulations are created that illustrate the dynamics of each model.

Competition in the Presence of a Virus in an Aquatic Environment

Dr. Gail Wolkowicz

Department of Mathematics and Statistics, McMaster University

Recent research has determined that viruses are much more prevalent in aquatic environments than previously imagined. We derive a model of competition between two populations of bacteria for a single limiting nutrient in a chemostat where a virus is present. It is assumed that the virus can only infect one of the populations, the population that would be a more efficient consumer of the resource in a virus free environment, in order to determine whether introduction of a virus can result in coexistence of the competing populations. Criteria for the global stability of the disease free and endemic steady states are obtained. It is also shown that it is possible to have multiple attracting endemic steady states, oscillatory behavior involving Hopf and homoclinic bifurcations, and a hysteresis effect. Mathematical tools that are used include Lyapunov functions, persistence theory, and bifurcation analysis.

Evolution of Schistosome's Drug Resistance and Virulence

Dr. Dashun Xu

Department of Mathematics, Southern Illinois University, Carbondale

Motivited by some recent empirical studies on Schistosoma mansoni, we use a set of ordinary differential and integral equations to investigate the role of drug treatments of human hosts in the evolution of drug resistant parasites. By studying evolutionarily singular strategies (ESS) of parasites, we found that high drug resistance (and low virulence) is likely to develop for high drug treatment rates, which usually tend to promote monomorphism as the evolutionary endpoint. Our study also shows that the coinfection of the intermediate host does not affect the drug resistance and virulence levels of parasites, but tends to destabilize ESS points and hence promote dimorphism or even polymorphism as the evolutionary endpoint.

The Impact of Periodic Proportional Harvesting Policies on TAC-Regulated Fishery Systems

Dr. Abdul-Aziz Yakubu

Department of Mathematics, Howard University

We extend the TAC regulated fish population model of Ang et al. to include stock under compensatory and overcompensatory dynamics with and without the Allee effect. We focus on periodic harvesting strategy, a subset of variable proportion harvesting strategies that includes constant harvesting as a special case. Both periodic and constant harvesting strategies have the potential to stabilize complex overcompensatory stock dynamics with or without the Allee effect. Furthermore, we show that both strategies force a sudden collapse of TAC fishery systems that exhibit the Allee mechanism. However, in the absence of the Allee effect, TAC fishery systems decline to zero smoothly under high exploitation. As case studies, we apply the TAC theoretical model framework to Gulf of Alaska Pacific halibut data from the International Pacific halibut Commission (IPHC) annual reports and Georges Bank Atlantic cod data from the North East Fisheries Science Center !(NEFSC) Reference Document 08-15. We show that TAC does a good job of preventing the collapse of halibut while cod is endangered. Furthermore, we observe that the likelihood of stock collapse increases with increased weather variability.

Geometric and Potential Driving Formation and Evolution of Biomolecular Surfaces

Dr. Shan Zhao

Department of Mathematics, University of Alabama

In this talk, I will present some geometrical flow equations for the theoretical modelling of biomolecular surfaces in the context of multiscale implicit solvent models. When a less polar macromolecule is immersed in a polar environment, the surface free energy minimization occurs naturally to stabilize the system. This motivates us to propose a new concept, the minimal molecular surface (MMS), for modelling the solventbiomolecule interface. The intrinsic curvature force is used in the MMS model to drive the surface formation and evolution. To further account for the local variations near the biomolecular surfaces due to interactions between solvent molecules, and between solvent and solute molecules, we recently proposed some new potential driven geometric flows, which balance the intrinsic geometric forces with the potential forces induced by the atomic interactions. High order geometric flows are also considered and tested for biomolecular surface modelling. Extensive numerical experiments are carried out to demonstrate the proposed concept and algorithms. Comparison is given to a classical model, the molecular surface. Unlike the molecular surface, biomolecular surfaces generated by our approaches are typically free of geometric singularities. (This is a joint work with Peter Bates and G.W. Wei, Michigan State University).

Dynamics of an Epidemic Model with Non-Local Infections for Diseases with Latency over a Patchy Environment

Dr. Xingfu Zou

Department of Applied Mathematics, University of Western Ontario

Assuming that an infectious disease in a population has a fixed latent period and the latent individuals of the population may disperse, we formulate an SIR model with a simple demographic structure for the population living in an $n$-patch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the latency and a non-local term caused by the mobility of the individuals during the latent period. Assuming irreducibility of the travel matrices of the infection related classes, an expression for the basic reproduction number \\mathcal{R}_0$ is derived, and it is shown that the disease free equilibrium is globally asymptotically stable if $\\mathcal{R}_0<1$, and becomes unstable if $\\mathcal{R}_0>1$. In the latter case, there is at least one endemic equilibrium and the disease will be uniformly persistent. When $n=2$, two special cases allowing reducible travel matrices are considered to illustrate joint impact of the disease latency and population mobility on the disease dynamics. In addition to the existence of the disease free equilibrium and interior endemic equilibrium, the existence of a boundary equilibrium and its stability are discussed for these two special cases.

Poster Presentations

Discrete Time Optimal Control of Species Augmentation: Augment then Grow

Ms. Erin Bodine

Department of Mathematics, University of Tennessee

Stability of the Endemic Coexistence Equilibrium for One Host and Two Parasites

Mr. Thanate Dhirasakdanon

School of Mathematics and Statistics, Arizona State University

For an SI type endemic model with one host and two parasite strains with complete cross protection between the parasite strains, we study the stability of the endemic coexistence equilibrium, where the host and both parasite strains are present. Our model assumes reduced fertility and increased mortality of infected hosts. The model also assume that one parasite strain is exclusively vertically transmitted and cannot persists just by itself. We find several sufficient conditions for the equilibrium to be locally asymptotically stable. One of them is that the horizontal transmission is of density-dependent (mass-action) type. If the horizontal transmission is of frequency-dependent (standard) type, then, under certain conditions, the equilibrium can be unstable and undamped oscillations can occur. We support and extend our analytical results by numerical simulations and by two-dimensional plots of stability regions for various pairs of parameters.

Optimal Control of Growth Coefficient in a Steady State Population Model

Dr. Heather Finotti

Department of Mathematics, University of Tennessee

(Joint work with Ding, Lenhart, Lou, and Ye)

We study the control problem of maximizing the net benefit in the conservation of a single species with a fixed amount of resources. The existence of an optimal control is established and the uniqueness and characterization of the optimal control are investigated. Numerical simulations illustrate several cases, for both one- and two-dimensional domains, in which several interesting phenomena are found. Some open problems are discussed.

An Age-Structured Setup of the Bacteria-Bacteriophage Model

Mr. Zhun Han

School of Mathematics and Statistics, Arizona State University

The mathematical model of the bacteria-bacteriophage interaction has been an interesting topic since 1960's. In this alternate setup, we introduce an age structure on the infected species and rewrite the model into a combination of delayed differential equations and integral equations. We show that this alternate setup coincides with an existing model. However, by employing this age structure, we may have a better view in the biological sense.

The Spreading Speed and Traveling Wave Solutions of a Spatial Competition Model

Ms. Kim Meyer

Department of Mathematics, University of Louisville

Integro-difference equations are used to model spatial spread of species with nonoverlapping generations. We look at a two species competition model with Ricker's growth functions in the form of integro-difference equations. We investigate spatial dynamics about how an introduced competitor spreads into a habitat pre-occupied by a resident species. We found a formula for the so called spreading speed at which the resident species retreats and the introduced species expands in space. We also obtained conditions under which the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. In addition, we conducted numerical simulations and showed that a traveling wave solution can have a complicated tail. (Joint work with Bingtuan Li)

Mathematical Modeling of SARS with a Focus on Super-Spreading Events (SSEs)

Mr. Thembinkosi Mkhatshwa

Department of Mathematics and Applied Sciences, Marshall University

"Super spreading events" (SSEs) have been cited to have been one of the major factors which were responsible for the spread of severe acute respiratory syndrome (SARS), the first epidemic of the 21st century. The understanding of these SSEs is critical to understanding the spread of SARS. We present a modification of the basic SIR disease model, an SIPR model, which captures the effect of the SSEs.

Modeling Antibiotic Resistant Bacteria in the Mud River, WV

Dr. Anna Mummert

Department of Mathematics, Marshall University

When antibiotics are used by humans, or for livestock or crop production, antibiotic resistant bacteria enters the environment. The antibiotic resistant bacteria wash into rivers, where the antibiotic resistance gene is transferred to naturally occurring bacteria in the river. In this poster we present and study a system of ordinary differential equations that model antibiotic resistant bacteria in rivers. The influence of bacteria entering the river due to nearby land use appears in the model as an external forcing term. The model is compared with data from the Mud River, WV.

Bacteriophage Infection Dynamics:Multiple Host Binding Sites

Mr. Roy Trevino

School of Mathematical and Statistical Sciences, Arizona State University

We construct a stochastic model of bacteriophage parasitism of a host bacteria that accounts for demographic stochasticity of host and parasite and allows for multiple bacteriophage adsorption to host. We analyze the associated deterministic model, identifying the basic reproductive number for phage proliferation, showing that host and phage persist when it exceeds unity, and establishing that the distribution of adsorbed phage on a host is binomial with slowly evolving mean. Not surprisingly, extinction of the parasite or both host and parasite can occur for the stochastic model.

Novel H1N1: Predicting the Course of a Pandemic

Dr. Sherry Towers

Department of Applied Statistics, Purdue University

In April, 2009 a new strain of H1N1 was identified, causing a spring epidemic in Mexico, and a summer wave of infection in the US and elsewhere. Because influenza is seasonal in nature (more infectious in winter than summer in the northern hemisphere), world health officials anticipate a second, larger fall wave, similar to that seen in 1918. We examine the prevalence of H1N1 in the US during summer 2009. In a unique study, we use this information, along with what we know about the seasonal behavior of influenza, to predict the prevalence of influenza during fall 2009, and examine the efficacy of the planned CDC H1N1 vaccination campaign.

Dispersal Behavior with Biased Edge Movement Between Two Different Habitat Types

Ms. Xiuquan Wang

Department of Mathmatics, Southern Illinois Univercity, Carbondale

(Joint work with Drs. Mingqing Xiao, John D. Reeve, Dashun Xu, and James T. Cronin)

we analyze the behavior of the reaction-diffusion model of organisms to study insect diffusion. The main idea is based on some set of hypotheses about the scale and structure of the spatial environment and the way that insects disperse through it to express how the insect responds to edges between two different habitats, therefore, its behavior can be analyzed according to the shape of corresponding distribution curves. By analyzing the reaction-diffusion equations, this also provides a simulation of occupancy mean times for insect in different habitats.

Mathematical Modeling of the Vancomycin-Resistant Enterococci

Dr. Mohammed Yahdi

Department of Mathematics and Computer Science, Ursinus College

A mathematical model of the Vancomycin-Resistant Enterococci (VRE) is introduced. It includes a system of three nonlinear differential equations with three variables connected and twelve parameters, such as fitness costs, rates of colonization, and hygiene compliance. Equilibrium point simulation and outbreak analysis are performed to visualize and measure the impact of the parameters on the spread of the antibiotic resistant VRE, and to provide optimal control strategies.

Effect of Host Heterogeneity on the Coevolution of Parasite and Host

Ms. Yiding Yang

Department of Mathematics, Purdue University

A system of differential equations which models the disease dynamics of schistosomiasis is used to study the evolution of parasite virulence. The model incorporates both the definitive human hosts and two strains of intermediate snail hosts. An age-structure of human hosts is considered to reflect the age-dependent transmission rate and age-targeted drug treatment rate. The basic parasite reproductive number R_i of strain i snail hosts is computed, and the invasion reproductive number R_{ij} for strain i snail host when type j snail hosts are at the equilibrium. We establish the criterion for strain i to invade strain j snail host, and the criterion is used to examine the evolutionary dynamics of snail hosts and the parasite.

Modeling Disease Transmission with Age and Spatial Heterogeneities

Dr. Feng Yu

Statistics and Epidemiology, Statistics Research Division, RTI International

We have developed a comprehensive disease epidemic model that takes both age-heterogeneity and spatial-heterogeneity into account. The model features an age structure that captures the dynamics of disease epidemic with seasonal effect for various age groups. This model also simulates the impact of migration among geographic locations on disease transmission. In addition, vaccination strategy has been built into the model for disease intervention and control. We implemented the model using the AnyLogicTM software package with graphical user interface for presentation of results and for changing parameters interactively. For illustration purpose, we will use Measles transmission as an example.

Different Orders of Harvesting an Integrodifference Model

Ms. Peng Zhong

Department of Mathematics, University of Tennessee

An optimal control harvesting problem for a population modeled by an integrodifference equation model is considered. The proportion to be harvested is taken to be control. The goal is to find the optimal harvesting control to maximize the profit. The effect of order on optimal control of harvesting on integrodifferential equations is studied.

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