Math Biology Seminar
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Time: Fridays 3:40 PM Contact: Yang Kuang Other Seminars Of Interest: SoLS Seminar - Fall 2009 |
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Date |
Speaker |
Topic/Abstract |
| August 28 |
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Organizational Meeting |
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September 4 |
Horst Thieme |
Title: Population persistence without compact attractor Abstract: The theory of persistence is designed to provide an answer to such ques- tions 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 pos- itive 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 assump- tion is the existence of a compact attractor of points. This assumption ex- cludes, 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. |
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September 11 |
Karl P. Hadeler |
Title: The Largest Basic Reproduction Number in Multi-type Epidemic Models Abstract: 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 $\rho(XA)$ 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 $A$. |
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September 18 |
Nicolas Lanchier |
Title: Coexistence in spatially explicit stochastic models Abstract: Interacting particle systems are stochastic processes that model populations evolving in spatially explicit environments through the inclusion of local interactions. I will review thirty five years of research on this topic, from the building blocks of the theory until works in progress. The emphasis will be on multispecies competition models and the mechanisms that promote coexistence in spatial environments. |
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September 25 |
Maia Martcheva |
Title: Vaccine-Induced Pathogen Strain Replacement Abstract: Host immune systems impose selection on pathogen populations which respond by evolving di erent antigenic signatures. Like many evolutionary processes, pathogen evolution re ects an interaction between different levels of selection: pathogens can win in between-strain competition by taking over individual hosts (within-host level), or by infection of more hosts (population level). Vaccination, which intensifies and modifies selection by protecting hosts against one or more pathogen strains, can drive the emergence of new dominant pathogen strains { a phenomenon called vaccine-induced pathogen strain replacement. In this talk reports of increased incidence of subdominant variants after vaccination are reviewed and the current model for pathogen strain replacement, which assumes that pathogen strain replacement occurs only through the differential effective- ness of the vaccines, is extended. Our theoretical studies suggest that a broader range of mechanisms is possible including pathogen strain replacement even when vaccines are perfect { that is, they protect all vaccinated individuals completely against all pathogen strains. Pathogen strain re- placement with perfect vaccination occurs when strains interact through super-infection or co-infection but does not seem to occur in the simplest models when the strains interact through cross-immunity. Super-infection, co-infection, and cross-immunity are some examples of coexistence mechanisms {mechanisms that lead to coexistence of pathogen variants. The question which coexistence mechanisms lead to strain replacement under perfect vaccination and which do not is also addressed. |
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October 2 |
Libin Rong |
Title: Modeling latently infected cell activation in HIV patients under effective drug treatment Abstract: Current combination therapy can suppress viral loads in HIV-1-infected individuals to below the detection limit of standard commercial assays. However, it cannot eradicate the virus from patients. HIV-1 can generally be identified in resting memory CD4+ T cells and persists in patients on potent treatment for a long time. These latently infected cells decay slowly, but can produce new virions when activated by relevant antigens. Many patients experience transient episodes of viremia, or blips, even though they have “undetectable” plasma viral loads for many years. Here, we develop a new mathematical model describing latently infected cell activation upon random antigenic stimulation. Using the model, we show that programmed expansion and contraction of latently infected cells upon activation can generate both low viral load persistence and viral blips. Occasional replenishment of the latent reservoir may explain the different decay kinetics of the reservoir observed in clinical practice. We also show that a model with homeostatic proliferation of latently infected cells can explain persistence of low-level virus, stability of the latent reservoir, and emergence of viral blips. These results provide novel insights into the long-term virus dynamics and could have implications for the treatment of HIV-1 infection. |
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October 9 |
Yalin Wang |
Title: Brain Mapping with Computational Conformal Geometry Abstract: Conformal Structure is a natural geometric structure on surfaces, which governs many physics phenomena. We develop general approaches that conformally parameterize brain anatomical surfaces with Riemann surface structure. With harmonic energy minimization, holomorphic 1-form and the disrete Ricci flow methods, we can conformally parameterize brain surfaces with various canonical surfaces such as sphere, Euclidean plane, and Poincare disk. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. It also induces conformal invariants - Teichmuller shape space coordinates - and multivariate tensor-based morphometry, which are stable shape indices for statistical shape analysis. Our research offers ways to apply harmonic analysis to compare anatomy and generate grids on surfaces for PDE-based signal processing. Various brain mapping applications will also be discussed. |
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October 16 |
Fred Brauer |
Title: Backward bifurcations in a simple disease transmission model Abstract: 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. |
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October 23 |
Todd Parsons |
Title: Some Results for Density-Dependent Population Genetics Abstract: Near the beginning of the century, Wright and Fisher devised an elegant, mathematically tractable Markov chain model of gene reproduction and replacement that, combined with Kimura's continuous approximation, laid the foundation for contemporary population genetics. The Wright-Fisher model and its many extensions have given biologists powerful tools of statistical inference that enabled the quantification of genetic drift and selection. Given the beauty of the results and the utility of these tools, it is often forgotten that the underlying model - for mathematical reasons - initially made assumptions that are violated in many real-world populations. In this talk, I discuss a complementary model that starts from an approach of Moran, and merges it with the density dependent model of resource-limited competition proposed by Gause, Lotka and Volterra. The model attempts to more closely approach a mechanistic description of the population genetics of microbial communities. I will show that the model is asymptotically analytically tractable, can be used to obtain generalisations of classical results in population genetics, and that it predicts novel possibilities that may have implications for empirical studies in genetics. |
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October 30 |
Justine Gunog Seo |
Title: The dynamics of a laissez-faire model with two predators Abstract: In my talk, I study the dynamics of a laissez-faire predator-prey model with specialist and generalist predators. I analyze the stability of an equilibrium by performing linearized stability analysis and by constructing a Lyapunov function. If I hold the generalist predator population constant, treating it as a bifurcation parameter, I show that my model can possess multiple (up to three) limit cycles that surround an equilibrium in the interior of the first quadrant. My model shows rich dynamics including transcritical, fold, pitchfork, Hopf, cyclic-fold, and Bautin bifurcations as well as heteroclinic connections. If I instead vary the generalist predator population slowly across bifurcations, the model exhibits bursting behavior as it alternates between a repetitive spiking phase and a quiescent phase. |
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November 6 |
David M. Bortz |
Title: Fragmentation and Aggregation of Bacterial Emboli Abstract: 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 novel approach to fragmentation modeling, and (time permitting) c) self-similar solutions to the equations. |
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November 13 |
Wenzhang Huag |
Title: The Minimum Wave Speed of Traveling Waves for a Lotka-Volterra Competition Model Abstract: Consider a reaction-diffusion system that serves as a 2-species Lotka-Volterra competition model with each species having logistic growth in the absence of the other. Suppose that the corresponding reaction system has one unstable boundary equilibrium E_1 and one stable boundary equilibrium E_2. Then it is well known that there exists a positive number C_*, called the minimum wave speed, such that, for each c larger than or equal to C_*, the reaction-diffusion system has a positive traveling wave solution of wave speed c connecting E_1 and E_2, and the system has no nonnegative traveling wave with wave speed less than C_*. It has been shown that the minimum wave speed for this system is identical to another important quantity - the speed of the population spread towards to the stable equilibrium. Hence to find the minimum wave speed C_* not only is of the interest in mathematics but is of the importance in application. Although much research work has been done to give an estimate of C_* and some partial results have been obtained, the problem on finding an algebraic or analytic expression for the minimum wave speed remains unsolved in general. In this talk we will introduce a new, more efficient approach that enable us to determine precisely the minimum wave speed algebraically under conditions weaker than those given previously. We also show that the minimum wave speed in general cannot be determined by the linearization at the unstable equilibrium point. The conjecture on the precise minimum wave speed is also given. |
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November 20 |
Yun Kang |
Title: Relative nonlinearity and permanence Abstract: We generalize the invasibility concept for species coexistence in discrete time two-species population models to include coexistence of a predators and prey in addition to the usual case of coexistence of competing species. We find sufficient conditions of invasibility and permanent coexistence, and apply these results to situations with nonpoint attractors. In this case, a key question is the role of population fluctuations in species coexistence. The invasibility criterion depends on the long-term growth rate from low density, i.e. the invasion rate of one species invading the attractor of another. We show that the concept of relative nonlinearity precisely identifies when the invasion rate depends on the fluctuating dynamics of the resident species. When relative nonlinearity is zero, fluctuations have no effect on this invasion rate, and the invasion rate can be found simply by substituting the resident equilibrium into the growth rate of the invader. In particular, fluctuations can undermine invasibility and also permanent coexistence when relative nonlinearity makes a negative contribution to invader growth. These results are illustrated with a two-species competition model and a predator-prey model. |
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