Events
Department of Mathematics and Statistics
Texas Tech University
Influenza A virus (IAV) has a high mutation rate and large inhost population size, and is thus capable of rapid evolutionary change. While
large-scale antigenic changes have been well-studied at the epidemiological
scale, we tackle the underlying small-scale question: how does the virus
received at the start of an infection in a single individual differ from the
virus transmitted from that individual to others? To address this, we couple a
system of ordinary differential equations, describing the in-host dynamics of
IAV, to a branching process describing the fate of de novo mutant lineages
within the host. This coupled approach is necessary because IAV can reach
extremely high copy numbers in a single infected host, but the transmission
bottleneck may involve as few as one to ten viral particles. We can then
answer the question: how different is the flu you pass on from the flu you
received?
In June 23, 2006's issue of Science magazine,
Pendry et al and Leonhardt independently published their
papers on electromagnetic cloaking with metamaterials. Since
then, there is a growing interest in using metamaterials to
design invisibility cloaks and other interesting devices. In
this talk, I will present some time-domain cloaking models
we studied in recent years. Well-posedness study and
various time-domain finite element methods will be
discussed. I will show numerical simulations of invisibility
cloaks and other interesting simulations such as optical
black holes. I will conclude the talk with some open
issues.
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Structural sensitivity is when the dynamics of a system is strongly dependent on the functional form used in model development. Recently, Cordoleani and colleagues have investigated a chemostat problem in which very similar metabolic response functions, all consistent with experimental data, could produce qualitatively different results (limit cycle or limit point). They introduced a derivative in model space that allowed quantification of local sensitivity to model structure.
We introduce a complementary method for quantification of structural sensitivity in a more global sense. Following Wiener and Feynman, we introduce a measure on the space of possible models. Integration in this measure -- a path integral -- allows assignment of probabilities to the qualitative outcomes. The path integrals can be weighted by a likelihood function based on experimental measurements. We analyze the chemostat problem of Cordoleani et al using this method, and discuss application of the method to model sensitivity problems in galactic dynamics.
This talk will be accessible to graduate and advanced undergraduate students, who will see how topics from analysis and several methods from numerical analysis (function approximation, finite elements for solution of boundary value problems, methods for solution of initial value problems, and numerical integration in high dimensional spaces) can be combined to solve complex problems.
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