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The basic reproduction number R_0 is the number of infections caused
by one infectious case when the population has no immunity to
infection, when infections are growing exponentially. In models of
unstructured populations, R_0 predicts many important epidemic
statistics like the total number of cases over time and the
vaccination coverage needed to achieve herd immunity.
However, in a structured population, where some epidemiological
parameters vary with some population characteristic like age, R_0
provides no information on these important epidemic statistics. I show
by example, using two mathematical models of populations that have 2
epidemiological groups, that if one group is small but has very high
transmission within that group, R_0 is driven by this group and can be
very large, while the total number of people infected by an epidemic
or the critical vaccination coverage can be tiny.
In this talk we overview mathematical and computational modeling of physical phenomena in which one can clearly distinguish more than one scale of variability; this feature is common in geophysical flows including porous reservoirs and icesheets with fractures, composite materials in materials science, and engineering applications such as reinforced concrete. Some involve the length scales from molecular to nanometer scales of microscopic imaging through the meter scale of laboratory experiments, to the practical kilometer scale of large scale simulation models; some others focus on the time scales, or both. Classical example is some stationary partial differential equation (PDE) whose coefficients or the domain geometry have periodic character, with a period much smaller than the size of the domain on which the PDE is posed. The goal is to find and analyze the "global behavior" of the solution. One popular technique is called homogenization: assuming periodicity, the process uses averaging to determine the asymptotic limit and the "effective coefficients" of a PDE model whose solution is the global limit. The simplest setting for linear problems is extended by mathematically sophisticated techniques which study the qualitative nature of convergence of the true varying solutions to the limit function with global behavior. Furthermore, in practical setting we use numerical homogenization aka "upscaling"; these allow to discover and understand the global limits for non-periodic setting as well as for transient and nonlinear behavior. Another avenue is to couple micro-scale and macro-scale models dynamically.
In this talk we aim for an overview from a personal perspective rather than technical details, and we illustrate the talk with results on modeling of highly heterogeneous porous media as well as modeling at porescale when the geometry is changing. This is joint work with many current and former students and collaborators to be named in the talk.