I will discuss how network models are helping us to understand (1) how diseases
spread through complex host populations and (2) how mutations shape evolutionary
dynamics.

In the early 20th century, two epidemiologists introduced a simple and powerful
deterministic model for predicting infectious disease transmission which tracks the
unidirectional movement of hosts among three states: susceptible (S), infected
(I), and recovered (R). This SIR model provides important insight into the temporal
progression of outbreaks and the efficacy of vaccination, and is the foundation for
a recent proliferation in predictive methods. Contact network epidemiology is a
particularly promising development in which bond percolation on random graphs is
applied to modeling disease transmission through heterogeneous populations. My
lecture will introduce the SIR model, explain its generalization to disease
propagation on graphs in which vertices and edges represent individual hosts and
disease-causing contacts, respectively, and link recent theoretical results to
issues of public health and conservation.

Evolution by natural selection is fundamentally shaped by the fitness landscapes in
which it occurs. Yet fitness landscapes are vast and complex, and thus we know
relatively little about the long-range constraints they impose on evolutionary
dynamics. Here, we describe the global structure of fitness landscapes for all RNA
molecules of lengths 12 to 18 nucleotides, based on an exhaustive computational
survey of these molecules. We find that phenotype abundance---the number of
genotypes producing a particular phenotype---varies in a predictable manner and
critically influences evolutionary dynamics. A study of naturally occurring
functional RNA molecules using a new structural statistic suggests that these
molecules are biased towards abundant phenotypes. This supports an "ascent of the
abundant" hypothesis, in which evolution yields abundant phenotypes even when they
are not the most fit.