| Popis: |
This dissertation contains two sections. In the first, we consider the contact process with avoidance, a model for epidemics on networks that models the idea that individuals might change their behavior in response to information about their neighbors. We take some graph $G = (V,E)$, on which vertices may be infected or healthy, a set of initially infected vertices, and allow the process to evolve as follows. Individuals become infected at a rate $\lambda$ times the number of infected neighbors they have via active edges. Individuals recover at rate $1$. Edges from infected to healthy vertices deactivate at a rate $\alpha$ and reactivate when the offending infected vertex recovers. We study this model in the cases where the graph $G$ is $\boldZ$, $\boldZ_n$, and the star graph. This work is an important first step in understanding this and other similar models where the graph co-evolves with infection dynamics on more general random graphs. The second section examines the performance of phylogenetic estimators that estimate species tree topologies from genomic data. We prove that two well known estimation methods, maximum likelihood and SVDQuartets, are statistically consistent estimators of the species tree topology in this setting. In addition, we derive a bound for the error rate of SVDQuartets. In addition, we consider the case of populations under selection. Most phylogenetic estimation methods assume the population is not under selection. We study the population dynamics under selection from the perspective of phylogenetic inference take some useful steps toward understanding how the presence of selection impacts the performance of phylogenetic estimators. |
