Abstract: Understanding how host mixing influences virulence evolution is important for predicting disease emergence and spread. Previous theoretical studies have shown that increased host mixing favors higher pathogen virulence, whereas vertical transmission tends to select for more prudent host exploitation. In this thesis, we investigate whether this prediction remains valid using a model with different pathogen transmission scales. Specifically, we assume that horizontal transmission occurs at a local scale within patches, while vertical transmission occurs at a global scale across the population. Our model predicts that the relationship between host mixing and virulence can be reversed when horizontal transmission is locally restricted. We propose that this reversal occurs because lower virulence allows pathogens to exploit local host populations for longer periods. Overall, our key assumptions emphasize that virulence evolution with host mixing depends not only on the pathogens' transmission mode, but also on the spatial structure of the host population.

Abstract: This thesis studies the recovery of an unknown permutation group $G \leq S_n$ from independent observations of an error-prone sampling process that samples uniformly from $G$ with probability $1-p$, and samples uniformly from $S_n$ with probability $p$. This model is motivated by the numerical computation of monodromy groups using homotopy continuation, where path-tracking errors may result in faulty monodromy permutation computations. We study the effects of these errors under the simplifying assumption that the distribution induced by path-tracking errors is the uniform distribution on $S_n$. We first establish the fragility of the standard approach for computing $G$ and, applying results from probabilistic group theory, present our own algorithms which can reliably recover $G$ even when the error rate $p$ is high.