Mathematical and statistical models of the biological world are powerful tools for working with these complex systems and distilling them down to interpretable phenomenon which we can use to develop our understanding of the dynamics at play. In this dissertation I put forward a pair of mathematical models, which advance our understanding of two crucial in- fectious disease systems. Firstly, I present a mathematical model of hyperparasitism, that is parasitism of a parasite, that allows us to understand guiding principles behind the evolution of these hyperparasites in nature. Among the results, I critically show that the proportion of hyperparasitized hosts that transmit both their parasite and hyperparasite has a core impact on the evolutionary outcomes of the system. This probability of co-transmitting or ”hitch- hiking” by the hyperparasite is central to the dynamics of a hyperparasite system. Second, I present a mathematical framework for more accurately modeling the way in which immunity from vaccination wanes over time in a host. With it I am able to derive an analytical rela- tionship between this waning process and the ability of a population to prevent infections via population level immunity. With this relationship I show that we must measure more precisely both the waning process and the immunity remaining after the waning process, in order to create better vaccine control strategies for infectious disease. Finally, this motivates the creation of a new non-parametric estimation method for naturally acquired immunity from infection. I use a negative control design and show we can create a new odds ratio esti- mator for the reduction in susceptibility associated with infection and show that for certain conditions it is unbiased and is robust to individual heterogeneity in susceptibility. I then compare it to other methods for estimating this quantity, and apply it to cohort data for rotavirus infection.