Over the past few decades, mathematical models have provided key insights into the spread and evolution of infectious diseases both within individual hosts and across populations. This thesis applies models to understand two topics in infectious disease dynamics at different scales: 1) the emergence of drug resistance within a patient taking antimicrobial therapy, and 2) the role of human contact networks in the transmission of SARS-CoV-2. Tens of millions of individuals around the world now receive drug therapy for chronic viral infections like HIV, Hepatitis B, and Hepatitis C, but outcomes are compromised by the need for long-term adherence to drug regimens and the evolution of drug resistance. In recent years, there has been an increased focus on developing long-acting therapies to improve patient adherence, but it is unclear whether changing drug kinetics will increase or decrease the risk of drug resistance. In Chapter 2 we show that the timescale of drug dosing affects the generation and selection of resistant strains in a complicated manner. Long-acting therapies may be better or worse for the emergence of resistance, depending on how much they improve adherence, the pharmacological properties of the drug, the degree of resistance, whether resistance is primarily pre-existing or generated de novo, and the role of persistent/latent infection. Next, in Chapters 3-5, we turn our attention to the COVID-19 pandemic, which has infected hundreds of millions around the world since the beginning of 2020. We examine how the structure of the transmission network over which SARS-CoV-2 spreads impacts the epidemic dynamics and the impact of control measures in three different contexts. In Chapter 3 we develop a stochastic epidemic model to study the effects of COVID-19 clinical progression and the transmission network structure on the efficacy of social distancing measures. We find that the strength of within-household transmission is an important determinant of success. Coupled with residual external transmission it governs the size of the epidemic, individual risks of infections and can lead to long delays before the effects of an intervention become apparent. In Chapter 4, we extend this model to quantify the effect of evictions, and policies preventing them, on SARS-CoV-2 epidemics in cities. We show that evictions always lead to higher levels of infections and that they reduce the effectiveness of social distancing measures. Our results are in favor of policies that stem evictions as a means to control epidemics like COVID-19. Finally, in Chapter 5, we show that a hierarchical metapopulation model of COVID-19 spread at multiple spatial or demographic scales can explain the counter-intuitive observed relationship between crowding and the temporal dynamics of epidemics in cities.