Explore the intricacies of mathematical modeling for infectious disease outbreaks in this Oxford Mathematics Public Lecture. Delve into the construction, assumptions, and accuracy of models that are shaping pandemic responses. Learn about compartmental modeling, the SI model, basic reproduction number, and herd immunity. Understand how mathematicians assess outbreak shapes, evaluate model effectiveness, and consider factors like symptomatic transmission and contacts in various settings. Discover how these models inform control strategies, initial policies, and vaccination efforts to mitigate epidemic risks and reduce case numbers.