Mathematical models, as Bob May wrote, "force[s] clarity and precision upon conjecture, thus enabling meaningful comparison between the consequences of basic assumptions and the empirical facts”.  Mathematical biology has been particularly useful at the intersection between demography and infectious disease, where colliding timescales and feedbacks drive non-intuitive outcomes of relevance from public health to evolutionary biology. I'll illustrate examples of such insights, ranging from development of policy around rubella vaccination to deepening our understanding of the evolution of immunity.