On the Road

This chapter introduces the SIR (susceptible, infected, resistant) model of infectious disease spread as an example problem in quantitative biology. The model allows us to several things, including computing the size and timing of a disease outbreak. The chapter next discusses how this model is stated in terms of rates of change and then defines the “inverse problem” when we know the rate of change of a quantity but not the quantity itself. We then introduce the concept of functions and computing the rate of change of functions with derivatives, using the examples of moving cars and falling balls. The chapter shows how the inverse problem can be approximately solved by treating a known derivative as constant over short intervals to compute an approximate value of the quantity of interest. It then relates this approach to the concepts of integration and antiderivatives, noting that taking the antiderivative is the formal solution to the inverse problem but is often not analytically soluble.