Students are aware of the impact of disease in their own lives, and consequently disease transmission problems grab interest and strongly motivate mathematical modelling. To take advantage of this we developed a disease transfer game for a second-term calculus class, with the aim of leading students to make discoveries and apply relationships and algorithms such as separation of variables, the logistic equation, and partial fractions integration. The design of the game and group objectives are outlined in appendix B.
Students implemented the discrete version of the game on a hex-grid system, using the rules in appendix B. Students leapt immediately to doing the simulation, joking about the diseases they were modelling, the `splat factor' and `sneeze radius' of the disease, the meaning of overlap of diseased and susceptible populations and so forth. With the initial set of basic rules students found the population moving from susceptible to infected in five or six turns (five or so minutes of real time).
In collecting various data sets students adjusted the `splat factor' and how long diseased individuals stay infectious. One group removed diseased individuals (due to death or medevac) after some number of turns. Students were enthralled with the data collection and simulation, and volunteered time outside of class to continue.
At first the theoretical end of the exercise was difficult. Even though the instructor had worked through the logistic differential equation using separation of variables and integration by partial fractions, students found it hard to do on their own, particularly because of the presence of free parameters. Since students were working in groups, however, only one student in each group had to get a glimmer of understanding to infect the rest of the group with it. In short order everyone appeared comfortable applying integration procedures to find the analytic form of the solutions. The next difficulty was `fitting' the analytic curves to simulated data. Although all of these students had done well with limits in the previous term, understanding the limiting behavior of their analytic solutions was difficult at first. Again, an enterprising few plotted sample solutions, and realized that the asymptotic behavior was constant-simply the eventual infected fraction. At that point students felt comfortable working out the asymptotics of their analytic solution, and were consequently able to remove one parameter from the fitting problem. They could then experiment with the remaining parameter to produce a good fit.
Students then exceeded their instructor's expectations, which wre only that they would tune the remaining parameter to get a visually acceptable fit. Almost all groups went further, finding an analytic representation for the parameter in terms of data and developing some sort of averaging procedure for their data to estimate the parameter. The group which removed diseased individuals were unable to fit the logistic curves, which they took as evidence falsifying the model. Their final report included a suggestion for an improved model which would, they argued, more closely match the observations. (This amounted to suggesting the SIR disease model .) In the end, students applied the expected techniques in ways which deepened their understanding and improved their abilities. They also learned a great deal about applying mathematics, and how modelling can shed light on other fields (not one of the students had initially believed that any mathematics other than statistics would apply to biological questions).