The purpose of this investigation is for students to see the real power of calculus to describe and explain the world around them. We have chosen the H1N1 flu since it is a topic that is much in the news. Introductory calculus gives some insight into the problems and possible solutions to epidemics of an infectious disease like H1N1 flu. Students will need to bring together topics and concepts from across the course and put them together in ways they may not have seen as a part of the standard first course in calculus.

The model that will be considered is the standard SIR model that is commonly used for many infectious diseases. The name of the model reflects the three populations that it models: Susceptible people, Infected people, and Recovered people. There are two essential parameters, and the current H1N1 flu mutation is sufficiently unstudied that we don't yet have a good value for the transmission rate. One of the things students will consider is how varying that rate affects the spread of the disease. There are a number of important thresholds in this model that we hope students will discover. Reaching, or failing to reach, these thresholds is a crucial feature of managing the spread of infectious diseases. The system is sensitive to some changes and not to others, so this gives insight into where (at what point) the problem should be attacked.

This investigation has four essential components. Teachers can use some or all of them, depending on the background of their students and the amount of time they would like to commit to the project. We generally try to give at least two weeks for projects like this one. Student material and teacher material, most with solutions, are provided so teachers can see what content is involved in each section.

The investigation includes:

Initial Flu Investigation. There are several pages of introductory material that describe the basic SIR model and how to think about the modeling process. Students will use tools of calculus to modify the defining differential equations and solve some components analytically to get closed form solutions. They will determine the crucial thresholds that determine if the disease becomes epidemic or fizzles out. Links to: student handout (or .doc) and teacher handout. AB teachers may want to teach a short section on separation of variables before students begin this component.
Excel Spreadsheet. We have set up the SIR model in Excel, so students can modify the equations by varying the parameter values and look for patterns and critical threshold values in a numerical investigation. The spreadsheet uses Euler's method. For teachers whose students are unfamiliar with Euler's method we have provided a short introduction to Euler's method & a corresponding Excel spreadsheet.
Project Extensions. The final component is also a computer-aided investigation, which considers the effectiveness of quarantine. There are three documents associated with this component. One is a text document describing how to extend the basic SIR model to incorporate quarantining some victims. The other two are Excel documents (1 2) that already contain Euler's method solutions to the two models that are described in the quarantine document. (In addition to the quarantine model, another model is described in which flu victims go through an asymptomatic but contagious "incubation period".)

Students may find that there is a threshold for the usefulness of quarantine. Unless you can quarantine at least k% of the infected people, there is no reason to quarantine at all. Following the description of the quarantine models are lists of questions that students may wish to explore. One that they may find particularly interesting is the question of mask effectiveness. They may find that an effective way to control the spread of the flu is to have those healthy people who are attending quarantined victims wear highly protective masks, such as "N95" masks.

The suggested extended explorations do not include solutions, since there are many directions in which the students can go.

Also, a second version of the Excel spreadsheet (& accompanying teacher and student - .pdf .docx - handouts) is slightly more accurate because it uses quadratic approximations rather than linear approximations. This gives an opening to discussing polynomial approximations and how they can be used to reduce error if teachers want to bring together implicit differentiation and polynomial approximations.

If you choose to have your students work through this investigation and find errors or have questions, please e-mail me at: teague@ncssm.edu. One of my colleagues or I will get back to you with support.

Note: Teacher handouts are password-protected. Please feel free to email teague@ncssm.edu for credentials.