Precalculus Group Project: Exponential, Log and Logistic Models

The Spread of A Communicable Disease

You will use your calculator to generate random numbers between 1 and 10.

Try it: Math>>PRB>>randInt(1,10)>> Enter

Keep pressing Enter to get a new random number.

For one minute, students greet other students with a handshake, write down each other’s name, and get a random number from their calculators for each encounter. If the number is a 1 or 2, then the encounter is "risky" and that name should be circled. A risky encounter means that if either of the students is a carrier of the disease, it is passed to the other student (i.e. you sneezed.)

These one minute "days" are repeated five times with students circulating around the room. It is possible to have an encounter with the same individual more than once.

Choose the first infected student by random draw. Write the name on the board. Find out who had a risky encounter (circled name) with that individual in the first stage. Write their names on the board. Write down the total of infected students at the end of this stage. Continue identifying risky encounters and writing names. Write the total at the end of each stage.

This is Fun! Can we do it again? Try it again with 2 minute "days". This time write down names of only the risky encounters.

In the PreCalc folder (either on the desktop or in the math folder), double click on runinfect.ide. Click on the lightening bolt to run the program. Run the program once, using 10000 (no comma) for the size of your society, 10 days, risk factor of 0.2 and 2 encounters per day. Write down your results. Close the finished simulation by clicking on the X in the simulation window. DO NOT CLOSE the proj window at the bottom!

Use Fathom to make a scatter plot of the total infected individuals over time (days).

Use the calculator to model the curve with exp, ln, and logistic curves. Plot these functions on your Fathom plot. Record each equation used in standard mathematical notation (not weirdo calculator-ese.)

Choose the best type of equation for modeling the spread of disease. Justify your choice.

Use the computer simulation to determine the impact of two factors: the size of the society (from 100 to 30,000) and the number of encounters per day (sociability index). Note: you may have to change the number of days for the simulation—you must run it until everyone is sick. Suggestion: each group member chooses one variable and runs computer simulations with different inputs for that variable. Using only the type of model you chose above as the best type, model each output with a graph and an equation. How do these factors affect the graph and model? Specifically address changes to the rate of the spread of the disease in the basic model. Be specific: how much faster or slower? How much more or less? Also address changes to the range and domain of the graphs.

What would be the impact of a cure (what if 50% of infected individuals became healthy after each stage?) Sketch your idea of what the graph would look like and describe how this idea would change the equation of the basic model.

What would be the impact of education (what if the risk factor lowered in the later stages?) Sketch your idea of what the graph would look like and describe how this idea would change the equation of the basic model.

What would be the impact of the rate the disease would spread to other cities with and without mass transportation? Sketch your idea of what the graph would look like and describe how this idea would change the equation of the basic model.

Note: I do not expect you to write the new equation for any of the predictions. You should explain how the original equation would change.

Turn in one report per group. Each group should have two members. Your report should include the graphs, equations and analyses for all three of the sections listed above. Your report does not address the class experiment. The class simulation was done so you would have a feel for how the computer simulation works.

Worksheet for Class Experiment on Spread of Disease

In the spaces below, write the name of each student you greet in that stage. Circle the name if it was a risky encounter.

Write in the total number of infected individuals at the end of each stage.

In the spaces below, write the name of each risky student you greet in that stage.

Write in the total number of infected individuals at the end of each stage.