A Risky Encounter by Sarah Grabowski

The logistic function can be modeled by an experiment about the spread of disease. Everyday a person encounters many other people and while most of these encounters are harmless, some can be risky. If a contagious disease is in existence, this disease can be transmitted through risky encounters and be spread throughout the entire population.

• Statement of Problem

Using your class as a population that encounters one another during five stages, document the risky encounters with other people. Then randomly choose one infectious person. For each stage, record the total number of newly infected people from risky encounters with previously infected people. Observe the collected data and develop a recursive formula based on this data. Compare the collected data to the formulated data.

• Description of Simulation

Each person will need a set of instructions, a data sheet, a die, and an identification number. Identification numbers are three digits, where all three digits are between the numbers one through six. There will be five stages, each being two minutes in length with thirty-second pauses between each stage. During a stage, exchange identification numbers with another person. Each person then rolls his or her die. If the sum of the two dice is smaller than or equal to five, the encounter was risky and both persons circle the identification number of the other person on their sheet.

After the five stages have been completed, the first infectious person will be chosen randomly by rolling a die three times to get and identification number. To track the disease, start in stage one and list the people would had risky encounters with the first infected person. Total the number of people infected at the end of stage one and record it in the data table. For stage two, list the people who had risky encounters with the people who were infected in previous stages. Total the number of people infected at the end of the stage and record it in the data table. Repeat these steps for each additional stage.

• Data Sheet and Graph

My original data sheet is included in this write-up towards the end. The data sheet includes the people I encountered in each stage, a table with the class results, and a sketch of the data on a graph. The graph is almost a smooth curve that continues upward and appears to nearly level off. Also, on the bottom of the sheet is the graphing calculator’s equation of best fit to the data,

This logistic equation gave a very close approximation for the data.

• Development of Recursive Formula

Let be the number of infected people at the end of stage n.

Let be the number of newly infected people at the end of stage n.

Let P be the probability of being infected.

In each stage, there are the people who were infected by the end of the previous stage plus the people infected during the current stage.

In the initial stage, before the encounters begin, there is one person infected.

In the first stage, there are the people infected in the initial stage plus the people who were infected during this stage, the newly infected. There is a probability P that the people not infected will be infected during this stage.

In the second stage, there all the people infected at the end of stage one plus the people who were infected during this stage.

Therefore,

For clarification let’s rewrite

We know that the total number of people infected by the end of n+1 stage is equal to the people infected in the previous stage plus the newly infected. The newly infected is equal to the probability of having a risky encounter with those already infected times the number of people not infected.

, where T is the total population.

However, this is not quite accurate. We need to divide the newly infected people by the total population to have the correct result. Remember that I is equal to the total possible interactions.

Therefore, the final equation is:

My spreadsheet includes the original data collected and the data generated by the formula .

Two graphs created on Excel are also included. The first graph shows the Stage versus the Collected Data. The second graph shows the Stage versus the Collected Data and the Generated Data. Notice the closeness in the two series of data.

The Department of Mathematics Education
University of Georgia