Afternoon session The Same Difference: the spread of rumors, catching the flu, and more math

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Mon, July 19 in FWO 208: (class observed)

• Intro to Excel: save a file; copy, paste; clear all, clear formats; format cell number, alignment, border, patterns; insert or delete a row or column; change column width or row height; relative addressing; entering a formula; dragging to create an arithmetic sequence; rename a tab
• Took some notes in a Word document.
• formula for interior angles of a regular polygon
  http://www.mathsisfun.com/geometry/polygons.html
• formula relating vertices, edges, and faces of a polyhedron - verified for the platonic solids.
  http://www.mathsisfun.com/platonic_solids.html
  http://plus.maths.org/issue43/features/kirk/

Tues, July 20 in FWO 208:

• crickets and temperature.
  Read equations from http://www.sciencebuddies.org/science-fair-projects/project_ideas/Zoo_p055.shtml
  Listened to multiple cricket songs from http://entnemdept.ufl.edu/Walker/buzz/585a.htm . Noticed general relationships between the speed of the chirps, the pitch, and the temperature.
  From here click the "Previous" link at the bottom to see the cricket whose data we'll use. Listened to these songs and those of other cricket species.
  Discussed the Sound spectrograms of each page.
  Download CricketData.xls that Shelton gleaned from Walker's site. Discussed what the entries are in the data, how the sounds were "captured", etc.
• Examined the temperature values in Celsius from the cricket data; a student found a website to convert Celsius to Fahrenheit. We used this to generate 2 F values from the first C values in the data. We entered these in a spreadsheet, along with (0, 32) and (100, 212) as the freezing point and boiling point of water, respectively. (Discussed the use of "respectively.")
  Performed algebra on the board to create the linear equation. Calculated the slope and the intercept values in the spreadhseet. Typed a bunch of C values. Coded the conversion using absolute reference in Excel.
  Created a graph of the ordered pairs. Added a trendline; noticed the points are exactly on a line.
• We copied temperature and "P/S" (pulses per second) from the cricket data into another worksheet. We graphed these, added a trendline with the equation and R^2 value.
• Returning to the C-F spreadsheet, we added a trendline with the equation and R^2 value.
• We discussed this and wrote up some of our findings in Word, including an interpretation of the R^2 value.

Wed, July 21 in FWO 208:

• Using the "F from C" equation we obtained last time, use algebra to create a "C from F" equation.
  Copy the values into another worksheet to have (F, C) values. Create a graph, add a trendline with the equation and R^2.
  Compare the equations from algebra and the trendline for temperature conversion.
• Copy and paste the cricket data to have ("P/S", "C") values. Graph, add a trendline with the equation and R^2. Use algebra to show that the trendline equations are not equivalent.
• Revisit the first equation from http://www.sciencebuddies.org/science-fair-projects/project_ideas/Zoo_p055.shtml.
  Discuss how this is different from our info.
  Copy and paste the cricket data to have ("P/S", "C") values.
  Insert a column and code the "N₁₄" values, meaning the number of pulses in 14 seconds. (Uses the idea of "rate" that we discussed in the morning class.)
  Insert a column and code the Fahrenheit values.
  Graph ("N14", "F"), add a trendline with the equation and R^2.
  What species of crickets were used?
• All of this serves as an introduction to modeling, a refresher of some algebra, calculations and graphs in Excel, trendlines, R^2 values, least squares fit to data.
• Each student tried their hand at fitting a trendline to find the "least squares" at least squares.
• We discussed this and wrote up some of our findings in Word.

Thurs, July 22 in FWO 208:

• Simulation of disease spread within our group of 5. (We began with different sizes of paper, and one had a red dot. The students decided to start over with indistinguishable papers so we could really randomize our choices.)
  Entered data into Excel and made graphs of x="number of steps", y="number sick".
  Wrote up discussion in Word: what we did, our data, our graph, what it means, general shape of the graph is non-decreasing.
• briefly looked at noaa.gov site on the current tropical depression: graphic of possible path, proabilistic paths; probability tables for wind speed; Mariner's Rule as an example of modeling. Discussed the meaning of the probability information.

Fri, July 23 in FWO 208: SU computers offline.

• Simulation of disease spread with Dr. T and the V2S2 group of 15. We did 2 runs by drawing paper names.
  Recorded in a matrix where "1" meant the person had the disease. Each column represented a single person; a new row was begun for each trial.
  Manually drew graphs of x="number of steps", y="number sick".
• Investigated various dice and chatted about equally likely, randomization.
• Performed a final simulation with each of us rolling a 20-sided die; ignored values 17-20 and just rolled again. Graph by hand on graph paper.
• Discussed the shape of the graphs -- "S" or "sigmoidal". Think about what they are always like this.
  Discussed how this is an "S-I" model, for "susceptible" and "infected." Mentioned other models that include "recovered" and "immune."
  Discussed assumptions for our model -- fixed population; no one leaves the group and no new people enter.
  If a person was sick in the last run and we chose that person again, s/he remained in contact with only him/herself or with other sick people.
• Showed a "loaded" set of dice.

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Mon, July 26 in FWO 208: (class observed)

• Discussed the random numbers we generated before. We needed the values to be equally likely; were they?
• Random number generation in Excel.
  Download DiseaseModel.xls and do further work. ( "randbetween(1,16)" didn't work without an add-in so we used "trunc(rand()*15)+1")
  Edit/MoveOrCopySheet/"create a copy"; coded our last simulation; Copied this and coded the 2nd simulation from last time.
• See sigmoidal math at http://mathworld.wolfram.com/LogisticEquation.html and the effect of "r".
  See the r values from DiseaseModel.xls and our additional work.
  We noted that the math in the equations is beyond their experience but the concepts of the graphs are not.
• Discussion and write-up of why the model and data are S-shaped. Related to the spread of rumors. Would this work if we posted gossip on the internet?
• good sigmoidal graph from flu data http://www.cdc.gov/flu/weekly/weeklyarchives2007-2008/weekly20.htm . This shows that our simulation reflects reality. Looked at the difference between some graphs which show number of new flu cases in a week and those which show the cumulative number of cases. Note that the time step is one week. Looked at the number of hospitalizations for different age groups and the seasonality of the flu.
• See http://www.cdc.gov/flu/weekly/ for "threshold", seasonality, graphs, and data. Copied the table from http://www.cdc.gov/flu/weekly/weeklyarchives2009-2010/data/AHDRA20.htm (using Safari) into Excel. Coded cumulative number of hospitalizations and graphed.

Tues, July 27 in FWO 208:

• Pan(dem)ic?, which includes an exponential model for the number of new cases that we coded in Excel.
• The Mathematics of Diseases. Both of these discuss the "reproductive number; the second article gives values for different diseases. The second mentions an "S-I-R" model, for "susceptible," "infected," and "recovered."
• A similar result for a different application: "The market for a new good grows according to a logistic curve if the number of buyers adopting it at a given time follows a normal distribution." http://demonstrations.wolfram.com/LogisticSigmoidMarketModel/. Same model, same math, different situation; "the same difference."
• See the temperature equation and graph or the cricket trendline. Discuss the rate of change.
• Yet another application which is modeled by a sigmoidal curve: ecology and population growth, http://ats.doit.wisc.edu/biology/ec/pd/t2.htm Answer the "quiz" question after watching the fish animation.
  Incorporated description into our Word document for our model.
• Coded another model of disease spread using a matrix and an age-structured model to predict the approximate number who stay well, become ill, or are infected but do not become ill. (McCown and Sequeira p 580-581)

Wed, July 28 in FWO 208:

• Write up: bell-curve; first matrix model.
• Markov Chain models and corresponding "flow" diagrams. Limiting values in the powers of the matrix.
• Look up the definition of Markov Chain.
• Write up a description of the discussion and results.

Thurs, July 29 in FWO 208:

• Difference equation models.
• If time permits, prepare for presentations.

Fri, July 30 in FWO 208:

• Prepare for presentations.
• Just one break today so students can leave early to prepare for Volente Beach trip!

- - - - - - - C E L E B R A T I O N ! ! ! - - - - - - -

Sat, July 31

• Presentation

References

"Ask a Cricket, 'What Is The Temperature?'" Science Fair Project Ideas, Answers, & Tools. Science Buddies, Inc. Web. 15 July 2010. http://www.sciencebuddies.org/science-fair-projects/project_ideas/Zoo_p055.shtml.

Baker, Samuel L. "Least Squares Demo." Web. 21 July 2010. http://hadm.sph.sc.edu/COURSES/J716/demos/LeastSquares/LeastSquaresDemo.html.

"CDC - Seasonal Influenza (Flu) - Flu Activity & Surveillance." Centers for Disease Control and Prevention. Web. 23 July 2010. http://www.cdc.gov/flu/weekly/fluactivity.htm. Also http://www.cdc.gov/flu/weekly/weeklyarchives2009-2010/data/AHDRA20.htm and http://www.cdc.gov/flu/weekly/.

Clapham, Christopher. The Concise Oxford Dictionary of Mathematics. Oxford: Oxford UP, 2005. Print.

Freiberger, Marianne . "Pan(dem)ic?" Plus.maths.org. Web. 23 July 2010. http://plus.maths.org/latestnews/may-aug09/swineflu/index.html.
The home site is edited by Freiberger and Thomas.

Kalman, Dan. Elementary Mathematical Models: Order Aplenty and a Glimpse of Chaos. Mathematical Association of America, 1997. Print.

"Logistic Growth." Connecting Concepts: Interactive Lessons in Biology. Web. 26 July 2010. (produced collaboratively at the University of Wisconsin-Madison) http://ats.doit.wisc.edu/biology/ec/pd/t2.htm

Matthew, Keeling. "The Mathematics of Diseases." Plus.maths.org. Web. 23 July 2010. http://plus.maths.org/issue14/features/diseases/.

McCown, Jack R., and Michael A. Sequeira. Patterns in Mathematics. Boston: PWS Pub., 1994. Print. p35, 55, 59-60

National Hurricane Center. NOAA - National Oceanic and Atmospheric Administration. Web. 22 July 2010. http://www.nhc.noaa.gov/

Schreiber, Michael. "Logistic Sigmoid Market Model." Wolfram Demonstrations Project. Web. 25 July 2010. http://demonstrations.wolfram.com/LogisticSigmoidMarketModel/.

Walker, T. "Snowy Tree Cricket (Oecanthus Fultoni)." University of Florida Entomology and Nematology Dept. Web. 15 July 2010. http://entnemdept.ufl.edu/Walker/buzz/585a.htm .