Math 122 Calculus for Biology II Fall Semester, 2012 Lab Help	08-Sep-12
	San Diego State University

This page is designed to provide helpful information about the laboratory questions. You should start with your Cover Page and be sure that you have all the important contact information about your lab partner for coordination later in the week. On the cover page you begin by typing in the name of each team member and your Computer numbers. For your WeBWork answers, it is important that you keep all calculations to 5 or 6 significant figures. Too few digits have been one of the leading causes of errors that we have found when approached by students.

This particular lab examines a discrete nonlinear dynamic model and two optimization problems. The first two questions examine how to create optimal volume objects, a box and a tent. The third question examines CDC data for influenza using a classical Discrete Dynamical system known as the SIR model.

Before discussing the problems for this lab, we want to remind students of an easy means of transferring data from the WeBWorK lab questions to Excel. One easy way to do this accurately (which is especially important for large data sets) is to first create a hardcopy of your lab in WeBWork (which you should probably do anyway for easy reference in the future). Open the hardcopy in Acrobat Reader (default setting for PDF file). Next find the data set that you want in Excel and highlight the appropriate data. Copy this data and paste it into Excel. In Excel, proceed to the tab labeled data and select text to columns.... A window pops up, in which you select Delimited, then click on Next and check the boxes for Tab and Space. The result is that your data should be expanded into columns and are ready for work in Excel.

Question 1: This problem is one of the classic problems in using Calculus to find an optimum. You may want to take a sheet of paper and cut squares out of the 4 corners to give you some intuition into this problem. Recall that the volume of a rectangular box is the product of the length times the height times the width. The surface area will clearly be the area of the sheet of cardboard minus the area of the pieces cut out of the cardboard. By trying to cut out square corners from a sheet of paper, you should be able to figure the limits of the domain. (Obviously, one is x = 0 by not doing any cutting from the paper.) For the graph of the volume as a function of the surface area, you may want to connect the answer in Part a (value of x) to the function in Part b to get the value of S at the maximum V.

Question 2: This problem is another standard problem in using Calculus to find an optimal volume. You will first want to look up the volume of a pyramid and use the diagrams provided to develop intuition on converting the variables into ones required for computing the volume of the pyramid. You may want to print the figures in the hyperlink for the problem, cut out the appropriate shape, and fold them to observe relationships between key variables. Once you have determined the volume, V, as a function of x, then the Calculus portion of the problem is easily handled by hand or Maple.

Question 3: This question expands the single population modeling concept as illustrated in the previous lab to a system of discrete dynamical equations. Data are provided from a flu season tracked by the Center for Disease Control (CDC). The Susceptible-Infected-Recovered (SIR) model is used, and the simulation is assumed to be over a closed population that doesn't grow. Thus, the model is assumed to examine a well-mixed population with a constant size, which is given. This easily allows you to eliminate one of the variables in the SIR model and reduce our modeling effort to the study of only the infected (I_n) and susceptible (S_n) individuals in the sample population. The data only tracks the infected individuals, so we only fit our data to this class. 

Begin by copying the data from the CDC into Columns A and B on an Excel Spreadsheet. Create named parameters, b, g, and N on the spreadsheet (say in Columns G and H) to use in the model with the initial values given in the Lab problem. In Columns C and D we insert our model., starting with the initial populations of susceptible (S₀) and infected (I₀) individuals. In the cells below the initial populations, you insert the equations given in the Lab problem using the named parameters. In Column E, you find the square error between the CDC data on infected individuals and the numbers given by the model. As we have done before, you sum the square error between the model and the data using Excel's Solver changing the parameters b and g. Note that you are only fitting the population of infected individuals to the data, but you use the information about the susceptible individuals to determine the total number of people getting the flu over the season.

The remainder of the problem has you simulate the model with changes in S₀, b, or g, to reflect treatment plans of vaccination, education/isolation, or antiviral medicine. You simply change the appropriate parameter, then simulate the model as you did in the first part of the problem. You will report quantitative information that you gather and discuss something about these different strategies and their efficacy.