(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 35727, 1313]*) (*NotebookOutlinePosition[ 36782, 1349]*) (* CellTagsIndexPosition[ 36738, 1345]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[ "Mathematics 162 Laboratory 8 \ Week of March 22, 1993\n\t \nName: _____________________________ Lab \ Partner: ___________________________\n\t \nConsulted with: \ ____________________________________________________________"], "Text", CellMargins->{{17, Inherited}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{1, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData["\[Copyright] Lafayette College, 1994."], "SmallText", Evaluatable->False, TextAlignment->Right, AspectRatioFixed->True], Cell[TextData["A Model for an Epidemic"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Suppose a contagious disease has appeared in a population. 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Suppose that ", Evaluatable->False, AspectRatioFixed->True], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 0.9 and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["d", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" = 0.01. 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What will happen to the proportion of people infected over the \ long run? (You may plot more terms if you wish, to support your answer.)"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Now consider the case when the disease is a little less mild; take ", Evaluatable->False, AspectRatioFixed->True], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " = 1.1. 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Over the long run?"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part 2: Analysis of the Model using Limits"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Show that if the sequence {", Evaluatable->False, AspectRatioFixed->True], StyleBox["d", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["n", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["} has a limit ", Evaluatable->False, AspectRatioFixed->True], StyleBox["L", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", then the limit must satisfy the equation:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[" L", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["k L", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" (1 \[Dash] ", Evaluatable->False, AspectRatioFixed->True], StyleBox["L", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[").", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Hint:", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" Take the limit of the recursive definition as ", Evaluatable->False, AspectRatioFixed->True], StyleBox["n", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" goes to infinity.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Use the equation you just derived to find the limit of the sequence in \ each of the two cases considered in Part 1. You may use ", Evaluatable->False, AspectRatioFixed->True], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", but write your answers below.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Public health officials often have some control over the parameters ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["b", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", and hence over ", Evaluatable->False, AspectRatioFixed->True], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[". For instance, they may be able to decrease ", Evaluatable->False, AspectRatioFixed->True], StyleBox["b", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" by giving proper treatment and they might be able to decrease ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" through quarantines, or education. For what values of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" will the disease tend to be eliminated from the population?", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part 3: Dependence on parameters"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["We have seen that given two constants: the value of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", and the initial value of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["d", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ", we can find a sequence to model an epidemic. We call these constants ", Evaluatable->False, AspectRatioFixed->True], StyleBox["parameters", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ". In this part we will examine how the evolution of an epidemic depends on \ these parameters.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Assuming ", Evaluatable->False, AspectRatioFixed->True], StyleBox["d", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[ " is still 0.01, investigate the long term effect of an epidemic on a \ population with each of the following ", Evaluatable->False, AspectRatioFixed->True], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " values: 2.8, 3.84, 4.0. In each case, write a few sentences describing \ your conclusion.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n\n\n\n\n\n\n\n\n\n\n\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Suppose that in the case when ", Evaluatable->False, AspectRatioFixed->True], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 4.0, the sequence had a limit. What would it be?", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Finally, consider the effect of changing the initial value ", Evaluatable->False, AspectRatioFixed->True], StyleBox["d", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[". Define a sequence using ", Evaluatable->False, AspectRatioFixed->True], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 4.0 and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["d", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" = 0.75, and plot its graph. Repeat this using ", Evaluatable->False, AspectRatioFixed->True], StyleBox["k", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" = 4.0 and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["d", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontSize->9, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[ " = 0.7500001. Compare the long term behavior of these two sequences.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n\n\n\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Suppose a public health official said \"The disease is not at all mild, \ the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["a", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[" value is about 3.1 and the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["b", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " value is 0.9, but we appear to have the epidemic under control. The \ proportion of the population that is infected has been staying near 58% for a \ few days now.\" What would you reply? Should the official expect the \ epidemic to remain under control for long? What can you tell her concerning \ the future of this epidemic? 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