(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing 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[ 50415, 1631]*) (*NotebookOutlinePosition[ 51153, 1658]*) (* CellTagsIndexPosition[ 51109, 1654]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[StyleBox["Spread of Disease; A project using differential \ equations", FontColor->RGBColor[1, 0, 0]]], "Subtitle"], Cell["\<\ Don't forget to drag the icon for this notebook to your own desktop \ before using.\ \>", "Subsubtitle"], Cell["\<\ Oral presentations are to begin on Thursday 9/11. The written \ reports are due in class on Tuesday, 9/16. Follow the project guidelines I \ handed out on the first day of class.\ \>", "Text"], Cell[CellGroupData[{ Cell[TextData[{ " solutions to differential equations. ", StyleBox["(Read this to get started, then delete it.)", FontColor->RGBColor[0, 0, 1]] }], "Subsection", FontFamily->"Geneva"], Cell[TextData[{ "The following example shows how to use DSolve to solve a differential \ equation. The equation in this example is one you've probably already \ studied: ", Cell[BoxData[ \(TraditionalForm\`dy\/dt\)]], "= -k y This is the equation for nuclear decay, for instance. Note that \ in the example I have replaced k with a value, 2, and that the initial \ condition is also specific, y[0]==10. Note also that the output is the \ function which solves the equation. You can solve this with paper and pencil \ too--this might be good review!" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[{\(y'\)[t] == \ \(-2\)*y[t], y[0] == 10}, y[t], t]\)], "Input"], Cell[BoxData[ \({{y[t] \[Rule] 10\ E\^\(\(-2\)\ t\)}}\)], "Output"] }, Open ]], Cell["\<\ If you want to plot the solution you can make the usual \ moves.\ \>", "Text"], Cell[BoxData[ \(Clear[f]; f[t_]\ := \ 10\ *E\^\(\(-2\)\ t\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(plot1 = Plot[f[t], {t, 0, 3}, \ PlotRange -> {0, 12}, AxesLabel -> {"\", "\"}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.31746 0 0.0515028 [ [.18254 -0.0125 -9 -9 ] [.18254 -0.0125 9 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", StyleBox[" ", FontWeight->"Plain"], StyleBox["(10 points)", FontVariations->{"CompatibilityType"->0}], StyleBox[" One model for the spread of a disease is the logistic equation. \ Remember, the logistic differential equation is", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`dp\/dt\)], FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["= k *p *(L - p ) where k is a constant and L is the limiting \ value, in this case the total number of people who can get sick, and t is \ measured in days. \n\nSuppose that 50 people in Rohnert Park have the flu on \ February 1 (we will say this is at t=0). Suppose that k=.0006 and that an \ official from the Public Health and Safety Department believes that 5000 \ people in Rohnert Park will eventually get the disease (she approximates this \ from data on past flu seasons). She would like you to develop a model of the \ number of people who have gotten sick as a function of time after t=0. \ (Hint: If the solution contains a giant number, 2.71828.... , remember that \ this is \"E\".)\n\nWhat does the model show? What does it look like? What \ is the rate at which people get sick -- that is, ", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ \(TraditionalForm\`dp\/\(\(dt\)\(\ \)\)\)], FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["(you won't get hints like this in the future)? What does it look \ like? What is the exact time at which the disease is spreading most rapidly? \ Use two techniques: make an appropriate plot and solve the appropriate \ equation. (Hint: Neither Solve nor NSolve can handle these equations. \ Hurray, it's FindRoot to the rescue! The syntax is FindRoot[lhs = = \ rhs,{t,bestguess}] where lhs is the \"left hand side\" of the equation, rhs \ is -- you guessed it -- and bestguess is a reasonable guess of the \ solution. Mathematica will use numerical techniques to find a solution.) \ What is the approximate number of people who get sick during the period from \ 12 midnight on day three to 1 AM on the same day?", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }], "Subsubsection"], Cell[TextData[{ "2. (20 points) ", StyleBox["Now here's the creative part. The logistics differential \ equation you've been using has k= 0.0006, a number which remains constant \ throughout the spread of the disease. It can be interpreted as indicating \ the \"communicability\" of the disease. (If k=0, no one will get the disease \ from the original 50 people.) It makes sense that as more people get sick \ other people will take measures to protect themselves. Therefore, we might \ regard k not as a constant, but as a function of the number of sick people, \ k(p). Further, we might reasonably expect that k declines in value as p goes \ up. What other assumptions about k(p) would be reasonable to make? \n\nYour \ last task is to replace k = 0.0005 with a function of p such that the \ function starts out at 0.0005. In fact, argue the pros and cons of three \ different possible functions k(p) which might be used. Can you solve these \ differential equations and plot the solutions? How do your new models \ compare with the original model? Which is your optimal solution and why?\n", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }], "Subsubsection"] }, FrontEndVersion->"4.0 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowToolbars->"EditBar", CellGrouping->Manual, WindowSize->{673, 871}, WindowMargins->{{12, Automatic}, {29, Automatic}}, PrintingCopies->1, PrintingPageRange->{1, 9999} ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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