Modeling Project 3 MTH 142 Spring 2004

Department of Mathematics, University of Rhode Island

 

The number of spam attacks seems to approximately follow a logistic model (see insert).  Here is a list of the number of attacks P (in millions), starting on January 2001 (t = 0) through  october 2002 (t = 21 ): 0.74, 0.81, 0.89, 1.01, 1.14, 1.301, 1.50, 1.74, 2.01, 2.31, 2.64, 2.98, 3.33, 3.67, 4.00, 4.30, 4.57, 4.80, 5.00, 5.16, 5.30, 5.41 Your job will be to find a suitable model for this data, and to use it to determine some trends.

Write a report that has the following parts.

1. Title, Author, Date, Course and Section, Instructor.
2. Calculate numerical estimates of $\frac{dP}{dt}$ for  t =1 through t = 20. For this, use the "central difference " approximation $$\frac{dP}{dt} \approx \frac{ f(t+h)-f(t-h) }{ 2 h }$$
3. Plot of $y=\frac{dP}{dt}$ (vertical axis) versus $P$ (horizontal axis) for the points you have information on (see previous question)
4. Use Maple's fit command to fit a polynomial curve
   $$y = a P^2 + b P + c$$
   to the points in the previous question by choosing suitable values of the parameters $a$, $b$, and $c$. Plot together the points and the curve that you have chosen, so that it can be verified (visually) that you have a ``reasonable fit''.
   
   
5. The differential equation
   $$\frac{dP}{dt} = a P^2 + b P + c \quad \quad {\rm (DE)}$$
   is satisfied approximately by  the number of spam attacks $P$ (in millions) as a function of time $t$(in months). Explain why.
   
   
6. Use algebra and Maple to determine if there exist equilibrium values of $P( t )$. Verify your answer by producing a slope field plot with 1 < t < 30  and 0 < P < 10.. Comment on whether the plot confirms or not your conclusion on equilibrium values.
7. Generate a slope field plot with 0< t < 100 and 0 < P < 10 .   What does the model predict the number of attacks will be in January of 2003? What will be tthe number of attacks in January of 2008?

Tips, comments, and additional information

$\bullet$

In this example, points and a curve are plotted together:

  with(plots):      

  p1 :=PLOT(POINTS([1,2],[2,4],[1.5,3]));     


  p2 := plot(x^2,x=0..2):      

  display([a,b]);

To use differential equations related commands in Maple you must first load the package DEtools. Here is an example of how one plots in Maple a direction field and a solution of the logistic differential equation $\frac{dx}{dt} = 3 x (1- x/300 )$. Note the x(t) term:

  with(DEtools):      

  de1 := diff(x(t),t) = 3*x(t)*(1-x(t)/300);     


  DEplot(de1,x(t),t=0..3,x=0..400);

Here is how to plot the slope field and a particular solution:

DEplot(de1,x(t),t=0..3,x=0..400,[[x(0)=50]]);

Maple can solve some differential equations, for example:

dsolve({de1,x(0)=50},x(t));

Curve fitting in Maple may be done with the fit command. Here is one example:

with(stats):

  xvals := [-2.,-1.,0.,1.,2.];

  yvals := [-3,0,1,0,-3];

  fit[leastsquare[[x,y], y=a*x^2+b*x+c]]( [xvals,yvals]);

$\bullet$

The final project should have only one author. You may discuss the project with your classmates, but what you turn in should contain your own original designs.

$\bullet$

Before each calculation, describe in English what you are about to do.

$\bullet$

Maple should be used in all calculations and plots.

$\bullet$

For basic information on Plotting, solving equations and calculating integrals in Maple: see the Maple worksheet `` Introduction to Maple in Calculus II''(intro142.mws) , located in www.math.uri.edu/Center/

$\bullet$

MAPLE HELP is be available. The schedule and location is announced in www.math.uri.edu.