See the attached fileThe link is the sample https://faculty.math.illinois.edu/~kleisman/course…Math 441 Differential Equations — Spring 2020 — Homework 4
• Write your name at the top of the first page, and write your class meeting time
at the top also. The assignment must be stapled in the top left corner.
• You may write your solutions to this assignment by hand, or else use LaTeX or other
software. If you submit hand-written solutions, they must be neat and well labeled,
and easy to follow. All graphs must be produced by software (not hand-drawn).
SIR model for the spread of disease, continued
In this homework assignment we use software to plot solutions for a system of differential
equations, and investigate the effect of varying the recovery parameter k and the infectionspreading parameter b.
Source Material (click to download)
1. Euler’s Method for Systems (https://www.maa.org/press/periodicals/loci/joma/thesir-model-for-spread-of-disease-eulers-method-for-systems)
2. Relating Model Parameters to Data (https://www.maa.org/press/periodicals/loci/joma/thesir-model-for-spread-of-disease-relating-model-parameters-to-data)
Direction field plotter (https://www.bluffton.edu/homepages/facstaff/nesterd/java/slopefields.html)
Read This First! We will use the “System” option in the plotter. Since the plotter displays
only two equations, we will handle the DEs for s(t) and i(t) and omit the equation for the
recovered fraction r(t). That equation for r(t) is not essential, because r(t) = 1 − s(t) − i(t)
and so if we know s and i then we can get r. Thus you will generate graphs for s(t) and i(t),
but not r(t).
The plotter will show a 2-dimensional “phase plot” with s- and i-axes, but no t-axis.
(We will study such phase plots later in the semester.) For this assignment, in order to get
solution graphs of s(t) and i(t), do the following:
• click on “System”,
• click on “Numerical solution tables, and timeplots (for systems)”,
• create solution curves by clicking in the direction field (or else enter initial conditions
in the boxes), noting that the two curves plotted may have different vertical axis limits,
shown in red and blue,
• make sure your plotted domains are appropriate for all variables (s, i and t), noting
that you can set the s and i domains just below where you input your DE, and you
can change the t domain with the Zoom + and – buttons to the lower right of the plots
of s(t) and i(t)),
• save the solution graphs as an image by taking a screenshot. (The plotter does not
seem to have a command for exporting solution graphs.)
Part A. Do Exercises 2–4 in Euler’s Method for Systems.
But Read This first:
For each exercise, your answer will consist of a plot and some answers to the written
questions. Choose appropriate viewing windows in order to get good graphs.
Ignore Exercise 1, because you are using the direction field plotter and not their “helper
application (CAS) worksheet”.
In Exercise 2, ignore the instruction to superimpose your solutions on the “exact” solutions from Step 1. Instead, First generate numerical solutions using the Euler method (use
the values for b, k, s(0), and i(0) given in exercise 1) as described in exercise 2. Then look
again at the solution graphs shown below for s(t) and i(t) (from the Differential Equations
Model) and compare with the plots you generated. Do they have roughly the same shape
and size, or not? Specifically comment on any similarities and differences.
In Exercise 4, to be confident that your step size is small enough to generate accurate
solution graphs, you should reduce the step size even further and see whether the solution
changes shape (it may help to look carefully at the trajectory in the phase plane, which plots
s vs i, and see if it changes as you change h). Once you have found a small enough step size,
keep using it for the rest of the assignment. (Do not go back to using step size 10.)
Part B. Do Exercises 1–7 in Relating Model Parameters to Data.
(Exercise 6 is optional, no credit.)
Read this first:
For Exercise 1, make one graph for each value of b that you use. In your homework
solution, label each graph with the problem number (B1) and the value b =. . . that you
used. (Ignore the instruction to vary colors and overlay graphs.)
For Exercise 3, make one graph for each value of k that you use. In your homework
solution, label the graph with the problem number (B3) and the value k =. . . that you used.
(Ignore the instruction to vary colors and overlay graphs.)
For Exercise 5, you should describe the change you see. Try changing k by increments of
0.05 and see which value is closest to the change. (You may want to add some plots to your
solution for Exercise 3 to illustrate this.)
Exercise 6 is interesting, but optional (no credit).
For Exercise 7, include three or four observations of how the model does or does not
reasonably correspond to the data. Be thoughtful, but brief.
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