Grading Guide for the

S-I-S Disease Endemic Limit Project

If you got help from someone, remember to acknowledge it in your report.  This could go right after your introduction.

General Remark on use of Help NoteBooks

The Help NoteBooks ARE intended to help you organize your project AS A MATHEMATICA NOTEBOOK, but we take offense to your leaving in the hints like:

Since a flu epidemic takes place in __ days, the number of births, deaths, and immigrations bla, bla, blah...

or

Therefore, the rate of of the fraction of the population

This sentence fragment is a HINT to help you conclude:

Therefore, the daily rate of creating new cases as a fraction of the population is  a s i,

daily rate of new cases = a s i  (in units of people per total population per day)

Please READ what you submit.  It is supposed to be clear and self-contained.  "Blah, bla, bla" is not clear and self contained.  Ask yourself, "Would my high school teacher understand this report?"

Use of Mathematica as an Editor

Use text cells for text, don't type prose in input cells (adding your own returns.)

Use separate text cells for each paragraph.

Use Section, Subsection, Subsubsection cells to group appropriate portions of your project and organize your thinking.

Lab monitors can help you with these technical details.  ASK - check the sign on the door for who's on duty or go to

http://www.math.uiowa.edu/reserv.html

Main Portions of Your Grade

Introduction (3 pts)

The Introduction should be a very clear statement of the problem and a non-technical and inviting statement of the conclusion.  For example,

In this paper we study "S-I-S diseases" that are spread by casual contact.  "S-I-S" means susceptible, infectious, susceptible.  S-I-S diseases do not confir immunity; when you recover from an infection you become susceptible and can get the disease again.  We develop a model that applies to diseases like strep throat in a large population like a college campus.  These diseases have the potential to become "endemic," in other words, to be perpetually passed around in the population and not die out.  Our model uses a parameter  c  that is a measure of the number of potentially disease spreading contacts an infectious person makes and this parameter determines when the disease will become endemic or die out.  The parameter  c  predicts the nature of the disease: If  c > ???, the population tends to a limit where ( 1 - ???)X100 % of  the population is always infected, whereas if  c < ???, the disease dies out in time and everyone is well (but susceptible).  Therefore, knowing  c  predicts whether the disease will become endemic.

PLEASE WRITE YOUR INTRODUCTION IN YOUR OWN WORDS.

Derivation of the model  (6 pts)

We are looking for a clear description leading up to:

daily rate of recoveries = b i  (in units of people per total population per day)

daily rate of new cases = a s i  (in units of people per total population per day)

BE CAREFUL WITH UNITS

These terms are used to describe the change in susceptible and infectious fractions:

the daily rate of change in s = new recoveries added - new cases subtracted

(ds/dt) =  b i  -  a s i

the daily rate of change in i = ...

((d ... )/dt) =   ...

Computer Experiments  (6 pts)

We are looking for ONE copy of the baisc code (the later experiments can have the code in a closed cell or cut) and lots of experiments with various sizes of  c.  We are also looking for a text description of what each individual experiment means.

Conjecture (2 pts)

By the end of your experiments you should have observed that when  c  is very large most everyone ends up sick and when  c is very small, everyone recovers in time.  HOWEVER, YOU CAN DO BETTER THAN THIS VAGUE STATEMENT.  When  c = 3/2,  s[t]  tends to  2/3, when  c = 3,  s[t]  tends to 1/3,  when  c = 4,  s[t]  tends to 1/4,  you should have a mathematical guess like:

When  c < 1, the disease dies out in time, (Underscript[limit, t → ∞]) s[
            t] = 1 and (Underscript[limit, t → ∞]) i[
            t]  =  0.  The limit  (Underscript[limit, t → ∞]) s[
            t]  =  1 means that when  t  is sufficiently large,   s[t] ≈ 1.  When  s[t] ≈ 1, nearly everyone is ...  and almost noone is  ....

When  c > 1, the disease has an endemic limit, (Underscript[limit, t → ∞]) s[
            t]  =  ?? and (Underscript[limit, t → ∞]) i[
            t]  = 1 - ?? , meaning that  (1-??)X100% of the population is always sick.

Proof (6 pts)

Show mathematically that your conjecture is true.  Suppose for starters that  c < 1.  You want to show that  (Underscript[limit, t → ∞]) s[
            t]  =  1, so just show that  s[t]  is increasing if it is less than 1.

A simpler form of the model: (ds/dt) =  b (1 - s) (1 - c s)

Here's some help with the form of the equation: (ds/dt) =  b i  -  a s i.  We know s + i = 1, so  i=1-s.

(ds/dt) =  b i  -  a s i  = 
          b (1 - s) - a s (1 - s)

We know  a = b c, so

(ds/dt) =  b (1 - s) (1 - c s), for positive constants  b  and  c.

This means we can study the single equation

(ds/dt) =  b (1 - s) (1 - c s)

rather than the pair of equations

(ds/dt) =  b i  -  a s i

(di/dt) =  a s i -  b i

Proof of our conjecture when  c < 1:

s < 1  because  s is a fraction of the total population, so  0 < 1-s  (subtract s from both sides), (1-s) is positive.

If c < 1, then  c s < s < 1, (multiplying both sides by s) and  0 < 1 - c s (subtracting cs from both sides of c s < 1)

In other words, we have shown that the terms  b, (1-s), and (1 - c s) are all positive when  0<c<1  and  s < 1.  This means  (ds/dt)is positive BECAUSE ...???  and  s[t]  keeps growing as long as it is less than  1, hence

(Underscript[limit, t → ∞]) s[t]  =  1

the limit of i[t] is ???  BECAUSE...  the two limits mean ...

Proof of our conjecture when  c > 1:

s < 1  because  s is a fraction of the total population, so  0 < 1-s  (subtract s from both sides), (1-s) is positive.

When  c > 1, 1/c <1 so we can have three cases of  s,  1/c < s < 1,  0 < s < 1/c and  s = 1/c.

Notice that if  s = 1/c,  (1 - c s) = (1 - (c/c))  = 0, and  (ds/dt) = 0.  If  s[t]  ever equals  1/c, it stays there because the slope of the curve is zero.

Next, suppose  s < 1/c...

...

Finally, suppose  1/c < s < 1...

Summarize these two cases with  c > 1 together so that no matter where  s[0]  starts (between 0 and 1), (Underscript[limit, t → ∞]) s[
            t]  = 1/c... meaning ...

Conclusion (2 pts)

We are looking for a somewhat more technical or detailed conclusion than you had in the last sentence of your introduction.