Logan Milliken, Connor Degenhardt, Joe Lang

   Use of SIR Models in Epidemics


   The title sounds a bit strange here because the SIR models are not
   being used to produce epidemics(!)  Maybe "... in the Study of Epidemics
   and Methods for Their Control."  Your project paper on SIR models and 
   applications is reasonably good, but I think your oral presentation was stronger. 
   You have an OK conclusion as well, rather than just sort of "petering out" 
   without tying together the points you have made.  I know it can be hard 
   to manage things like this when you are "writing by committee." Making 
   sure you are communicating and coordinating your efforts is an important 
   part of working in a team and that aspect of the paper was pretty 
   successful.  

   The major criticism I have is that the paper is somewhat skimpy.  It would 
   have been good to include some more examples and a more explicit discussion
   of how quarantines could be modeled to illustrate the models you are discussing and 
   to show more concretely what the effects of vaccination and/or quarantine
   measures would be.  In particular, the questions of how many people must
   be vaccinated to control an outbreak are interesting and they could have
   been attacked with some of the same tools used in the discussion from
   the text on page 148 (the basic reproduction number).  

   Your first reference in the Works Cited does not have a title.

   Some specific comments

   (1) Page 2:  We have not yet developed effective vaccines for all diseases 
                and there also needs to be a functioning public health 
                infrastructure in place to administer the vaccine and monitor
                progress of the disease and the countermeasures.

   (2) Page 3:  You're presenting a continuous-time SIR model because the 
                left sides of your equations (dS/dt, dI/dt, dR/dt) are
                time derivatives.  This is different from the discrete-time
                difference equation models we looked at in class.  Where you 
                have dS/dt, for instance, we had S(n+1) - S(n), and your value 
                of is different from what ours represented.  What you have is 
                OK, but be sure you understand the difference.  It would also
                have been good to point out that there is a difference.

   (2) Page 4:  I think there's an issue with your SIR vaccination model.
                With the -c term in the equation for dS/dt, you're essentially
                vaccinating a fixed number of people in each unit time period.
                Of course, that cannot go on forever because eventually you
                will have vaccinated everyone and the model becomes unrealistic.
                I think you actually mean something like 

                         dS/dt = -aSI - cS, with 0 < c < 1,

                where you remove some fixed proportion of the susceptible 
                population in each unit period of time.  The last equation 
                in the system also needs to have a plus sign on the term
                in the equation representing the additions to the removed
                group:

                         dR/dt = bI + cS

                It would have been really good to present a numerical example
                showing what you need to do to prevent an outbreak from turning
                into an epidemic.  

    (3) Page 5: What is the "threshold parameter" that you are discussing here?
                I think it's a version of the basic reproduction number treated
                in the textbook, but you need to say more and say how it 
                relates to c, the vaccination rate.  

    (4) Page 6: You are right that public health responses are not usually designed
                to eradicate a disease completely.  The reason is that that is 
                an expensive and difficult thing to do in many cases.  It is usually
                considered to be sufficient to take health care measures so that
                disease outbreaks are controlled and "die out" on their own.  Only
                a very few diseases have been successfully eradicated.

    (5) Page 7: What does an SIQR model look like?  How are the quarantined 
                individuals built into the model equations?  Also, avoid using 
                technical terms MSRA or SIQR unless you define them fully.  
                
    Grade:  90 (A-)