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Contemporary Mathematics: Contemporary Mathematics at Nebraska

Michelle Homp, Alyssa Seideman, Sean Gravelle, Andrew Hayes, The Mabel Elizabeth Kelly Fund

Section 13.2 Chapter 13 Exercises

Subsection 13.2.1 Section 13.1 Homework

  1. Describe two ways that the game could be changed to incorporate the idea of quarantine.
  2. Describe two ways that the game could be changed to incorporate the idea of vaccination.
  3. In its current form, the game assumes students come into contact with one another in school each day.
    1. How would the game change if you took weekends into account? Is this realistic? Why or why not?
    2. Some schools will close for a day to prevent further spread of a prevalent illness. Do you think this is a useful strategy? Explain.
  4. In its current form, the BITS game assumes that the incubation period and the symptomatic period each last one day. Go to https://www.everydayhealth.com/flu/guide/how-long-does-the-flu-last/#stagesoftheflu
     1 
    www.everydayhealth.com/flu/guide/how-long-does-the-flu-last/#stagesoftheflu
    , scroll past the information about the common cold and read about stages of the flu.
    1. How long do each of the stages last? (Note: you will need these answers for #2 on the Section 8.2 homework.)
    2. How might the BITS game be changed to better simulate an outbreak of the flu?
  5. Some illnesses, such as Influenza A, are known to be highly contagious. If we consider Influenza A to be more contagious than BITS, how might we change the simulation game to accommodate this?
  6. A norovirus (which has flu-like symptoms, but is different from the flu), is also known to be highly contagious. Unlike some illnesses, because there are so many different kinds of noroviruses, being infected once does not prevent you from becoming infected again.
    1. How could the BITS game be changed to simulate the spread of noroviruses?
    2. What impact do you think this will have on the game?

Subsection 13.2.2 Section 13.2 Homework

  1. An illness known as SPAM, which standards for Sudden Penchant for Advanced Mathematics, has been infecting Math 203 students. In addition to having an insatiable desire to solve math problems, symptoms include insomnia, distraction, and brief episodes of euphoria at inappropriate times. It is a highly contagious condition with students showing symptoms three days after becoming infected. Students generally are contagious for two days before showing symptoms and for two days after they begin showing symptoms. The symptoms last for approximately 5 days, after which students (sadly) recover.
    Starting with “Healthy” and ending with “Recovered”, build the row descriptions in the first column of a table (similar to the top half of the table in Question 1 in Section 8.2 of the course manual) that could be used to build a spreadsheet to model SPAM. You will need the following terms: Infected, Infected + contagious, Sick + contagious, Sick, and Recovered.
  2. Often it is not so easy to give a specific number of days for which each of the symptoms last. Once again visit https://www.everydayhealth.com/flu/guide/how-long-does-the-flu-last/#stagesoftheflu
     2 
    www.everydayhealth.com/flu/guide/how-long-does-the-flu-last/#stagesoftheflu
    , scroll past the paragraphs about the common cold and find the information on the flu. (You may also reference #4 on the Section 8.1 homework.) If you had to choose a number for each of the categories below, what numbers would you choose and why?
    1. Infected
    2. Infected and contagious
    3. Sick and contagious
    4. Sick
  3. The flu virus now known as H1N1, was originally introduced to the human population in June 2009 when it was known as the "Swine Flu" and became a global pandemic until August 2010. The CDC estimates that nearly 61 million people in the United States were infected with the illness resulting in 12,469 deaths. There were 575,400 deaths from the swine flu worldwide. It is now considered a "normal" flu and included in standard flu vaccines. Suppose the H1N1 flu has been spreading in a classroom for 7 days. Use the table to answer the questions below.
    Table 13.2.1. Stages of H1N1
    Stage Day 7
    H Healthy 22
    1 Infected 8
    2 Infected 6
    3 Infected 6
    4 Infected & Contagious 4
    5 Sick & Contagious 4
    6 Sick & Contagious 3
    7 Sick & Contagious 3
    8 Sick & Contagious 2
    9 Recovered & Contagious 2
    10 Recovered & Contagious 1
    11 Recovered & Contagious 0
    R Recovered 0
    1. Calculate the total number of spreaders on Day 7 (i.e. those who can transmit the illness to others).
    2. Suppose that H1N1 has a transmission rate of 2.4%. (It is actually believed to be much higher.) Calculate the expected number of new transmissions that take place on Day 7 (and would therefore begin Day 8 as newly infected).
    3. Now suppose that 75% of the students who are sick (i.e. showing symptoms) stayed home on Day 7. How many spreaders are there in this case?
    4. Based on the number of spreaders you found in part (c), and still using the 2.4% transmission rate, how many new transmissions of the illness do you expect will occur on Day 7 now?
  4. Using your table for the SPAM illness in problem #1, suppose on a particular day there are 43 healthy people in some combined sections of Math 203 and 3 students in each of the other categories (in rows 1-8). Also suppose that in any given interaction between a healthy person and an infected individual, there is a 3% chance that the illness will be transmitted.
    1. Calculate the total number of spreaders for this day.
    2. Calculate the number of new transmissions that will be passed along to the next day.
    3. Suppose that half of the students exhibiting symptoms end up staying home to read advanced math books. Recalculate the number of spreaders and the number of transmissions in this case.
  5. You are at a sandwich shop that offers six types of sandwiches (Italian, turkey, chicken salad, Rueben, ham and vegetarian) with five different “bread” options: whole wheat bun, multi-grain bun, flatbread, corn tortilla and lettuce wrap.
    1. How many different sandwich combinations can be made by choosing one of the types and one of the bread styles?
    2. Suppose you are on a gluten-free diet so only the corn tortilla and lettuce wrap options are available to you.
      1. Determine how many of the sandwich options are gluten-free by determining what percentage of “breads” you can eat from those found in your answer to part (a).
      2. Check your answer by finding the number of gluten-free options another way.

Subsection 13.2.3 Section 13.3 Homework

Now that we have considered quarantine and vaccination scenarios in class, study what happens to the outbreak when we introduce a combination of both quarantine and vaccination. Your job is to come up with a plan to minimize the spread of a very serious variation of BITS at your school. You will share the plan with the school board. You will need data (in the form of graphs, etc.) to explain your recommendation. Your recommendation should include suggested rates for both quarantine and vaccination along with your reasons for choosing these rates.
Guidelines
  • Be sure that you are on the “and with Vaccination” page of the spreadsheet. Set the total number of people to 100 and the transmission rate is to .01, and always begin each simulation with one infected individual.
  • Use your results from Question 2 in Section 8.3 of the course manual to identify a reasonable quarantine rate that helps to minimize the outbreak of BITS. (Remember, 100% quarantine is not realistic!) Explain why the rate you chose is reasonable.
  • Enter this quarantine rate into the spreadsheet. Then, without changing this fixed quarantine rate, run the simulation using vaccination rates of 10%, 20%, up to 100% to build a table and graph similar to the one found in Question 3. (Note that the values in the table and the graph should be different from those in Question 3 in Section 8.3 of the course manual.)
  • Now, in a similar manner, use your results from Question 3 in Section 8.3 of the course manual to identify a reasonable vaccination rate that helps to minimize the outbreak of BITS. Explain why the rate you chose is reasonable.
  • Enter your chosen vaccination rate into the spreadsheet. Then, without changing this fixed vaccination rate, run the simulation using quarantine rates of 10%, 20% up to 100% to build a table and graph similar to the one found in Question 2 of Section 8.3 of the course manual. (Note that the values in the table and the graph should be different from those in Question 2.)
  • Make some observations about your graphs from bullet points 3 and 5, and compare them with the results from Questions 2 and 3 in the course manual. Based on the information from these four graphs, make a recommendation as to what combination of quarantine and vaccination rates are the most effective. Explain why you are making this recommendation. Include 2-3 graphs in your explanation.
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