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How Long Should a Town Be Locked down to Eliminate an Infectious Disease?

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Title: How Long Should a Town Be Locked down to Eliminate an Infectious Disease?
Language: English
Authors: Nelson, M. I.
Source: International Journal of Mathematical Education in Science and Technology. 2023 54(6):1153-1167.
Availability: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 15
Publication Date: 2023
Document Type: Journal Articles
Reports - Descriptive
Education Level: Higher Education
Postsecondary Education
Descriptors: Mathematics Education, COVID-19, Pandemics, Disease Control, Relevance (Education), Reports, Writing Skills, College Students, Epidemiology, Computation, Models, Student Attitudes
DOI: 10.1080/0020739X.2022.2088419
ISSN: 0020-739X
1464-5211
Abstract: In the past, my mathematics students have frequently complained at any suggestion that they should communicate ideas through the medium of a written report. This article discusses student responses when they were asked to write a short report for the mayor of a (hypothetical) small town in response to the mayor's plan to eliminate a contagious disease by locking the town down for three weeks. I discuss the approaches that students took in constructing their reports and summarize some of the great ideas that they had. Many students could see the parallel between what they were asked to do in the assignment and concurrent discussions in the communities that they came from with regard to the spread of COVID-19. The idea that mathematicians might have to communicate ideas in the form of a written report was not dismissed out of hand. I also reflect on ways in which the learning experience could have been improved. This hinges on providing a mechanism by which an individual student has the opportunity to read the reports of all the other students.
Abstractor: As Provided
Entry Date: 2023
Accession Number: EJ1386898
Database: ERIC
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  Value: <anid>AN0163409173;imt15may.23;2023May03.03:22;v2.2.500</anid> <title id="AN0163409173-1">How long should a town be locked down to eliminate an infectious disease? </title> <p>In the past, my mathematics students have frequently complained at any suggestion that they should communicate ideas through the medium of a written report. This article discusses student responses when they were asked to write a short report for the mayor of a (hypothetical) small town in response to the mayor's plan to eliminate a contagious disease by locking the town down for three weeks. I discuss the approaches that students took in constructing their reports and summarize some of the great ideas that they had. Many students could see the parallel between what they were asked to do in the assignment and concurrent discussions in the communities that they came from with regard to the spread of COVID-19. The idea that mathematicians might have to communicate ideas in the form of a written report was not dismissed out of hand. I also reflect on ways in which the learning experience could have been improved. This hinges on providing a mechanism by which an individual student has the opportunity to read the reports of all the other students.</p> <p>Keywords: Modelling; SIR epidemic; units; writing</p> <hd id="AN0163409173-2">1. Introduction</hd> <p>A disease, the dynamics of which are governed by a susceptible, infectious, recovered model, is spreading through a small town (population 3000). Presently 10% of the individuals in the town are infected.</p> <p>The mayor proposes to eliminate the disease by locking down <emph>all</emph> households, so as to reduce the value of the infectious contact rate to zero. The streets will be patrolled by the police and citizens will only be allowed to leave their homes for medical emergencies.</p> <p>As it is known that, on average, it takes fourteen days for an individual to recovery from the disease the major proposes to implement this measure for three weeks, 'just to make sure'. The mayor believes that his plan must lead to the eradication of the disease since the lockdown will ensure that there will be no contacts.</p> <p>As you are the only member of this community to have studied advanced mathematics the mayor asks you for your opinion of their plan. (Question on assignment)</p> <p>In years gone by, my undergraduate mathematics students have frequently been aghast at any suggestion that they need to know how to communicate ideas using sentences rather than stringing together a sequence of mathematical symbols. Indeed, echoing across the years is the refrain 'If I wanted to write essays, I would have taken an English degree. The beauty of mathematics is that it does not require you to write sentences!' I have never enquired where it is that these students have picked up the idea that communication skills are unimportant to mathematics graduates. Sometimes, students that advocate strongly that writing has no place in mathematics tell me, perhaps not in the same breath, that 'I do not understand Dr X's lecture notes'...</p> <p>The quotation at the start of this paper comprises part of an assignment from a second-year subject on mathematical modelling. The assignment contributed 10% to a student's final mark in the subject, with this particular question worth 4 marks from a total of 18 on the assignment.</p> <p>There were 20 students enrolled in the class, including two master's students. Seventeen of the 18 undergraduate students were enrolled in some flavour of a mathematics degree. The assignment was given to students during the second week of the session. They had one week to submit their solutions.</p> <p>The first half of the subject covers mathematical modelling in medicine, with a heavy emphasis on mathematical epidemiology. A strong emphasis is placed on the assumptions underlying the models and what the terms in the models represent physically. The epidemiological content includes the susceptible, infectious, susceptible, infectious, susceptible, and SIR epidemic models. When the assignment was released, the standard SI and SIS models had been covered in class.</p> <p>I have found that students are often more interested in epidemiology than other areas of mathematical modelling because they can see an almost instant connection between mathematical ideas and practical consequences. The question that starts this paper was an attempt to persuade students that there is an intermediate step between mathematical ideas and their 'practical consequences'; communicating the ideas in a medium that non-mathematicians can understand, i.e. the written word. In writing this question, I hoped that students would see a parallel between the question and discussions that have been very prominent in the (Australian) media for much of 2020.</p> <p>The contributions of this paper are as follows. First, it introduces a 'new' application (?) of a well-known model: exponential decay. Second, it shows how this application can be embedded into a question where students are required to communicate the consequences/interpretation of their solution. Finally, it is hoped that the author's reflections on how this question worked, or did not work, in conjunction with the inclusion of comments made by the students will be helpful to teaching staff who are considering introducing writing questions into their classes.</p> <p>The idea that mathematics students find writing challenging is not new. Neither is the idea that many mathematics staff find the prospect of setting and/or marking writing assignments challenging. Meier and Rishel ([<reflink idref="bib9" id="ref1">9</reflink>]) discuss how to create writing assignments for mathematics classes as a way to engage students in mathematics. Crannell et al. ([<reflink idref="bib3" id="ref2">3</reflink>]) provide a collection of ready-made writing projects covering a wide range of mathematics subjects. Despite these, both these books being user-friendly, eschewing pedagogical terminology, writing assignments remain an under-utilized pedagogical strategy (Latulippe & Latulippe, [<reflink idref="bib7" id="ref3">7</reflink>]).</p> <p>Latulippe and Latulippe ([<reflink idref="bib7" id="ref4">7</reflink>]) work to overcome the reluctance of teaching staff to engage with writing assignments by providing a detailed framework for embedding such assignments into any subject covering calculus, differential equations, or mathematical modelling. This includes strategies for reducing workload, a common objection to the inclusion of such assignments. DeDieu and Lovric ([<reflink idref="bib4" id="ref5">4</reflink>]) discuss the use of writing assignments in a differential equations course. Their focus is on evaluating whether students view writing as an effective learning strategy. In addition to the aforementioned resources, the curated collection of articles published in <emph>PRIMUS</emph> on assessment contains much of relevance (Katz, [<reflink idref="bib5" id="ref6">5</reflink>]) regarding the use of writing assignments.</p> <p>I now outline the structure of this paper. In Section 2, the standard SIR epidemic model is given. Standard mathematical results for this model are not presented since within the context of the question, the SIR model collapses to a single linear differential equation (the 'radioactive decay' model). Two simple mathematical calculations are detailed in Section 3. It was hoped that students would realize the need for these calculations, so that their policy advice would be informed by their 'advanced mathematical knowledge'. Connected with these calculations I raise some issues that students found problematic. In Section 4, I discuss the reports written by the students, collectively they made many excellent points. In Section 5, I reflect upon two points. First, how did students perceive the question? Was it successful in its aim of convincing them that there is a place for mathematicians to write reports using the English language? Second, was the structure of the assignment appropriate to help students improve their writing skills? Finally, in Section 6, I draw some conclusions.</p> <hd id="AN0163409173-3">2. The SIR model</hd> <p>This section is provided for a reader who does not have a background in mathematical epidemiology. The SIR epidemic model can be used to predict the progression of a contagious disease through a population provided that it spreads sufficiently rapidly that changes in the population size due to births and deaths can be ignored. The model splits the population into three categories or compartments. These are <emph>susceptibles</emph>, individuals who have never been infected, <emph>infectives</emph>, individuals who have caught the disease and are infectious, and <emph>recovereds</emph>, individuals who have caught the disease and recovered. It is assumed that recovery provides indefinite immunity to reinfection. The recovered category may be subdivided into two further compartments: individuals that are recovered and living and those that are dead. At any moment in time every member of the community belongs to one, and only, compartment. The differential equation model for the spread of the disease is given by Equations (<reflink idref="bib1" id="ref7">1</reflink>)–(<reflink idref="bib3" id="ref8">3</reflink>).</p> <p>The rate of change of the number of susceptible individuals</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>S</mi></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mrow><mo>=</mo><mo>−</mo><mtext fontfamily="times">β</mtext><mi>I</mi><mi>S</mi><mo>,</mo><mspace width="1em" /><mi>S</mi><mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><msub><mi>S</mi><mi>i</mi></msub><mo>.</mo></math> </ephtml> (<reflink idref="bib1" id="ref9">1</reflink>)</p> <p>The rate of change of the number of infective individuals</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>I</mi></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mrow><mo>=</mo><mrow><mphantom><mo>−</mo></mphantom></mrow><mi>I</mi><mi>S</mi><mo>−</mo><mi>b</mi><mi>I</mi><mo>,</mo><mspace width="1em" /><mi>I</mi><mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><msub><mi>I</mi><mi>i</mi></msub><mo>.</mo></math> </ephtml> (<reflink idref="bib2" id="ref10">2</reflink>)</p> <p>The rate of change of the number of recovered individuals</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>R</mi></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mrow><mo>=</mo><mi>b</mi><mi>I</mi><mo>,</mo><mspace width="1em" /><mi>R</mi><mrow><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo>=</mo><msub><mi>R</mi><mi>i</mi></msub><mo>.</mo></math> </ephtml> (<reflink idref="bib3" id="ref11">3</reflink>)</p> <p>The model contains two parameters, <emph>β</emph> and <emph>b</emph>, which are linked to two processes. The first process represents infection of susceptibles by infectives. This is modelled using 'mass action' kinetics. The constant of proportionality (<emph>β</emph>) is known as the infectious contact rate and has units</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">s</mi></mrow><mspace width="thinmathspace" /><msup><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">e</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> </ephtml> . The second process represents recovery of infectious individuals, which is modelled as an exponential process. The constant of proportionality (<emph>b</emph>) has a very appealing interpretation: it is the reciprocal of the average duration that an individual is infectious (Martcheva, [<reflink idref="bib8" id="ref12">8</reflink>], chapter 2.2.1). Consequently, it has units</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">e</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> </ephtml> .</p> <p>A key parameter in understanding the behaviour of the SIR model is the basic reproduction number (</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>0</mn></msub></math> </ephtml> ). This is the number of secondary infections caused by the introduction of a single infectious individual into a population otherwise comprised of susceptibles. If the basic reproduction number is greater than one, then an epidemic sweeps through the population with, in the early stages, exponential growth in the number of infectious individuals. If the basic reproduction number is less than one, then the number of infectives decreases to zero. For the SIR model, the basic reproduction number is given by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>0</mn></msub><mo>=</mo><mtext fontfamily="times">β</mtext><mi>K</mi><mrow><mo>/</mo></mrow><mi>b</mi></math> </ephtml> , where <emph>K</emph> is the population size (</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi><mo>=</mo><msub><mi>S</mi><mi>i</mi></msub><mo>+</mo><msub><mi>I</mi><mi>i</mi></msub><mo>+</mo><msub><mi>R</mi><mi>i</mi></msub></math> </ephtml> ).</p> <p>When a new contagious disease arises key concerns include: identifying the correct modelling framework, determining the definition of the basic reproduction number, and then estimating the basic reproduction number. The SIR epidemic model is a standard starting point in such endeavorus. However, for COVID-19, it has been found that a better representation of the disease dynamics is to use a susceptible, exposed, infectious, recovered epidemic model in which an 'exposed' compartment is added between the susceptibles and infectious. This compartment represents individuals that have been infected but are not infectious, i.e. it introduced an incubation period into the disease dynamics.</p> <p>The SIR model is covered in many textbooks, a small selection of which includes Brauer and Castillo-Chávez ([<reflink idref="bib1" id="ref13">1</reflink>]), Britton ([<reflink idref="bib2" id="ref14">2</reflink>]), Martcheva ([<reflink idref="bib8" id="ref15">8</reflink>]), and Murray ([<reflink idref="bib10" id="ref16">10</reflink>]). Ketcheson ([<reflink idref="bib6" id="ref17">6</reflink>]) provides an extremely accessible and non-technical overview of the SIR model which includes simple ways to model intervention strategies.</p> <hd id="AN0163409173-4">3. Mathematics calculation</hd> <p>The mayor has put forward the proposition that locking down their community will ' reduce the value of the infectious contact rate to zero'. Our first calculation is therefore to investigate the consequences of this reduction. (Many students identified problems with this assumption, their concerns are identified in Section 4.2.)</p> <p>Assuming that the effect of lockdown is to reduce the value of pairwise infectious contact rate (<emph>β</emph>) to zero, the differential equation for the rate of change of the number of infectious individuals simplifies to</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>I</mi></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mi>b</mi><mo>⋅</mo><mi>I</mi><mo>,</mo><mspace width="1em" /><mi>I</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mn>300</mn></math> </ephtml> </p> <p>Thus,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mrow><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>=</mo><mn>300</mn><mi>exp</mi><mo>⁡</mo><mrow><mo>[</mo><mo>−</mo><mi>b</mi><mi>t</mi><mo>]</mo></mrow><mo>.</mo></math> </ephtml> </p> <p>Using the information provided (' it is known that, on average, it takes fourteen days for an individual to recover'), we have</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>1</mn><mn>14</mn></mfrac><mspace width="thinmathspace" /><msup><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">y</mi><mi mathvariant="normal">s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>.</mo></math> </ephtml> </p> <p>Thus the number of individuals who are infected at the end of the lockdown is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mrow><mrow><mo>(</mo><mn>21</mn><mo>)</mo></mrow></mrow><mo>=</mo><mn>300</mn><mi>exp</mi><mo>⁡</mo><mrow><mo>[</mo><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>]</mo></mrow><mo>=</mo><mn>66.9</mn><mspace width="thinmathspace" /><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">s</mi></mrow><mo>.</mo></math> </ephtml> </p> <p>There are two points regarding this calculation. First, the information provided gives the recovery time in unit of days whereas the mayor proposes to lockdown the town in units of weeks. Some students did not notice this discrepancy. This led to a discussion as to whether it is 'fair' or 'unfair' for a question to contain parameters with inconsistent units. Students taking mathematics in conjunction with a degree in either science or engineering did not see a problem, commenting that it is always necessary to check the consistency of units. Some of the students taking some flavour of a 'straight' mathematics degree stated that they had not been exposed to 'units' before. This is not entirely true, because the importance of units is stressed in the subject. Perhaps a more accurate statement is that they had not previously been exposed to a situation where they have to check the consistency of the units.</p> <p>The second point is whether the answer 66.9 individuals should be rounded up to 67 or down to 66. If this were a purely numeric question, there would be no question about rounding up to 67. However, in applied mathematics before deciding how to round a number, we should think about the context in which it is being used. For example, in financial mathematics, rounding favours the lender rather than the borrower.</p> <p>In this context, we should ask the question, what is the condition for the disease to be eliminated? This condition is <emph>not</emph> that there are no infectives in the community, this only happens after an infinite amount of time. A practical definition is that the disease is eradicated if less than one person is infected. This suggests that we should round the number of infectives down. So our total of 66.9 infectives becomes 66 infectives (some students rounded up to 67). In either case, we should tell the mayor that his plan will not lead to eradication of the disease.</p> <p>Under your proposed lock-down plan, there would be approximately 66 infective individuals remaining upon cessation of the lock-down. If your aim was to completely eradicate the disease over the course of the lock-down, my calculations indicate that the restrictions you propose would be inadequate. (Student answer)</p> <p>(I like the wording in this passage, because it says 'approximately 66' indicating uncertainty in the real number of infectives in the population, i.e. a difference between what the model says and what we would find if we could test everyone in the population at the end of the lockdown. To skip ahead of myself, in retrospect I should have devoted some of my teaching time to discussing the use of language in the report).</p> <p>If the disease is not eradicated after a lockdown of 21 days, how many days are required? Let</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi></mrow></mrow></msub></math> </ephtml> be the time at which there is one infectious individual in the population. Then the lockdown is successful if the lockdown period is greater than this value. We must solve the equation</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mn>1</mn></mtd><mtd><mi /><mo>=</mo><mn>300</mn><mi>exp</mi><mo>⁡</mo><mrow><mo>[</mo><mo>−</mo><mi>b</mi><mo>⋅</mo><msub><mi>t</mi><mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi></mrow></mrow></msub><mo>]</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mo>⇒</mo><msub><mi>t</mi><mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi></mrow></mrow></msub></mtd><mtd><mi /><mo>=</mo><mfrac><mn>1</mn><mi>b</mi></mfrac><mi>ln</mi><mo>⁡</mo><mn>300</mn><mo>,</mo></mtd></mtr><mtr><mtd><mo>⇒</mo><msub><mi>t</mi><mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi></mrow></mrow></msub></mtd><mtd><mi /><mo>=</mo><mn>14</mn><mi>ln</mi><mo>⁡</mo><mn>300</mn><mo>≈</mo><mn>79.85</mn><mspace width="thinmathspace" /><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">y</mi><mi mathvariant="normal">s</mi></mrow><mo>.</mo></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib4" id="ref18">4</reflink>)</p> <p>Should the quantity 79.85 days be rounded down to 79 days or to 80 days? The quantity</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi></mrow></mrow></msub></math> </ephtml> is the time at which there is one infectious individual. If we take</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo><</mo><msub><mi>t</mi><mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi></mrow></mrow></msub></math> </ephtml> , there is at least one infectious individual in the community and it is not safe to end the lockdown. If we take</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><msub><mi>t</mi><mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">e</mi></mrow></mrow></msub></math> </ephtml> , there is less than one infectious individual in the community, round down to zero, and it is safe to end the lockdown. Thus, we should round 79.85 days to 80 days. Similarly, if the calculated number had been 79.15 days then this would similarly have been rounded up to 80 days.</p> <hd id="AN0163409173-5">4. The report for the mayor</hd> <p>In the previous section, I discussed two calculations which I hoped would underpin the report that students wrote for the mayor. In this section, I discuss issues relating to this report. I start in Section 4.1 by discussing the varying approaches that students took in structuring their report. In Section 4.2, I report issues identified by students as reflecting a conflict between the idealization of the model, e.g. assuming that the lockdown prevents contacts between susceptibles and infectives, and what is likely to happen in reality. Having identified that the proposed plan would be unsuccessful, some students saw a need to provide policy advice to the mayor for the post-lockdown period. This is discussed in Section 4.3. Several students stressed that their recommendations were based on considering a simplified model and that more detailed policy recommendations could be justified by a more detailed model. They therefore suggested ways in which the model could be improved. These are discussed in Section 4.4.</p> <hd id="AN0163409173-6">4.1. The structure of the report</hd> <p>Students structured their reports in a variety of ways. Some students believed that advice could only follow from concrete calculations. They therefore carefully explained their assumptions, presented their models (not always defining the symbols in the equations nor the assumptions leading to the equations), carried out calculations, drew conclusions, and finally presented advice to the mayor at the end of their report. Other students thought that the last thing the mayor wanted to see was mathematical calculations. They therefore presented their recommendations first and relegated mathematical assumptions and calculations to latter. A third group of students went for an approach between these extremes in which advice, modelling calculations, and discussions of the limitations of the calculations were presented as a unified whole.</p> <p>A small group of students decided that mathematics of any kind would be too horrifying for the mayor. Consequently they presented their advice without providing any discussion of models or calculations. A typical comment made by these students was ' I think that the ideas sounds like it would work'. One student's answer revealed an important misunderstanding: ' because</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>0</mn></msub></math> </ephtml> is reduced to zero the disease will be eradicated after the three week lockdown period' – not realizing that this needs to be checked. Some students in this group provided well-argued policy positions, without substantiating their argument in mathematical calculations.</p> <p>Half-way between students who provided calculations and those that did not was a student who gave the mayor an Excel spreadsheet to graph the number of infectious individuals in the town on a weekly basis. The mayor had the flexibility to change parameter values, such as the initial number of infectious individuals in the community and the recovery rate, and then seeing the consequences. (There was also an option to have a non-zero value for the infectious contact rate.)</p> <p>Finally, one student declined to play the game and argued that it is impossible to evaluate the lockdown plan without simultaneously considering its economic consequences.</p> <hd id="AN0163409173-7">4.1.1. Explaining the model to the mayor</hd> <p>As noted in the previous section, one divide between students was between those who included differential equations in their report, on the grounds that failing to do so would be mathematically irresponsible, and those who did not include differential equations, on the grounds that the mayor would be not familiar with them. Some students in the latter camp suggested that the mayor would neither enjoy seeing differential equations nor understand them.</p> <p>A subset of students replaced the differential equation model with an ' input–output' figure containing the compartments in the model and the transitions between them; similar to Figure 1. Typically they supplemented this figure with a description of the assumptions behind the model. The figure provides a visual demonstration to the mayor that reducing the value of the infectious contact rate (<emph>β</emph>) to zero severs the link between susceptibles and infectives. (The use of a graphical representation of a compartment model, such as Figure 1, is stressed in lectures as being highly beneficial when presenting epidemiological models to a non-mathematical audience.) This approach aims to explain the essential parts of the model to the mayor in a non-confronting manner, i.e. without differential equations.</p> <p>Graph: Figure 1. The three compartments and two transitions in the SIR epidemic model.</p> <p>One student explained to the mayor that the recovery time of 14 days is an <emph>average</emph> recovery time, with some individuals recovering quicker and some slower. (A graph showing the expected distribution times around the mean would have been a great visual element to include, but the distribution of recovery times is <emph>not</emph> discussed in this subject.)</p> <hd id="AN0163409173-8">4.2. Model assumptions: suggestions from students</hd> <p>'<emph>You can only reduce <emph>β</emph>, you can not completely eliminate it</emph>'. (Student comment)</p> <p>A number of students pointed out to the mayor that their calculations would under-estimate the number of infectious individuals at the end of the lockdown period. This follows from the assumption that the lockdown eliminates interactions between susceptible and infectious individuals.</p> <p>A variety of problems were identified with this assumption.</p> <p></p> <ulist> <item> The model does not allow for the spread of the disease within households if the family unit contains an infectious individual. One student recommended that the mayor encourage infected individuals to self-isolate within households, recognizing that it would not be possible to police such a policy.</item> <p></p> <item> In connection with the previous point, one student suggested that during the lockdown period infectious individuals should be identified and ' isolated from the rest of the community'.</item> <p></p> <item> Not all individuals will adhere to the rules of the lockdown, hence mixing of susceptibles and infectives cannot be eliminated. Thus, there is the possibility of individuals contracting and spreading the disease.</item> <p></p> <item> Members of the police will come into contact with any infected individuals who disobey the lockdown rules. If they become infected themselves, they can spread the disease both to their family and to their colleagues.</item> <p></p> <item> Workers in hospitals and in the emergency services are at risk of contracting and then spreading the disease, both to their family and members of the public who have a medical emergency.</item> <p></p> <item> Several students recommended that citizens allowed out of their homes during the lockdown, such as police officers and medical workers, should be required to wear masks.</item> <p></p> <item> Finally, one student recommended that because of the risks associated with their jobs all police officers and medical workers should be regularly tested during the lockdown period.</item> </ulist> <p>In conjunction with such comments, one student suggested that during the lockdown period it was essential to educate the public on basic hygiene rules and associated measures to minimize the chance of individuals catching and spreading the disease, both during the lockdown period and afterwards.</p> <hd id="AN0163409173-9">4.3. Policy recommendations after lockdown</hd> <p>'The mayor's plan of a 21 day lockdown will reduce <emph>β</emph> to 0, making it harder for an infective to transmit the disease to susceptibles through limiting contact during their infective period. My calculations show that the lockdown will not eliminate the disease. To eradicate the disease after lockdown the mayor must reduce the basic reproduction number to less than one. To achieve this we must reduce <emph>β</emph> by following social distancing and lockdown.' (Student comment)</p> <p>A number of students correctly determined that although the lockdown would reduce the number of infectious individuals in the community, it would not eliminate the disease from the population. Consequently some students suggested that the mayor must be willing to implement policies to reduce the infectious contact rate in the post-lockdown period. One student suggested that the lockdown period ' will give the mayor a breathing space to formulate policy to apply when the lockdown is removed' whilst another commented that post-lockdown measures would ' help keep society partially functional as efforts are made to eradicate the disease'. Several students pointedly commented that failure to implement post-lockdown measures would inevitably lead to a second wave of infections.</p> <p>A number of comments dealt with what should happen if a test for the disease is developed during the lockdown period.</p> <p></p> <ulist> <item> ' If a reliable test is developed before the end of lockdown, then the prevalence of the disease within the community should be estimated before any decision is made on removing the lockdown'.</item> <p></p> <item> A more specific piece of advice was to test the entire population before the end of lockdown, isolating infectious individuals.</item> </ulist> <p>Other post-lockdown recommendations are identified below.</p> <p></p> <ulist> <item> A number of students commented to the effect that in the absence of a test ' anyone still showing symptoms remain in lockdown...a caution not to accidentally spread the disease'.</item> <p></p> <item> One student recommended that rather than lifting the lockdown unilaterally, it would be safer to lift it ' slowly, so as to keep the infectious contact rate to a minimum as the remaining infected are isolated and recover'.</item> <p></p> <item> One student pointed out that even if the disease was eradicated from the local community, the risk of it being reintroduced by a visitor could not be ruled out. Along similar lines, one student pointed out that a member of the community could be infected if they travelled outside their local area. They therefore recommended that the mayor ' encourage reduced travel and less physical contact with others...to lower the infectious contact rate after the lockdown'.</item> </ulist> <p>One student was so concerned about the lack of concrete policy proposals that they were led to comment ' in the absence of extra policies to run in conjunction with the lockdown there are too many risks for me to confidently recommend the plan'. It seems unduly harsh to throw away the advantages of the lockdown because of a lack of concrete policy proposals. I feel that this student would have benefited from having the opportunity to reflect upon their report, a point that I return to in Section 5.2.</p> <hd id="AN0163409173-10">4.4. Improving the model</hd> <p>'More information on the disease and how it affects individuals and groups within the community, i.e. any age groups that are more susceptible, determining if it effects males or females more etc. would give more accurate estimates of the time it will take to eradicate the disease, and if there are certain groups that need to be isolated more than others.' (Student comment)</p> <p>Several students noted that the SIR model is the simplest plausible model to predict the spread of a contagious disease. However, as more understanding and data becomes available, it is possible to develop a more detailed model which can deliver more specific policy guidance, as outlined in the previous quote. Students putting forward this view mostly concentrated on the need to be able to estimate the value of the infectious contact rate ' so as to be able to estimate what measures will be required after the lockdown is removed'. For example, one student commented that value of the infectious contact rate can be used to</p> <p>'</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>[</mo><mo>...</mo><mo>]</mo></mrow></math> </ephtml> estimate the number of individuals which can be allowed out-of-lockdown so as to keep the effective value of the basic reproduction number below one. This would allow more essential workers to remain in the work force, reducing the strain on the town's economy'.</p> <p>A number of students recognized limitations in the accuracy of their predictions based upon the limited information that was available to them. One student recognized that deficiencies in current knowledge cannot be used as justification not to provide advice.</p> <p>' However saying "I don't know, there isn't enough information" is not good advice and not what the Mayor wants'.</p> <hd id="AN0163409173-11">5. Discussion</hd> <p>I noted at the beginning of this paper that in the past I have found that many students are not convinced that there is a need for them to write reports to communicate mathematical ideas. In Section 5.1, I discuss feedback from students regarding their perception of this question. In Section 5.2, I discuss my own reflection on the assignment question. Did it achieve its aim of opening the eyes of students to the need to communicate mathematical ideas in print? I think it did. Could a better learning environment have been provided to help students improve their writing skills? There is the rub. In Section 5.3, I discuss how some of the modelling issues that were identified in various reports were discussed in a subsequent lecture.</p> <p>One of the reviewers asked me if I had used this assignment before 2020 and if so what did the students make of phrases such as 'essential workers' or 'the efficacy of testing'? I have taught the 'application of mathematical modelling to medicine' component of this subject yearly since 2017. It had not occurred to me to ask a question along these lines in 2017–2019, it was only through the events of 2020 that I saw how a connection might be made between the classroom model and the associated discussion in the media. I am sure that before 2020, phrases such as 'essential workers' and the 'efficacy of testing' would have baffled most of my students. As perhaps they will in a generation's time.</p> <hd id="AN0163409173-12">5.1. Student perceptions of the question</hd> <p>'I believe that question 2 was easily the most important question of this assignment. Q 2 is incredibly fascinating as it forces us to apply our knowledge of mathematics into a practical written way that is usually uncommon in mathematics. Too often in advanced level mathematics I feel like the content has no practical application, but this question phenomenally highlights the benefits of advanced mathematics through a simple narrative. Short essays and longer written answers are also great ways to highlight the application of the mathematics and also improve the writers skills at communicating complex topics.</p> <p>I also really liked from this question the immediate real world application and parallels. The situation of the question mirrors a small scale simplified model of a response to COVID-19. When I initially believed the plan was very suitable, as in theory three weeks of isolation should eliminate all cases it does not solve the fundamental problem. It will definitely lower the amount of cases, but not all of them as seen through calculation. It also does not solve the fundamental issue of the infectious contact rate after lock-down which if high enough will cause the disease to continue spreading despite the mayor's effort. This highlighter the importance of measures like "social distancing" to lower the infectious contact rate '. (Student comment)</p> <p>The instructions for this assignment asked students to identify what was the most important topic covered on the assignment. (The answers to this question are not marked.) In their answers, a number of students selected the 'mayor's question' as being the most important topic, with some identifying this as an example of what a professional mathematician might have to do. Typical comments included ' applying theoretical knowledge to a "real" problem' and ' this question simulated an example that could be a briefing given to a mathematician in the workplace'. Many students saw the parallel between the question and the ongoing COVID-19 pandemic. Significantly, no students complained about having to write! However, as alluded to in Section 4.1, some students gave no quarter to the mayor's lack of expertise in mathematics and provided mathematics heavy answers. Sometimes these were little more than the two calculations discussed in Section 3. Did the assignment structure provide an opportunity for these students to see that other approaches were possible?</p> <p>The calculations required are very simple. However, some students found that the context that they were embedded into endowed them with extra weight. They were ' important because it is a real scenario which could help people and not just doing calculations'. A number of students commented that the question had forced them to consider how post-lockdown policy decisions had implications in reducing the value of the infectious contact rate, though noting that the precise effect of any particular policy on this parameter was very difficult, if not impossible, to predict.</p> <p>Some students commented that the best learning experience from this question was appreciating that factors not considered in a model can affect its effectiveness and usefulness. A typical comment was ' given the ongoing COVID19 pandemic I am more aware & interested in how models predictions fail due to unforeseen or difficult to control circumstances'. This led to considerations as to whether such factors can or should be included in the model and to what extent model deficiencies should be revealed to the mayor.</p> <p>Not all students had a positive perspective on this assignment. I finish this section with a negative comment.</p> <p>'I found that while the question was engaging & required some thinking, it was not clear what exactly was required when answering.</p> <p>Overall I found question 2 to be frustrating as the question does not make it clear what things I should be focusing on to consider the question, in fact it barely hinted at it. Question 2 was too vague & broad to be on assignment in my opinion'.</p> <p>Could the assignment question had been better designed to make it more apparent to all students how to approach it?</p> <hd id="AN0163409173-13">5.2. Reflections on the reports written by students</hd> <p>Was the assignment question a good one? No student complained about having to write a short report – though some students turned in an answer based upon calculations with no supporting structure. As evidenced from the quotations given in earlier sections, many students could envisage a professional mathematician being asked to analyse the proposed scenario. So, was the exercise a success? Only partially, as the student learning experience on this question was only the tip of what it could have been.</p> <p>Students came up with many different ways to approach their report. But only the lecturer was exposed to this cornucopia. A small handful of students had truly outstanding ideas on how to write the report, or how to express a mathematical idea or concept in a way that the mayor would best understand. But only the lecturer saw this. Some students had a very good turn of phrasing, ' if we can sever the link of susceptibles becoming infected then we can break the cycle', and had evidently given careful though about their use of language. But only the lecturer saw this. Several students stressed the need to improve the model so as to provide better policy recommendations. But only the lecturer saw this.</p> <p>As noted in Section 4.1, although some students provided well-argued policy positions, they did so without substantiating their arguments in mathematical calculations. The learning experience would have been improved if an opportunity had been provided for the students to read all the submissions for this question. For example, students who provided mathematical calculations without structuring them into a report might then have realized that this was not the way to do it. Students who had written a report might have seen ways to improve their report.</p> <p>I noted in Section 4.3 that many students recommended introducing ' policies to reduce the infectious contact rate'. Evidencing thought about the use of language, one student phrased it slightly differently; recommending ' long-term measures to reduce the amount of contact between individuals, this will reduce the infectious contact rate'. This rewording of the idea is appealing, not least because it is probable that the mayor will not remember what the 'infectious contact rate' is. Reminding the mayor that it is related to contacts between susceptible and infectious individuals is therefore an appealing idea. This reinforces my earlier suggestion that after the assignments had been returned some class time should have been devoted to discussing language issues.</p> <p>In hindsight I should have carved this question off the main part of the assignment and asked students to submit a solution to the assignment proper and a separate submission in the form of a report. But how best to utilize the submissions for the mayor?</p> <p></p> <ulist> <item> There could be two deadlines, one for submission of a first draft and one for submission of the final draft. After the first deadline, all the (anonymous) submissions could be made available. Potentially, this could lead to an improvement of a large number of reports. This would particularly help those students who were not clear what they should be focusing on. On the downside, this approach could antagonize the better students who might feel that they have lost some of the distinctiveness of their answers as other students cull their ideas.</item> <p></p> <item> (Although some students scored full marks for this question, no student had a perfect report, so all students could potentially have improved their answer to this question).</item> <p></p> <item> All the submissions could have been made available to students after they had been marked. Although students would not benefit from seeing the other reports on this assignment, they could read through them and learn lessons for the next time they are asked to write a report.</item> <p></p> <item> As a variation on the previous idea, the reports could be made available to students taking this subject in the next academic year. They could be asked to read through these before answering a similar question.</item> </ulist> <p>A collection of student reports is potentially a good learning resource for future students to consult.</p> <p>The question discussed in this paper was worth four marks. When writing the question, I had in mind that there would be two marks for the mathematical analysis and two marks for the written opinion. This subdivision of the marks was not indicated to students – it seemed to me that the calculations were a sine qua non of providing advice to the mayor. I considered using a rubric for the written part of the question, but discarded the idea as it did not seem worthwhile for a question that was only worth two marks.</p> <p>In retrospect, I should have provided a rubric for this question. This would have provided guidance for students in structuring their answer, including the importance of basing recommendations upon calculations – at the time it did not occur to me that marks for the mathematical analysis could be built into the rubric. Not only should a rubric have been used but in view of the detailed answers provided by students, the marks for the written opinion should have been increased. Had I know about them, the example rubrics provided by Crannell et al. ([<reflink idref="bib3" id="ref19">3</reflink>]) and Latulippe and Latulippe ([<reflink idref="bib7" id="ref20">7</reflink>]) would have been very helpful.</p> <hd id="AN0163409173-14">5.3. Reflections on the model</hd> <p>The SIR model assumes that relatively large groups of individuals are mixing with each other. The model no longer applies once individuals are forced to remain in their households. One student pointed out that the SIR model could be applied if the community was partitioned into three palatial mansions with 1000 people in each. Similarly, it could be applied if the community self-isolated into three thousand households each containing one individual. However, the SIR model cannot predict transmission within small households containing more than one individual.</p> <p>A different modelling approach is required to model the spread of a disease at the household level. This provided an opportunity to informally discuss in a subsequent lecture different modelling approaches, outside the scope of this subject, that are applicable at the household level. It was also possible to make the observation that a modelling approach that works at the level of individuals that are contained in household should approach (in some mathematical way) the SIR model in an appropriate limit – perhaps as the number of individuals in the household becomes very large.</p> <p>One student commented that: 'This question obviously echos similarities to COVID19, particularly how a model should work if everyone involved does the right thing but can drastically fail if even a few individuals ignore advice/information'. A potential avenue for exploration with future students is a model in which the population is split into two components: one faction following social-distancing and taking precautions and one faction not following social-distancing and failing to take precautions. How does the size of the latter component affect the spread of the disease?</p> <hd id="AN0163409173-15">6. Conclusions</hd> <p>Throughout my career I have been faced by cohorts of students who have claimed that mathematics is a discipline where it is not important to be able to communicate ideas using the written word. (Perhaps I should have asked them what makes a good set of lecture notes...) In 2020, I asked students to write a short policy report for a mayor in a context similar to the COVID-19 outbreak. This proved to be successful, in that no students complained about having to write a report; though not all students realized the necessity for writing a report. Positive feedback was received from a number of students, who saw a connection between the assignment question and how they might be expected to work as a professional mathematician.</p> <p>Although the mathematical model is nominally based on the SIR model, the assumption that the infectious contact rate is zero reduces the model to a single linear differential equation. Consequently, the report question can be used for other groups of students; even for students on an introductory calculus subject.</p> <p>Many students had excellent ideas, but only the lecturer saw these ideas. Thus students learning on what makes a good report was restricted to learning from the mistakes that they made themselves (more accurately, my comments on their mistakes), rather than learning from the excellent ideas of their peers. The student learning experience would have been improved by developing a mechanism that allows them to catch good ideas and to use them to improve reports that they write in the future.</p> <p>When I wrote this question I thought of it as being a small component of the assignment – it was worth 4 marks from a total of 18. Having reflected on the outcomes, and through exposure to the references put forward by the referees, I see that I had it the wrong way round. This question should have been a much larger component of the assignment. Perhaps even it should have been the assignment.</p> <p>I noted in Section 4.1 that one student provided the mayor with a spreadsheet implementing the SIR model so that they could graphically see the consequences of changing numbers in the model. Whilst writing this article, I asked this student how they came up with the idea. I preface their response by noting that they are a master's student who has re-entered tertiary education after a long period away. The student explained that at one time they had been employed as a 'data analyst' in which they had be analyse sales data, produce simple models, and communicate their findings to senior management. As their senior management did not have a mathematical background, they provided the models in the form of an Excel spreadsheet: the graphs provided the information that management needed, the spreadsheet allowed them to investigate the effect of changing parameters.</p> <p>That would have been an excellent insight to share with the other students taking this subject. If only I'd gathered my wits together at the time...</p> <hd id="AN0163409173-16">Acknowledgments</hd> <p>The author thanks the referees for their careful reading of this paper, their detailed comments, and the suggestion of various references.</p> <hd id="AN0163409173-17">Disclosure statement</hd> <p>No potential conflict of interest was reported by the author.</p> <ref id="AN0163409173-18"> <title> References </title> <blist> <bibl id="bib1" idref="ref7" type="bt">1</bibl> <bibtext> Brauer, F., & Castillo-Chávez, C. (2001). Mathematical models in population biology and epidemiology (1st ed., Vol. 40). Texts in applied mathematics. Springer-Verlag.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref10" type="bt">2</bibl> <bibtext> Britton, N. F. (2003). Essential mathematical biology (1st ed.). Springer-Verlag.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref2" type="bt">3</bibl> <bibtext> Crannell, A., LaRose, G., Ratliff, T., & Rykken, E. (2004). Writing projects for mathematics courses: Crushed clowns, cars, and coffee to go. Classroom Resource Series. MAA Press.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref5" type="bt">4</bibl> <bibtext> DeDieu, L., & Lovric, M. (2018). Student perceptions of the use of writing in a differential equations course. PRIMUS, 28 (2), 166 – 185. https://doi.org/10.1080/10511970.2017.1337659</bibtext> </blist> <blist> <bibl id="bib5" idref="ref6" type="bt">5</bibl> <bibtext> Katz, B. P. (2021). Curated collection: Assessment. PRIMUS, 32 (5), 636 – 649. https://doi.org/10.1080/10511970.2021.1879333</bibtext> </blist> <blist> <bibl id="bib6" idref="ref17" type="bt">6</bibl> <bibtext> Ketcheson, D. I. (2020). Modeling the spread of COVID-19. SIAM News, 53 (4), 6 – 7.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref3" type="bt">7</bibl> <bibtext> Latulippe, J., & Latulippe, C. (2014). Reduce, reuse, recycle: Resources and strategies for the use of writing projects in mathematics. PRIMUS, 24 (7), 608 – 625. https://doi.org/10.1080/10511970.2013.876794</bibtext> </blist> <blist> <bibl id="bib8" idref="ref12" type="bt">8</bibl> <bibtext> Martcheva, M. (2015). An introduction to mathematical epidemiology. Springer.</bibtext> </blist> <blist> <bibl id="bib9" idref="ref1" type="bt">9</bibl> <bibtext> Meier, J., & Rishel, T. (1998). Writing in the teaching and learning of mathematics. Number 48 in MAA Notes. The Mathematical Association of America.</bibtext> </blist> <blist> <bibtext> Murray, J. D. (1989). Mathematical biology (2nd ed., Vol. 19). Biomathematics. Springer-Verlag.</bibtext> </blist> </ref> <aug> <p>By M. I. Nelson</p> <p>Reported by Author</p> </aug> <nolink nlid="nl1" bibid="bib10" firstref="ref16"></nolink>
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  Data: How Long Should a Town Be Locked down to Eliminate an Infectious Disease?
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  Data: <searchLink fieldCode="SO" term="%22International+Journal+of+Mathematical+Education+in+Science+and+Technology%22"><i>International Journal of Mathematical Education in Science and Technology</i></searchLink>. 2023 54(6):1153-1167.
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  Data: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
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  Data: In the past, my mathematics students have frequently complained at any suggestion that they should communicate ideas through the medium of a written report. This article discusses student responses when they were asked to write a short report for the mayor of a (hypothetical) small town in response to the mayor's plan to eliminate a contagious disease by locking the town down for three weeks. I discuss the approaches that students took in constructing their reports and summarize some of the great ideas that they had. Many students could see the parallel between what they were asked to do in the assignment and concurrent discussions in the communities that they came from with regard to the spread of COVID-19. The idea that mathematicians might have to communicate ideas in the form of a written report was not dismissed out of hand. I also reflect on ways in which the learning experience could have been improved. This hinges on providing a mechanism by which an individual student has the opportunity to read the reports of all the other students.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2023
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1386898
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=eric&AN=EJ1386898
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1080/0020739X.2022.2088419
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 15
        StartPage: 1153
    Subjects:
      – SubjectFull: Mathematics Education
        Type: general
      – SubjectFull: COVID-19
        Type: general
      – SubjectFull: Pandemics
        Type: general
      – SubjectFull: Disease Control
        Type: general
      – SubjectFull: Relevance (Education)
        Type: general
      – SubjectFull: Reports
        Type: general
      – SubjectFull: Writing Skills
        Type: general
      – SubjectFull: College Students
        Type: general
      – SubjectFull: Epidemiology
        Type: general
      – SubjectFull: Computation
        Type: general
      – SubjectFull: Models
        Type: general
      – SubjectFull: Student Attitudes
        Type: general
    Titles:
      – TitleFull: How Long Should a Town Be Locked down to Eliminate an Infectious Disease?
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Nelson, M. I.
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2023
          Identifiers:
            – Type: issn-print
              Value: 0020-739X
            – Type: issn-electronic
              Value: 1464-5211
          Numbering:
            – Type: volume
              Value: 54
            – Type: issue
              Value: 6
          Titles:
            – TitleFull: International Journal of Mathematical Education in Science and Technology
              Type: main
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