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enMaths in a minute: "R nought" and herd immunity
https://plus.maths.org/content/maths-minute-r0-and-herd-immunity
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/herd_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="max-width: 300px;"><img src="/content/sites/plus.maths.org/files/news/2020/herd/chart_exponential.png" alt="Exponential growth" width="300" height="438" />
<p>The number of new infections after <em>n</em> weeks for <em>R</em><sub>0</sub>=2.</p>
</div><p>Two things many of us will have heard about over the last few weeks are the concept of herd immunity and a number called <img src="/MI/5779b19ea24f8c78cce5c212d3466c21/images/img-0001.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" /> (which people say as "R nought"). </p><h3>The basic reproduction number</h3><p>Given an infectious disease, such as COVID-19, <img src="/MI/5779b19ea24f8c78cce5c212d3466c21/images/img-0001.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" /> is the <em>basic reproduction number</em> of the disease: the average number of people an infected person goes on to infect, given that everyone in the population is susceptible to the disease. For COVID-19 this is currently estimated to lie between 2 and 2.5. For seasonal strains of flu, it lies between 0.9 and 2.1. And for measles it is a whopping 12 to 18. </p>
<p>You can see how a large enough <img src="/MI/5779b19ea24f8c78cce5c212d3466c21/images/img-0001.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" /> leads to a rapid spread of the disease. For example, if <img src="/MI/5779b19ea24f8c78cce5c212d3466c21/images/img-0001.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" /> is equal to 2 and an infected person infects their victim within a week of becoming infectious, then a single infected person sparks the following growth of new infections:</p>
<p>After 1 week: <img src="/MI/967e8bf551622b6beb697d0ea46d3930/images/img-0001.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> new infections</br>
After 2 weeks: <img src="/MI/6b0230fa1b5ff31888db12e13787e49a/images/img-0001.png" alt="$4$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> new infections</br>
After 3 weeks: <img src="/MI/8929140ad5da5b0ea43c1a8281d6fef2/images/img-0001.png" alt="$8$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> new infections</br>
After 4 weeks: <img src="/MI/324b4bd16b8830a963c286db64dc58f4/images/img-0001.png" alt="$16$" style="vertical-align:0px;
width:15px;
height:12px" class="math gen" /> new infections.</br></p>
<p>Generally, after <img src="/MI/8f069d777c270870c20f004d237b3936/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> weeks there are <img src="/MI/8f069d777c270870c20f004d237b3936/images/img-0002.png" alt="$2^ n$" style="vertical-align:0px;
width:16px;
height:12px" class="math gen" /> new infections. At this rate the entire world population (7.8 billion) would be infected after slightly under 33 weeks. </p>
<p>When the basic reproduction number <img src="/MI/5779b19ea24f8c78cce5c212d3466c21/images/img-0001.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" /> is less than 1 a very different picture emerges. As an illustration, imagine we have <img src="/MI/e88576aed2a8b24a6720d0eaae750b52/images/img-0001.png" alt="$R_0=0.5.$" style="vertical-align:-2px;
width:67px;
height:15px" class="math gen" /> Now obviously, an infected person can't go on to infect half a person, but remember that this is an average: it means that 10 people can be assumed to go on to infect 5 others, or that 100 people can be assumed to go on to infect 50 others. As before let's assume there is 1 infected person to start with, and that an infected person infects their victim within a week of becoming infectious. Then the number of new infections behaves like this:</p>
<p>After 1 week: <img src="/MI/544a1196f93b89c9494ff79b03c31905/images/img-0001.png" alt="$0.5$" style="vertical-align:0px;
width:21px;
height:13px" class="math gen" /> new infections</br>
After 2 weeks: <img src="/MI/e3c141ae1219b65e88265ed272d5ac69/images/img-0001.png" alt="$0.25$" style="vertical-align:0px;
width:29px;
height:13px" class="math gen" /> new infections</br>
After 3 weeks: <img src="/MI/8db82853acf6d206d2b0915e89f0e22d/images/img-0001.png" alt="$0.125$" style="vertical-align:0px;
width:37px;
height:13px" class="math gen" /> new infections</br>
After 4 weeks: <img src="/MI/2bd0a60a008ba174d8b890c505bcb034/images/img-0001.png" alt="$0.0625$" style="vertical-align:0px;
width:45px;
height:13px" class="math gen" /> new infections.</br></p>
<p>Generally, after <img src="/MI/799e970f4502fd8cc1afb4a054c7b531/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> weeks there are <img src="/MI/799e970f4502fd8cc1afb4a054c7b531/images/img-0002.png" alt="$(0.5)^ n$" style="vertical-align:-4px;
width:40px;
height:18px" class="math gen" /> new infections. This number becomes smaller and smaller as the number <img src="/MI/799e970f4502fd8cc1afb4a054c7b531/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> of weeks becomes larger. A dead end for the disease. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/news/2020/herd/chart_0.5.png" alt="Growth for R0=0.5" width="600" height="331" />
<p style="max-width: 600px;">The average number of new infections after <em>n</em> weeks for <em>R</em><sub>0</sub>=0.5.</p>
</div>
<p>What if <img src="/MI/b730f03e4692fcde4724b542bf153374/images/img-0001.png" alt="$R_0=1$" style="vertical-align:-2px;
width:49px;
height:14px" class="math gen" />? In this case the disease will be <em>endemic</em>: always present in the population, but not an epidemic.</p>
<div class="rightshoutout"><p>See <a href="/content/tags/covid-19">here</a> for all our coverage of the COVID-19 pandemic.</div>
<h3>The effective reproduction number</h3>
<p>So, given that the <img src="/MI/5779b19ea24f8c78cce5c212d3466c21/images/img-0001.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" /> of measles, or some strains of seasonal flu, is greater than 1, how come the whole world hasn't been infected with these diseases a long time ago? The reason is that <img src="/MI/5779b19ea24f8c78cce5c212d3466c21/images/img-0001.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" />
is the average number of people an infected person goes on to infect, given that <em>everyone in the population is susceptible</em>. In real life, this might be the case if someone who has become infected with a disease elsewhere enters a part of the world where the disease has never been seen before, so people don't have immunity and there isn't a vaccine to protect them. An <img src="/MI/5779b19ea24f8c78cce5c212d3466c21/images/img-0001.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" /> of <img src="/MI/967e8bf551622b6beb697d0ea46d3930/images/img-0001.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> then means that, at the beginning, the number of infected people will grow wildly, as we've described above. </p>
<p>However, once a person has recovered from the disease they will (hopefully) gain some immunity. This means that after a while we're not dealing with a totally susceptible population anymore. Indeed, there may be other reasons why some people in the population aren't susceptible: they may be immune for other reasons, or if there's a vaccine, they may have received it, or they may be isolated from the rest of the population. </p>
<p>In most real life situations we should be looking at the <em>effective reproduction number</em> of the disease, sometimes denoted by <img src="/MI/8376109178459a0807de4bee577f02c7/images/img-0001.png" alt="$R$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" />: the average number an infected person goes on to infect in a population where <em>some people are immune</em> (or some other interventions are in place). Of course <img src="/MI/5779b19ea24f8c78cce5c212d3466c21/images/img-0001.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" /> and <img src="/MI/8376109178459a0807de4bee577f02c7/images/img-0001.png" alt="$R$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> are related. Writing <img src="/MI/cb0077b76c2240181213f901fe7beba5/images/img-0001.png" alt="$s$" style="vertical-align:-1px;
width:7px;
height:9px" class="math gen" /> for the proportion of the population that is susceptible to catching the disease, we have</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/4767fc4589ff614fae3326713ee3b484/images/img-0001.png" alt="\[ R=sR_0. \]" style="width:67px;
height:13px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
As an example, if only half the population is susceptible, so <img src="/MI/755c7495ef8a73dc882a00431f6ccea5/images/img-0001.png" alt="$s=0.5$" style="vertical-align:0px;
width:51px;
height:13px" class="math gen" />, we have <img src="/MI/755c7495ef8a73dc882a00431f6ccea5/images/img-0002.png" alt="$R=0.5R_0$" style="vertical-align:-2px;
width:75px;
height:15px" class="math gen" />. In this case, if <img src="/MI/755c7495ef8a73dc882a00431f6ccea5/images/img-0003.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" /> is less than or equal to <img src="/MI/755c7495ef8a73dc882a00431f6ccea5/images/img-0004.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />, then <img src="/MI/755c7495ef8a73dc882a00431f6ccea5/images/img-0005.png" alt="$R$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> is less than or equal to <img src="/MI/755c7495ef8a73dc882a00431f6ccea5/images/img-0006.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" /> and the disease won't turn into an epidemic. The ideal aim of any intervention, be it vaccination or social distancing, is to get the effective reproduction number down to under 1.
<h3>Herd immunity</h3>
<p>What does all this have to do with herd immunity? The general idea behind herd immunity is that in a population where many people are immune a disease can't take hold and grow into an epidemic, thereby protecting people who aren't immune. The population (perhaps unfortunately called a herd ) protects vulnerable individuals.</p>
<p>So how many people in a population need to be immune to have herd immunity? Imagine a disease has a basic reproduction number <img src="/MI/5779b19ea24f8c78cce5c212d3466c21/images/img-0001.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" />, which is greater than 1 so an epidemic threatens. As we have seen, if the <em>effective</em> reproduction number <img src="/MI/8376109178459a0807de4bee577f02c7/images/img-0001.png" alt="$R$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> is less than 1, then the disease will eventually fizzle out. So to achieve herd immunity we need to somehow get the effective reproduction number <img src="/MI/8376109178459a0807de4bee577f02c7/images/img-0001.png" alt="$R$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> to under 1. Since <img src="/MI/e8194180ae11769a5a82b3af26d3b5d8/images/img-0001.png" alt="$R=sR_0$" style="vertical-align:-2px;
width:62px;
height:13px" class="math gen" />, where <img src="/MI/cb0077b76c2240181213f901fe7beba5/images/img-0001.png" alt="$s$" style="vertical-align:-1px;
width:7px;
height:9px" class="math gen" /> is the proportion of the population that is susceptible, we need</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e548c421e6b197af124d22cd3998996b/images/img-0001.png" alt="\[ sR_0<1. \]" style="width:62px;
height:14px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> Rearranging, this gives </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e548c421e6b197af124d22cd3998996b/images/img-0002.png" alt="\[ s<1/R_0. \]" style="width:71px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>In other words, we need to get the proportion of susceptible people in the population to under <img src="/MI/e548c421e6b197af124d22cd3998996b/images/img-0003.png" alt="$1/R_0.$" style="vertical-align:-4px;
width:39px;
height:16px" class="math gen" /> How many people need to be immune to achieve this? If the proportion of susceptible people is <img src="/MI/e548c421e6b197af124d22cd3998996b/images/img-0004.png" alt="$s$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" />, then the proportion of people who are not susceptible, in other words immune, is <img src="/MI/e548c421e6b197af124d22cd3998996b/images/img-0005.png" alt="$1-s$" style="vertical-align:0px;
width:34px;
height:12px" class="math gen" />. Now </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e548c421e6b197af124d22cd3998996b/images/img-0006.png" alt="\[ s<1/R_0 \]" style="width:65px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> means </p><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e548c421e6b197af124d22cd3998996b/images/img-0007.png" alt="\[ 1-s>1-1/R_0. \]" style="width:127px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>So, to achieve herd immunity we need to make sure that at least a proportion of <img src="/MI/e548c421e6b197af124d22cd3998996b/images/img-0008.png" alt="$1-1/R_0$" style="vertical-align:-4px;
width:62px;
height:16px" class="math gen" /> of the population is immune. For an <img src="/MI/e548c421e6b197af124d22cd3998996b/images/img-0009.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" /> of 2.5, the higher end of the estimates for COVID-19, this means that we need to get at least a proportion of <img src="/MI/e548c421e6b197af124d22cd3998996b/images/img-0010.png" alt="$1-1/2.5=0.6$" style="vertical-align:-4px;
width:108px;
height:17px" class="math gen" /> of the population immune. This translates to at least 60%. </p>
<div class="rightimage" style="width: 425px;"><img src="/latestnews/sep-dec09/vaccines/iStock_crowd.jpg" alt="A crowd" width="425" height="282" /><p>The herd can protect the individual.</p>
</div>
<p>How do we do this? Well, ideally we would do it by vaccinating at least 60% of the population. In the absence of a vaccine, we can hope that this level of immunity will be achieved naturally, by people becoming sick and then immune. But because a lot of people die of COVID-19 we can't just let the disease wash over the population, confident in the knowledge that more infections mean more immunity. </p><p>It's because we need to protect vulnerable people and our health care systems that much of the world is currently in lockdown. Ironically, lockdowns mean that many of us are not gaining immunity by having been infected, so the epidemic may spike again once social distancing measures are lifted. </p>
<p>So what are we to do in this worst case scenario? One option would be to remain in lockdown until there is a vaccine, but that could be over a year. Another is to go into intermitted lockdowns to keep successive spikes of the epidemic below the critical capacity of health care systems.</p>
<p>The truth is that at this moment nobody knows exactly what is going to happen in the future. Our most educated guesses come from mathematical models which try and predict the course of the pandemic. You can find out more about these models <a href="/content/how-can-maths-fight-pandemic">here</a>. An <a href="/content/call-action-covid-19">urgent call</a> has gone out to the scientific modelling community to help find the best exit strategies from our current predicament.</p>
<p>In general, our calculations above also send an important message about vaccination: it does not only protect the individual who is being vaccinated against the disease, but also those people who for some reason or other won't be vaccinated and are therefore vulnerable. Vaccination isn't just for you, it's for the whole "herd"!</p>
<hr/>
<p>This article is based on a chapter from the book <em><a href="https://amzn.to/2G6v65L">Understanding numbers</a></em> by the <em>Plus</em> Editors <a href="/content/people/index.html#rachel">Rachel Thomas</a> and <a href="/content/people/index.html#marianne">Marianne Freiberger</a>. </p>
</div></div></div>Thu, 02 Apr 2020 10:46:30 +0000Marianne7277 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-r0-and-herd-immunity#commentsTaking the pandemic temperature
https://plus.maths.org/content/taking-pandemic-temperature
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/chart3_icon.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Rarely before have mathematics and science played such a prominent role in public life. Over the last few weeks we have all become familiar with epidemiological curves, exponential growth, percentage changes, and biological details about viruses we previously never thought about. And not since WWII have we seen our personal freedoms curtailed the way they are right now.</p>
<div class="rightimage" style="width: 400px;"><img src="/content/sites/plus.maths.org/files/news/2020/Winron/cases.png" alt="A map indicating COVID-19 cases world wide" width="400" height="242" /><p>The number of confirmed cases of COVID-19 per million people on March 28, 2020. Source: European Centre for Disease Prevention and Control. Figure: <a href="https://commons.wikimedia.org/wiki/File:Total-confirmed-cases-of-covid-19-per-million-people.png">Our World Data</a>, <a href="https://creativecommons.org/licenses/by/4.0/deed.en">CC BY 4.0</a>.</p>
</div>
<p>Governments and their scientific advisors face a tricky situation. It's essential to keep people on site, as social interventions will only be successful if people stick to them. Whether people do depends, at least in part, on how seriously they take the threat posed by COVID-19 —and this depends on how well governments communicate the science and mathematics behind it.</p>
<p>So are the people on site? How do they feel about the pandemic and the measures taken by their governments? That's what a new study, conducted by the <a href="https://wintoncentre.maths.cam.ac.uk/">Winton Centre for Risk and Evidence Communication</a>, is trying to find out.</p>
<p>The Winton Centre is based at the Mathematics Department at the University of Cambridge. "We are interested in how best to communicate scientific evidence, and how different ways of doing that affects people's understanding, worry, and ability to make decisions," says <a href="https://wintoncentre.maths.cam.ac.uk/about/people/dr-alexandra-freeman/">Alexandra Freeman</a>, Executive Director of the Centre. </p>
<div class="leftshoutout"><p>See <a href="/content/tags/covid-19">here</a> for all our coverage of the COVID-19 pandemic.</div>
<p>A recent <a href="https://www.pnas.org/content/early/2020/03/17/1913678117">Winton Centre study</a>, published rather fittingly on the day the UK prime minister announced a lock down, addressed an important question when it comes to communicating complex scientific issues: how should one deal with the fact that scientific evidence often comes with uncertainties. This is particularly pertinent at the moment, as there is still a lot scientists don't know about COVID-19 and the way that social interventions, might affect the spread of the disease (see the feature article <a href="/content/how-can-maths-fight-pandemic">Fighting COVID-19</a>). The big question is whether being told about the uncertainty surrounding a particular issue might cause people to lose trust, not only in the facts and numbers they are being told about, but also in the sources these come from.</p>
<p>The results of the study were reassuring: they suggest that being told about uncertainty only causes a small decrease in trust in numbers and trustworthiness of the source — and when it was done as precisely as possible, with a numerical range, this change was negligible. As part of their COVID-19 work, the Winton team have repeated the same experiment using uncertainty around the mortality rate from the virus and confirmed the results even in this very present, emotional context. This should help reassure communicators, including those who are currently informing us about the COVID-19 pandemic, that they can be open and transparent about the limits to what they currently know.</p>
<h3>Attitudes to the COVID-19 crisis</h3>
<p>The new study into people's attitudes to the COVID-19 crisis builds on the Centre's work on uncertainty. Around the end of the third week of March the Centre collected data from people in seven countries: the UK, US, Italy, Spain, Germany, Mexico and Australia. The aim is to find out what people think and feel about this unprecedented situation, the severe limitations imposed on their freedoms, and the way their governments are dealing with the crisis and keeping them informed.</p>
<p>The initial analysis of the results gives a first glimpse of the attitudes and make fascinating reading. "Perhaps the most striking thing is the similarity across countries (with Mexico as the most different)," writes Freeman on the Centre's <a href="https://medium.com/wintoncentre/how-different-countries-are-reacting-to-the-covid-19-risk-and-their-governments-responses-63e35f979111" target="blank">blog</a>. Worry about the virus is high in all countries, but perhaps surprisingly, people from all countries felt their governments' response in the past few weeks had "not been firm enough". Even Italians, who had been living with strict social distancing measures for at least ten days before the survey and Spaniards, who had been under lock-down for a week, felt this way. "Mexico, with the fewest restrictions at the moment, is most united that the response is definitely 'not firm enough'."</p>
<div class="centreimage"><img alt="Chart 1" src="/content/sites/plus.maths.org/files/news/2020/Winron/chart1.png" style="max-width: 600px; height: 308px;" />
<p style="max-width: 360px;">Chart produced by the Winton Centre for Risk and Evidence Communication.</p>
</div>
<p>But there were also interesting differences "One thing that we thought might affect people's attitude to their government's strategy was how much they felt that we as individuals should give things up for the benefit of society, and here the cultural differences are, perhaps, surprising," writes Freeman. "Italy seems incredibly pro-social whilst Germany — and the UK — much more individualistic."</p>
<div class="centreimage"><img alt="Chart 2" src="/content/sites/plus.maths.org/files/news/2020/Winron/chart2.png" style="max-width: 600px; height: 308px;" />
<p style="max-width: 600px;">Chart produced by the Winton Centre for Risk and Evidence Communication.</p>
</div>
<p>Variation was also found in the level of trust people in different countries have in their government to deal with the crisis. Germans are leading the way here, with levels of trust much higher than in Mexico, the UK and the US.</p>
<p>This may partly be down to how much people feel they understand their governments' strategies, which will in turn depend on how well they feel they understand the science and mathematics that informs them. Here there is again a lot of variation. "Germany seems particularly confident about their understanding of the government's strategy, whilst the UK and US much less so," says Freeman. (This data was collected before the UK prime minister's broadcast on the 23rd March which started a lock-down).</p>
<div class="centreimage"><img alt="Chart 3" src="/content/sites/plus.maths.org/files/news/2020/Winron/graph4.png" style="max-width: 600px; height: 308px;" />
<p style="max-width: 600px;">Chart produced by the Winton Centre for Risk and Evidence Communication.</p>
</div>
<div class="centreimage"><img alt="Chart 3" src="/content/sites/plus.maths.org/files/news/2020/Winron/chart3.png" style="max-width: 600px; height: 308px;" />
<p style="max-width: 600px;">Chart produced by the Winton Centre for Risk and Evidence Communication.</p>
</div>
<p>Trust in scientific advisors, however, was high in all countries.</p>
<p>The initial summary of the findings, shown in more detail on the <a href="https://medium.com/wintoncentre/how-different-countries-are-reacting-to-the-covid-19-risk-and-their-governments-responses-63e35f979111" target="blank">Winton Centre blog</a>, is only the tip of the ice berg. Over the next few weeks the Winton Centre will be subjecting the statistics to a more extensive analysis. They are planning to run their survey in two more countries, as well as surveying the UK several times over the next few months. The results will not only be interesting for policy makers and communicators during the current crisis, but hopefully also inform the way they will communicate with us in the future.</p>
<p>In the mean time, governments and their officials might be well advised to heed the advice of David Spiegelhalter, Chair of the Winton Centre, on communicating during a crisis. (You can read a longer interview with Spiegelhalter <a href="/content/communicating-corona-crisis">here</a>).</p>
<p>"You should be communicating a lot, consistently and with trusted sources," he says. "You have to be open and transparent. You have to say what you do know and then you have to say what you don't know. You have to emphasise, and keep emphasising, the uncertainty, the fact that there is much we don't know. Then you have to say what you are planning to do and why."</p>
<p>"Finally, you have to say what people themselves can do, how they should act. The crucial thing to say is that this will change as we learn more."</p>
<p>If there is anything positive about this pandemic then it's perhaps the fact that we are all getting to witness science in action. This may influence our future attitudes to the uncertainties involved and make the job of communicators a little easier. If such a positive impact results, then Winton Centre will be sure to find out.</p>
</div></div></div>Wed, 01 Apr 2020 09:20:14 +0000Marianne7276 at https://plus.maths.org/contenthttps://plus.maths.org/content/taking-pandemic-temperature#commentsA call to action on COVID-19
https://plus.maths.org/content/call-action-covid-19
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_15.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>An urgent call has gone out to the scientific modelling community to help fight against the COVID-19 pandemic. The call is coordinated by the Royal Society and led by a small group of academics including two Cambridge mathematicians, Michael Cates, the Lucasian Professor of Mathematics, and Julia Gog, Professor of Mathematical Biology.</p>
<div class="rightshoutout">"The response has been overwhelming with hundreds of researchers offering their services so far." – Michale Cates</div>
<p>The UK is lucky to have world-class experts in epidemiology and pandemic modelling. These researchers are working round-the-clock on the evolving pandemic, providing evidence to inform government policy, but they are fully stretched and this is where researchers from other fields can help. The Rapid Assistance in Modelling the Pandemic (RAMP) taskforce has been set up to harness the valuable skills of the wider scientific community in the UK, such as those experts in computer modelling who do not have direct experience working on pandemic models. Cates says that this is the first time in his lifetime that he's encountered such an urgent call to arms for mathematicians and scientists to help save lives.</p>
<div class="leftimage" style="max-width: 350px;"><img src="content/sites/plus.maths.org/files/blog/042016/crowd.jpg" alt="Shibuya crossing" width="350" height="234"/>
<p>Agent based models are used widely in simulating systems in biology, sociology, economics and even management – including modelling the movement of people. Photo: <a href="https://www.flickr.com/photos/andreweland/4087246655/in/photolist-7j9nJ1-gPGF9-88aReK-f4RHRA-88e77G-5XAduS-7ebd2V-wrQYr-FPicn-8ajFZe-osCa8-942vS5-pkcq5x-8m1Kzg-7Fw5L2-88aUsP-ePDBQs-DuPXcd-7MkYu3-6fckzt-9osr9d-7UNuXU-39KJPX-dTPwqZ-ossoh5-baviBe-dTtXZZ-97oAVg-5LxrNC-vhQ4q-7NjQ69-B9VgZH-9fBmg3-4FFohK-B7JhMQ-d1KxE-7wFsJj-Yx4jhJ-G7PvY-bavdcH-nWPuBT-76cZU3-7UNvaN-dW9kur-2DCzAJ-8anWdy-6fcm76-97oDBv-arD1BW-oUBSLq/">Andrew Eland</a>, <a href="https://creativecommons.org/licenses/by-sa/2.0/">CC BY-SA 2.0</a>.</p>
</div>
<p>Scientists from both academia and industry, working in research areas such as urban traffic planning, financial market modelling, dataflow optimization across communications networks, and individualized marketing on social media, have vital skills that could support the pandemic modelling effort. As the RAMP call explains, for example, some existing epidemic models, called individual based models, are closely related to <a href="/content/agm">agent based models</a> used in some of these other research fields. The fact that expertise from such a wide range of areas can help is testament to the effectiveness and interconnected nature of mathematics, with some of the epidemiological models being closely related to models in these other research fields.</p>
<p>"Julia Gog has been involved in the epidemic modelling all along and will play a pivotal role connecting the new 'volunteer' workforce with the existing modelling teams," says Cates. The taskforce could provide advice on how research from these different areas can be applied to modelling the pandemic, build the software needed to integrate the vast datasets into the pandemic models, analyse the data, and add to the human and computing resources required to tackle this massive task.</p>
<div class="rightshoutout">"We are all humans first and mathematicians or scientists second – faced with a human catastrophe, everyone's first instinct is to do whatever they can to help out." – Michael Cates</div>
<p>A primary aim of RAMP is to understand how possible exit strategies from the lockdown conditions currently in place in the UK, and around the world, could work, says Cates. In order to do this, RAMP will coordinate the efforts of this new volunteer workforce to build the computing and modelling capacity needed to explore possible exit strategies, refine the predictions of these models in response to live data from around the world, and try to understand our best way forward. "Understanding possible exit strategies is one primary aim; other [aims] may emerge as the situation evolves," says Cates.</p>
<p>"The response has been overwhelming with hundreds of researchers offering their services so far," says Cates. "They have been incredibly helpful. It shows that we are all humans first and mathematicians or scientists second – faced with a human catastrophe, everyone's first instinct is to do whatever they can to help out. The goal of RAMP is to marshal this goodwill into the most productive channels possible."</p></div></div></div>Tue, 31 Mar 2020 11:18:30 +0000Rachel7275 at https://plus.maths.org/contenthttps://plus.maths.org/content/call-action-covid-19#commentsMaths in a minute: Voronoi diagrams
https://plus.maths.org/content/maths-minute-voronoi-diagrams
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/voronoi_icon.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Imagine a city with a number of hospitals. When someone has an emergency you'd like them to always go, or be taken, to the hospital that's nearest to where they currently are. What you need is a map showing each hospital's catchment area: for anyone in that region, that hospital is closer than any other. How do you do that?</p>
<p>It's not hard, only possibly a little tedious if you do it by hand. Start with two hospitals at points A and B on the map. Draw the line segment that connects them, find the midpoint of that line segment, and then draw the line that passes through that midpoint and is perpendicular to the segment from A to B. That line divides the city into two regions. One of them, the one containing A, contains all the points closer to A than to B. The other contains all the points closer to B than to A. The points on the line are the same distance from A and B.</p>
<div class="centreimage"><img alt="Two points and bisector" src="/content/sites/plus.maths.org/files/articles/2020/Miam/2points.png" style="max-width: 500px; height: 379px;" />
<p style="max-width: 500px;"></p>
</div>
<p>Now look at the third hospital, at point C. Repeating what we did above you work out the region of points that are closer to A than to C, and the region of points that are closer to B than to C. The region of points that is closer to A than to both B and C is now the intersection of the region of points closer to A than to B and the region of points closer to A than to C. </p>
<div class="centreimage"><img alt="voronoi diagrams with three regions" src="/content/sites/plus.maths.org/files/articles/2020/Miam/3points.png" style="max-width: 500px; height: 389px;" />
<p style="max-width: 500px;"></p>
</div>
<p>You continue like this, intersecting regions, until you have accounted for all the hospitals. The picture you get at the end, the division of the map into regions of points that are all closer to one of the given points than any other, is called a <em>Voronoi diagram</em>. It's named after the Russian mathematician <a href="http://mathshistory.st-andrews.ac.uk/Biographies/Voronoy.html">Gregory Voronoi</a> (1868-1908).</p>
<div class="centreimage"><img alt="voronoi diagrams with three regions" src="/content/sites/plus.maths.org/files/articles/2020/Miam/voronoi.png" style="max-width: 350px; height: 350px;" />
<p style="max-width: 350px;">A Voronoi diagram (created by <a href="https://commons.wikimedia.org/wiki/File:Euclidean_Voronoi_diagram.svg">Balu Ertl</a>, <a href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a>.</p>
</div>
<p> As you can imagine Voronoi diagrams are useful in all sorts of areas. For example, they can be used to study the growth patterns of forests, or help robots find clear routes through a set of obstacles. <a href="https://en.wikipedia.org/wiki/Voronoi_diagram#Applications">Wikipedia lists many other applications</a>.</p>
<div class="rightimage" style="max-width: 300px;"><img src="/sites/plus.maths.org/files/news/2010/broadst/varonoimap_frerichs.jpg" alt="John Snow's map." width="300" height="304" />
<p>John Snow's map. Each bar represents a death at an address. The curve marks points at equal distance from the Broad Street pump and another pump.</p>
</div><!-- image in public domain -->
<p>But we chose the medical analogy for a reason. In the 1850s a cholera outbreak was decimating Soho in London, killing 10% of the population and wiping out entire families in days. It was thought at the time that the disease was caused by "bad air", but physician John Snow had another idea: he thought that cholera came from contaminated water supplies, which in those days came from pumps positioned throughout the city.</p>
<p>He was able to convince others of his theory by first marking the number of deaths at each address on a map of Soho. He then also identified the "catchment area" of a particular water pump at 40 Broad Street (now Broadwick Street). Points within this catchment area were closer to the Broad Street pump than any other pump — only that, unlike in our example above, Snow didn't use the direct distance a crow would fly, but the walking distance along streets and alleys. It turned out that almost all the deaths marked on the map lay inside the catchment area of the Broad Street pump and anecdotal evidence explained the few cases that did not.</p>
<p>This demonstrated very convincingly that contaminate water was indeed the cause of cholera. Today the spot where the pump once stood is marked by a memorial and there's a pub right next to it named in John Snow's honour. You can find out more in <a href="/content/uncovering-cause-cholera">this article</a>.</p> </div></div></div>Mon, 30 Mar 2020 14:57:19 +0000Marianne7274 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-voronoi-diagrams#commentsHow can maths fight a pandemic?
https://plus.maths.org/content/how-can-maths-fight-pandemic
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_38.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/packages/2017/Women/Julia_small.jpg" alt="Julia Gog" width="350" height="233" />
<p>Julia Gog.</p>
</div>
<p>"Life is not going to be the same for a long time," says <a href="https://www.infectiousdisease.cam.ac.uk/directory/jrg20@cam.ac.uk">Julia Gog</a>, an epidemiologist at the University of Cambridge. Gog's own life changed abruptly in early February, when she dropped her normal duties at the Centre for Mathematical Sciences and started to devote all her efforts to SPI-M, a modelling group which feeds its results into the
<a href="https://www.gov.uk/government/groups/scientific-advisory-group-for-emergencies-sage-coronavirus-covid-19-response">Scientific Advisory Group for Emergencies</a> (SAGE). SPI-M has run for some time in preparedness for influenza pandemics, but now has been turned over to focussing on the pandemic of COVID-19. Gog is also on the steering committee of a <a href="https://epcced.github.io/ramp/">national consortium</a>, led by the Royal Society, to deal with the pandemic.</p>
<p>SPI-M's job is to develop and use mathematical models that can help us predict what will happen next and how different interventions might change that: how the COVID-19 pandemic is going to evolve and what effect the social interventions we are all living through now are likely to have. But what are these models and are they accurate?</p>
<h3>The models</h3><div class="rightshoutout"><p>See <a href="/content/tags/covid-19">here</a> for all our coverage of the COVID-19 pandemic.</div>
<p><p>You may not have noticed it, but there’s a good chance you’ve already dabbled in some mathematical modelling yourself. If you have heard that the number of infections with COVID-19 doubles every three days, you may well have worked out that, if today we have <img src="/MI/57b2a9c154131d1eaaeaebc11ac22964/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> cases and this trend continues, then we will have </p><p><img src="/MI/57b2a9c154131d1eaaeaebc11ac22964/images/img-0002.png" alt="$2x$" style="vertical-align:0px;
width:17px;
height:12px" class="math gen" /> cases in 3 days, </p><p><img src="/MI/57b2a9c154131d1eaaeaebc11ac22964/images/img-0003.png" alt="$4x$" style="vertical-align:0px;
width:17px;
height:12px" class="math gen" /> cases in 6 days, </p><p><img src="/MI/57b2a9c154131d1eaaeaebc11ac22964/images/img-0004.png" alt="$8x$" style="vertical-align:0px;
width:17px;
height:12px" class="math gen" /> cases in 9 days </p><p>and, generally, <img src="/MI/57b2a9c154131d1eaaeaebc11ac22964/images/img-0005.png" alt="$2^ nx$" style="vertical-align:0px;
width:26px;
height:12px" class="math gen" /> in 3<img src="/MI/57b2a9c154131d1eaaeaebc11ac22964/images/img-0006.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> days. That’s the steep growth which has led to this disaster. </p>
<p>This extrapolation, simple as it may be, highlights the basic ingredients of a model: a mathematical expression which describes the general nature of the change that will happen over time and parameters which pin down the exact shape of the change. In our example we have exponential growth over time, and the steepness of this growth is decided by the parameter of the doubling time, which is 3 days.</p>
<div style="max-width: 340px; float: right; border: thin solid grey;
background: #CCC CFF; padding: 0.5em; margin-left: 1em; font-size:
75%; margin-bottom: 1em; ">
<h3>
The SIR model</h3><p>
<p>Let <img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0001.png" alt="$S$" style="vertical-align:0px;
width:10px;
height:11px" class="math gen" /> denote the number of susceptible people, <img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0002.png" alt="$I$" style="vertical-align:0px;
width:9px;
height:11px" class="math gen" /> the number of infected people, and <img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0003.png" alt="$R$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> the number of recovered people. The equations for the SIR model are </p><table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray">
<tr id="a0000000003">
<td style="width:40%"> </td>
<td style="vertical-align:middle; text-align:right"><img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0004.png" alt="$\displaystyle \frac{dS}{dt} $" style="vertical-align:-11px; width:20px; height:33px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0005.png" alt="$\displaystyle = $" style="vertical-align:2px; width:11px; height:4px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:left"><img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0006.png" alt="$\displaystyle - \beta SI $" style="vertical-align:-3px; width:42px; height:14px" class="math gen" /></td>
<td style="width:40%"> </td>
<td style="width:20%" class="eqnnum"> </td>
</tr><tr id="a0000000004">
<td style="width:40%"> </td>
<td style="vertical-align:middle; text-align:right"><img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0007.png" alt="$\displaystyle \frac{dI}{dt} $" style="vertical-align:-11px; width:18px; height:33px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0008.png" alt="$\displaystyle = $" style="vertical-align:2px; width:11px; height:4px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:left"><img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0009.png" alt="$\displaystyle \beta SI - \nu I $" style="vertical-align:-3px; width:68px; height:14px" class="math gen" /></td>
<td style="width:40%"> </td>
<td style="width:20%" class="eqnnum"> </td>
</tr><tr id="a0000000005">
<td style="width:40%"> </td>
<td style="vertical-align:middle; text-align:right"><img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0010.png" alt="$\displaystyle \frac{dR}{dt} $" style="vertical-align:-11px; width:22px; height:33px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:center"><img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0005.png" alt="$\displaystyle = $" style="vertical-align:2px; width:11px; height:4px" class="math gen" /></td>
<td style="vertical-align:middle; text-align:left"><img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0011.png" alt="$\displaystyle \nu I $" style="vertical-align:0px; width:17px; height:11px" class="math gen" /></td>
<td style="width:40%"> </td>
<td style="width:20%" class="eqnnum"> </td>
</tr>
</table><p> Here <img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0012.png" alt="$\beta $" style="vertical-align:-3px;
width:10px;
height:14px" class="math gen" /> is the transmission rate and <img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0013.png" alt="$\nu $" style="vertical-align:0px;
width:8px;
height:7px" class="math gen" /> is the recovery rate. The expressions <img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0014.png" alt="$d/dt$" style="vertical-align:-4px;
width:32px;
height:16px" class="math gen" /> represent the rate of change over time, so <img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0015.png" alt="$dS/dt$" style="vertical-align:-4px;
width:41px;
height:16px" class="math gen" /> means the rate of change of the number of susceptibles over time. </p><p>The number <img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0016.png" alt="$R_0= \beta /\nu N,$" style="vertical-align:-4px;
width:88px;
height:16px" class="math gen" /> where <img src="/MI/c5a1e16295d96b146b1a629cfae91ca4/images/img-0017.png" alt="$N$" style="vertical-align:0px;
width:15px;
height:11px" class="math gen" /> is the size of the population, is called the <em>basic reproduction number</em> of the disease.</p> <p>You can find out more about the SIR model in <a href="/content/mathematics-diseases">this article</a>.</p></div>
<p>If you want longer term predictions and to simulate the impact of interventions in detail, you need a more sophisticated model. Different models are designed for different purposes, such as making short term predictions, long term predictions, or simulating the effect of particular interventions such as school closures. But although models differ, they tend to be built around an approach that has been around since the 1910s: the <em>SIR model.</em> </p>
<p>To get the general idea behind the SIR model, imagine a population of people in which everyone is either susceptible to a disease (S), infected (I), or recovered (R) and therefore immune. The way people pass from the S class into the I class, and then from the I class into the R class, is described by mathematical equations. These equations depend on the <em>transmission rate</em> for the disease and also the <em>recovery rate</em>. You start the model off with only a small proportion of the population in the I class, and then let it evolve over time, seeing how the disease spreads and then subsides as people recover and become immune.</p>
<p>Although simple, the SIR model gives good predictions for simple populations, such as students at a boarding school. When it comes to more complex populations you can link up many individual SIR models representing different geographical locations and sub-populations, including for example individual towns or schools.</p>
<h3>Contacts are key</h3>
<p>What is hugely important in this context are people's contact patterns: who meets whom and how frequently. Information on this comes from social mixing studies. An example is a large-scale <a href="/content/very-useful-pandemic">citizen science project</a> which ran in 2018 as a collaboration between the BBC and Gog's team. Here people were asked to download an app which tracked their movements and asked them to keep track of the people they met (all suitably anonymised). Such contact data is represented mathematically by arrays (<em>matrices</em>) of numbers (see the figure below) which are built into the model.</p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2020/Gog/contact_matrix.jpg" alt="Contact matrix" width="350" height="352" /><p>A contact matrix displaying average contacts between different age groups. Darker shades indicate more contacts (here the colours are used instead of numbers to make the matrix easier to understand). The figure is from the paper <a href="https://www.sciencedirect.com/science/article/pii/S1755436518300306">Contagion! The BBC Four Pandemic – The model behind the documentary</a>. Used by permission.</p></div>
<p>To see how a particular social intervention, such as school closures, might affect the spread of the disease, you adapt the contact data accordingly by removing or scaling down parts of it that relate to some intervention. </p><p>But you have to be careful. "Is entirely switching off the school component realistic, well no, it should just be reduced because key worker kids are still going to go to school," says Gog. "And if there isn't clear guidance, then the children off school might end up mixing together in other ways, or with their grandparents, which means there's additional contacts happening that need to be taken into account. Though we can only guess at the extent of them." Existing data, such as information on what happened during teacher strikes, can help you calibrate your contact data to reflect interventions and predict their effect on the epidemic. This, put simply, is how epidemiological modelling works. </p>
<h3>But are the models right?</h3>
<p>There isn't just one epic model designed to represent the whole of the UK, Europe, or even the entire world. Instead, there are many different ones designed to do different things and, although the compartmental approach of the SIR model is a ruling paradigm, models can still be different in their nature. Some are completely deterministic, others contain a degree of randomness, some are designed to run just once to demonstrate the role of one particular factor, others are run many times to get ranges of predictions in the face of uncertainty.</p>
<p>The big question is whether the models are realistic. One problem is that COVID-19 is a new disease and existing models were developed for seasonal flu. "We all started from influenza, no one had a corona pandemic model," says Gog. "So at the [start of this pandemic] we had to ask what might be systematically different between COVID-19 and the typical models we had built for influenza."</p>
<p>Although the pandemic dynamics are similar for 'flu and the coronavirus, there are differences. One is that for COVID-19 there's a substantial latent period: a person can have the infection without showing any symptoms. "For 'flu maybe you get a few hours of that, but for this coronavirus it can be a few days," says Gog. In terms of the model this means moving into the world of SEIR, where E stands for "exposed": people in this class have the infection but no symptoms yet. The E class can be split further into those who are infectious to others and those who aren't. "All models are approximations, and for 'flu you can often get away with SIR, depending on what you are trying to address with the model. But ignoring the latent period for this virus would be a much worse approximation, especially if making short-term predictions, so usually it does need to be considered," says Gog.</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2020/Gog/sirsys-p9_0.png" alt="Output from a SIR model" width="350" height="263" /><p>This is the typical output from a simple SIR model. The number of susceptible people is shown in blue, the number of infected in green, and the number of recovered in red. </p></div><!-- Image in public domain -->
<p>There are also plenty of other things we don't know about COVID-19. "There are some details we just do not know, for example exactly how infectious you are on day one, or on day two," says Gog. "Inferring that from imperfect data is very hard, though we do now have some data from China and other countries, and the data from cruise ships have been very interesting. You just try and do your best to extrapolate from limited information." When there isn't enough information, modellers decide which of the unknowns are most important to resolve in order to remove uncertainty — making such calls is a critically important aspect of modelling.</p>
<p>The importance of any one parameter can depend on exactly what you are trying to do. "To predict how many cases there are going to be tomorrow you don't need to know many things. It's almost exponential at the moment," says Gog. "But to make a prediction about whether there is going to be a second wave you need to know some really very different things."</p>
<p>An important number for longer term predictions is the <em>basic reproduction number</em> of a disease (often denoted as <img src="/MI/5779b19ea24f8c78cce5c212d3466c21/images/img-0001.png" alt="$R_0$" style="vertical-align:-2px;
width:19px;
height:13px" class="math gen" />): the number of people an infectious person will infect on average, assuming that everyone in the population is susceptible to the disease (it is related to the transmission rate, see the box above). For COVID-19 this is estimated to lie between 2 and 2.5. Modellers will often run their model for each of a range of possible values, coming up with a range of corresponding predictions. </p>
<h3>Infected but not ill</h3>
<p>Another important number, for which many epidemiologists would currently give a limb, is the number of true cases of the infection in the population, which includes the number of cases of people who have had the disease but shown no symptoms. "The number of asymptomatics is the number that I lose sleep over at the moment. Knowing this is crucial for our exit strategy," says Gog. </p>
<p>This is because there are basically two things that can reduce the current exponential growth. "The first is that the contact rates change under interventions such as school closures and physical distancing: this is the entire point of these," says Gog. "The second thing that changes the exponential process is the depletion of susceptibles." Since having had the disease will make you immune for some time, knowing the true number of cases including asymtpomatics, will tell you how fast the class of susceptible people will become smaller. The much-discussed mechanism of herd immunity means that as the number of susceptibles reduces, the exponential growth of the disease flattens out, and then becomes exponential decay.</p>
<p>Luckily there is hope as far as knowing data about the asymptomatics is concerned. Antibody tests which can tell whether a person has had the disease, knowingly or not, do now exist, although the first wave of these will understandably <a href="https://www.theguardian.com/world/2020/mar/24/matt-hancock-35m-coronavirus-test-kits-are-on-the-way-to-the-nhs">only be rolled out to NHS staff</a>.</p>
<p>But even with imperfect information, modelling predictions are not just stabs in the dark especially if they account for and present the range of uncertainty. Good models comprise all the relevant information we do have. Good modellers keep careful track of the limitations of their models and the uncertainties within them, often including ranges of possible parameter values, which then lead to ranges of predictions: ranges of possible future scenarios that may occur under different intervention strategies. The predictions won't be perfect, but they are the best we can do with the information we have. </p>
<h3>So what <em>is</em> going to happen?</h3>
<p>Nobody can tell exactly what is going to happen. A big hope on the horizon is the arrival of a vaccine, another means by which herd immunity can be built with the options of protecting the most vulnerable as priority, expected in about twelve to eighteen months. The question is how to get ourselves to that point with the least amount of damage done.</p>
<p>One thing that everyone agrees on is that this will involve long term sacrifices. "We can't just shut down for a week and expect this thing to be gone," says Gog. "It's going to still be here [if social distancing measures are lifted too soon] and we'll have no herd immunity. At the moment we've got no choice but to shut down to ensure our health care system is not pushed over capacity, but we are very aware that this isn't a permanent strategy."</p>
<p>
A <a href="https://dash.harvard.edu/bitstream/handle/1/42638988/Social%20distancing%20strategies%20for%20curbing%20the%20COVID-19%20epidemic.pdf?sequence=1&isAllowed=y">paper published last week</a> by a former PhD student of Gog's, <a href="http://www.stephenkissler.com/">Stephen Kissler</a>, and colleagues at Harvard had a detailed look at the problem of resurgence, taking account of seasonal variations too: outbreaks of respiratory diseases tend to be worse in autumn and winter, putting an even greater strain on health care systems as they coincide with seasonal flu outbreaks. Kissler and his team used an SEIR model that included components reflecting such seasonal variations. The effect of social distancing measures is reflected in the model by a reduction of the basic reproduction number of COVID-19 by up to 60%, on par with what was observed in China.</p>
<p>The conclusions of this latest study aren't exactly cheerful. A single period of social distancing isn't going to be sufficient to stop critical care capacity from being overwhelmed (the study looked at critical care capacity in the US, rather than the UK, but similar results will apply in the UK). "[According to this study] we are looking at a period of serial lock downs," says Gog. "The idea is that you lock down when critical care is about to keel over. But what is happening in the UK is that the NHS is expanding its provisions, so hopefully our lock downs can get shorter and less severe and not too frequent."</p>
<p>Just how long and frequent these intermittent periods of social distancing are likely to be under different assumptions (based on US figures) is shown in the figures below, taken from the paper.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2020/Gog/kissler.png" alt="Prediction from model by Kissler et al." width="500" height="650" /><p style="max-width: 500px;">These graphs show the prevalence of the virus (black curves) and critical cases (red curves) under intermittent social distancing
(shaded blue regions). The first and third graphs are without seasonal forcing; the second and fourth with seasonal forcing. Critical care capacity is depicted by the solid horizontal black bars; the first two graphs are the scenarios with
current US critical care capacity and the third and fourth graphs are the scenarios with double the current critical care capacity. The
maximal value of the basic reproduction number is 2 in wintertime, and for the seasonal scenarios it is 1.4 in summertime. The figure is from the paper <a href="<a href="https://dash.harvard.edu/bitstream/handle/1/42638988/Social%20distancing%20strategies%20for%20curbing%20the%20COVID-19%20epidemic.pdf?sequence=1&isAllowed=y">Social distancing strategies for curbing the COVID-19 epidemic</a> by Kissler et al. Used by permission.</p></div>
<p>Within all this gloom there are, however, a couple of rays of hope. One is that medication and better treatment protocols for the severe cases of COVID-19 may arrive at some point. This would mean that people would get less sick for shorter periods of time, reducing pressure on the NHS. Since much of the severity of social distancing measures is down to our need to keep the NHS from collapsing, so that people with severe disease can be effectively cared for, this may then also mean less draconian measures for shorter periods of time.</p>
<p>The other ray of hope goes back to that unknown total number of mild and asymptomatic infections. If this is much higher than assumed in the model, if many more people have been sick and are now immune, then the outlook isn't anywhere near as bad as the figures above suggest. We can only hope that this is the case. Until we find out, all we can do is stick to the rules and stay at home.</p>
<hr/>
<h3>About this article</h3>
<p> <a href="https://www.infectiousdisease.cam.ac.uk/directory/jrg20@cam.ac.uk">Julia Gog</a> is Professor of Mathematical Biology at the University of Cambridge. She is a member of SPI-M, a modelling group which feeds its results into the
<a href="https://www.gov.uk/government/groups/scientific-advisory-group-for-emergencies-sage-coronavirus-covid-19-response">Scientific Advisory Group for Emergencies</a> (SAGE). She is also a member of the steering committee of a <a href="https://epcced.github.io/ramp/">national consortium</a>, led by the Royal Society, to deal with the COVID-19 pandemic.</p>
<p><a href="https://plus.maths.org/content/people/index.html#marianne">Marianne Freiberger</a>, Editor of <em>Plus</em>, interviewed Gog on March 24, 2020.</p>
</div></div></div>Mon, 30 Mar 2020 14:47:44 +0000Marianne7272 at https://plus.maths.org/contenthttps://plus.maths.org/content/how-can-maths-fight-pandemic#commentsMaths in a minute: Social distancing
https://plus.maths.org/content/maths-minute-social-distancing
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/hexagon_icon.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Imagine lots of people in a park, sitting in the Sun. Each person wants to keep at least 2m distance to everyone else, but they want to have as many people as possible at the minimum 2m distance. How should the people arrange themselves to make this happen? And what's the number of people a person will have exactly 2m away from themselves in this arrangement?</p>
<div class="rightshoutout"><p>See <a href="/content/tags/covid-19">here</a> for all our coverage of the COVID-19 pandemic.</div>
<p>Let's start with a single person whom we will call person A. People who want to be 2m away from person A should position themselves somewhere on the circle that has A as its centre and a radius of 2. Let's look at two people on this circle, B and C. Since B and C also want to be 2m away from each other, the triangle formed by A, B and C should be an equilateral triangle with side length 2. </p>
<div class="centreimage"><img alt="Equilateral triangle and circle" src="/content/sites/plus.maths.org/files/articles/2020/Miam/triangle.png" style="max-width: 300px; height: 298px;" />
<p style="max-width: 300px;"></p>
</div>
<p>How many such triangles can you fit around A so that they don't overlap? Well, the angles in an equilateral triangle are all 60 degrees. There are a total of 360 degrees available around A, which means that you can fit exactly 360/60=6 equilateral triangle around A. In other words, the number of people that can position themselves 2m away from A and also be 2m from each of their two neighbours on the circle is six. </p>
<div class="centreimage"><img alt="Hexagon on circle" src="/content/sites/plus.maths.org/files/articles/2020/Miam/hexagon.png" style="max-width: 300px; height: 298px;" />
<p style="max-width: 300px;"></p>
</div>
<p>Now exactly the same argument holds for any other person on A's circle. The number of its closest neighbours, exactly 2m away is six, and they should be positioned around the person in the shape of a regular hexagon made up of six equilateral triangles. </p>
<div class="centreimage"><img alt="Two hexagons" src="/content/sites/plus.maths.org/files/articles/2020/Miam/2hexagons.png" style="max-width: 350px; height: 295px;" />
<p style="max-width: 350px;"></p>
</div>
<p>Since this works for every person the answer is that people should arrange themselves on a triangular lattice made up of equilateral triangles. Every person in the lattice is also at the centre of a regular hexagon.</p>
<div class="centreimage"><img alt="Triangular grid" src="/content/sites/plus.maths.org/files/articles/2020/Miam/grid.png" style="max-width: 400px; height: 348px;" />
<p style="max-width: 400px;"></p>
</div>
</div></div></div>Fri, 27 Mar 2020 17:09:36 +0000Marianne7273 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-social-distancing#commentsMyths of maths
https://plus.maths.org/content/myths-maths
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/myth_icon_0.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>When we think of mathematics we tend not to think about myths. Myths are the stuff of legend and wonder, while maths is logical and has no room for doubt and error. This is certainly true, but it doesn't mean that there is no room for mythology in the study of mathematics. A mathematical truth, through retelling and a lack of some understanding, enters the public consciousness as a myth rather than a truth. This is especially the case if the story satisfies some underlying need for some order and pattern in life, the Universe, and everything. Unfortunately, it is often mathematical myths that are reported in the press and on TV, rather than the underlying truth. This is a great shame as mathematical truths are often far more exciting and surprising than any myth, and give us insight into the way Universe works. </p>
<p>Here are three mathematical myths. In each article we will look both at the myth and the underlying truth. We hope that we will convince you that the truth is often much stranger than fiction. </p>
<div style="float: left; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/myth_icon.png" alt="" width="100" height="100" /> </div>
<p><a href="/content/myths-maths-golden-ratio">The golden ratio</a> — The golden ration is a number with many amazing mathematical properties. But is it really a secret of nature and the epitome of beauty? </p></div>
<div style="float: left; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/world_map4_icon.png" alt="" width="100" height="100" /> </div>
<p><a href="/content/myths-maths-four-colour-theorem">The four colour theorem</a> — It's one of mathematics' most famous results: every "map" can be coloured using at most four colours. What it doesn't usually apply to, however, are real maps. </p></div>
<div style="float: left; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/doors_icon.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/myths-maths-monty-hall-problem">The Monty Hall problem</a> — This puzzle is famous because the accepted answer is counter-intuitive. But is it always correct?</p></div>
<hr/>
<p>These articles are based on a talk in Chris Budd's ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-in-the-21st-century/">Gresham
College lecture series</a>. Below is a video of the talk.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/fwtVlcO6s-Y" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
</div></div></div>Mon, 23 Mar 2020 11:31:50 +0000Marianne7267 at https://plus.maths.org/contenthttps://plus.maths.org/content/myths-maths#commentsMyths of maths: The Monty Hall problem
https://plus.maths.org/content/myths-maths-monty-hall-problem
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/doors_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightshoutout"><p>This article is based on a talk in an ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-in-the-21st-century/">Gresham
College lecture series</a>. You can see a video of the talk <a
href="/content/myths-maths-monty-hall-problem#video">below</a>.</div><p>The Monty hall problem is one of the most famous problems in mathematics and in its original form goes back to a game show hosted by the famous <a href="https://en.wikipedia.org/wiki/Monty_Hall">Monty Hall</a> himself. The contestants on the game show were shown three shut doors. Behind one of these was a high value prize, such as a car. Behind the other two was a low value prize, such as a goat. If the contestants opened the correct door then they won the prize, otherwise they won nothing.</p>
<p> The contestants were then asked to choose a door, and to tell the host which door they had chosen. This door remained shut for the time being. The host then opened a different door to reveal a goat behind it. The contestants were then given a choice. They could stay with the door that they have chosen, or they could swap to the remaining unopened door. The door they finally ended up with was then opened, to reveal the prize car, or maybe just a goat. </p>
<p>The question is: should the contestant change their choice of door or not?</p>
<p>The accepted answer is "yes". In fact, by swapping doors you double your chances of winning the prize. This is surprising, which is why the problem has become so famous. This answer was given in the gambling film <em>21</em> and has also been <a href="https://www.bbc.co.uk/news/magazine-23986212">advocated as a reason why you should make changes in your choice of love</a>.</p>
<div class="leftimage" style="max-width: 400px;"><img src="/content/sites/plus.maths.org/files/articles/2020/myths/doors.png" alt="Three doors and a goat" width="400" height="223" /><p>Car or goat?</p></div><!-- Image in public domain -->
<p>However, it turns out that the accepted answer is not always correct and is an example of loose thinking. The answer to whether you switch doors or not depends entirely upon the host (and to some extent the contestant) and what they know, or don't know. </p>
<p>
First let's assume that the host knows behind which door the car is. When you, the contestant, have picked a door the host will always choose to open a door with a goat, and you know that this is the case. This is a reasonable assumption: if the host chose to open the door with the car, the game would be over. If you decide to stick with your door then you win if your initial guess was correct. The chance of this was 1/3 and the host opening a door with a goat doesn't change this: whatever your initial choice, they would always have responded accordingly, making sure they open a door with a goat. If you swap doors, then you win if your initial guess was wrong. The probability of this was 2/3, and again this probability hasn't changed by the host revealing a goat. What has changed, however, is the fact that this probability of winning is now tied solely to the third remaining door, the one you didn't pick initially and which the host didn't open. </p>
<p>This reasoning confirms the accepted answer to the Monty Hall problem: it pays to swap because your probability of winning is higher if you do.</p>
<p>But what if the host doesn't know themselves where the car is and opens a door at random after you have picked your door? The the fact that the door opened by the host reveals a goat does give you new information about your door. Since one door has been eliminated it is now more likely that the door you picked is the one with the car, and this extra likelihood is shared with the third door. The probability of your door concealing a car is now 1/2; the same as the other remaining door concealing the car. It now doesn't make a difference whether you swap or not.</p>
<p>Thus when confronted with a situation that reminds you of the Monty Hall problem, don't just assume that the best thing to do is embrace change. It all depends on the underlying assumptions on who knows what.</p>
<p>We can make the reasoning above more explicit using <em>Bayes' theorem</em>, which tells you how to work out the probability of an event A occurring <em>given that </em> an event B has occurred. In our example, we take A to be the event that the door you have picked has a car behind it and B the event that the host opens a door with a goat. </p>
<p>Bayes' theorem tells us that the probability A <em>given</em> B, which we write as P(A|B) is equal to</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/b237ff65a8aef999b81d66ef517983d6/images/img-0001.png" alt="\[ P(A|B)=\frac{P(B|A)P(A)}{P(B)}. \]" style="width:181px;
height:40px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Now <img src="/MI/b237ff65a8aef999b81d66ef517983d6/images/img-0002.png" alt="$P(A)$" style="vertical-align:-4px;
width:36px;
height:18px" class="math gen" /> in our example is the probability of you having picked the door with the car, so <img src="/MI/b237ff65a8aef999b81d66ef517983d6/images/img-0003.png" alt="$P(A)=1/3$" style="vertical-align:-4px;
width:84px;
height:18px" class="math gen" />. If the host knows where the car is and always opens a door with a goat, then <img src="/MI/b237ff65a8aef999b81d66ef517983d6/images/img-0004.png" alt="$P(B)=1$" style="vertical-align:-4px;
width:68px;
height:18px" class="math gen" />. </p><p>What about <img src="/MI/b237ff65a8aef999b81d66ef517983d6/images/img-0005.png" alt="$P(B|A)$" style="vertical-align:-4px;
width:54px;
height:18px" class="math gen" />? It’s the probability the host opens a door with a goat <em>given</em> that you have picked the door with the car. Since the host always opens a door with a goat, irrespective of your choice, <img src="/MI/b237ff65a8aef999b81d66ef517983d6/images/img-0006.png" alt="$P(B|A)=1$" style="vertical-align:-4px;
width:85px;
height:18px" class="math gen" /> as well. This means that </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/b237ff65a8aef999b81d66ef517983d6/images/img-0007.png" alt="\[ P(A|B)=\frac{P(B|A)P(A)}{P(B)}=P(A)=1/3. \]" style="width:288px;
height:40px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Therefore, the probability that the car is behind the door you haven’t picked and the host hasn’t opened, is <img src="/MI/b237ff65a8aef999b81d66ef517983d6/images/img-0008.png" alt="$1-1/3=2/3$" style="vertical-align:-4px;
width:98px;
height:16px" class="math gen" />. </p>
<p>If the host doesn’t know where the car is and opens a door at random, then <img src="/MI/c09e50a5ca14785b7cc220d1ae468ec3/images/img-0001.png" alt="$P(B),$" style="vertical-align:-4px;
width:42px;
height:18px" class="math gen" /> the probability that the host opens a door with a goat, is equal to <img src="/MI/c09e50a5ca14785b7cc220d1ae468ec3/images/img-0002.png" alt="$2/3$" style="vertical-align:-4px;
width:24px;
height:16px" class="math gen" />. Since the host is not going to pick the same door as you, the probability <img src="/MI/c09e50a5ca14785b7cc220d1ae468ec3/images/img-0003.png" alt="$P(B|A)$" style="vertical-align:-4px;
width:54px;
height:18px" class="math gen" /> (that the host picks a door with a goat, <em>given</em> that you picked the door with the car) is equal to <img src="/MI/c09e50a5ca14785b7cc220d1ae468ec3/images/img-0004.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" />: the host is bound to pick a goat since you have the car. In line with our reasoning above, Bayes’ theorem now tells us that </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/c09e50a5ca14785b7cc220d1ae468ec3/images/img-0005.png" alt="\[ P(A|B)=\frac{P(A)}{P(B)}=\frac{1/3}{2/3}=1/2. \]" style="width:223px;
height:40px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>There is also another instance of this problem, which involves a mean host set on fooling you, who knows where the car is. The host banks on the fact that you know the usual answer to the Monty Hall problem. If you pick the door with the car (probability 1/3), the host will open a door with a goat and challenge you to swap doors. Of course you do, and end up with a goat with probability 1.</p>
<p>If you first choose a door with a goat (probability 2/3), then the host asks you to pick one of the other two doors which will then be eliminated from the choice of three. (No door will be opened by the host.) Having done that, you are then allowed to either stick with your original door or swap to the remaining door. In this case, there is a chance of 1/2 that the door you excluded was the one with the car, so whether you swap or not, your chance of getting a goat is 1/2.</p>
<p>Overall, with these rules the chance of getting a goat is <img src="/MI/a36bc3eecf21626b7f35527a121c26c9/images/img-0001.png" alt="$1/3 \times 1 + 2/3\times 1/2 = 2/3$" style="vertical-align:-4px;
width:189px;
height:16px" class="math gen" />. The mean host is likely to win! </p>
<p>(I am indebted to Rob Eastaway, the Director of <a href="http://www.mathsinspiration.com/">Maths Inspiration</a>, for telling me about this way of playing the Monty Hall problem.)</p>
<a name="video"></a>
<hr/>
<iframe width="560" height="315" src="https://www.youtube.com/embed/fwtVlcO6s-Y" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2020/myths/chris_budd.jpg" alt="Chris Budd" width="332" height="321" />
<p>Chris Budd.</p>
</div>
<p>This article is based on a talk in Budd's ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham
College lecture series</a> (see video above). You can see other articles based on the talk <a href="/content/myths-maths">here</a>.
</p>
<p>Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the <a href="http://www.ima.org.uk/">Institute of Mathematics and its Applications</a>, Chair of Mathematics for the <a href="http://www.rigb.org/registrationControl?action=home">Royal Institution</a> and an honorary fellow of the <a href="http://www.britishscienceassociation.org/">British Science Association</a>. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.</p><p>
He has co-written the popular mathematics book <em><a href="/content/mathematics-galore">Mathematics Galore!</a></em>, published by Oxford University Press, with C. Sangwin, and features in the book <em><a href="https://global.oup.com/academic/product/50-visions-of-mathematics-9780198701811?cc=gb&lang=en&">50 Visions of Mathematics</a></em> ed. Sam Parc.
</p>
</div></div></div>Mon, 23 Mar 2020 11:28:12 +0000Marianne7264 at https://plus.maths.org/contenthttps://plus.maths.org/content/myths-maths-monty-hall-problem#commentsMyths of maths: The four colour theorem
https://plus.maths.org/content/myths-maths-four-colour-theorem
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/world_map4_icon.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="rightshoutout"><p>This article is based on a talk in an ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-in-the-21st-century/">Gresham
College lecture series</a>. You can see a video of the talk <a
href="/content/myths-maths-four-colour-theorem#video">below</a>.</div><p>Following Brexit we are faced with the worry of the possible break-up of the United Kingdom. Suppose that Scotland and Wales become independent, but the Northern Island does not? How will this alter the map? Well, one of the things that will happen is that it will no longer be possible to colour the map of the British Isles with four colours.</p>
<p>If you know a bit of maths, then this may surprise you. There's a very famous result which states that every map can be coloured with at most four colours. However, the result depends upon what you mean by a map. This is not so much a myth, more of a misquotation. </p>
<p>Maps showing different countries have been produced since the 1800s. To distinguish between the countries it was useful to colour them in different colours. A simple rule for doing this was that any two countries which shared a border, other than meeting at a point, should have different colours. Now, it costs money to print a coloured map, so map makers aimed to find the smallest number of colours needed to colour the map with the "no touching condition". It was found experimentally that all of the maps considered only needed four colours to colour them in. Here is an example of a map of the world coloured with exactly four colours with no adjacent countries having the same colour.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2020/myths/world_map4.png" alt="Map of the world." width="586" height="277" /><p style="max-width: 400px;">A map of the world in which the continents are coloured using four colours. If we take the oceans into account, however, we need five colours (see below for an explanation). Image <a href="https://commons.wikimedia.org/wiki/File:World_map_with_four_colours.svg">Fibonacci</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a>.</p></div>
<p>This discovery led the mathematicians of the day to conjecture that every map on the plane needed at most four colours to colour it with the above rules. This conjecture was first proposed in 1852 by <A href="https://en.wikipedia.org/wiki/Francis_Guthrie">Francis Guthrie</a>, who was trying to colour the map of counties of England. A first purported proof was given by <a href="http://mathshistory.st-andrews.ac.uk/Biographies/Kempe.html">Alfred Kempe</a> in 1879. This proof was later shown to be incorrect, but was modified at the time to show that any planar map could be coloured with at most five colours. However the original problem, despite the attempts of many mathematicians, remained open. The four colour conjecture rapidly became one of the most celebrated problems in mathematics. </p>
<p>The four colour theorem was finally proved in 1976 by <a href="http://mathshistory.st-andrews.ac.uk/Biographies/Appel.html">Kenneth Appel</a> and <a href="https://en.wikipedia.org/wiki/Wolfgang_Haken">Wolfgang Haken</a>. The proof itself was remarkable and gained a great deal of notoriety because it was the first major theorem to be proved using a computer. (Essentially a mathematical analysis reduced the problem to a large, but finite, number of maps, each of which was then checked by a computer to see if it could be four coloured). Initially, this proof was not accepted by many mathematicians, because it was impossible to check by hand. However I think quite the opposite. I believe that this proof has ushered in a new way of doing mathematics. Indeed it has led the way to many other proofs by computer, including some of my own work. (See <a href="/content/future-proof">this article</a> for more on computer proofs.)</p>
<p> The result is also important in modern WiFi technology. Imagine different WiFi transmitters, for example in an office block, all using different frequencies. To avoid interference we have a rule that no two adjacent WiFi transmitters should use the same frequency. The question we then ask is, "how many frequencies are needed to give a non-interfering network"? </p>
<p>It doesn't take much imagination to see that this problem is exactly the same as the four colour problem. So we only need four frequencies. Easy! Well, no, in fact we need more. Possibly many more. The problem arises, for example, in an office block, when different WiFi transmitters are assigned to different companies, and the company wants all of its transmitters to share the same frequency. This rapidly increases the number of frequencies that we need.</p>
<p>The same issue arises when we try to colour a map. By this I mean exactly what I say. A map. The sort of map that you would find in an Atlas. The issue arises in maps when countries have regions which are separated, or are possibly part of an Empire. These regions introduce an extra thing to consider when colouring the map, as they all have to have the same colour. The British Empire for example had all of its territories coloured red on the map. It also arises when the map has lakes, seas, or oceans. Not unreasonably all of these should be coloured blue. Below is an example of a map with two lakes. These I have coloured blue. The boundaries between the countries are indicated by black lines.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2020/myths/map_5colours.png" alt="Map with lakes" width="264" height="189" /><p style="max-width: 264px;"></p></div><!-- Image provided by author -->
<p>I challenge you to find a way to colour this map in four colours, whilst keeping the lakes blue. It can't be done — five colours are needed. The same applies to the map of the world above, if we take the oceans into account and realise that the colour white is also a colour, giving a total of five colours.
The map can be coloured using four colours but only if some countries have
the same colour as the sea. This makes them looks like lakes, and leads to a very
confusing map indeed.</p>
<p>So, how does this affect the map of the British Isles? If there is independence of the home nations, then they will no doubt adopt their traditional colours of white for England, red for Wales, and dark blue for Scotland. The British Isles is surrounded by the light blue sea. Although Wales and Scotland do not touch they need different colours as the Welsh (and English) will need an embassy in Scotland which will need to be coloured the same colour as the nation. The question is: what is the colour of Ireland? If the home nations all have an embassy in Ireland with their own colour, then Ireland must have a different colour again (green of course). So we need five colours (at least).</p>
<p>So what is the myth? The four colour theorem as carefully stated (for <em>non-contiguous planar graphs</em>) is certainly true. But one thing it does not necessarily apply to is an actual map. </p>
<a name="video"></a>
<hr/>
<iframe width="560" height="315" src="https://www.youtube.com/embed/fwtVlcO6s-Y" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2020/myths/chris_budd.jpg" alt="Chris Budd" width="332" height="321" />
<p>Chris Budd.</p>
</div>
<p>This article is based on a talk in Budd's ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham
College lecture series</a> (see video above). You can see other articles based on the talk <a href="/content/myths-maths">here</a>.
</p>
<p>Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the <a href="http://www.ima.org.uk/">Institute of Mathematics and its Applications</a>, Chair of Mathematics for the <a href="http://www.rigb.org/registrationControl?action=home">Royal Institution</a> and an honorary fellow of the <a href="http://www.britishscienceassociation.org/">British Science Association</a>. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.</p><p>
He has co-written the popular mathematics book <em><a href="/content/mathematics-galore">Mathematics Galore!</a></em>, published by Oxford University Press, with C. Sangwin, and features in the book <em><a href="https://global.oup.com/academic/product/50-visions-of-mathematics-9780198701811?cc=gb&lang=en&">50 Visions of Mathematics</a></em> ed. Sam Parc.
</p></div></div></div>Mon, 23 Mar 2020 10:57:13 +0000Marianne7266 at https://plus.maths.org/contenthttps://plus.maths.org/content/myths-maths-four-colour-theorem#commentsCommunicating the coronavirus crisis
https://plus.maths.org/content/communicating-corona-crisis
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/rss-djs_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em><a href="http://www.statslab.cam.ac.uk/~david/">David Spiegelhalter</a> is Chair of the Winton Centre for Risk and Evidence Communication, based at the Centre for Mathematical Sciences in Cambridge. What better person to ask about the UK government's communications regarding Covid-19? Here is what he told us.</em></p>
<p><strong>How should one communicate in a crisis?</strong></p>
<div class="rightimage" style="max-width:350px;"><img src="/content/sites/plus.maths.org/files/news/2020/spiegelhalter/rss-djs.jpg" width="300" height="383" alt="David Spiegelhalter" /><p>David Spiegelhalter.</p> </div>
<p>There are some basic principles, which I learnt from John Krebs, former Chair of the Food Standard Agency, who had to deal with many crises. The first thing is that you should be communicating a lot, consistently and with trusted sources. You have to be open and transparent. You have to say what you do know and then you have to say what you don't know. You have to emphasise, and keep emphasising, the uncertainty, the fact that there is much we don't know. Then you have to say what you are planning to do and why. Finally, you have to say what people themselves can do, how they should act. The crucial thing to say is that this will change as we learn more.</p>
<p>One of the problems here is scientific disagreement, which always exists and, through social media, is much more public now than it used to be. There can be disagreement for various reasons: people might disagree about values, or they might have access to different information. They can even have the same information and come to different interpretations. It is difficult to play this out in public because most people don't realise just how much disagreement there is in science. The media still present an idea of science as a monolithic body of "facts". This is complete nonsense because scientists argue all the time. Though there are of course some things they don't argue about, where there is an agreed body of knowledge.</p>
<div class="leftshoutout"><p>See <a href="/content/tags/covid-19">here</a> for all our coverage of the COVID-19 pandemic.</div>
<p>Closing the schools in the current crisis is a classic example of real uncertainty. We do not know what the impact will be, you can predict positive impacts and you can predict negative impacts. It might mean that the children play more with their grandparents, putting them at risk, or it might be they end up playing less with them, because the parents are home too. We just do not know what is going to happen except that it's going to happen. Even though closing schools will cause massive disruption, it's probably not feasible to keep schools open with so many teachers absent.</p>
<p><strong>How well is the UK government doing communicating policy and evidence?</strong></p>
<p>I think government communication started off really quite well. Patrick Vallance, Chief Scientific Advisor, and Chris Whitty, Chief Medical Officer, who are extremely good people of the highest integrity, were put forward to front the communications and they were doing fine. However, around March 12th it all fell apart. Suddenly things were coming out in interviews by Vallance and some of the modellers, and Matt Hancock was giving interviews to favoured journalists on what might potentially happen.</p>
<p>There was an eruption of anger and complaint by scientists about the way things were being communicated: the lack of transparency about the basis on which the decisions had been made. The big difference between now and the swine flu epidemic in 2009 is social media, in particular Twitter. It's easy on social media to organise vocal opposition. This then makes its way into the mainstream media, because the mainstream media follow Twitter and use it to identify people they would like to speak to.</p>
<p>In mid-March there were at least three letters circulating, complaining about the lack of transparency. There was a <a href="https://sites.google.com/view/covidopenletter/home" target="blank">letter from 600 behavioural scientists</a> asking for the basis for the decisions being made. In particular for evidence for the idea of "behavioural fatigue"; that people will not stick to the interventions. The evidence on this has been questioned and produced a big response by senior psychologists in this country.</p>
<p>Then there were calls, one of which <a href="https://www.thetimes.co.uk/article/coronavirus-modelling-must-be-made-clear-3zcl7x578">I signed</a>, asking for the models that informed the policies. These should be open to scrutiny. Government communication continually referenced how policy was based on the best science — but then why were we doing things so differently from other countries? I wasn't saying this was necessarily wrong, but I wanted to know why.</p>
<p>The fascinating thing is that, as soon as the results of a model by researchers from Imperial College <a href="https://plus.maths.org/content/how-best-deal-covid-19" target="blank">were released on March 16th</a> the whole policy changed because the new report strongly contradicted the policy that had been in place up until the previous Friday.</p>
<p><strong>Many people were shocked by the idea of herd immunity, worrying that vulnerable people were being thrown to the wolves. Should this have been communicated differently?</strong></p>
<p>This is the idea that if a lot of people get the disease and develop some immunity, then this will help suppress resurgences. It was bizarre that Patrick Vallance, who is extremely good, mentioned this idea in interviews. If you look at his words carefully, he never said that achieving herd immunity was a target. Only that it might be a consequence and a by-product of the UK government policies. The idea of herd immunity is perfectly reasonable, but it was not a target, it wasn't mentioned in any of the policy documents. </p>
<p>We feel very strongly at the Winton Centre that we should be communicating openly and transparently, but we also need to test our messaging, we need to be sure that it is not going to be misunderstood, as this was. Of course this doesn't mean that we shouldn't say things. We should be communicating more, and consistently, and openly, and admitting uncertainty.</p>
<p><strong>Were we purposefully misled regarding the severity of the crisis?</strong></p>
<p>I would not accuse anyone of maliciousness. But if last week the UK government did have models that were telling them what the total number of deaths would be, they certainly were not telling us that.</p>
<p>Now we have an estimate of around 20,000 deaths, which I think is very plausible. My own guess for some time has been that it would be between 5,000 and 20,000 deaths. If it's at the lower end of this range then it's like a moderate flu year! Flu kills 6,000 to 8,000 on average every year, and in a bad year it's between 13,000 and 15,000. The difference with flu is that cases tend to be spread over time, it's not as severe a disease as COVID-19, and that it doesn't attract any publicity at all.</p>
<p>A very delicate question is how you treat old people with respiratory problems. Many old people with flu will get pneumonia; it's almost the traditional way to die. Most hospitals will not start ventilating in that case, partly because old people, even if they do recover, may be seriously damaged by the treatment. This has always gone on behind the scenes, the idea that not everyone will go on a ventilator is not new. The problem with Covid-19 is that younger people could die too and that many cases will occur all at once, potentially overwhelming the service.</p>
<p><strong>Trust is obviously very important. How can governments retain (or re-earn) the public's trust?</strong></p>
<p>Trustworthiness has been characterised quite well by <a href="https://en.wikipedia.org/wiki/Onora_O%27Neill" target="blank">Onora O'Neill</a>, a philosopher of Kant. She describes what she calls <em>intelligent transparency</em>. This involves making sure that information is <em>accessible</em>, which means repeating it again and again making sure it's available from many sources. The information has got to be <em>comprehensible</em>, people have to understand it and you should check that they are getting the right impression. And the information has to be <em>useable</em> — answer people's questions and concerns: you have got to listen.</p>
<p>The final and crucial point, which people fall down on all the time, is that the information must be <em>assessible</em>: you shouldn't just have to take it on trust. Most people will take it on trust, but there are people out there who know a lot about what is going on, and these people should be able to check your working. Otherwise we have pure paternalism. We are not that sort of population — we want to know what is being done to us. People are making huge sacrifices and they have to know that it's worthwhile. This sort of assessibility means you have to be vulnerable, you have to open yourself up to criticism. That is trustworthiness and I haven't seen much of that at all, although now the scientific advice is being <a href="https://www.gov.uk/government/news/coronavirus-covid-19-scientific-evidence-supporting-the-uk-government-response">published</a>.</p>
<p><strong>How important and reliable are mathematical models in predicting the future of this crisis?</strong></p>
<p>This is an area where mathematical modelling can be important, but one has to be cautious about it because, with such limited data, it is all based on assumptions. In theory you have to change, not just the parameters, but also the structure of a model for every new disease coming along, and there isn't the time for that.</p>
<p>This doesn't mean you shouldn't have mathematical models going, you definitely should. But you should also work very closely with clinicians and real public health people, and people who have experienced this disease early on.</p>
<p><strong>What would you most like to know about Covid-19?</strong></p>
<p>The true number of cases. If we knew how many people have had it, and maybe didn't even know they had it, we might get some idea what is really going on.</p>
<p>The other thing I would like to know is whether the disease will recur. Will it rebound again after we stop clamping down so we have to clamp down again? Or, when it comes back next year, will the virus have mutated, so we have to start again? Is this going to go the way of smallpox, and be wiped out, or is it going to be like seasonal flu which keeps coming back in a different form? This would be very useful to know, but we won't know for a long time.</p>
<hr/>
<h3>About this article</h3>
<p><a href="http://www.statslab.cam.ac.uk/~david/">David Spiegelhalter</a> is Chair of the Winton Centre for Risk and Evidence Communication, based at the Centre for Mathematical Sciences at the University of Cambridge. He was interviewed by <a href="/content/people/index.html#marianne">Marianne Freiberger</a>, Editor of <em>Plus</em>, on March 18, 2020.</p>
</div></div></div>Mon, 23 Mar 2020 10:27:16 +0000Marianne7271 at https://plus.maths.org/contenthttps://plus.maths.org/content/communicating-corona-crisis#comments