tag:theconversation.com,2011:/id/topics/k-number-88472/articlesK number – The Conversation2021-07-28T01:34:57Ztag:theconversation.com,2011:article/1648582021-07-28T01:34:57Z2021-07-28T01:34:57ZWe’ve heard of R numbers and moving averages. But what are k numbers? And how do they explain COVID superspreading?<figure><img src="https://images.theconversation.com/files/412589/original/file-20210722-15-jdqje7.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://unsplash.com/photos/ZzOtl6FSpLs">Forest Simon/Unsplash</a></span></figcaption></figure><p>First thing in the morning, or come 11 o'clock, countless Australians anxiously wait for the daily COVID-19 case numbers, trying to understand whether their outbreak is under control, and how much longer they will be in lockdown. </p>
<p>As well as daily case numbers, people want to know what proportion of cases were infectious in the community, and whether there were any unlinked or “mystery” cases. </p>
<p>People have also been following the daily Reff, or effective reproduction number, hoping it will get below 1, showing public health measures are working to halt the spread. </p>
<p>However, to have a good understanding of the dynamics of an outbreak, it is also necessary to understand k, which shows how much variability there is in daily case numbers. </p>
<h2>COVID-19 superspreaders</h2>
<p>Many superspreading events have occurred in the current pandemic. An infectious volunteer dressed as <a href="https://www.nature.com/articles/d41586-021-00460-x">Santa Claus</a>, for example, visited a care home in Antwerp in December 2020, and infected 40 staff members and more than 100 residents. </p>
<p>Even more drastic is a <a href="https://graphics.reuters.com/CHINA-HEALTH-SOUTHKOREA-CLUSTERS/0100B5G33SB/index.html">South Korean woman</a> who caused a superspreading event resulting in more than 5,000 cases in the South Korean city of Daegu. </p>
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Read more:
<a href="https://theconversation.com/how-to-prevent-covid-19-superspreader-events-indoors-this-winter-147439">How to prevent COVID-19 ‘superspreader’ events indoors this winter</a>
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<p>Meanwhile in <a href="https://www.abc.net.au/news/2021-01-09/how-a-single-case-of-covid-19-forced-a-lockdown/13042612">Australia</a>, we have seen many examples of cases being detected, but not infecting a single other person. </p>
<p>So, how can this disparity be explained?</p>
<h2>Remind me, what’s the Reff?</h2>
<p>The effective reproduction number Reff, also called Re or R(t), tells us, on average, how many people an infected person will pass it on to. Unlike the basic reproduction number, <a href="https://theconversation.com/r0-how-scientists-quantify-the-intensity-of-an-outbreak-like-coronavirus-and-predict-the-pandemics-spread-130777">R0</a>, Reff takes into account that some people will be vaccinated or immune, and social distancing is in place. </p>
<p>So, if a virus has a Reff of 2, each infected person (primary case) will on average infect two others (secondary cases). </p>
<p>However, this average hides a huge amount of variability. Most infected people simply <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7338915/">infect no one</a>, whereas others (the superspreaders) infect many people. </p>
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Read more:
<a href="https://theconversation.com/a-few-superspreaders-transmit-the-majority-of-coronavirus-cases-139950">A few superspreaders transmit the majority of coronavirus cases</a>
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<p>We’re <a href="https://www.nature.com/articles/d41586-021-00460-x">unsure</a> why this is the case. It could be some people are naturally social animals, or fail to maintain social distancing, mask-wearing, or hygiene. </p>
<p>Alternatively, it could simply be that some people have a much higher viral load than others or tend to emit virus particles as aerosol clouds more than others.</p>
<h2>Daily case numbers can vary substantially</h2>
<p>During periods of outbreaks, health authorities report daily case numbers. Here they are for <a href="https://www.covid19data.com.au/victoria">Victoria</a> when the fifth lockdown began: </p>
<p><iframe id="CLQch" class="tc-infographic-datawrapper" src="https://datawrapper.dwcdn.net/CLQch/3/" height="400px" width="100%" style="border: none" frameborder="0"></iframe></p>
<p><strong>Average daily count</strong> </p>
<p>The average (mean) daily count over these ten days is 10.7 cases per day (you can calculate it yourself by adding up all the cases and dividing by ten). </p>
<p>However, there is a lot of variability, with numbers going up and down like a yo-yo from zero to twenty. Because of this variability, we often use moving averages to try and smooth things out. </p>
<p><strong>7 day moving average</strong></p>
<p>For a seven-day moving average, we add up the cases from July 12 to the 18 and divide by 7, to get 8.4. Then we do the same for July 13 to the 19 to get 10.3. </p>
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<p>This way, we end up with a much smoother series of numbers without all the up and down jags, that allows us to see trends much more easily. Importantly, I also use the moving average to calculate the Reff. </p>
<p><strong>Variance</strong></p>
<p>We measure the amount of variability in the daily case numbers by a statistic called the variance. This measures how far apart the daily counts are from their average value of 10.7. For most count data (for example, the number of days each month you exercise), the average and variance are the same. So, if the average count is 10.7, the variance is 10.7. </p>
<p>However, for this epidemic, because of the superspreaders, the variance is much greater – we call this overdispersion. </p>
<h2>So what is the k?</h2>
<p>An estimate of how much extra variability or overdispersion there is, is measured by a statistic called k. A small k means the variability is higher than the average daily count, whereas a large k means the variability is closer to the average daily count. </p>
<p>So, with a high value of k (say 2), and a Reff of 2, most infected people would typically infect two others, but it could of course be higher or lower than this. </p>
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<img src="https://cdn.theconversation.com/static_files/files/1731/Diagram-1.gif?1627370869" width="100%">
<figcaption>Source: The Conversation/Adam Kleczkowski (CC-BY-ND)</figcaption>
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<p>In the above diagram, the number of people a case infects is shown in each circle. The original maroon (primary) case infects two others (red). Each of these secondary cases infects three or four others (pink), and so the outbreak continues. Typically, most infected people, infect at least one other person.</p>
<p>However, with k close to 0 and a Reff of 2, most people would infect no one else, and there would be one or more superspreaders. </p>
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<img src="https://cdn.theconversation.com/static_files/files/1730/Diagram-2.gif?1627370611" width="100%">
<figcaption>Source: The Conversation/Adam Kleczkowski (CC-BY-ND)</figcaption>
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<p>In the above diagram, the primary case (maroon) is a superspreader, infecting 16 other people. Although most of these secondary cases do not infect anyone else, one of the tertiary cases is also a superspreader, infecting 11 others. </p>
<p>In both diagrams the Reff was 2. So, you can see that knowing the Reff is only part of the story.</p>
<p><a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7338915/">Estimates</a> of COVID-19’s k range from 0.1 to 0.5. These are very small values, and indicate 80% of secondary infections are caused by around 10% of primary cases. This means the majority of infectious people do not infect anyone. </p>
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Read more:
<a href="https://theconversation.com/is-the-k-number-the-new-r-number-what-you-need-to-know-140286">Is the K number the new R number? What you need to know</a>
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<h2>Why is it useful to know the k?</h2>
<p>When an infected person is diagnosed, contact tracers immediately try and find their close contacts. These are then tested and put into isolation. This is called forward contact tracing. </p>
<p>However, in the context of superspreaders, it’s equally important to find out who infected the original diagnosed case, as that person could potentially be a superspreader. </p>
<p><a href="https://www.theatlantic.com/health/archive/2020/09/k-overlooked-variable-driving-pandemic/616548/">Forward contact tracing</a> of that potential superspreader would likely lead to many more cases being detected. In fact, <a href="https://www.bbc.com/news/health-54648734">modelling</a> has found looking backwards as well as forwards could prevent two or three times as many infections. This is known as backward contact tracing and is now <a href="https://www.nature.com/articles/d41586-020-03518-4">widely used</a> in Australia. </p>
<p>The k number shows us the importance of backwards as well as forwards contact tracing.</p><img src="https://counter.theconversation.com/content/164858/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Adrian Esterman does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>The k number tells us whether the spread of a disease is steady or comes in big bursts, with a small proportion of people infecting many others. The latter is know as superspreading.Adrian Esterman, Professor of Biostatistics and Epidemiology, University of South AustraliaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1472882020-10-12T14:23:42Z2020-10-12T14:23:42ZCoronavirus: R number does not measure risk – financial disaster modelling offers a better alternative<p>The most important measure on the state of the pandemic is the R number. Also known as the reproduction rate, this is the average number of people an infected person will pass the virus on to. The R number guides government decisions, and there is no doubt it is a very intuitive way to measure the state of the pandemic. </p>
<p>However, the R number has come in for <a href="https://theconversation.com/is-the-k-number-the-new-r-number-what-you-need-to-know-140286">much criticism</a> because it is an average, meaning it ignores useful information about individuals and therefore does not account for uncertainty. This includes the fact that there is a sizeable variation in the incubation time of the virus and a large number of infected but asymptomatic people who are hard to detect, plus superspreaders who infect many more people than the average. </p>
<p>To better estimate the threat to our populations from the virus, governments should look to the statistical models used for financial markets. I say this because the public health shares an important characteristic with financial markets: they are both made up of many interacting parts that can be exposed to rare, pervasive shocks, with potentially critical consequences that can spread across borders. </p>
<h2>Tail events and downside risk</h2>
<p>The global financial crisis of 2007-09 was in many ways the financial equivalent of a pandemic. It started when a housing bubble burst in the US and quickly spread internationally through the complex system that existed for trading mortgage debt between financial institutions. This led to everything from banking collapses to national debt defaults to the great recession. </p>
<p>To try to prevent such a catastrophe happening again, the global financial system developed a regulatory system for stress testing banks and the investment portfolios of major firms. These tests assess the fragility of banks and investment portfolios and improve their immunity to shocks by asking questions such as, “how much could they lose from a rare event?” and “how severe must that shock be before the bank would collapse?” </p>
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<a href="https://images.theconversation.com/files/361082/original/file-20201001-21-1s32nac.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Piggy bank with knife sawing under it" src="https://images.theconversation.com/files/361082/original/file-20201001-21-1s32nac.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/361082/original/file-20201001-21-1s32nac.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=515&fit=crop&dpr=1 600w, https://images.theconversation.com/files/361082/original/file-20201001-21-1s32nac.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=515&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/361082/original/file-20201001-21-1s32nac.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=515&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/361082/original/file-20201001-21-1s32nac.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=648&fit=crop&dpr=1 754w, https://images.theconversation.com/files/361082/original/file-20201001-21-1s32nac.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=648&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/361082/original/file-20201001-21-1s32nac.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=648&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">Pig in a poke.</span>
<span class="attribution"><a class="source" href="https://www.bankofengland.co.uk/-/media/boe/files/monetary-policy-report/2020/august/monetary-policy-report-august-2020">Mopic</a></span>
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<p>These tests rely on financial predictive risk modelling. This technique doesn’t focus on what is expected to happen next, but rather on the probability of rare events like the ones that precipitated the 2007-09 crisis. Such outcomes are referred to as tail events because, unlike the average outcome (the “central value”), they arise from the “tail” of the distribution of probabilities. Such tests focus on tail events associated with losses, whose probability is known as the “downside risk”.</p>
<p>This modelling is also used to look at the downside risk of what might happen to the macroeconomy, which is another highly uncertain environment. For instance, how much GDP will be lost if a rare negative shock hits the financial markets. Similar techniques could be used to improve our predictions about COVID-19.</p>
<h2>Financial modelling and COVID-19</h2>
<p>To get around the problems with the R number being an average, an <a href="https://www.theatlantic.com/health/archive/2020/09/k-overlooked-variable-driving-pandemic/616548/">alternative indicator</a> known as the <a href="https://theconversation.com/is-the-k-number-the-new-r-number-what-you-need-to-know-140286">K number</a> is used alongside it when the R is very low. The K number measures the dispersion of probability around the average to show how many people are passing on infections. A higher K number means that more individuals are responsible for the observed infections, while a lower number means the virus is being spread by fewer people – in other words, superspreaders. </p>
<p>Measuring uncertainty with dispersion is also very common in economics. For example, economists produce charts that show what they estimate is the most likely to happen (the central value) and then illustrate the uncertainty as different shaded areas of other possible outcomes that fan out from the main area. The Bank of England has used these fan charts to predict inflation since the 1990s, for instance (see below). </p>
<p><strong>Bank of England inflation projections, August 2020</strong></p>
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<a href="https://images.theconversation.com/files/361080/original/file-20201001-13-9krsid.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Bank of England inflation fan chart" src="https://images.theconversation.com/files/361080/original/file-20201001-13-9krsid.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/361080/original/file-20201001-13-9krsid.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=384&fit=crop&dpr=1 600w, https://images.theconversation.com/files/361080/original/file-20201001-13-9krsid.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=384&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/361080/original/file-20201001-13-9krsid.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=384&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/361080/original/file-20201001-13-9krsid.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=483&fit=crop&dpr=1 754w, https://images.theconversation.com/files/361080/original/file-20201001-13-9krsid.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=483&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/361080/original/file-20201001-13-9krsid.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=483&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="attribution"><a class="source" href="https://www.bankofengland.co.uk/-/media/boe/files/monetary-policy-report/2020/august/monetary-policy-report-august-2020">Bank of England, August 2020</a></span>
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<p>Superspreaders are certainly a major issue with coronavirus. For example, the current R number in the UK <a href="https://www.gov.uk/guidance/the-r-number-in-the-uk#history">is estimated</a> to be within a range of 1.2 to 1.5 – meaning the average infected person will infect between 1.2 and 1.5 people. Yet a superspreader has been linked to <a href="https://www.bbc.co.uk/news/uk-wales-54359674">32 secondary infections</a>, and even more extreme cases have <a href="https://www.theguardian.com/us-news/2020/sep/17/maine-wedding-superspreader-event">been reported</a>.</p>
<p>But the K number’s ability to alert us to this phenomenon has two important limitations. First, it is less accurate when the R number is higher. This is because K is not able to distinguish between variation above and below the average.</p>
<p>When R is low, there can’t be much dispersion below average since zero is the minimum number of infections that anyone can transmit. This means that when K is showing a wide variation from the average, it must be due to superspreaders. On the other hand, when R is higher, there is more scope for below-average spreaders to explain a wide variation from the average, so K is less useful. </p>
<p>Second, even if the R is low enough, in statistics it is well known that when extreme events like superspreading are sufficiently probable, dispersion becomes a rather poor indicator of the risk associated with such tail events.</p>
<p>To get a more accurate picture of COVID-19 risk that could be used alongside the information from the R and K numbers, I would therefore argue that governments should use stress testing for adverse tail events. This would make it possible to estimate the likelihood of a particular rise in cases in a given span of time – say a twofold increase in a week. </p>
<p>And given that most cases are mild or asymptomatic, it is arguably not contagion that is the ultimate concern for society but the burden on the health system. As several countries have seen, the pandemic can overwhelm hospitals with large numbers of incoming patients – way too high for the resources available. </p>
<p>Stress testing can help measure the vulnerability of the health system by providing answers to questions such as, “how likely is it today that the available intensive-care beds, personal protective equipment and ventilators reach a critical level, say 90% of capacity?”; and “how much can contagion rise before that level is reached?”</p>
<p>After the 2007-09 financial crisis, regulators introduced stress testing to restore confidence in the banking system. With <a href="https://www.theguardian.com/world/2020/sep/24/10000-more-deaths-than-usual-occurred-in-uk-homes-since-june">fading trust</a> in the health system and people avoiding hospitals due to COVID-19 fears, a more transparent and detailed way of measuring the risks using the same kind of predictive modelling might produce similar benefits.</p><img src="https://counter.theconversation.com/content/147288/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Stefano Soccorsi does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>How the lessons learned from the global financial crisis can transform our view of COVID risk.Stefano Soccorsi, Lecturer in Financial Economics, Lancaster UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1402862020-06-16T09:57:09Z2020-06-16T09:57:09ZIs the K number the new R number? What you need to know<p>Just a few months ago, no one, aside from epidemiologists and their ilk, had heard of the R number. Now, thanks to the coronavirus, everyone has heard of it and most people can tell you that it’s the reproduction number, an indicator of whether the number of infected people is increasing or decreasing.</p>
<p>The R number is regularly referred to by governments around the world and by news anchors and their guests when discussing the pandemic. Yet no sooner had the public wrapped their head around one mathematical symbol than another cropped up. This time, the <a href="https://www.theguardian.com/world/2020/jun/01/k-number-what-is-coronavirus-metric-crucial-lockdown-eases">letter K</a>. So what do we need to know about K and why has it suddenly become <a href="https://news.sky.com/story/coronavirus-what-is-the-k-number-and-how-could-it-help-end-lockdown-11999741">the focus of interest</a>?</p>
<p>The <a href="https://www.bbc.co.uk/news/health-52473523">R number</a> represents the average number of people an infected person goes on to infect. If R is <a href="https://theconversation.com/coronavirus-what-a-second-wave-might-look-like-138980">larger than one</a>, the number of people with the disease is increasing. The target for control strategies, including lockdown, self-isolation and masks wearing, is to bring R below one and thereby reduce the number of people with the disease. </p>
<p>At the start of the coronavirus outbreak, R in the UK was <a href="https://www.ft.com/content/033745f3-2d78-4869-9690-ea46fcc9cb3d?shareType=nongift">around three</a>. If every infected person infected exactly three people, the epidemic would have spread as in the figure below.</p>
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<img alt="" src="https://images.theconversation.com/files/341536/original/file-20200612-153849-1ko8h3x.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/341536/original/file-20200612-153849-1ko8h3x.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=316&fit=crop&dpr=1 600w, https://images.theconversation.com/files/341536/original/file-20200612-153849-1ko8h3x.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=316&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/341536/original/file-20200612-153849-1ko8h3x.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=316&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/341536/original/file-20200612-153849-1ko8h3x.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=397&fit=crop&dpr=1 754w, https://images.theconversation.com/files/341536/original/file-20200612-153849-1ko8h3x.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=397&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/341536/original/file-20200612-153849-1ko8h3x.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=397&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Epidemic spread with R=3; four generations are shown from the first person marked in red, through yellow, green and blue. Numbers indicate how many new infections originate from each case.</span>
<span class="attribution"><span class="source">Adam Kleczkowski</span></span>
</figcaption>
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<h2>The average is not enough</h2>
<p>The R mentioned in the daily press briefings represents an average of the whole country or region, involving millions of people. But its single value hides <a href="https://www.sciencemag.org/news/2020/05/why-do-some-covid-19-patients-infect-many-others-whereas-most-don-t-spread-virus-all">many differences</a> between individuals and their impact on virus transmission. </p>
<p>Rather than assuming that every infected person and every contact they make follows the same pattern (as with the R number), scientists working on epidemic models allow for the number of new cases caused by each infected person to <a href="https://www.nature.com/articles/nature04153#MOESM1">vary randomly</a>. </p>
<p>Some people might have <a href="https://wwwnc.cdc.gov/eid/article/21/11/15-0764_article">high viral loads</a> or might simply <a href="https://www.nature.com/articles/s41598-019-38808-z">cough more</a> and hence spread the virus more effectively. </p>
<p>Many people, although ill and highly infectious, <a href="https://www.acpjournals.org/doi/10.7326/M20-3012">don’t show any symptoms</a>. They <a href="https://doi.org/10.1093/cid/ciaa424">might make many contacts</a> without realising they pose a danger to others. An example from history is the infamous <a href="https://en.wikipedia.org/wiki/Mary_Mallon">Mary Mallon</a> (“Typhoid Mary”), a cook in New York City in the early 1900s. Although she carried typhoid bacteria, she didn’t show any symptoms and is believed to have infected more than 50 people over seven years. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/341662/original/file-20200614-153808-8cdo8c.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/341662/original/file-20200614-153808-8cdo8c.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=445&fit=crop&dpr=1 600w, https://images.theconversation.com/files/341662/original/file-20200614-153808-8cdo8c.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=445&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/341662/original/file-20200614-153808-8cdo8c.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=445&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/341662/original/file-20200614-153808-8cdo8c.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=559&fit=crop&dpr=1 754w, https://images.theconversation.com/files/341662/original/file-20200614-153808-8cdo8c.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=559&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/341662/original/file-20200614-153808-8cdo8c.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=559&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Mary Mallon in hospital (foreground).</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/w/index.php?curid=689801">Wikimedia Commons</a></span>
</figcaption>
</figure>
<h2>Super-spreaders</h2>
<p>People also differ in the way they interact with others. For some, contacts might involve just the immediate family or a small group of colleagues at work or friends. The disease will then only have a chance to be transmitted to a few people. But if an infected person goes to <a href="https://www.cdc.gov/mmwr/volumes/69/wr/mm6919e6.htm">choir practice</a>, a <a href="https://www.independent.co.uk/news/world/coronavirus-italy-champions-league-atlanta-valencia-milan-bergamo-a9426616.html">football match</a> or visits <a href="https://www.cbsnews.com/news/south-korea-coronavirus-cluster-linked-to-seoul-nightclubs-fueling-homophobia-fears-gay-men/">several pubs or nightclubs</a>, the number of people who might catch the disease becomes large. Scientists call such <a href="https://arxiv.org/abs/2005.13689">massive and rapid outbreaks</a> caused by one or a few infected individuals, <a href="https://www.nature.com/articles/nature04153#MOESM1">super-spreading events</a>, and their initiators are known as <a href="https://www.journalofhospitalinfection.com/article/S0195-6701(20)30177-8/fulltext#secsectitle0015">super-spreaders</a>. In many cases, 80% of the new disease cases are caused by only 20% of such super-spreading individuals.</p>
<h2>Dispersion parameter, K</h2>
<p>Different pathogens will have different ways in which they spread and statisticians use K, the so-called <a href="https://www.huffingtonpost.co.uk/entry/what-is-the-r-number-and-how-is-it-different-to-the-k-number_uk_5ed8ae41c5b6f3c22ef014e9">dispersion parameter</a>, to describe how variable the infection can be. For some diseases, the variation will not be large, as shown below.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/341549/original/file-20200612-153817-nh8d82.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/341549/original/file-20200612-153817-nh8d82.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=329&fit=crop&dpr=1 600w, https://images.theconversation.com/files/341549/original/file-20200612-153817-nh8d82.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=329&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/341549/original/file-20200612-153817-nh8d82.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=329&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/341549/original/file-20200612-153817-nh8d82.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=414&fit=crop&dpr=1 754w, https://images.theconversation.com/files/341549/original/file-20200612-153817-nh8d82.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=414&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/341549/original/file-20200612-153817-nh8d82.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=414&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Epidemic spread with a distribution of secondary cases with low dispersion and value of K much larger than 1.</span>
<span class="attribution"><span class="source">Adam Kleczkowski</span></span>
</figcaption>
</figure>
<p>Simply put, a low K value suggests that a small number of infected people are responsible for large amounts of disease transmission. For the 1918 influenza, the number K is thought to be around 1, and perhaps 40% of infected people might not pass on the virus to anybody else. But for diseases like <a href="https://www.nature.com/articles/nature04153#MOESM1">Sars</a>, <a href="https://www.sciencemag.org/news/2020/05/why-do-some-covid-19-patients-infect-many-others-whereas-most-don-t-spread-virus-all">Mers</a> and <a href="https://wellcomeopenresearch.org/articles/5-67">COVID-19</a> with K as low as 0.1, this proportion rises to 70%. In contrast, large outbreaks will be initiated by only few super-spreaders, as shown below. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/341544/original/file-20200612-153858-1oxuf6a.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/341544/original/file-20200612-153858-1oxuf6a.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=320&fit=crop&dpr=1 600w, https://images.theconversation.com/files/341544/original/file-20200612-153858-1oxuf6a.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=320&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/341544/original/file-20200612-153858-1oxuf6a.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=320&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/341544/original/file-20200612-153858-1oxuf6a.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=402&fit=crop&dpr=1 754w, https://images.theconversation.com/files/341544/original/file-20200612-153858-1oxuf6a.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=402&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/341544/original/file-20200612-153858-1oxuf6a.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=402&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Epidemic spread with a distribution of secondary cases with high dispersion and value of K around or below 1.</span>
<span class="attribution"><span class="source">Adam Kleczkowski</span></span>
</figcaption>
</figure>
<h2>Why is K so important?</h2>
<p>There are two reasons why scientists are looking into the <a href="https://www.ft.com/content/2418ff87-1d41-41b5-b638-38f5164a2e94">role of variability</a> in controlling coronavirus transmission. First, super-spreading events are critical in the late stages of the epidemic when the virus is almost eradicated. Small values of K mean that one infected person can trigger many new cases in a very short time. If this happens, the epidemic can quickly rebound, even if locally eradicated. </p>
<p>Outbreaks in <a href="https://www.cbsnews.com/news/south-korea-coronavirus-cluster-linked-to-seoul-nightclubs-fueling-homophobia-fears-gay-men/">Seoul nightclubs in South Korea</a>, <a href="https://blogs.lse.ac.uk/covid19/2020/05/05/covid-19-contributes-to-the-crisis-of-occupational-safety-and-health-in-meatpacking/">meatpacking plants in the US</a>, and <a href="https://uk.reuters.com/article/us-health-coronavirus-poland-coal/poland-will-temporarily-close-a-dozen-mines-to-stop-coronavirus-idUKKBN23F0TK?rpc=401">coal mines in Poland</a> show how damaging super-spreading events can be. So governments need to be <a href="https://www.sciencedirect.com/science/article/pii/S0262407920309477">diligent in identifying the risks</a> associated with the reopening of industries and entertainment. A way to identify and track potential super-spreaders is fundamental to prevent future outbreaks.</p>
<p>But there is also a glimmer of hope. If indeed <a href="https://wellcomeopenresearch.org/articles/5-67">K is as low as 0.1</a>, <a href="https://europepmc.org/article/ppr/ppr165671">70% of infected individuals fail to pass on the virus</a>. As a result, most cases arriving from outside the country or region might recover without starting a new outbreak. It might, therefore, be easier to eradicate the disease and to maintain the disease-free status than suggested by the average reproductive number, R. </p>
<p>While R is not going to be replaced by K in the daily press briefing, both are needed to understand how COVID-19 spreads.</p><img src="https://counter.theconversation.com/content/140286/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Adam Kleczkowski has received funding from BBSRC, NSF, Academy of Medical Sciences, and the Scottish Government.</span></em></p>K is all about the super-spreaders.Adam Kleczkowski, Professor of Mathematics and Statistics, University of Strathclyde Licensed as Creative Commons – attribution, no derivatives.