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.
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 letter K. So what do we need to know about K and why has it suddenly become the focus of interest?
The R number represents the average number of people an infected person goes on to infect. If R is larger than one, 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.
At the start of the coronavirus outbreak, R in the United Kingdom was around three. If every infected person infected exactly three people, the epidemic would have spread as in the figure below.
Average is not enough
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 many differences between individuals and their impact on virus transmission.
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 vary randomly.
Some people might have high viral loads or might simply cough more and hence spread the virus more effectively.
Many people, although ill and highly-infectious, don’t show any symptoms. They might make many contacts without realising they pose a danger to others. An example from history is the infamous Mary Mallon or “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.
Super-spreaders
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 choir practice, a football match or visits several pubs or nightclubs, the number of people who might catch the disease becomes large. Scientists call such massive and rapid outbreaks caused by one or a few infected individuals, super-spreading events, and their initiators are known as super-spreaders. In many cases, 80% of the new disease cases are caused by only 20% of such super-spreading individuals.
Dispersion parameter
Different pathogens will have different ways in which they spread and statisticians use K, the so-called dispersion parameter, to describe how variable the infection can be. For some diseases, the variation will not be large, as shown below.
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 one, and perhaps 40% of infected people might not pass on the virus to anybody else. But for diseases like Sars, Mers and Covid-19 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.
Why K is important
There are two reasons why scientists are looking into the role of variability 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.
Outbreaks in Seoul nightclubs in South Korea, meatpacking plants in the US, and coal mines in Poland show how damaging super-spreading events can be. So governments need to be diligent in identifying the risks associated with the reopening of industries and entertainment. A way to identify and track potential super-spreaders is fundamental to prevent future outbreaks.
But there is also a glimmer of hope. If indeed K is as low as 0.1, 70% of infected individuals fail to pass on the virus. 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.
While R is not going to be replaced by K in the daily press briefing, both are needed to understand how Covid-19 spreads.
Adam Kleczkowski, Professor of Mathematics and Statistics, University of Strathclyde.
This article first appeared on The Conversation.
