Read on Twitter
If the reproduction number drops below 1, an epidemic won't disappear immediately. But how many additional infections will there be before it declines to very low levels? Fortunately there's a neat piece of maths that can help... 1/
Let's start by considering a situation where one person is infectious. On average they'll infect R others, who in turn will infect R others, and so on. We'd therefore get the following equation for average outbreak growth: 2/
If R is above 1, the outbreak will continue to grow and grow until something changes (control measures, accumulated immunity, seasonal effects etc.) But if R is below 1, above transmission process will decline & converge to a fixed value. (Proof here: https://en.wikipedia.org/wiki/Geometric_series) 3/
For example, if we have a single case and R=0.8, we'd expect an overall outbreak size = 1/(1-0.8) = 5 (i.e. 4 additional infections if we don't count the initial one). If R=0.9, we'd expect 9 additional infections. 4/
So far we've assumed 1 initial infection, but if we start with more we can just multiply the outbreak size accordingly. So if we start with 10,000 infections and R=0.9, we'd expect an additional 90,000 infections before the outbreak fades to very low levels. 5/
Here's a plot to show the number of additional infections per initial infection for different values of R between 0.5 and 0.999 (note log scale on y-axis). In other words, even if R is below 1, amount it's below one makes a big difference to subsequent number of infections. 6/
"What if superspreading events can occur?" you ask... Well, the above method happens to work regardless of how much overdispersion there is: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1002993 7/
More on the origins of these ideas: https://theconversation.com/the-ancient-greek-riddle-that-helps-us-understand-modern-disease-threats-25612 8/8
