Last week Prime Minister Boris Johnson assured Britons that, come 21 June—at least, if all goes according to plan—we will “re-open everything up to and including nightclubs, and enable large events such as theatre performances.” Life will return to normal, or so he says.
Alas, Johnson is fooling himself, and it takes only a modest understanding of exponential growth to see why.
The fundamental number governing the behaviour of any epidemic is the reproduction number R: how many people, on average, one infected person will infect. If R is greater than 1, the epidemic will explode. If R is near 1, the epidemic will stay roughly constant. And if R is less than 1, the epidemic will decline.
For the original variant of SARS-Cov-2—the one prevalent throughout the world in early 2020—the evidence shows that, under conditions of normal social interaction, R is somewhere between 2 and 3. Let’s be optimistic and say 2. And the incubation period of the virus—how long it takes, on average, for an infected person to become infectious—is about 5 days. So one infected person will generate two cases after 5 days, then four after 10 days, then eight after 15 days, then 16 after 20 days, then 32 after 25 days, then 64 after one month. Then about 4000 after two months, and about 250,000 after three months, from just one initial case.
This is all approximate, of course, but you get the idea: exponential growth means a very rapid explosion, as anyone who has taken out a loan at 700% interest will sadly attest.
But the B.1.1.7 variant—the one now dominant in the UK—is believed to be about 50% more transmissible. That means an R, under normal social conditions, of somewhere between 3 and 4.5. Let’s again be optimistic and say 3. One infected person will therefore generate about 700 after one month, and about half a million after two months.
But now we have vaccines; let’s factor those in.
The Astra Zeneca vaccine—the one predominantly used in the UK—is probably 50–70% effective in reducing transmission. That’s far from guaranteed—the data are not very clean—but it does seem likely. Let’s be optimistic and say 70%.
Note also that this estimate applies to the original variant of SARS-Cov-2, for which the vaccine was developed. The efficacy against B.1.1.7 may be somewhat lower; indeed, the efficacy against the South African variant seems to be nearly zero. But let’s still be optimistic and stick with 70%.
How many people in the UK will be vaccinated by late June?
After so many screwups, the vaccination programme is the one thing that this country is doing right. (Maybe because it is run by the NHS, not outsourced to Serco.) Boris Johnson predicted that the first jab will be offered to all adults by the end of July—and in that, at least, he may well be right. Let’s hope so. Let’s even go one better and move the date up to late June.
How many people will take the vaccine when offered? Here, the data are less encouraging. Vaccine hesitancy is surprisingly high: surveys suggest that only 65–85% of UK adults will agree to be vaccinated (though this willingness may be increasing over time). Let’s again be optimistic and say 85%. And let’s also be optimistic and forget about all the unvaccinated children—despite the fact that children do get infected, and presumably infect others as well.
So we have 85% of the adult population receiving a 70% effective vaccine. Eighty-five percent times 70% equals a 60% reduction in transmission. That brings our assumed R=3 down to about 1.2.
That’s a huge reduction, but 1.2 is still bigger than 1: it means that the epidemic will double every 3 weeks. And this estimate is based on the most optimistic assumptions about the R factor, vaccine efficacy and vaccine uptake.
Right now in the UK, we have 9,000 new cases and 300 deaths a day. If we don’t want to keep having that many deaths forever, we need to keep R down well below 1 for a while (right now, under lockdown, it is about 0.8), and then keep it at or below 1 permanently. Vaccines alone aren’t likely to do that.
What could bring R below 1, short of a full-scale lockdown? The answer is straightforward: vaccines, plus all the social distancing measures that we know well, but have used much too inconsistently.
We need mandatory universal mask use in all indoor public spaces, including classrooms and restaurants; better ventilation in shops, schools and public transport; adequate compensation for people who are self-isolating, so that they don’t have to choose between feeding their families and protecting the rest of us; and those of us who are fortunate enough to be able to should continue to work from home.
How long will this have to last? Until the number of cases has been reduced to a small enough number that further outbreaks—and there will be many—can be contained by contact tracing and brief local lockdowns.
Australia, New Zealand, China, Taiwan and other countries—many of them much poorer than the UK—have shown how to do this. We should learn from them. But, in the meantime, we have to get R below 1, and keep it there.
Boris Johnson may have fooled himself and his fellow Tory politicians into thinking that the coronavirus epidemic will soon be behind us. But viruses can’t be fooled—and the rest of us shouldn’t be fooled either.