If mathematicians have one tiny fault – a point which I am not ready to concede – it is that they are entirely unintelligible.
If you ask a mathematician to explain the spread of the coronavirus, they may say things like,
“well, with a simple SIR model, you can just solve the differential equations, ds/dt = -b s(t) i(t) and dr/dt = k i(t), while remembering of course that ds/dt + di/dt + dr/dt = 0”
and that may not be exactly what you wanted to know!
But now is a time when it is actually quite helpful to understand the way the virus spreads. So this article aims to help you do that without having to look again at those differential equations.
The key to understanding it is this picture.
In the diagram, the population is represented by water distributed across six glass vessels. The largest vessel is the ‘uninfected’ population, which feeds into a vessel for those who have been infected but are ‘asymptomatic.’
Most of those who are asymptomatic will recover and end up in the other large vessel (‘Recovered’) at the bottom of the picture. But a proportion will go on to develop mild symptoms.
Again, most of the people who develop mild symptoms will recover and end up in the ‘recovered’ vessel, but a proportion will develop severe symptoms.
Most of these will also recover, but a proportion will die, and end up in the small ‘dead’ vessel on the right side.
At the moment, there is only one tap in this system: the one between the uninfected and the asymptomatic vessels. If we had a cure, we could put a tap between ‘mild’ and ‘severe’ so that everyone who develops the symptoms would end up in the ‘recovered’ vessel. But at the moment we do not have that. So to the best of our knowledge, roughly 1% of those who enter the asymptomatic vessel will end up dead. (Of course, governments could easily make things worse — for example by not having sufficient ventilators for those with severe symptoms — but there is no way to make things better right now other than turning-off the tap).
So this picture can help us to understand what to expect from the virus. Specifically, it can tell us three very important things:
- in the long run, since we don’t have a cure yet, the total number of deaths is determined by the rate of new infection;
- in the short run, it is the other way round: it is the number of people already infected that will determine the death rate; and therefore
- we should not expect the lockdown strategy to have an immediate impact on the death rate.
In the long run deaths are determined by the rate of new infection
If we assume that no cure is found, and that the 1% estimate of fatalities is reasonable, then as the disease progresses, even though 99% of the people who enter the ‘asymptomatic’ vessel will end up – via various routes – in the ‘recovered’ vessel, 1% of them will end up ‘dead.’ Therefore, to minimise long run fatalities, we need to do what we can to turn off the tap.
What determines how open the tap is? Three things:
- how infectious is the disease? If I’m infected and I come into contact with 20 uninfected people per day (for example), how many of those 20 will I infect?
- How many people per day do I in fact come into contact with?
- How many of those people are uninfected (and therefore susceptible) and how many are recovered (and hopefully immune)?
How infectious is the disease?
For the first question, we do have quite a lot of data. In most countries, in the early stages, the virus seemed able to double roughly every three days. That means that, with normal rates of social contact, each person is able to infect roughly 0.25 other people per day (because 1.25×1.25×1.25 is roughly 2).
How many people do I come into contact with?
And the second question brings us to the reason for the lockdown – if we can reduce the rate of social contact by a factor of three, say, then the number of people that each infected person infects per day will drop from 0.25 to around 0.08, and at that level, people are recovering faster than they are infected and the virus goes into retreat.
How many of those people are uninfected?
The third question highlights the notion of ‘herd immunity.’ If I have contact with 20 people per day, but 19 of them are immune, odds are that most days I won’t infect anybody. Again, the number of newly infected people is less than the number recovering, and the virus is in retreat. So, even without any preventive measures, the virus will eventually burn itself out.
But as we pointed out before, the problem is that this means sacrificing the lives of approximately 0.6% of the population.
So, thinking about the long run makes it clear that we need to turn off the tap. And right now, the only effective way to do that is 100% commitment to the lockdown strategy.
In the short run ‘people infected’ determines the death rate
But the short run is entirely different. Even if we could turn off the tap completely today, which is not realistic, there are many people already infected, especially in the ‘asymptomatic’ vessel. It may take several weeks for them all to end up in their final destination – which will either be ‘recovered’ or ‘dead’ – and there is not very much we can do to influence which of those vessels they end up in.
We should not expect lockdown to have an immediate impact
For that reason, we should not be unduly concerned by the continued upward movement of the death count – tragic though it is. And we should certainly not take it as evidence that lockdown is not working.
So when I see Tweets like these, I think it is important not to panic.
We are now (2/4/2020) less than two weeks from when UK lockdown started to be effective, and as the Imperial College team has pointed out, there is a “.. lag of 2-3 weeks between when transmission changes occur and when their impact can be observed in trends in mortality”
The good news, of course is that many of the trajectories in countries which implemented lockdown earlier are now clearly showing signs of positive impact.
And as Reuters reports,
“Scientists used an online survey to ask 1,300 people in Britain to list their contacts for the previous day – and found that the average number of contacts now is more than 70% lower than before the lockdown. ‘If we see similar changes across the UK population, we would expect to see the epidemic to start to decline,’ said John Edmunds, who led the study at the London School of Hygiene & Tropical Medicine (LSHTM).
He added, however, that the findings were very preliminary and should not be seen as suggesting ‘job done.’ “Rather, they should be used as motivation for us all to keep following UK government instructions,” Edmunds said. ‘It’s imperative we don’t take our foot off the pedal.’”
We now have an animation of this article here.
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