In virology they have a key figure which they call "R0" which is the number of people who each infected person infects. A common quoted figure for R0 was 2.6. This compares with what actually happened which is that in most countries the number of cases increased exponentially by about 10x in 6 to 9 days. From this we can work out the average time between someone getting infected and passing it on to those 2.6 people. 2.6^2.4 =10. Which means the average time for a 2.6x increase is the time for a 10x increase over 2.4. So the average time to pass it on is about 2.5 to 4 days. However, there is a problem with this figure. Because the average time for someone to start showing symptoms such as a cough is about 5 days. So for an R0 value of 2.6, people must be spreading the virus BEFORE they start to cough, which is one of the main ways to pass it on.

So, let's approach this the other way. We know it is increasing by 10x in about a week. We also know that symptoms first appear in 5 days and evidence suggest peak virus shedding (coughing) another 4 days on. So, peak infection is about 9 days after infection. Because the virus is spreading by 10x in a week, that means that R0 must be at least 9 and probably nearly 20, or 8x the figure that virologists believed it was.

# Proof

At the simplest level the exponential growth can be modelled as N = exp(A . (d-d0)) When d0 is the day where  N is 1 and d is the time (i.e. days) and A is a constant.

If we use a simple concept that people infect everyone on day d1, then we know that if the number they infect is R0 then, on day d1 the number of newi infections is:

N = R0

so

R0 = exp (A . d1-d0)

So

A = ln (R0) / (d1-d0)

If we take do as being "day 0" then this becomes simply:

A = ln (R0) / d1

Thus the rate of growth is set by two factors: R0 (the number each person infects) and d1 (the time for them to infect others).

We have data for the growth rate (about 10x in about a week). If we also know the average time to infect other people then we should be able to calculate R0. Symptoms appear on average on day 5 after infection. The peak infectivity is on day 4 after symptoms appear, so the time from infection to peak infection is 9 days.

For ease of calculation, let's say the rate of growth was 10x in nine days, then we know on day 9 (if day 0 has one case) that the number of total cases is 10.  On day nine d1 = 9 and N = 10, so:

R0 = 10

But because we used a 10x increase in nine days rather than the more typical faster increase of 10x in a week, we can say that:

R0 >9

# Why the difference - the tip of the iceberg

CV19, whilst caused by one virus, has behaved in extremely different ways in extremely different people. In the vast majority of people, at least 90% but perhaps as high as 98%, the virus has been so mild that most people didn't think it any different from a normal cough. These people did not feel very ill and kept on with their normal lives. But in a small group (2-10%), the virus was severe enough that they sought medical attention. Amongst this group, the tip of the iceberg, the virus behaved in a very different way, causing them to feel very unwell with a high number (of this small group) going into hospital and a substantial number of those going into hospital ending in Intensive care (about 20%) and a large number of those dying. These are the group the medics saw. The differene is that in this group, despite it being just as contagious as the rest, because they felt very ill they stopped their normal lives and to some extent isolated. This, I think, is why the virologists got the R0 value so terribly wrong. The group the medics saw, had a low R0, because they stopped their normal lives so the R0 value was about 2.6. But the vast bulk of those with the virus, so the vast bulk of the spreading was amongst those who continued their normal lives who had an R0 value over 10.

# Implications

### Timing of the epidemic

Early on we were being told that the virus would not sprad quickly. A peak in 3 to 4 months was being suggested and that different parts of the UK would have peaks at up to a month apart. That was totally at odds with the evidence that showed the virus was going to peak in about 1-2months and that there was less than a day difference between Scotland and the rest of the UK. Clearly the models were dependent not only on a very low R0 of 2.6, but there must also have been a presumption that there was over a week delay beween infection on one person and the next. This explains why several  times the UK PM gave a figure of "doubling in a week", when the evidence showed about 10x in a week. It also explains the initial slow response.

### Severity of the epidemic

Bizarrely as it may seem, the higher the R0 value, the less concerned we should be about the epidemic. Because a very high R0 value is symptomatic of a very mild disease, whereas diseases that tend to have very bad outcome, tend to have a low R0 because both the patient and the society around them, tends to take very strong measures to contain the outbreak. This all adds to the growing evidence that the Infection Fatality Rate (the number of those infected who die) is around 0.15%. Of these around 90% are over 50. So, for the majority of young people the IFR is around 0.015% or 1 death in 7000 infections. This is about the same as an under 50s yearly risk of dying from a road accident.

### Ending the epidemic

If on average each person who gets the virus passes it on to less than 1 person (i.e. many do not pass it on to anyone), then the number of people with the virus will decrease, until (it is hoped) that after a long enough time there is only one person with the virus and they happen to be one who doesn't pass it on and then no one has the virus.

That strategy seems realistic if the R0 valuei is 2.6. Because, to put it simply, if instead of meeting 100 people a week, we drop the number of people we get close enough to infect to less than 38 people a week, then as there are <1/2.6 fewer oportunities to infect peple the R0 value  should drop below 1. So, modest social distancing could kill off the virus.

But if the R0 value is 10, then instead of cutting our social interactions from 100 to 38, we need to cut them from 100 to 10. But that only reduces the R0 value to 1. We actually need it to be less than 1, so the real cut is from 100 per week to perhaps 5x a week. Given the number of people we come into contact just supermarket shopping, let alone if we are a nurse or a bus driver, this reduction is very extreme social distancing which would be extremely difficult to enforce.But worse, even if R0 falls to 0.5, if we start with 16,000 cases, the time to get to one case is 14 x 9 days (18 weeks). There are very likely more than 16,000 cases, so, we require almost the total shut down of society for much of the year. There is no way on earth that is feasible, for the simple reason that we'da all starve or if we didn't the economy would be so wrecked the miniscule numbers of deaths from the virus would be far far outstripped by the horrendous death toll from the economic collapse (when e.g. we couldn't afford the NHS).

That means anyone trying to end the epidemic by "social distancing" is wasting theirs and our time and our money. The only way this virus is going to end, given the earliest anyone is talking about a vaccine is September, long after the virus will have peaked, is for the majority of us to get "herd immunity". And, to make it worse, if the R0 value is 10, that requires an infection rate of about >90%. As half the deaths occur in over 70s who are 10% of the population, on the face of it, that suggest we could protect all this group, but only if everyone else has got immunity. Realistically not everyone will get the virus which means that some over 70s will have to have had it to get herd immunity. On the good side, the rate of death even in this group is less than we thought earier on. But it is still quite horrendous as the deaths in care homes amongst the most vulnerable elderly have shown.