**I am getting heartily fed up **with the government’s daily ‘Number Theatre’ as David Spiegelhalter calls it. So many numbers, but so little meaningful insight into what is happening.

**We are all repeatedly informed that** we need the reproduction number to be less than unity (i.e. *R*< 1) in order to drive down the prevalence of the disease. But it is generally not stressed how extraordinarily hard this is, and how we really need* R *to be *very much less* than unity i.e. *R*<<1

*Individuals are integers*

** R is expressed as a decimal fraction, **but for any individual ill person, the number of people they infect is an

*integer*not a fraction i.e. it could be:

- 0 – ideally.
- 1 – almost inevitably
- 2 or more – quite easily

**So to achieve R = 0.9** for ten ill people, the number of people infected could be:

- 1,1,1,1,1,1,1,1,1,0 i.e. 9 people infect one other person, but one hero manages to infect no-one else.
- 2,1,1,1,1,1,1,1,0,0 i.e. 1 person infects 2 other people – it’s easily done. Now to achieve
*R*= 0.9 we need two zero-infection heroes. - 3,1,1,1,1,1,1,0,0,0 i.e. 1 person infects 3 other people – it’s easily done. Now to achieve
*R*= 0.9 we need three zero-infection heroes.

**In all of these cases –** there are still many infectious people after a single ‘generation’ of transmission.

**For a disease which can be infectious** *before people know they are ill*, it is extraordinarily hard not to infect at least one other person. It has taken the disruption of our “lockdown” to achieve *R *in the range 0.7 to 0.9.

**The situation above** describes only transmission across one ‘generation’ of the virus. What happens when we trace infections down a chain? When we do this we find that * chance fluctuations* really matter.

*Fluctuations really matter*

**To see the effect of fluctuations** I created a simplified spread sheet model to track infections across 4 generations.

**I started with a single infected person**and tried to calculate the number of people who would be infected after 4 generations of transmission.**I simplistically assumed**each infected person had a chance of infecting two other people, each with a probability*R*/2 to make a total transmission probability of*R*.**I then used Excel’s random number generator**to produce a random number between 0 and 1.**If the random number generator**produced a number less than*R*/2, then infection took place, otherwise no infection took place.

**I then repeated this **for three more generations and at the end I asked – *how many people are infected now? *I repeated the experiment 50 times for different values of *R* and the results are shown below.

**On average**, the numbers of people infected after 4 generations (**red dots**) were close to what would be expected i.e. *R*^{4}, shown as a grey line. But the fluctuations were very significant. On the graph above I have shown **in green** the *maximum *number of people infected in at least 1 of the 50 simulations

**During 50 experiments at**, I twice observed that there were 4 infected patients after 4 generations of transmission when we would have expected (on average) only 0.65 of an infection.*R*= 0.9**During 50 experiments at**, I observed on one occasion that there were 7 infected patients after 4 generations of transmission when we would have expected (on average) only a single infection .*R*= 1.0

**This shows that even** when the * average R value* is below unity, and infection clusters might be expected to die away

*. Fluctuations mean that it is possible – indeed likely – that some single infections can persist for many generations of transmission and in fact grow into clusters and seed a new outbreak.*

**on average***Where we are now?*

**As Louis Armstrong might have said, **I don’t get around much anymore. But on my expeditions out, I see many more people than at the height of the “lockdown”. It is also not hard to see some groups of people who do not appear to be social distancing.

**If the “lockdown” achieved R = 0.7,** then I would estimate that the

*R*value now must be higher, probably close to 1. The point of this article is to say that that even for

*R*values below 1

*, fluctuations can allow a single infection to grow into a significant cluster over several generations.*

**on average****The ONS estimates **that there are roughly 133,000 people currently infected with the virus. Even with *R*=0.9 it will take until the end of the year to reduce the number of infections to 10,000 – still a massive number. Over that length of time, and as economic and social activities resume, there will be thousands of opportunities for small outbreaks to grow significantly.

**In my opinion, **with this infection rate and a likely transmission rate *R* close to unity, there is a strong likelihood of significant further outbreaks. 93% of our population is still susceptible and if the death rate were similar to what we have seen already we could potentially lose 10 times more people than we have already.

**I understand the economic and social imperatives** that are driving us towards re-opening the economy. But personally I am skeptical that the virus is sufficiently under control to allow re-opening without many further outbreaks. **I do hope I am wrong**.