We want to know the probability of contracting coronavirus, especially from the viewpoint of someone for whom this poses a significant risk. We try a top-down and a bottom-up approach.

**Top down**

The ONS gives an incidence rate of 0.04% per week **over the course of their study **(from 8 June to 21 June). So the probability of not contracting CV in a week is (1- 5×10^{-4})

The probability of not contracting CV in N weeks is (1- 4×10^{-4})^{N}

The probability of contracting CV in N weeks is [1 – (1- 4×10^{-4})^{N}]

So this gives results like:

Of course, we would not expect the incidence rate to stay stable in reality, and the probability of any individual becoming infected will depend on their own circumstances. However, this does indicate that it would be highly desirable to attain a technical solution vaccine, etc to COVID-19 within 100 weeks/2 years.

**Bottom-up**

We start from contact between a susceptible and an infected person.

The prevalence of COVID is about 10^{-3 }ONS data 25 June.

The probability of infection given contact with an infected person is ~ 10^{-1 }[Suggested 5 x 10^{-2} here]

- Prob of not being infected in 1 contact ~ (1-10
^{-4}) - Prob of not being infected in N contacts ~ (1-10
^{-4})^{N} - Probability of being infected in N contacts ~[1- (1-10
^{-4})^{N}]

Or we can look at the lower range of this in more detail:

So this obviously raises a number of questions, such as:

i) what level of risk of infection is acceptable?

ii) what level of uncertainty is there in the parameter estimates?

iii) what exactly is a contact?

*What level of risk of infection is acceptable?*

This is obviously not merely a question for individual judgement, give the possibility of transmitting infection directly or at many removes to vulnerable individuals.

Confining ourselves to the consideration of individual risk as a first step, we consider the case of a male in his early seventies without relevant underlying health conditions. Data from China gives an infection fatality rate of 4.3% for this group. UK data shows something like a 2% risk for this group dying in a year. So let us say our subject wants the risk of dying from COVID to be no more than one tenth of this normal risk. So we require the risk of infection to be less than 0.05 in round terms, which means our subject should confine himself to about 500 qualifying contacts in a year at the current prevalence.

*What level of uncertainty is there in the parameter estimates?*

We can consider the question of who the contacts are. On the one hand, you are more likely to have contact with a supermarket cashier than with a shelf-stacker, and cashiers are likely to have higher prevalence for that very reason—they are in contact with more people. (Compare ONS treatment of US data.) On the other hand, it may well be that people with [relevant] symptoms have a higher prevalence than those without and are also likely to withdraw from contact. (0NS 12 June). This could do with further and more systematic treatment.

*What exactly is a contact?*

This is of course a crucial question. The latest official UK advice with regard to contact tracing as of 17 June says:

What do we mean by a ‘contact’?

*A ‘contact’ is a person who has been close to someone who has tested positive for COVID-19 anytime from 2 days before the person was symptomatic up to 7 days from onset of symptoms (this is when they are infectious to others). For example, a contact can be:*

*people who spend significant time in the same household as a person who has tested positive for COVID-19*

*sexual partners*

*a person who has had face-to-face contact (within one metre), with someone who has tested positive for COVID-19, including:*

*being coughed on*

*having a face-to-face conversation within one metre*

*having skin-to-skin physical contact, or*

*contact within one metre for one minute or longer without face-to-face contact*

*a person who has been within 2 metres of someone who has tested positive for COVID-19 for more than 15 minutes*

*a person who has travelled in a small vehicle with someone who has tested positive for COVID-19 or in a large vehicle or plane near someone who has tested positive for COVID-19*

*Where an interaction between 2 people has taken place through a Perspex (or equivalent) screen, this would not be considered sufficient contact, provided that there has been no other contact such as any of those indicated above.*

So we can take this as a characterisation of the qualifying contacts between two people which might lead to transmission of infection.

Or we can say that R_{0 }= (contacts per day)*(transmission probability per contact)*(number of days case is infective).

So we take R_{0 }as about 3, transmission probability per contact 0.1 and infectious period 10 days, so

3 = (contacts per day)*(0.1)*(10)

So there are **something like** 3 qualifying contacts per day in a state of nature before imposition of social distancing, which implies a fairly substantial kind of contact.

**Conclusions**

On the basis of the discussion above, and indeed of common sense, we would say that if you are in a group where COVID-19 poses a significant risk, you should limit your qualifying contacts (with contacts as defined in the contact tracing guidance) as much as possible unless or until this threatens to do tangible harm to yourself or others.