To ask you to do some Corona related maths

[mathdoc]

I've been following some of the (many) corona threads here and I've realised that I've been doing a virtual eye-roll everyone says that they have been doing their own risk assessments, because I think assessing risk is incredibly hard. However, on reflection this might just be me being very arrogant (it has been known!) so I thought I'd get some actual evidence by asking you to try this little maths question. (See the disclaimers at the bottom for people who are even more pedantic than me). I'm not claiming that I'm right, because I'm just as liable as everyone else to make mistakes with this, so I'd be very happy to be corrected. I'm not expecting people to do calculations for this (although feel free if you like that sort of thing) - it's more about checking people's intuition.

The question is this. Suppose that 3 people meet up and you are one of them. Let's suppose that there is a particular probability of you getting the virus in this situation, and there's also a particular probability of the virus being transmitted between two people attending.

Now suppose that instead 6 people meet up.

1. By approximately how much has your risk of catching the virus increased?

A. Basically no change B. Doubled C. Increased by a factor of 2.5 D. Increased by a factor of 12 E. Increased by a factor of 20 F. Increased by a factor of 80

1. By how much has the risk to society of a "virus transmission event"increased.

A. Basically no change B. Doubled C. Increased by a factor of 2.5 D. Increased by a factor of 12 E. Increased by a factor of 20 F. Increased by a factor of 80

Disclaimers for those interested in the fine print.
I'm assuming:

• that these people are all chosen at random (e.g. not already in the same bubble)
• that everyone in the meeting spends roughly the same time with everyone else and in the same way, so that the probability of transmission for each interaction is approximately the same.
• that the probability of transmission in any individual interaction is very low, and independent of any other interaction

Obviously some of these assumptions are dubious, but as long as the probability of transmission is low I don't think that they have a large impact on the comparison I'm looking at.

The numbers 3 and 6 have been chosen to be illustrative. I'm not suggesting this is anything to do with the current rules - in particular whether or not they are sensible. This is just to see how good people are at assessing risk.

Op I think it is very reasonable to try to determine what people are actually talking about when they say they have assessed to risks of xyz. I’ve also found the discussions about effective R really interesting because a lot of people have clearly latched into it without understanding what it can and cdnt tell us

If say the op has an excellent point actually in that even in this incredibly simplistic scenario that would never exist in real life, noone here has even tried to answer the question . I'd hazard a guess that for most it is because they don't know the maths I include myself in that ). If we don't know the answer to this simple question how the can we asses our risk in the much more complex real world?

But I think when people say "I'm doing my own risk assessments" they don't mean stats. They mean looking at the rules and applying common sense and logic. For example, in June my DD (now 12 months old) went back to nursery, no social distancing of course because of the age of the kids. One of the staff there is a woman approx late 50s. She is looking after my DD, picking her up, helping her eat, changing her nappy etc. This is all allowed. However my colleague with a DS a couple of months older than my DD wasn't allowed to go back to using her mum for childcare as she did pre-coronavirus - her mum is healthy, and a similar age to the woman looking after my DD. There is no logical reason the woman at DD's nursery can look after my child with no social distancing to allow me to work, but my colleague's DM cannot look after her child to allow her to work. My colleague followed the rules but I bet a lot of people did their own risk assessments and used grandparents for childcare. They don't need to look deeply into the actual stats.
I did a degree in a very stats-heavy course and have worked as a data analyst for years, and at no point did I sit down and do the actual maths for risk.

Several people has pointed out that, no matter what the factor of increase is, when it is applied to a small number the answer will still be small. This is true for any individual meeting but when scaled up many millions of people each increasing their risk substantially can have massive population level effects.

Population level is not the same, since you're also ignoring other deficiencies in your scenario, the chance of a transmission is not constant between a group of 4 and a group of 7 - If social distancing, it is not possible for all people to be 2m apart from each of other, some pairs need to be further away, therefore the actual risk of transmission between pairs change - so whilst your 3 and 6 people could possibly have had the same chance of transmission (if everyone remained 2m from you, with you talking at them the same amount) although even that is unlikely as you would normally of course stand in a circle in such meetings.

Even if the distance was dropped the relative transmission between pairs would differ.

Your model doesn't work - it doesn't work at all as a model to think about it for most people who don't see the numbers (those who think a doubled risk is scary when it's actually irrelevant even at a population level) and it doesn't work as a model for you either as it's too simplistic, there isn't a simple "risk of transmission" in a group. It completely depends on behaviour of the group as well as them getting together.

@sirfredfredgeorge

I certainly wouldn't be using this model to work out actual values of the risks - it is just a vastly oversimplified toy situation to look at comparative risk - i.e. by how much does the risk increase when going from one situation to another comparable one. So I've assumed that if you're meeting with 2 other mask wearers, then the comparable situation is meeting with 5 other mask wearers etc.

If I've understood your critique correctly you're suggesting that all the people can't be equidistant (which is of course correct at any given time) but I was imagining more of a situation where the people were moving around and spending a certain fraction of the time chatting to the other people in a roughly comparable way. I'm sorry this wasn't clear.

However, irrespective of this I think (if I've understood it correctly) that your second point about doubling risk being irrelevant at the population level if the original risk is small is incorrect. When there are tens of billions of interactions between people every day, small risks very quickly accumulate. I believe that in (again simplistic, but still fairly useful) SIR models the R number is proportional to the risk of transmission in any given interaction. If it doubles, the R number doubles and it is very easy to get back into the exponential growth phase of the disease.

I freely confess I can't do the maths to answer yoru question.

But I can and do risk assess.
As others have said, that is based on lots of things
eg

1. the town I am in has had a low infection rate and few deaths, therefore the chance of me meeting someone with Covid is low.
2. we were all sticking to the rules, and only visiting shops for essentials and using click and collect etc, so our house had alow chance of getting or passing on the virus.
3. Once my kids went back to school (teens who go on the train) I was aware that our risk increased, and so didn't want them to mix with my parents for example.
4. We are all healthy in our house and no-one is shielding. But on th eother hand I have been collecting shopping for shielding families.
5. We have decided not to go clothes shopping/into town/out to the pub/out for a meal, but we are going to the supermarket and kids are meeting friends in the park at a distance (but I don't think they stick to it) and I meet friends in their bakc garden

So with each thing we are making a judgement call. How safe am I? How likely is it that I have been exposed to the virus? How likely is it that I will pass it on in these circumstances? How likely is it that I will catch it from my kids who've been on the train? and so on.
None of those are done on numbers or statistics except the numbers for how many are ill in our town.

Another interesting question for you as you are obviously a whiz at maths .

I have been interested in the comparison of death rates between countries . Mainly because they are wrought with demographic variations .

If you were doing comparisons say on covid deaths between countries how would you factor in the density of population.

So for example the uk has a dense population per sq km so because of this you would expect death rates per 100000 to be higher . Where as the USA has a huge population but average pop density must be low so in proportion their death rate should be lower .

But how do you reach that factor figure in a calculation.?

Haven’t RTFT but is it c and d.
My chance of getting it increases by 2.5 when there are 5 people I could get it from instead of 2
But the chances of anyone of the 6 getting it from anyone else in the 6 increase by 12

But I think when people say "I'm doing my own risk assessments" they don't mean stats. They mean looking at the rules and applying common sense and logic

Which is what most people here are saying - though whenever I see the words 'common sense' it tends to make my hackles rise. Common sense is quite often not a particularly good way of judging the right course of action in my opinion

I did a degree in a very stats-heavy course and have worked as a data analyst for years, and at no point did I sit down and do the actual maths for risk

As the op has said, this isn't the point. Very few people would work out the actual maths but most people know that two loaves of bread costs twice as much as one loaf (maths in the most pure sense but most people would call it 'common sense') ,.

Very few people however seem to instinctively understand the difference in the type of risk in the original two scenarios and the big difference in them.

This seems to me be to be one of the reasons behind why a lot of people fail to understand why people should wear face coverings. They argue about about how people don't wear them properly or how they have 'barely any effect on an individual's chance of catching Covid' unless they are medical grade MN99/P3/ff3 ones.

They seem to sometimes fail to understand that if enough people wear them, even if not always correctly they will reduce the risk of transmission at a population level. Even if the reduction is moderate, it will have a not insignificant effect on transmission levels overall.

But people seem mostly to make a risk benefit judgement based on ' How likely they are as individuals to catch the illness or to pass it on and how bad it would likely be for them, against whether the mask is inconvenient to them. They don't take into account that transmission overall would likely be decreased if most people wore them (thus making them safer, albeit indirectly)

Anyway I've found this thread fascinating (I am obviously of the geeky variety ). Thanks OP

So my answer, for what it's worth, is that by going from a group of 3 to a group of 6 the risk to an individual increases by a factor of 2.5 but the population risk increases by a factor of 5. In general, the risk to one particular individual grows far more slowly than the risk to the population. I don't know if people find this an interesting or surprising result.

My argument can be made more rigorous, using the excellent method outlined by @TeenPlusTwenties, but the rough idea is that in a group of 3 you are interacting with 2 people, whilst in a group of 6 you are interacting with 5 people, so from your perspective the risk increases by a factor of 5/2.

In a group of 3 people there are 3 possible interactions (think about the sides of a triangle) but in a group of 6 people there are 15 possible interactions (think about the sides and diagonals of a hexagon). So the probability of the disease being transmitted goes up by a factor of 5.

Obviously there are many factors to consider when doing a risk assessment, so I would not be trying to make claims about absolute risk but I've seen a few comments along the lines of "if a group of 6 is safe, then surely a group of 10 is ok". Based on arguments like this, going from 6 to 10 increases the individual risk by a factor of 1.8 but increases the population risk by a factor of 3. I don't know whether or not these increases appear significant.

Having said all that, I'm clear that numerical risk assessment is not the be all and end all (although we try to capture some of the psychological aspects of risk assessment using something called a utilty function for those interested in the geeky details). If my children were in a burning building, I'd be running in and sodding the maths. I know that many people are in situations where the alternatives are all bad and I try (and occasionally fail!) to not judge people for evaluating the risk differently to me. I think I now understand a bit better what people mean when they say they are doing a risk assessment.

OP, your answer is completely incorrect! The risk only increases by that factor if everyone has Covid!

The current population level of Covid is around 1/4000.

Your risk ONLY increases if anyone has COVID. Which in a group of 6 the odds are only 6/4000.

@SusieOwl4

Another interesting question for you as you are obviously a whiz at maths .

I have been interested in the comparison of death rates between countries . Mainly because they are wrought with demographic variations .

If you were doing comparisons say on covid deaths between countries how would you factor in the density of population.

So for example the uk has a dense population per sq km so because of this you would expect death rates per 100000 to be higher . Where as the USA has a huge population but average pop density must be low so in proportion their death rate should be lower .

But how do you reach that factor figure in a calculation.?

I'm not sure I'm a whizz at maths - I love it but the more I learn the more I realise I don't know!

However, the question you posed is actually one I've been working on a bit recently. Comparing covid deaths is actually remarkably difficult, and I haven't yet come up with a measure that I'm satisfied with. As you say population density is a factor, but using just one population density for the whole country doesn't quite work. (e.g. Australia as a whole has a tiny population density, but some of it's cities are much more dense). There are lots of other factors which also confound any comparison of excess deaths. For example, there are several countries which superficially look like they are doing great, but their life expectancies are very low so a disease that has a high morbidity in older people does not have such a big effect.

What I'm currently doing is using a regression model (think "line of best fit") with the excess deaths as a proportion of the whole population as the dependent variable and the independent variables being the overall population density, the population density of the largest city, the percentage of the population aged over 80 and a measure of freedom called the CATO human freedom index. Once we have the regression line, points which lie above it correspond to countries which have done worse than might be expected. However, as with most statistical models garbage in leads to garbage out so until the waves die down it's quite hard to make sense of it.

@mathdoc

Yes I agree . There are many variables which is why I think they stopped comparing figures . Especially as reporting of deaths with covid is not consistent . So excess deaths is the closest comparison. I was surprised that New York with a population of 8.9 million I think had 30000 deaths compared with the uk , which had about 40000 deaths with a population of 60 to 70 million. I also would be interested to know why belgium has such a high death rate ? Is it to do with transient population?

But if there were figures with pop density and a loading for average age of population it would be interesting.