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March 18

How many lives would preventing a single additional COVID-19 infection today save in all ? (Galton-Watson / Poisson branching question)

I've tried asking this on Stackoverflow and Mathoverflow, but it seems to be getting voted down. I'd be so grateful if somebody here could have a look:

Consider a tree process with Poisson branching with mu = 0.9, to simulate an epidemic decaying under strong social isolation.

How many total infections would stopping a single infection at the start of the tree save in all ?

In we assume an average CFR of 3.4%, how many lives is that in all ?

Bonus points for an image of what typical trees look like with mu = 0.9

So to try to make the model clearer: Start with a node. That node will have n forward branches, where n = 0 or 1 or 2 or 3 .... according to a Poisson distribution with mean 0.9 . This generates 0 or 1 or 2 or 3 ... new nodes. Repeat over all the new nodes, until an iteration produces no further new nodes. What do the trees that are the result of such a process look like? (ie a sample of trees). And counting up the total of new nodes, what is the average of that number?

I think the answer is 10 new infections, so that each single new infection would cost an average 0.34 lives.

But if somebody could either confirm the maths for me, or run a simulation to show me what the trees look like, I would be enormously grateful.

Motivation: Last night I managed to persuade my rabbi to cancel her service this Saturday morning that might have had 120 people in it. How many new transmission events would there be in that sanctuary and in that kiddush hall? I don't know, I'm not an epidemiologist. But I think that the effect if each transmission event would potentially be an average of 0.34 lives(*), of people who were never there, and never had any choice about it.

Have I got this right ?

I'm frantic to get a second pair of eyes on this, because if it's right, it's something everyone needs to know: the cost in lives of each single additional infection they fail to stop.

(*) In reality it would actually be more than 0.34 lives, because the UK is not yet doing strong social isolation, so there's still going to be n rounds of full-speed doubling on top of the 0.34

And yes, I know this is a spherical cow problem. But sometimes spherical cows are useful.

Thanks in advance, Jheald (talk) 08:29, 18 March 2020 (UTC)

Given the expected number of infections, we can compute the expected number of fatalities by simply multiplying it with the CFR – provided that we overlook the nicety that the latter is usually defined in relation to the number of diagnosed cases. This is not an issue if we replaced "infected" by "clinically ill". With μ representing the expected number of sprouts from any given tree node (where it is not relevant whether that number has a Poisson distribution or some other distribution as long as this is independent for different nodes), and E being the expected number of nodes in the tree, if E < ∞, it has to satisfy the equation E = 1 + μE. It follows that E = 1 / (1 − μ). Using the value μ = 0.9, we get E = 10. Multiplication with a CFR of 0.034 gives, indeed, 0.34. However, these nodes represent human individuals who in reality run the risk of getting infected from many sides, and what this tree model does not account for is that an individual surviving on one contact tree, so to speak, may be a fatality on another tree. The number obtained is at best an upper bound. This simple tree model only works at the onset of a potential epidemic, when virtually no one is infected yet. In the midst of an epidemic, it may not be adequate for this problem; you need to consider networks.  --Lambiam 12:05, 18 March 2020 (UTC)

The question makes an assumption that if X caught the virus from W, W caught it from U and so on to B catching it from A, then if you could have isolated A at the start then X would not get the disease. And while it's true that the disease would have had to take a different path to get to X, that doesn't mean B couldn't have gotten the virus from A' instead and X would still get it anyway, or that C couldn't have gotten it from B', etc. I think the only person who would have truly made a difference was the very first person to get the virus; if we could somehow go back in time and somehow figure out who that person was and isolate him or her before they passed the virus onto anyone else then the whole epidemic could have been avoided in the first place. But once the genie was out of the bottle I doubt that what would happen to a single individual would make much difference. Slowing the rate of infection, when averaged over an entire population, can make a difference, but individuals not that much. Even then, the relationship between rate of infection and the number of people to be infected eventually is highly non-linear and difficult to model; assuming it's a differentiable or even continuous function may not be realistic. --RDBury (talk) 13:44, 18 March 2020 (UTC)