One of my children has just recovered from swine flu, as a result of which I now have a clearer idea of what British policy is towards outbreaks. Much of it was perfectly sensible, but not quite all. Since there’s a small amount of mathematics involved, and since I wanted to get this off my chest, I thought I’d blog about it.
The good part was that everyone who had been in close contact with the child who had swine flu was immediately put on Tamiflu, which seems to have stopped any of the rest of us getting it. (It’s now been long enough that we can be almost certain of this.) The less good part was the piece of advice that I mainly want to discuss. The main question I had was, of course, to what extent I and my family should avoid contact with other people. The advice I was given, which, it was made clear to me, was the official policy and not just the whim of the public health official I spoke to, was that we should continue to lead our lives as normal for as long as we did not show any symptoms.
When you bear in mind that leading our lives as normal meant going to work or school and coming into contact with many people, and also that if one gets swine flu then one is highly contagious just before symptoms appear, this amounts to taking a smallish risk of infecting quite a lot of people. I decided to disregard the advice and stay at home, and here is why.
Having lectured probability for the last two years, I am well up on branching processes. They are not a perfect model for the spread of disease, but at least in the early stages one could imagine that each infected person has a certain probability distribution associated with the number of people they go on to infect, and that if A and B are infected and A infects C then it would almost certainly not also have been the case that B would have infected C. In that case, one has something pretty similar to a standard branching process, so a rough rule of thumb would be that if on average each infected person goes on to infect more than one other person then the disease has a positive probability of spreading to a very large number of people, whereas if on average they infect at most one other person then the disease will be contained with probability 1.
As I said, that is not a perfect model for the spread of disease (which is of course something that people have thought hard about), but it’s good enough to make it clear that there is almost certainly a phase transition: below a certain level of risk and only a few people will get the disease; above it and there’s a fair chance that huge numbers of people will get it. (And more realistic models do indeed have these phase transitions too.)
Now to the justification of the advice I was given: “If everyone who came into contact with someone with swine flu stayed away from work then the whole economy would shut down.” Well, there have been about 500 cases in Britain so far. Let’s suppose that 10000 people decided as a result to stay away from work. If that reduced the expected number of infections per infected person to below the critical probability, then there is a reasonable chance that it would stop millions of people getting the disease later. Hmm … I wonder which would be worse for the economy. (Or rather, I wonder which leads to more expected damage.)
Let me put that more mathematically. If we don’t know what behaviour leads to what probability of infecting somebody else, then what policy should be adopted? This seems to me to be an instance where the uncertainties involved do not lead to an uncertain conclusion: if there is a reasonable chance that being very very cautious takes us to the right side of a phase transition, then the potential payoff of that caution is huge, so the expected gain (counting the inconvenience of the caution as a loss) is huge.
It seems that in France they have a much stricter policy towards people who get swine flu. And it also seems that it is spreading much more slowly there. It will be interesting to see whether that continues. I’d also be interested to know what the policy is in the US, where there have been many more cases. But that could have been because the US was confronted with swine flu at an earlier stage when there was less time to formulate a policy.