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.

June 5, 2009 at 10:12 pm |

I fully agree: the advice given is a clearly suboptimal one. Let me try a translation of your argument in economics terms:

1) There surely is a trade-off between the inconvenience of more or less strict quarantine regimes for the “at risk” people and the danger of spreading the infection.

2) The advice to “continue to lead our lives as normal” can only be optimal if one believe that we are at a corner solution (i.e., zero inconvenience) – which is prima facie highly implausible.

3) An advice to limit contact with other people as much as possible (even leaving the choice to the “at risk” person) is a clearly superior policy compared to the current one (providing strong incentives to stay at home would probably be even better, but if the bureaucracy feels uncomfortable quantifying the incentives it should at least use zero-cost “nudging”).

Best wishes to your son and to all your family!

June 5, 2009 at 11:29 pm |

I live in Wisconsin. We top all US States in reported cases, but that’s because do our own testing.

Original recommendations seemed to support regional schools when they decided to close if a case was reported. But when it became clear that the virus was not a virulent as originally feared, recommendations became clear: do not close a school if a case is verified.

My intuition is that public health authorities calculated that the “herd immunity” effect should be pursued at this point, since the virus is not currently virulent. Maybe this will head off the spread of virulent mutation next Fall or Winter.

June 6, 2009 at 1:55 am |

Perhaps the call to maintain normality is more about not causing a panic than anything else. What’s the value of public peace of mind?

June 6, 2009 at 3:05 am |

I’ll echo some of the commenters’ opinions that the actual threat posed by this flu is currently quite low. Seen in this light, their recommendation seems reasonable, if not correct.

June 6, 2009 at 8:30 am

That’s possibly the case (though in Scotland there are now cases of people in their thirties and forties with no previous health problems ending up in intensive care), but the argument about number of working hours lost to the economy still stands. Perhaps one could turn your argument round and suggest that tougher measures should be taken not just for swine flu but also for the more normal bugs that go around each winter.

June 6, 2009 at 2:11 pm |

What the health authorities do regarding a pandemic threat must be , i guess, advised by statisticians specialising in epidemiology. It would be interesting to see their actual calculation in this case. Are the public given access to this sort of information?

A note off the side (this may be too applied for some)

This raises the question about how authorities use (and probably to some extent misuse) mathematical models and how they could be used. I for one would prefer a calculation with huge uncertainty bounds rather than just decisions based on previous experience. Reading the PCM should be mandatory for government officials (…). They probably hire consultants all the time who do good work, but the overall understanding…

I figure there must be huge discussions about these issues when we have had banks become insolvent , derivatives creating havoc (at least according to the news) etc. etc. But on what forums do they take place?

June 6, 2009 at 5:27 pm |

Wow, what a dramatic post. Best wishes for much health for all the family!

Based just on the analysis in the post, the current policy is justified if we are well below the critical probability, and apparently this is what the Health authority thinks. (And it seems reasonable.) Of course, there are uncertainties involved but the policy itself can be flexible. So baring in mind that the policy can be adjusted, the idea that being *very very cautious now* can lead us to the right side of the phase transition is not entirely meaningful. (There are also various delicate issues (some were mentioned in the comments) that are neglected in the post. Even the statement *damage to the economy* is much oversimplified.)

June 6, 2009 at 7:17 pm

Gil, I’m not sure I agree with everything you say. The virus seems to spread very easily, at least to judge from the number of people at my son’s school who caught it, so I would estimate that there was a non-negligible probability that I had it. And if I had, say, had lunch in my college, then I would have had opportunities to spread it to several other people. That’s not quite a proof that the expected number of people I would infect was more than one, but neither did it leave me confident that the expected number was at most one. And the fairly rapid growth of the number of cases in this country suggests to me that with the current policy we are in fact above the critical probability.

The question of whether it is better for people to get swine flu now rather than wait till a more virulent strain develops is a more complicated one and I haven’t formed a considered view about it.

As for the oversimplifications concerning the economy, I’m not sure they are problematic. Let’s just assume that staying away from work in order to reduce the chances of infecting people has a certain cost C_1, and getting swine flu has a cost C_2. I think it’s reasonable to take both C_1 and C_2 to be positive (if viewed as costs, or negative if viewed as benefits). If it is true that changing to a more cautious health policy has a chance of avoiding an explosion in the number of cases, then the expected gain of this policy will be large unless C_2 is much smaller than C_1, which looks sufficiently implausible to me that I don’t feel the need for a more sophisticated model.

I do take your point about flexibility though, and my family has had blood tests (results not yet known), which are apparently not so much for our sake as for the sake of the gathering of information. So perhaps there are people constantly monitoring what is going on and attempting to make judgments about whether the current policy really is sensible, and perhaps the recommendations will change in the light of new information. But even if that is the case, I still think it would be better to err on the side of caution, just because the possible cost of not being cautious enough seems to be much higher than the cost of being too cautious.

June 6, 2009 at 11:01 pm |

Dear Tim,

I feel a little awkward debating this matter since for you it was probably an ordeal and for me it is just a discussion but I assume that you are indeed interested to discuss it thoroughly. I think we can take it for granted that the authorities do want to confine the disease and not to let it pass the “critical probability”. (The fact that the illness is fairly benign suggests that while inside the subcritical regime there will not be an extreme effort to minimize the number of infections. Had the disease was more dangerous this was a different story.)

The reason I said I think we are well within the subcritical domain is that this is what the number of overall infected people in the UK suggests. Probably this means that the expected numbers of people that will catch the disease from a family member of an ill person, that get the required treatment but continue their life as usual is below one. (In spite of what you estimate.)

I do not know what is the right mathematical model here but it looks that we are well inside the subcritical domain in all countries. If this is true and as long as this is true there is no need to be very overly cautious. When I said the policy can be adjusted I mainly meant that it can be adjusted as a function of how quickly the flu spread. (But of course adjusting the instructions to individuals based on their condition also makes sense.)

Among the delicate things that you do not take into account in your analysis is the ratio between ill people who are known to carry the illness and people who have the disease without this being known. Imposing unnecessary (or even necessary) restrictions on family members may lead to higher percent of people who are ill but not known to the health authorities. The overall effect may be in the wrong direction.

Regarding the economic damage. Again, if we are indeed in the subcritical domain then it may well be that the expected gain from forcing family members to be home is negative. But in any case it is difficult to say what “damage to the economy” means and also it was not clear who will carry this damage if confinement to family member will be imposed. A little point regarding this issue is that reducing expected damage (to the extent it can be defined) is not necessarily an optimal policy and we should examine “damage for whom”. For example, if you are offered to save the overall economy from a damage of 2 billions pounds by spending 1 billion pounds from the government’s budget this may be a very bad deal.

I think my two meta points are: First being very very cautious to account for uncertainty in estimating huge but unknown risks cannot be a good policy on individual or larger scale; one has to be reasonably cautious (and keep gathering and evaluating information to reduce uncertainty). Second, a main reason not to be overly cautious as to instruct people to stay home is the necessity that they will stay home when it will be indeed necessary to so instruct them.

June 7, 2009 at 3:53 am |

Best wishes to your family.

I think it is also worth observing that between telecommunication and the increasing number of ‘knowledge workers’ increasingly large percentages of the populations can productively work from home if not every day, almost any day.

While you (presumably) can’t lecture from home (yet), there are any number of productive activities that you could have (did?:) engage in (research, reading papers, marking, electronically communicating with students, watching day time television:), … ).

Regardless of the disease spreading statistics, having people who can productively work from home doing so has got to reduce the likely spread of the disease with minimal productivity loss (heck I’m more productive at home). This clearly should be advised.

June 7, 2009 at 5:26 am |

I heard a panel discussion on the radio (probably from BBC World or NPR), and the experts on epidemics clearly made the point that simulations have shown that quarantining can only slow the spread of the disease, and not change the ultimate number of cases. This seemed really odd to me, because if we put every infected person on a remote island, then it would not spread to anyone else. They didn’t elaborate, but the only explanation I could think of is that we’re so far from being about to set up perfect quarantines that any measures we take will have a small impact. The panel did point out, though, that slowing the spread could be import if it gives you enough time to develop vaccines and so on.

June 8, 2009 at 12:07 pm

The only explanation I can think of for what the experts say is similar to yours: that quarantining reduces the probability but not to below the critical probability. In that case one would indeed expect that the spread of the disease would merely be slowed. Perhaps that is indeed the case: that however strict you are with people who have come into contact with known cases, there will always be enough people who have the virus without realizing that there’s any chance that they’ve got it for the probability to remain above the critical probability, and it just won’t be feasible to chase them all up. If that was the case, then even perfect quarantining wouldn’t stop the spread.

By the way, Gil I’ve no objection to your arguing the opposite case to mine: as ordeals go, this one was thankfully very minor indeed.

June 10, 2009 at 9:28 am |

“so I would estimate that there was a non-negligible probability that I had it. And if I had, say, had lunch in my college, then I would have had opportunities to spread it to several other people. That’s not quite a proof that the expected number of people I would infect was more than one, but neither did it leave me confident that the expected number was at most one.

This is indeed an interesting case to demonstrate the advantage and shortcommings of mathematical thinking. The intuition based on Branching processes is not great but probably reasonable as a first approximation. (In reality, statistical dependencies may push the spread of the desease down compared to a model based on independence.)

But the main issue is the question:

“What is the expected number of people infected by a family member of an ill person who takes preventive medications but lives his life as ususal.

Tim’s hunch (which does not seem to be based on any mathematics) is that this number is in the neighborhood of 1.

My hunch is that this number is lower than 1/100.

Of course, if I am correct then the probability that changing policy will have any effect regarding reaching the threshold is simply zero.

Now, there may be a way to check matters by looking at the statistics.

How many relatives of identified people with the illness that were instructed to take preventive measures are there, and among the five hundred or so people infected in the UK how many were infected by a relative of an ill person who took preventive measures. I would expect that now or over time in the UK or in other countries the 1/100 would be a good upper bound.

(Another interesting question related to the ratio between ill people and those who are identified is if we can estimate what is the real number of people who had the illness in Mexico. It is quite possible that the number in the early stages of the desease is much higher than the number of identified patients.)

June 10, 2009 at 9:53 am |

With so many snouts in the trough at the moment, best to take a cautious approach

June 10, 2009 at 10:15 am |

Gil, it’s true that my hunch is based on no mathematics, but it’s still my hunch. Last night we were rung up by a public health official, who sounded surprised that none of the rest of my family had caught swine flu off the child who did have it. And it spread round his school at a rate that simply doesn’t seem to be consistent with a number such as 1/100. Actually, that second point is not so relevant because they weren’t taking Tamiflu. But the public health official presumably had some intuition about the probabilities, and given that there were five people who might have become infected, and that she was surprised that none of them were (even given that we were taking Tamiflu), the probability for each of us would surely have to be quite a bit bigger than 1/100. And the rapidity of the spread in schools and other institutions where there have been outbreaks suggests to me that the expected number of people you infect if you are infected and live a normal life in confined spaces with several others is larger than one over this probability.

An indirect piece of evidence is what James said — that the official line is that it is not possible to contain the disease. This suggests that the probability is in fact significantly higher than the critical probability.

Having said all that, I admit that I could be wrong. But my secondary point is that the decision about how cautious to be is being made on the basis of an unknown parameter, so we’re in the realms of statistics rather than probability. Even if there is a smallish but non-zero chance that the parameter, then the expected payoff of caution is very large. But it could be that if we knew the actual value of the parameter, then the expected payoff would be small.

It would be very interesting to know a bit more about the actual statistics and what use is being made of them: I’d love to be able to discuss this in a more informed way.

June 10, 2009 at 11:03 am |

The question is not what is the probability that a person who is ill will infect others (this is relevant to the spread in your son’s school) but what is the probability that a person who have a family member ill and take preventive medications will catch the ilness and infect others (before he is symptomatic).

These numbers are essentially known. It is possible to tell how many family members were treated, how many treated family members did develop the desease, What is their percentage, and also to estimate how many people were infected by these people. These are not unkown numbers.

June 11, 2009 at 12:59 am |

Not to bring down the level of discussion, but entering “swine flu in england” at Wolfram|Alpha (nice segway to the other post!), and then clicking for “Show Plots” gives a picture that looks like increasing rates of infection. Same for France, but only a constant rate in US. I’ve no idea, of course, of the quality of the data lurking beneath those pics.

June 12, 2009 at 1:05 am |

“aggressive isolation of infectious individuals, quarantine of exposed latent individuals, in combination with other community intervention aimed to reduction transmission per contact, can be effective at reducing the overall impact of an outbreak by simultaneously fulfilling objectives 1-3 in Figure 10. More importantly, these common sense measures have been proven mathematically at qualitative level in this chapter, without relying on specific mathematical models.”

(Ping Yan, NON-IDENTIFIABLES AND INVARIANT QUANTITIES IN INFECTIOUS DISEASE MODELS, in MATHEMATICAL UNDERSTANDING OF

INFECTIOUS DISEASE DYNAMICS, S. Ma and Y. Xia (eds.), World Scientific 2009)

June 12, 2009 at 3:39 pm |

[…] ter tido uma ocorrência na família, Timothy Gowers debateu a questão, há uma semana, em Swine flu and British public health policy , considerando, entre outros […]

June 12, 2009 at 5:32 pm |

In the U.S., I think they’re not testing as many sick people for the swine flu as they were when it first started circulating, so this may be the difference (or alternatively they may have a handle on it … the authorities are closing quite a number of schools where a substantial number of students have the flu, although our elementary school has remained open despite one student with the flu).

June 13, 2009 at 6:20 pm |

I’ve had a further thought about the argument that if the expected number of people you infect is supercritical, then reducing it to another supercritical probability merely slows the spread of the disease.

The following crude model strongly suggests that this idea is wrong. Let’s suppose that a proportion of the population have had swine flu and have therefore developed an immunity to it, and let’s suppose that was the expected number of people you would infect in the early stages of the spread (when the proportion of people who are immune is so small that it can be ignored). Then the expected number now ought to be something like , since of the people you infect on average if nobody is immune, on average are in fact immune.

What this suggests is that the disease will spread exponentially until drops below , which means that the proportion of the population who have to get swine flu before it starts to die out is . So if you can change people’s behaviour so that is reduced, then the proportion of people who end up getting it is also reduced. And that could make a difference to millions of people.

Of course, if lots of people get swine flu, then the total inconvenience of any measures taken to reduce will also be very high, so there is a balance to be struck. But the point I am making is that reducing does not just slow the spread: it reduces the number of people who actually get the disease. (One way of looking at it is this: if you reduce to , then the number of people who escape swine flu increases by a factor of , until goes below , at which point it starts to die out without spreading all that much.)

I repeat an earlier point in a new context: the above model is very crude, but it seems to me adequate for a qualitative understanding of what is going on. (An example of why it is crude is that the population is clustered in important ways, so that the kind of progression I’ve just described actually happens at a more local level in places like schools.)

June 15, 2009 at 5:07 pm |

Indeed this look a good example to try to see the advantages and disadvantages of mathematical instincts, modeling and insights (by mathematicians who are non-experts in this area).

Let’s go back to the branching process model which, while very crude, served us well in this discussion. If the number of people infected by an ill person is supercritical than the epidemic spread exponentially, if it is subcritical then it dies out. We took the value 1 as a rough approximation of the crirical value.

What seems to be missing from this model (including in the previous comment) is that the expected number of infected persons per ill person seems to depends on many factors – some controlled (like the policy issue that we discuss) and some uncontrolled (e.g., the weather(?)).

Therefore, even if we fix our policy, the expected number of infected people per ill person flactuates quite a bit. It can be at times supercritical and at other times subcritical. I think most mathematical insights in the discussion were based on implicitely assuming the expected number of infected people being fixed, apart from the policy we want to change, while the reality may well be that the expected infected people flactuates a lot.

It looks to me that if a costly change of policy that changes the expected number of infected person per ill person in an amount well below the natural flactuation of this number, then the overall effect can be small.

Beside the mathematical model we identified two important matters:

1) We have very different (non mathematical) hunches on matters that can be studied statistically (much better than by deep pure thought). This is the case on what is the number of infected people per ill person by treated family members that continues to go to work.

My hunch is that this number is below 1/100 per family member or 1/20 per ill person over all his treated family members.

2) The alternative policy regarding treated family members was very vague: Will they be asked to stay home? required to do so? who will pay for it? Will this apply to family members of confirmed ill people or also of suspected ones, etc. etc.

It is very questionable what the overall effect in terms of infected persons per ill person of such a policy change will be and even if the effect will be positive or negative.

June 16, 2009 at 6:40 pm |

A mathematical way to describe the last sentence in the last remark is this. Let us divide the expected number E of infected people from an ill person to where is the expected number of people infected before the patient was identified as a swine flu case and all measured and treatement started, and after. We really want to minimize . It is reasonable to suspect that an unidentified ill person is more contangous before he is identified so one effort should be to try to devote many resources to shortened the expected time an ill person is unidentified and this include giving much incentives for people to be tested as early as possible.

The suggested policy change will decrease for treated family members who got nevertheless infected.

But concerning incentives for early testing for the entire infected population, the present policy seems to be very good. Identified patients are getting good treatement, their family gets preventive treatements and they are monitored but not isolated.

A change of policy which will make it harder for patients and their families may have a much stronger effect on on eancreasing for the entire infected population.

I am not talking about swine flu patients going underground, but about people who are making the decision: “is it a regular flu or should I go be tested?” delay their decision to get checked even by a few hours.

June 16, 2009 at 6:58 pm |

Hi Gil,

I don’t see any easy way of obtaining the data for the number of people infected by treated family members. Sure, you can count the number of sick people at workplaces where treated family members go, but since it is quite possible that there will be people at these workplaces infected by somebody else, and you expect a large deviation on these numbers anyway.

You’d have to collect a massive amount of data to get significant results on this, and I’m not sure anybody has.

June 18, 2009 at 6:50 am |

Dear Peter, I think it would be possible to estimate this number (lets call it z) in various ways, especialy in an early stage where the total number of infected people is not large and the authorities try to find out the source of illness in every new case.

One thing you can certainly do is to see what is the number y of treated family members who developed the illness.

If you have an estimate to the overall number x of infected people per ill person than xy is an upper bound on z (although we can expect that z will be substantially smaller because treated family members are monitored). You can get a better upper bound by estimating the ratio between the time an average ill person has the opportunity to infect others and the time a treated family member who develop the illness has.

So, for example, it will be useful to know say in Tim’s son school: how many children were infected and how many treated family members were infected.

June 18, 2009 at 7:49 am

As a matter of interest, I think they are going to considerable efforts to find out the answer to that last question.

July 10, 2009 at 9:29 am

It looks from the numbers and wikipedia article that the flu is actually “supercritical” in the UK (and probably many other countries) “It is estimated that on average each person who contracts flu passes it on to between 1.4 and 1.6 other people”. One aspect which seems unclear and important is if the number of people who contracted the flu abroad (mainly in north america) is still substantial and is a major factor for the difference between the spread of the flu in different countries.

http://en.wikipedia.org/wiki/2009_flu_pandemic_in_the_United_Kingdom

July 12, 2009 at 10:33 am |

Even if the total number of infected people is unchanged by following a certain protocol, slowing down the rate of infection is very important.

It buys time until a vaccine has been tested.

And the health services cannot cope with everybody being ill at once, but can cope better if people’s illness are queued up.

December 19, 2009 at 7:05 pm |

One thing that I find amazing (and rather irational), is that now that the swine flu vaccination is offered few people (both in teh UK and in Israel, for example) are taking it.

December 19, 2009 at 7:37 pm |

It’s especially amazing if what is stopping them taking it is anxiety about the risks of the vaccination. For some reason those seem to outweigh the anxiety about dying of swine flu, which has a much higher (though still small) probability.

December 18, 2011 at 5:07 pm |

[…] readers of this blog may remember a post about swine flu from a couple of years ago. When I wrote my first reply, I was more or less confined to the house […]