As I lay in bed in a state of high fever I couldn’t help googling “swine flu” for any other telltale symptoms that might add to my state of hypochondria. With the increase of cases in Australia, the World Health Organisation declared last week that swine flu, or the H1N1 virus as it’s also called, was officially a pandemic.

The biologists are analysing the composition of the virus and trying to create a vaccine to combat the disease, but it is mathematics that now allows the scientists to understand how virulent it is.

Each virus is assigned a number, which is a measure of how quickly it spreads. The number for Aids is between 2 and 5. Measles is much more contagious with a number between 16 and 18. Estimates for the number corresponding to swine flu is somewhere between 1.2 and 1.6. This is much smaller than normal influenza, which is between 1.5 and 3.

Even with a small infection rate, it is remarkable how quickly the disease can spread. If a contagious person infects 1.2 people each day then it takes only 125 days to infect the whole population of the Earth. Such is the power of exponential growth. Of course we don’t expect everyone in the world to contract swine flu, and that is why you need a more subtle mathematical model that takes into account other factors in the spread of such a disease.

For a start, once people have the disease they will either die or recover and become immune. So they will remain contagious only for a fixed period rather than continually infecting people. As the disease spreads, more of the population will become immune. To run the model it is important to know the proportion of the population who are likely to be susceptible to the disease. This means that the number corresponding to a spreading virus will change over time. Once it drops below 1, the mathematics implies that the disease will die out. If it stays above 1 then the model shows that eventually the number of infected people stabilises to a fixed amount. In this case we say the disease has become endemic. For example, chicken pox in the UK is an endemic virus.

It is by using such mathematical models that you can assess the effect of immunising the population against the spread of a virus. A vaccine decreases the proportion of the population susceptible to the disease. Immunise enough people and you can see the infection rate drop below 1 leading to the eradication of the disease. Smallpox had an infection rate of 4 but by immunising at least three quarters of the population the disease has died out.

The reason that measles has started to spread again in the UK is because the mathematical balance has been upset between the proportion of the population immunised and the high infection rate for measles at between 16 to 18. On the other hand, the model reveals that only 5 per cent of the population need be treated with antiviral drugs to counter the pandemic effect of a flu virus with contagion rate of 1.9.

The model is also helpful in assessing the effect of introducing travel restrictions to try to contain the spread of diseases. Research last year on the possible spread of avian flu revealed that drastic travel limitations would have the effect of delaying the pandemic’s evolution only by a few weeks with almost no effect on the mortality rate. Compared with the huge economic disruption that such travel restrictions would inflict, the mathematics has led to the conclusion that travel restrictions are of little use in combating viruses such as swine flu.

So it was reading about the mathematics of the spread of the virus that proved the best antidote to my bout of hypochondria last weekend. And sure enough, by the beginning of the week, the fever had dropped and I was feeling fine.

A spoonful of maths is all it took to see me through the worst of it.

Conundrum

An abbot visits a monastery and says that at least one of the monks there has a deadly but non-contagious disease. The only symptom is a red dot that will appear the following day on the forehead of those infected. As they will die an agonising death within months, he urges those infected to kill themselves as soon as they know. But they have sworn an oath not to communicate with each other and there are no mirrors and they meet only once a day, at lunchtime. One month later, the abbot returns to find that all the infected monks killed themselves on the same day. How did they know they were infected and why did they die on the same day?

Answer

Suppose there was only one monk who had the disease. On the day after the abbot leaves the monk sees no one else with a red dot on his forehead. Since at least one person is infected he knows it is him and he kills himself. Suppose two monks are infected. The infected monks see only one other monk with a red dot. But since they are still alive on the second day, they realise that there must be another monk infected, or else the infected monk they can see would already have killed himself. Since the infected monk can see only one other monk infected he deduces that he must also be infected. So on the second day they both kill themselves. The logic extends from 2 to 3 to N monks. It takes N days to realise that N monks are infected.

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