# Build your own disease

08/12/2009

### Some guidance on Build your own disease

In simulation 1 the infection will keep on spreading, as every infected person goes on to infect at least one other person. In simulation 2 the infection dies out as soon as you get heads. While it's theoretically possible that you only get tails, so that the infection keeps on spreading, this is unlikely if you trace the infection over many generations. So in general the disease is likely to die out of its own accord.

In simulation 1 an infected person infects 3/2 others on average, and it simulation 2 they infect 1/2 others on average. These numbers are the respective basic reproduction ratios.

Suppose there are n infected people to start with, then on average there are nR newly infected people in the next generation, nR2 in the third generation, nR3 in the fourth generation, and so on, up to nRk newly infected people in the k+1st generation. If R is greater than one, then Rk gets larger and larger as k gets larger, so the disease escalates. If on the other hand R is less than one, then Rk gets smaller and smaller as k gets larger. In this case the disease peters out. So any intervention programme should aim to reduce the basic reproduction ratio to an effective reproduction ratio which is less than 1.

Now suppose that R=3/2, as in simulation 1, and that you have vaccinated over 1-2/3=1/3 of the population. So over 1/3 of the 3/2 people an infected person goes on to infect on average are now immune. Since a third of 3/2 is 1/2, this leaves an effective reproduction number of just under 3/2-1/2=1.

The simulations assume that every infected person falls into one of two classes in terms of how many people they go on to infect. But this is unrealistic, as people's contacts patterns vary widely: a teacher may come into contact with hundreds of people a day, while an office worker may come into contact with only around ten, and someone working from home only with their immediate family. A more realistic model should take this into account, for example by dividing the population into different classes reflecting their contact patterns.

The simulations also assume that a person is either infectious or healthy. In reality, people can be infected but not infectious, they can recover from the disease (or, god forbid, die), or they can be immune. Standard epidemiological models divide the population up into classes (for example susceptible, infectious and recovered) and then use mathematical rules to describe the way in which people pass from one class to the next. See the rest of this package on epidemiology for more information.

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