infectious diseases are much on everybody’s mind at the moment, as frantic efforts are going into stopping the spread of the coronavirus and developing a vaccine. Medical research is obviously important in this, but so is mathematics. It is used extensively in modelling infectious diseases: finding out how rapidly they can be expected to spread, how many people will be affected, and also what proportion of a population should be vaccinated, if a vaccine exists.
A basic mathematical model, developed back in the 1920s but still used today, is called the SIR model. To understand how it works, imagine you are playing a computer game. In it there’s a population of people (of a city, country, continent or the world) divided into three classes: those that are not yet sick, but susceptible to the disease (class S), those that are sick and infectious (class I), and those that have been removed from the disease (class R), either because they have recovered and become immune, or because they have died. There also is a set of equations which describes how many people pass from one class to another in a given time step, say in a day, or in a month. You now click “go” and watch the computer simulate the disease developing over time. This, in essence, is how scientists use the SIR model.
Of course, everything depends on the equations that govern the transition from one class into the other. In the basic SIR model, these equations depend crucially on the likelihood that an infected person infects someone else and on the average length of time someone is sick for before they recover or die. When scientists use the SIR model to predict the evolution of a disease, they estimate these parameters by observing how the disease behaves in real life. By figuring out the impact interventions, such as travel restrictions or improvements in hygiene, have on the important parameters, they can also predict how useful those interventions are likely to be.
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