We are reminded each year to get vaccinated against the influenza virus. The severity of the annual outbreak is not known with certainty in advance, but a major pandemic is bound to occur sooner or later. Mathematical models play an indispensable role in understanding and managing infectious diseases. Models vary in sophistication from the simple SIR model with just three variables to highly complex simulation models with millions of variables [TM134 or search for “thatsmaths” at irishtimes.com].

The basic reproduction number, R_{0} or R-nought, is a key parameter determining the rate of spread of an epidemic. This number measures how many people are infected by someone who has already been infected. Obviously, a single quantity like R_{0} cannot reflect all the complexities of an epidemic, but it is a crucial factor governing the rapidity with which a disease spreads through a population.

Public health authorities planning vaccination programmes need to estimate how many people are at risk of infection, the gravity of this risk and the percentage uptake of vaccination required to prevent a major epidemic. To limit the spread of an infective virus, they aim to reduce the basic reproduction number R_{0} to the smallest possible value. Widespread vaccination can prevent epidemics by replacing the exponential rate of spread by a linear growth rate.

**Linear and Exponential Growth**

Growth is linear when there is an equal increase for equal time intervals. A tree that grows by 50cm each year is undergoing linear growth. In contrast, the rate of increase of a population is proportional to its size, and the number doubles within a fixed period; this is exponential growth. The difference between linear and exponential growth is dramatic. Let us consider how news spreads. Suppose that each person who hears a rumour tells just one other person. If one person knows the rumour today, then two know tomorrow, three the next day and so on. Each day another person hears the gossip and after a month about thirty people are in the know.

Now suppose each person who hears the rumour tells two other people. Two new people hear it tomorrow, four the next day, eight the next, and so on. The number of new people doubles each day so that, after a month, about one billion people have heard it. Within three more days, the entire population of the world is in the picture.

**Reducing R**_{0}** to unity**

Viral infections spread exponentially when the basic reproduction rate is greater than 1. For influenza, R_{0} is typically 2 or 3, for mumps it is about 5 and for measles between 12 and 18. How can we lower R_{0} to 1? Suppose that initially it is 3. If an average infected person has contact with 30 others while infective, three will become infected, each person having a 10% risk. Now suppose that 20 of the 30 are vaccinated. Then 10% of the remaining 10, or a single person, will become infected. With a two-in-three uptake of the vaccination programme, the basic reproduction rate has been reduced to 1, and the spread of the disease is no longer exponential but linear.

It is clear that vaccination provides protection for the individual recipient, but it also has community benefits. The risk of infection is reduced for everyone, including those who cannot be vaccinated. When a critical uptake is reached – the heard immunity threshold – the rate of spread is greatly reduced, potentially averting catastrophe.