Daniel Bernoulli was born in 1700 into a remarkably talented family. He excelled in mathematics, but also studied and lectured in medicine, botany and physics. In 1760, he submitted a paper to the Academy of Sciences in Paris dealing with the effects of inoculation on morbidity.

The practice of inoculation, the deliberate introduction of a small amount of a less virulent form of smallpox to provide protection against a more virulent strain, had been known in India and China for a long time. It was introduced in England in 1718, but it was very controversial, as *it proved fatal in about one percent of cases*.

Bernoulli studied inoculation from a mathematical perspective. His goal was to demonstrate that the long-term benefits outweighed the immediate risks. He considered the total population of a city at a particular time, and examined how that group survived as time went on. He applied the techniques of calculus, which had been devised by Newton and Leibniz towards the end of the seventeenth century. He made a number of simplifying assumptions that enabled him to formulate the problem using two coupled ordinary differential equations (odes).

One equation described how the population changed with time; the other gave the number of people who were susceptible to smallpox. Bernoulli combined the two equations to get a single nonlinear equation. Normally, such equations are difficult, if not impossible, to solve but Bernoulli was in luck: this one was of a special form previously studied by Jakob Bernoulli, Daniel’s uncle, and the method of solution was known.

To test his theory, Bernoulli used a table of statistics compiled by Edmond Halley, a contemporary of Newton, who is remembered for the comet named in his honour. He showed that, under his simplifying assumptions, *if the entire population were inoculated at birth, life expectancy would increase by more than three years*. He concluded his memoir by urging that, in a matter of such vital importance, no decision should be made “without all the knowledge that a little analysis and calculation can provide.”

Despite Bernoulli’s analysis, inoculation was not widely practised in France. But all changed utterly after 1796, when Edward Jenner showed that vaccination – inoculation with cowpox – protected against smallpox *with negligible risk*. Vaccination came rapidly into widespread use throughout Europe. In 1979 the WHO declared that smallpox had been eradicated, following extended vaccination campaigns over two centuries.

The mathematical techniques introduced by Daniel Bernoulli are still in use today, in developed and refined forms, for calculating the influence of vaccination programmes and for more general epidemiological studies.