Bernoulli, Jurin and the math behind smallpox

By ALLISON CHEN | December 6, 2018

In December of 1694, Mary II of England fell ill. Physicians who attended the stricken queen, half of the pair known as William and Mary, argued over a diagnosis, but before long it became clear that she had contracted a severe form of smallpox. She died three days after Christmas, and the news was carried across London by tolling bells.

By the winter of Queen Mary’s death, smallpox was a disease feared across Europe — and for good reason. Infecting indiscriminately, smallpox had gruesome symptoms, including characteristic fluid-filled sores, and killed about 30 percent of those it infected. The disease had been increasing in prevalence and virulence throughout the 1600s, and by the end of the century smallpox was a leading cause of death. 

People naturally sought ways to protect against the disease. It had been observed that those who had had smallpox would never contract it again, and so one preventative method involved the deliberate infection of an individual, called inoculation. 

Inoculation predates vaccination, and the principle is similar, though with an important distinction. While modern vaccines rely on agents, such as killed microbes or surface proteins that resemble the pathogen, to stimulate an immune response, inoculation used live smallpox. Materials such as scabs were taken from sick individuals and directly used to infect healthy ones, such as by inserting it into a wound.

Inoculation tended to result in milder infections. The procedure could be performed using material from a patient who had a mild case of smallpox and was generally done to those in good health. It involved introducing the virus through a small wound rather than through smallpox’s natural respiratory route of transmission. Nevertheless, the inoculated patient was still given smallpox, and there remained a real risk of developing a serious case of the disease. 

As a result, when the procedure was introduced into England from the East, primarily Constantinople, there was skepticism among the general populace and physicians alike. Major concerns included fears that smallpox contracted through inoculation would not protect against a natural infection and worries regarding how safe deliberately infecting people really was. 

Proponents of inoculation, recognizing that they needed some empirical way to shed light on the procedure’s uncertainties and demonstrate its benefits, turned to mathematics. 

One study was conducted by English scientist James Jurin and was presented to the Royal Society in 1723. Jurin sought to determine two points — whether inoculation successfully protected individuals and whether it was less dangerous than contracting smallpox naturally — and to this end he included data from the United States, looked through the London bills of mortality and compiled information from other practicing physicians. 

Jurin calculated that while the chance of dying from natural smallpox was between two in 17 and two in 11, the risk of dying from smallpox received through inoculation was one in 50. 

Jurin’s paper, eventually turned into a pamphlet, was so convincing that he published similar reports in four subsequent years. Each year it was shown that inoculation had a fatality rate eight to 10 times less than contracting smallpox naturally. 

In continental Western Europe, where early skepticism of inoculation was more intense than in England, physicist and mathematician Daniel Bernoulli similarly sought to examine inoculation from a mathematical point of view. 

Bernoulli, better known for his study of fluid dynamics and the resulting principle that bears his name, approached his analysis of inoculation from a different angle — seeking to determine how much life expectancy would increase if smallpox were no longer a cause of death. His work was first presented in 1760.

Bernoulli made use of differential equations, and in his model he considered populations to be divided into two groups: those susceptible to smallpox and those who had been previously infected and were thus immune. Comparing a hypothetical population in which smallpox was eradicated to real-world data, he determined that in a population without the disease, life expectancy would be increased by about three years. 

He further calculated that as long as deaths due to inoculation remained less than approximately 1 in 10, the procedure would be a net benefit.

In the end, it is difficult to say just how much Jurin and Bernoulli’s analyses contributed to popularizing inoculation, especially the latter’s esoteric model. Furthermore, the issue of establishing the procedure’s worth lost relevance within a century due to the development of the first vaccine. When, in 1796, Edward Jenner used cowpox to protect a young boy against smallpox (a procedure that conferred the same benefits as inoculation without the risks) he changed medical history by introducing the idea of a vaccine.

This does not, however, trivialize the mathematicians’ work. Bernoulli’s model, in particular, is likely the first compartmental epidemiological model and is considered to be one of the first attempts to use a mathematical model to influence public health policy. His contributions continue to have influence in the present day.

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