In 1760, Daniel Bernoulli turned his attention to the inoculation of smallpox, also known as the pox. Smallpox was an extremely serious disease, accounting at the time for around one-thirteenth of all deaths. A preventive technique originating in the East and introduced into Europe only in the 18th century involved inoculating healthy people with smallpox germs (through pus, for example) taken from people with mild cases of the disease. This gave them lifelong immunity to smallpox. Unfortunately, in rare cases, the procedure went wrong and the inoculated person died. The question, then, was whether the benefits of inoculation outweighed the risks. Voltaire and La Condamine supported inoculation.
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Bernoulli's model
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Daniel Bernoulli proposed the following mathematical model. In a stationary population, let x denote an individual's age, treated as a continuous variable, and P (x) the density of individuals aged x; in other words, the integral of P (x) over x > 0 gives the total population. This population is divided into two groups: S (x) is the density of individuals aged x who have never been infected with smallpox and are therefore susceptible to it, while R (x) is the density of those who have contracted the disease, survived and are therefore immune. Thus, P (x) = S (x) + R (x). The disease lasts for so short a time that the number of people currently ill can be neglected in comparison with these other two groups. Let m (x) denote mortality at age x, excluding smallpox. Daniel assumed that between ages x and x + dx (with dx infinitesimally small), every individual who was not yet immune had a probability q dx of contracting smallpox, where the rate q was independent of age. He thus obtained the differential equation:
(\) dS / dx = – m (x) S – q* S.
