Daniel Bernoulli's epidemiological model revisited

Abstract

The seminal paper by Daniel Bernoulli published in 1766 is put into a new perspective. After a short account of smallpox inoculation and of Bernoulli’s life, the motivation for that paper and its impact are described. It determines the age-specific equilibrium prevalence of immune individuals in an endemic potentially lethal infectious disease. The gain in life expectancy after elimination of this cause of death can be explicitly expressed in terms of the case fatality and the endemic prevalence of susceptibles. D’Alembert developed in 1761 an alternative method for dealing with competing risks of death, which is also applicable to non-infectious diseases. Bernoulli’s formula for the endemic prevalence of susceptibles has so far escaped attention. It involves the lifetime risk of the infection, the force of infection and the life expectancy at birth. A new formula for the basic reproduction number is derived which involves the average force of infection, the average case fatality and the life expectancy at the time of infection. One can use this estimate to assess the gain in life expectancy if only a fraction of the population is immunized.