The 300th anniversary year of Swiss mathematician Daniel Bernoulli's birth is an appropriate time to reveal that he was, by a long way, the first to express the proportion of susceptible individuals of an endemic infection in terms of the force of infection and life expectancy1.
His formula is valid for arbitrary age-dependent host mortality, in contrast to some current formulas which underestimate herd immunity (one minus the proportion of susceptible individuals), the vaccination threshold that has to be exceeded to eliminate an infection and the basic reproduction number (the inverse of the proportion of susceptible individuals)2.
Bernoulli's main objective was to calculate the adjusted life table if smallpox were to be eliminated as a cause of death. He clearly defined the two epidemiological parameters, which nowadays are called the force of infection λ (the annual rate of acquiring an infection) and the case fatality c (the proportion of infections resulting in death). If L is life expectancy, then the proportion of susceptible individuals u can be written as u = r/(λL), where r is the cumulative incidence (the proportion of a cohort that will be infected). As the proportion c of all infected individuals die, the proportion q of all deaths due to smallpox is q = cr.
Hence Bernoulli estimated the susceptible proportion using the expression u = q/(cλL) . From several large cities (especially London) which recorded cause-specific numbers of deaths, estimates of q were known to be about 1/13 = 7.7%. Bernoulli used Halley's life table for the city of Breslau and came up with the estimates λ = 1/8 per year and c = 1/8 = 12.5%. For Paris he assumed a life expectancy of 32 years which yields a proportion of susceptible individuals of 15% ( = 107,000/700,000), or a herd immunity of 85%.
If one assumes a constant death rate (an exponentially distributed survival time), then u = 1/(1 + λL) (ref. 3). This formula gives 80% as the estimate for herd immunity. The formula most frequently used today2 assumes a rectangular survival function in which all individuals live until a maximum age L. Then u ≈ 1/(λL) , which gives only 75% for herd immunity. We therefore suggest that it is more accurate to use the more general formula derived by Bernoulli.
Jean le Rond d'Alembert, who had been in conflict with Bernoulli over several issues previously, violated the unwritten rules of the day by publishing a critique of Bernoulli's approach in 1761 (ref. 4), five years before Bernoulli's essay eventually appeared in print. Bernoulli's motivation to add this formula in the final version (prepared in 1765) was explicitly to show the superiority of his 'exact' approach over the crude estimate of d'Alembert, according to whom the proportion of susceptible individuals was “much less than half”.
The reason why Bernoulli's important formula has escaped notice for so long may be its cryptic presentation: one has to recover it by substituting his numerical values with their general symbols. In formulating his laudable objectives, however, he is admirably clear: “I simply wish that, in a matter which so closely concerns the well-being of mankind, no decision shall be made without all the knowledge which a little analysis and calculation can provide.”
Bernoulli, D. Mém. Math. Phys. Acad. R. Sci. Paris 1– 45 (1766); English translation by Bradley, L. in Smallpox Inoculation: An Eighteenth Century Mathematical Controversy (Adult Education Department, Nottingham, 1971).
Anderson, R. M. & May, R. M. Infectious Diseases of Humans — Dynamics and Control (Oxford University Press, Oxford, 1991).
Dietz, K. Stat. Meth. Med. Res. 2, 23–41 (1993).
D'Alembert, J. Opuscules Mathématiques, t. II (Paris, David, 1761).
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