Smallpox

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Smallpox

• circa 1718 method of variolation brought to
  England by Lady Mary Wortley Montagu, wife of the
  Turkish ambassador, Edward Wortley Montagu
• 1721 epidemic in London
• Royal family tested variolation on condemned
  prisoners and when found successful they
  inoculated themselves
• variolation slow to be accepted
• 1796 Edward Jenner introduced a smallpox vaccine
  using cowpox fluid

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Daniel Bernoullis Smallpox Analysis

• Assumptions
• p, the risk of contracting smallpox is the same
  at every age (p 1/8)
• p, the risk of dying from smallpox is the same at
  every age (p 1/8)

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Notation
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Bernoullis Differential Equation
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The Competing Risk Model
Assuming a 1/8 probability of contracting
smallpox among those at risk and a 1/8
probability of death among those contracting
smallpox Bernoulli obtained
The solution to the differential equation yields
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Halleys Life Table, 1693
Bernoullis Life Table Analysis with Smallpox
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Gain Due to Inoculation

• Bernoulli noted that the asymptotic gain was
  approximately 7 to 8
• argued that if Paris produced 7000 20 year olds
  per yield with smallpox present then it would
  produce 8000 20 year olds under an inoculation
  system
• likewise if the civil births for the whole of
  France was 175000 then it would be 200000 without
  smallpox so that France would gain 25000 persons
  every year, all useful to the State and to
  society

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Laplaces Summary of DAlemberts Objections to
Bernoulli