From the record
Real cases, written from the published record. Predict first; the record and its sources come after.
1760 · Paris
Bernoulli's ledger
Smallpox killed many children. Inoculation, giving a mild case on purpose, protected for life but killed about one person in a few hundred within weeks.
Daniel Bernoulli built the first mathematical model of an epidemic to answer one question: on balance, does inoculating everyone save lives? His answer: yes, it would add about three years to the average life.
Jean d'Alembert objected. A parent might still say no. What would you advise?
What happened. Bernoulli argued from averages over a lifetime. D'Alembert replied that a small risk of dying next month is not the same to a person as a gain spread over decades. Both were right about something. A much safer method arrived in 1796 with Jenner's cowpox vaccine.
What we don’t know. Inoculation in 1760 (variolation) gave a real smallpox infection. Modern vaccines are made very differently and their serious side effects are far rarer; this case is about how to weigh a risk, in 1760.
The first epidemic model was written to win an argument, and the argument it started, about how to weigh risk, is still going.