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\noindent{\bf Daniel BERNOULLI}\\
b. 8 February 1700 - d. 17 March 1782
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\noindent{\bf Summary.} Daniel Bernoulli, well known as a mathematician,
provided the earliest
mathematical model describing an infectious disease. In 1760 he modelled the
spread of smallpox, which was prevalent at the time, and argued the advantages of
variolation.
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One of the purposes of modelling the spread of an infectious disease
mathematically
is to provide a framework within which predictions can be made about its
progress
within a population susceptible to infection. Perhaps the first such model
for an
infectious disease was that for smallpox due to Daniel Bernoulli.
Daniel, who was born in Groningen in 1700, was a member of the famous
Bernoulli
mathematical family, the second son of Johann I (Jean) Bernoulli, brother of
the
older and more famous Jakob I (Jacques). He studied mathematics under his
father
and medicine at Heidelberg receiving his M.D. in 1721. In 1724 he published
``Exercitationes Mathematicae", solving Riccati's equation, and in 1725, he was
appointed to the Academy of Sciences of St. Petersburg, which had just been
founded
by Peter the Great. He remained there until 1732.
After Peter's death in 1725, there followed a period of unrest until the
imperial
succession was settled. Bernoulli returned to Basel in 1733, to accept a
post in
Anatomy and Botany; this was followed in 1743 by one in Physiology and in 1750
by
another in Physics. His
interests remained very diverse, and he pursued his research into mathematics
(including probability), medicine, physics, botany, anatomy and philosophy,
writing
several treatises on these subjects. In 1734 he published his ``Memoire sur
l'Inclinaison des Orbites Planetaires", followed in 1738 by his ``Traite
d'Hydrodynamique", and a paper on the measurement of risk (1738). In 1740 his
``Traite sur les Marees" appeared, and shortly after, his work on vibrating
strings
and rods. He was involved in quarrels with his friend Euler on the
mathematical
theory of the vibrating string, and with D'Alembert (q.v.)
on his work on risk and the
merits of variolation.
Basel was very close to Alsace and the French frontier, and while the Swiss
cantons maintained their independence, intellectual and particularly
scientific life
was centered on France and its Royal French Academy of Sciences in Paris.
Between
1725 and 1749, Bernoulli won 10 Prizes from this Academy, including the Prize
of
Honour for his clepsydra (water clock) to measure time at sea.
For most of Bernoulli's mature life, the intellectual climate was dominated
by
the ``philosophes" who believed in rationalism and enlightenment; their views
were
presented in the {\it Encyclopedie} of Diderot (1713-1784), for which D'Alembert
wrote the
Introduction. Diderot himself was the author of an article on probability and
inoculation commenting on D'Alembert's critiques of Bernoulli's papers on these
topics. Although France suffered from its defeat in the Seven Years War
(1756-1763),
its population had grown rapidly in the later half of the 18th century, as
famines
and epidemics became rarer; it reached some 26 million by 1789. The size and
health of the population was of vital concern in the recruitment of soldiers
for the king's army.
Bernoulli was interested in the ravages caused by smallpox, which was
prevalent
at the time. He devised a simple deterministic model to relate the size of a
cohort
$w(t)$ of individuals at time $t$ after birth, with the number of susceptibles
$x(t)$ among
them who had not been infected with smallpox. To make his equations easier, he
assumed that individuals infected by smallpox died immediately, or recovered
immediately and became immunes $z(t)$, so that $ w(t) = x(t) + z(t)$. By
setting up and
solving two ordinary differential equations he obtained the relation
$$ x(t) = w(t)/[(1-a)e^{bt} + a] $$
where $b$ is the rate of catching smallpox and $a$ the proportion of those
infected with
smallpox who die instantaneously. Setting $a = 1/8$ and $b = 1/8$, and using
Halley's
(1693) life tables he was able to estimate the numbers of susceptibles in a
cohort
subject to smallpox at time $t >0$, as well as the increased expectation of
life if
smallpox were eliminated. He calculated that this expectation would increase
from
26 years 7 months to 29 years 9 months. If smallpox could be eliminated, then
by
age 26, the population would be some 14\% larger. He used his results to argue
the
advantages of variolation; in an earlier popular article (1760), he wrote
``The two
great motives for inoculation are humanity and the interest of the State". It
should
be mentioned that the Bernoulli family in Basel had several of its younger
members
variolated.
Variolation, or the inoculation of susceptibles by live virus from patients
with a
mild form of the disease, was becoming popular in Europe at that time. The
practice
had originated in China as early as 1,000 AD; Chinese doctors had ground the
scabs
of dried smallpox pustules into powder, this being inhaled by individuals
wishing to
acquire immunity. The custom had migrated to India and Turkey in a modified
form
involving inoculation with live virus from smallpox pustules. In the early
18th century,
travellers to Turkey had reported on the success of variolation, and some
doctors in
Europe began to practise it. Unfortunately, even virus from a patient with a
mild
form of smallpox did not guarantee a mild attack in the new host.
In a paper presented to the Royal French Academy of Sciences in Paris in
1760,
Bernoulli set out to present the advantages of variolation. He became
involved in a
quarrel with D'Alembert on the interpretation of the relative risks of death
from
smallpox and variolation; D'Alembert maintained that the real risks of the
latter
were 17 times greater than those of smallpox itself. However, he eventually
moderated
his objections, and the paper was finally published in 1766.
The potential benefits of variolation became irrelevant within 3 decades, as
Jenner
demonstrated the safety of the cowpox vaccine against smallpox in England
in 1796.
Among his many mathematical works, Daniel Bernoulli also made important
contributions to probability theory, for example to what became known
as the St. Petersburg paradox, to which his cousin Nicholas
Bernoulli (q.v.) also contributed usefully. Sheynin (1972) refers to 8 memoirs on
probability published between 1738 and 1780, also briefly described in
Todhunter
(1865). Possibly the most interesting are the memoirs on the normal law, the
measurement of risk (1738), the theory of errors, and various urn problems.
In his
``Traite d'Hydrodynamique", Bernoulli studied the effect of pressure and
temperature
on gases; assuming that a gas consisted of small particles, he treated the
problem
by the probabilistic methods of Pascal (q.v.)
and Fermat (q.v.). This was the first attempt
to develop a kinetic theory of gases, a feat later achieved by Maxwell and
Boltzmann in the nineteenth century.
Bernoulli died in Basel in 1782, laden with honours, a polymath whose works
on
mathematics, probability and infectious disease modelling were recognized for
their originality, depth and practical applicability.
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\noindent{\it Acknowledgement}
My thanks are due to Professor K. Dietz for his help in preparing this
article.
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\begin{thebibliography}{3}
\bibitem{1} Bernoulli, Daniel (1738). Exposition of a new theory on the
measurement of risk.
Translation from the original Latin. {\it Econometrica}
{\bf 22}(1954), 23-36.
\bibitem{2} Bernoulli, Daniel (1760). Essai d'une nouvelle analyse de
la mortalite causee
par la petite verole et des avantages de l'inoculation pour la prevenir. {\it
Mem. Math. Phys. Acad. Roy. Sci., Paris}, 1-45. In {\it Histoire de l'Academie
Royale des Sciences}, 1766.
\bibitem{3} Bernoulli, Daniel (1760). Reflexions sur les avantages de l'inoculation.
{\it Mercure de France}, June issue, 173-190.
Gani, J. (1978). Some problems of epidemic theory (with discussion).
{\it Journal of the Royal Statistical Society Series A},
{\bf 141}, 323-347.
\bibitem{4} Halley, E. (1693). An estimate of the mortality of mankind, drawn
from curious
tables of the births and funerals at the city of Breslaw; with an attempt
to ascertain the price of annuities upon lives. {\it Philosophical
Transactions of the Royal Society of
London}, {\bf 17}, No.196, 596-610.
\bibitem{5} Seal, H. (1977). Studies in the history of probability and statistics. XXXV,
Multiple decrements or competing risks. {\it Biometrika}, {\bf 64}, 429-439.
\bibitem{6} Shafer, G. (1982). The Bernoullis. {\it Encyclopedia of Statistical
Sciences},
Wiley, New York, Vol. 1, 214-219.
\bibitem{7} Sheynin, O.B. (1972). D. Bernoulli's work on probability. {\it RETE
Strukturgeschichte der Naturwissenschaften} {\bf 1}, 273-300. Reprinted in
{\it Studies
in the History of Statistics and Probability}, Vol. II (1977), M.G. Kendall
and R.L. Plackett, eds. Griffin, London, 105-132.
\bibitem{8} Todhunter, I. (1865). {\it A History of the Mathematical Theory of
Probability}.
Reprinted by Chelsea, New York, 1949.
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\hfill{J. Gani}
\end{thebibliography}
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