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GT MATH 3770 - LECTURE NOTES

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Rev. Med. Virol. 2004; 14: 275–288.Published online in Wiley InterScience (www.interscience.wiley.com).Reviews in Medical Virology DOI: 10.1002/rmv.443An attempt at a new analysis of the mortalitycaused by smallpox and of the advantagesof inoculation to prevent ityDaniel BernoulliReviewed by Sally Blower*AIDS Institute and Department of Biomathematics, David Geffen School of Medicine at UCLA,1100 Glendon Avenue, PH2, Los Angeles, CA 90024, USAAccepted: 15 May 2004‘I simply wish that, in a matter which so closelyconcerns the wellbeing of the human race, no deci-sion shall be made without all the knowledge whicha little analysis and calculation can provide’Daniel Bernoulli 1760.INTRODUCTIONShould the general population be vaccinatedagainst smallpox (Variola Major)? Would the bene-fits of mass vaccination outweigh the risks? Howmany deaths would occur as the result of a massvaccination campaign against smallpox? Canmathematical models of smallpox vaccination beused to determine health policy? Although small-pox was declared eradicated by the World HealthOrganization in 1979, these questions have all beenrecently debated based upon the premise thatsmallpox may be used as a weapon of bioterror-ism. Hence, a series of analyses has recently beenpublished that use mathematical models to try todetermine the most effective public healthresponse in the event of such an attack [1–4]. How-ever, these same controversial public health ques-tions were debated in the 18th century whensmallpox was endemic and Reviews in Medical Vir-ology has published two classic papers describingthe natural history of smallpox in 1902 and 1913to help inform these discussions [5,6]. We nowpublish an even earlier paper.In 1760 Daniel Bernoulli (1700–1782), one of thegreatest scientists of the 18th century, wrote amathematical analysis of the problem in order totry to influence public health policy by encoura-ging the universal inoculation against smallpox;his analysis was first presented at the Royal Acad-emy of Sciences in Paris in 1760 and later pub-lished in 1766 [7]. Here, we republish anddiscuss both the historical and the current signifi-cance of Bernoulli’s classic paper. A detailed dis-cussion of the mathematics of Bernoulli’s analysishas previously been presented by Dietz andHeesterbeck [8].According to Creighton [9] smallpox firstappeared in England in the 16th century. Smallpoxwas known in Western Europe in medieval times,but a particularly virulent strain emerged in theearly 17th century and gradually the case fatalityrate increased [10]. By the 18th century smallpoxwas endemic. Bernoulli calculated that approxi-mately three quarters of all living people (in the18th century) had been infected with smallpox[7]. One-tenth of all mortality at that time wasdue to smallpox, although there was considerableannual variation in smallpox mortality due to epi-demic outbreaks overlaying the endemic smallpoxmortality rate. For example, in London during theperiod 1761–1796 the annual number of deathsdue to smallpox varied from 3000 to 15 000. Wheresmallpox was endemic it was almost wholly a dis-ease of childhood, with a case-fatality rate of 20%–30%; the mean age of death due to smallpox hasbeen estimated as 2.6 years [10] or 4.5 years [11].CC L A S S II CCPP AA PP EE RRCopyright # 2004 John Wiley & Sons, Ltd.*Corresponding author: Dr S. Blower, AIDS Institute and Depart-ment of Biomathematics, David Geffen School of Medicine atUCLA, 1100 Glendon Avenue, PH2, Los Angeles, CA 90024, USA.E-mail: [email protected] from a translation of Mem Math Phy Acad Roy SciParis 1766, published by Adult Education Department, Nottingham,1971.Almost all adults were immune to smallpox asthey had been infected as children and had eitherdied or survived and developed immunity.China and India began the practice of transferringinfectious material from a smallpox pustule to anuninfected individual in order to induce a mildinfection (sometimes called ‘artificial smallpox’)that would then be followed by lifelong immunity;this practice was called variolation. In 1721 vario-lation was introduced into England. The practiceof variolation generated heated debates as ‘artifi-cial smallpox’ was sometimes fatal; Bernoullidecided that the best way to evaluate the publichealth consequences of variolation was to usemathematical reasoning [7]. Over time the contro-versy was resolved and inoculation became wide-spread in England; by the end of the 19th centurysmallpox was no longer endemic in England.The main purpose of Bernoulli’s analyses was toencourage universal inoculation against smallpox.He argued his case by calculating the gain in lifeexpectancy that would be achieved if smallpoxwere eradicated. To calculate this gain he assumedthat: (i) the risk of catching smallpox (given that theindividual has never had smallpox) was the sameat any age (and was 1 in 8), and (ii) the case fatalityrate of smallpox was independent of age (and was12.5%). He derived an equation (for any year ofage) for calculating the number of people whohad never had smallpox as a fraction of the peoplecurrently alive. He then used a second equation tocalculate how many lives would be saved if small-pox were completely eliminated. The central part ofBernoulli’s analysis is summarised in the form oftwo tables that show the number of individualsthat survive each year (up to the age of 25), begin-ning with a cohort of 1300 newborns, with andwithout smallpox mortality [7]. Bernoulli used aLife Table drawn up by Edmund Halley (theastronomer who named the famous comet) as thebasis for expected survivorship (i.e. the expectedsurvival curve including smallpox mortality), andhe then used these numbers in conjunction withhis equations to derive the expected number ofinfections and deaths due to smallpox each year(Table 1). Thus, he was able to calculate (seeTable 2) the expected number of individualssurviving each year if smallpox was eradicated.The benefits in eliminating smallpox could beviewed either as the increase in the number of sur-vivors per year or as the increase in average lifeexpectancy. Only 565 out of 1300 newborns reachedthe age of 25 in the 18th century when smallpoxwas endemic (Table 1); Bernoulli’s calculationsrevealed that 644 individuals (out of 1300) wouldsurvive to age 25 if smallpox was eliminated(Table 2). His calculations also showed that univer-sal inoculation against smallpox would increaseexpectation of life at


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