10

THE EPIDEMIC CURVE OF SMALLPOX.

BY W. J. MARTIN, B.SC.Of the Medical Research Council's Statistical Staff.

From the Division of Epidemiology and Vital Statistics, London Schoolof Hygiene and Tropical Medicine.

(With 3 Graphs in the Text.)

WHEN a disease assumes epidemic proportions, it is now generally recognisedthat certain conditions govern the rise and fall of the epidemic wave. Farr wasprobably the first person to attempt to describe these conditions in quantitativeterms. His theory was "the real law (i.e. of the epidemic) implies that the ratioof increase goes on rapidly decreasing until the ratio itself is decreasing." Inthe Appendix to the second Annual Report of the Registrar-General he discussesthe progress of the smallpox epidemic which had spread through England andWales in 1837-9, causing the deaths of over 30,000 persons. " Five die weeklyof smallpox in the metropolis when the disease is not epidemic....Why do thefive deaths become 10, 15, 20, 31, 58, 88 weekly and then progressively fallthrough the same measured steps?" He suggests, "amidst the apparentirregularities of the epidemic of smallpox and its eruptions all over the king-dom, it was governed in its progress by certain general laws." He found thatthe deaths from smallpox in the quarters of the year during the epidemic in-creased up to the third quarter very nearly at the ratio of 30 per cent. " Therate of increase is retarded at the end of the third period, and only rises 6 percent. in the next, where it remains stationary, like a projectile at the summit ofthe curve which it is destined to describe. The decline of the epidemic was lessrapid than its rise." He showed that the fall of mortality took place at auniformly accelerated rate and calculated a "regular series of numbers" (suchthat the second differences of the logarithms are constant) for the decline ofthe epidemic. He compared these with those actually recorded and found " onthe whole the agreement is remarkable." Farr did not put his law in the formof an equation nor did' he explain the method by which his regular series wereobtained, but it has since been shown that his epidemic law is a function of thenormal curve. From the fact that the first, third and fifth of the calculatedvalues agree exactly with the observed, for the two regions-Metropolis, Walesand the western counties-it seems probable that some method of differenceswas employed to find his "constant rate of acceleration." The calculated seriesfound by Farr can be obtained by taking logarithms of the first, third and fifthof the observed values and finding the second difference. This value divided by

four gives the constant second difference for the series, and taken with positivesign is the logarithm of the rate of acceleration. The logarithm of the secondcalculated value is obtained by adding to the logarithms of the first and thirdobserved values the constant second difference and dividing by two. The ratioof the first to the second calculated value gives the first rate of decrease andthe subsequent rates of decrease are found by multiplying the preceding rateby the "rate of acceleration." This method does not apply to the series for thewhole kingdom; in this series only the third value of the observed and calcu-lated are identical and no simple method of obtaining Farr's smoothed seriessuggests itself. The observed values, Farr's series and the calculated obtainedby the method of least squares, for the Metropolis are:

Mean quarterly Farr's -131X2+O*811x+6*8965*Quarter deaths registered series e

1 1103 1103 11242 959 967 9403 611 611 5794 240 278 2635 91 91 88

* Origin at 0.

Some years later Farr applied his "law" to a prevalent epidemic of cattleplague, which according to a member of the House of Commons would increasefrom thousands to tens of thousands until all the cattle would probably die of it.He modified his previous mathematical expression by making the third dif-ferences of the logarithms constant and predicted the maximum and course ofthe epidemic. This forecast met with hostile criticism, but later eventsjustified the accuracy of his main contentions. In 1873 Evans tried to extendFarr's method to epidemics of cholera and scarlet fever but without success.

Since Farr's time various studies have been made on the theory of the courseof the epidemic curve. Brownlee put forward the hypothesis that the degree ofinfectivity of the organism decreased, as the epidemic progressed, according tothe law of the monomolecular reaction in physical chemistry, i.e. in geometricalprogression. He maintained that there was no evidence that an epidemicdeclined because of the lack of susceptibles but he thought the decline was muchmore likely to be due to the loss of infectivity on the part of the infecting agent.

Since 1919 Greenwood, Topley and their co-workers have been endeavouringto study, under laboratory conditions, the genesis and development of epi-demics. These experiments are being carried out under conditions inwhichmanydisturbing factors, such as varying environment, nutrition, density, etc., canbe standardised.

VARIATION IN TYPE OF SMALLPOX.

The epidemic curve of any disease may be modified by certain factors, chiefamong these being (a) changes in virulence of the organism and (b) changes inthe resistance of the population. The epidemic curve of smallpox may havebeen influenced by the first factor; as for the second, smallpox is the onlydisease for which there was any metho-d of preventive treatment by artificial

W. J. MARTIN 11

Epidemic Curve of Smallpoximmunisation a hundred years ago. There is no question that the type of small-pox has changed from the classical type of the eighteenth and early nineteenthcenturies, both in fatality and age incidence, to the mild form which weexperience, in this country, to-day.

Jenner's discovery of vaccination was made public in 1798, and early in thenineteenth century an appreciable proportion of the population had beenvaccinated against smallpox. It was not until 1853 that the first vaccinationlaw was passed, and vaccination was made compulsory in 1871. It has beencontended that the general decrease in smallpox throughout the nineteenthcentury was a consequence of the introduction and dissemination of the methodof vaccination. For example, Guy, writing in 1882, on the statistics of smallpoxfor 250 years in London, showed that smallpox was the most formidableepidemic disease in the seventeenth and eighteenth centuries, and yet in thenineteenth century had decreased much more than any of the other importantinfectious diseases such as measles, scarlet fever, etc. From this he argued, onthe assumption that an equal improvement had been wrought by sanitarymeasures in all epidemic maladies, smallpox included, that the remarkableexcess of improvement in smallpox must be attributed to some cause or causesother than sanitary reforms, and that there was only one cause to which it wasreasonable to attribute this excess, namely, vaccination. A more recent studyby Greenwood is less confident. He concludes from a study of the vaccinationproblem that vaccination has "saved lives and diminished suffering underconditions which have prevailed in England and may prevail again," but theuse of vaccination is " not the sole, perhaps not the most important, factor inmodifying the epidemiological history of smallpox during the last hundredyears."

We have numerous and fairly accurate records of the time sequences inepidemics of smallpox both before and since the introduction of vaccinationand before the present phase of inordinately mild smallpox. It has seemed worthwhile to inquire whether the form of the epidemic in pre-vaccination days, thatis, its evolution in time, contrasts with its form in days when a proportion atleast of the exposed to risk had been protected. Even if we assumed that norecently vaccinated persons can take smallpox at all, the susceptible populationin post-Jennerian days must differ in some respects from those of the earlyeighteenth century. Some of them will be persons vaccinated many years beforethe epidemic and they will be intermingled with persons completely protected.Even if these considerations were ignored it would still be of. interest to learnwhether the form of an epidemic had changed over the long period of obser-vation.

For descriptive purposes, the family of frequency curves developed byProf. Karl Pearson has in point of flexibility and ease of computation few rivalsand naturally attracted the attention of those who wished to graduate curvesof epidemics. Pearsonian curves have been fitted by Brownlee, Greenwood andothers to data of epidemics; Brownlee in particular used the method exten-

12

W. J. MARTIN 13sively. From the standpoint of graduation the results were often satisfactory,but it did not prove possible to classify epidemics on the basis of the types ofcurves found appropriate.

As is well known, the Pearsonian curves were derived from the integrationof the equation 1 dy a-x

y dx CO+cLx+c2x2'where y is the frequency, x the abscissal value, and a, co, cl and c2are constants.The frequency distribution is taken to be unimodal. Integration leads totwelve types of curves. In order to use the method on the data of smallpoxepidemics, the following preliminary adjustments were made:

(1) The small subsidiary rises at the tails of the epidemics of 1716-17,1770-2 and 1881-2 were omitted from curve fitting, to conform with the factthat the theoretical curve is unimodal.

(2) The abscissa has, in some cases, been arbitrarily shifted to make theobserved data tail off at the ends. This was necessary since smallpox had avarying but fairly high endemic level, when the deaths of the eighteenth andearly nineteenth centuries were summed in four-week periods. In other wordsthe epidemic has been taken as a phenomenon superimposed on the endemiclevel.

LONDON BILLS OF MORTALITY, 1700-1800.

When the deaths from smallpox are summed in four-week periods they givean irregular curve. The time between successive maxima varies from one tothree years, but there is on the average a two-year period throughout thiscentury. There is no definite seasonal trend such as is shown by most epidemicdiseases, half the number of minima occur in spring and early summer and themaxima tend to be scattered over the rest of the year. When only the largerepidemics (where the maxima were 300 or more deaths in a month) are con-sidered, the peaks are, on the average, distributed fairly evenly over three-quarters of the year. The distribution is as follows:

No. ofFour-week No. of No. of maxima

period maxima minima (over 300)1-4 3 4 25-8 39-12 1 713-16 1 7 -17-20 10 -21-24 3 3 225-28 6 2 529-32 4 333-36 6 3 337-40 7 2 441-44 3 1 145-48 4 1 149-52 10 4 3

Totals 48 47 24

Epidemic Curve of Smallpox

LONDON, 1840-1931, REGISTRAR-GENERAL'S REPORT.

When the deaths from smallpox are summed in four-week groups, for thisperiod, the suggestion of periodicity shown by the data in the Bills of Mortalityfor the eighteenth century is absent. The interval between successive maximavaries from 14 to 6i years during the period 1840-85. Between 1800 and 1870the maximum and minimum values are below those of the eighteenth centuryand the peaks generally are not so definite. London, in common with mostEuropean cities, experienced a severe epidemic in 1870-1. This, the worst of thenineteenth century, rose to a maximum of 1048 deaths in the 17th-20th weeksof 1871. A period of low mortality followed this outbreak, only 246 deathsbeing recorded in the three years 1873-5. Epidemics of smallpox occurred in1877, 1878, 1881, 1885, the maxima being 381 in the lst-4th weeks of theyear, 241 in the 13th-16th weeks, 330 in the 17th-20th weeks and 239 in the17th-20th weeks respectively. By the end of 1885 smallpox had practically dis-appeared as a cause of death, and during 1886-91 only 55 deaths were recordedin London. A slight rise in the number of deaths followed, and from 1892 to1895 the annual deaths were 41, 206, 89 and 55. For the next five years only33 deaths were due to smallpox. The last fatal epidemic made its appearancein the summer of 1901, reached a maximum of 289 deaths in the 9th-12thweeks of 1902 and disappeared by the late summer. For the twenty-nine years1903-31 the deaths from smallpox were 113. The distribution of the maximaand minima for 1840-1902 is:

Weeks of No. of No. ofyear maxima minima1-4 4 15-8 29-12 1 113-16 1 117-20 5 221-24 1 125-28 129-32 233-3637-40 - -41-44 345-48 1 249-52 2

Totals 15 16

CURVE OF EPIDEMIC FOR LONDON.

The epidemics have been smoothed by fitting Pearson's Type Curves to thedeaths summed in four-week periods. The important constants and equationsare given in Table I. While these curves give a good description of the epi-demics, they do not "fit" in a statistical sense. Type II, a symmetrical curve,graduates the epidemics for the years 1751-3, 1870-1, 1876-7, 1877-8 and1880-1. Type I, an asymmetrical curve, describes the outbreaks for 1716-17,1780-2 and 1862-3; the epidemic of 1780-2 declined faster than it rose, whilst

14

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Epidemic Curve of Smallpoxthe reverse is the case for 1716-17 and 1862-3. The outbreak in 1901-2 con-forms to a Type IV and is the only distribution that approaches a statisticalfit. Most of the epidemics which have been smoothed are irregular, but some ofthe epidemics experienced in London give such a ragged distribution that nounimodal curve will smooth them, neither does a parabolic curve of the 4th orderdescribe the distribution (see Table III). In this class fall the outbreaks of1722-4, 1795-7 and 1884-5; the last exhibits three distinct peaks. A possibleexplanation of this seemed to be that the disease had made its appearance inhitherto unaffected districts, so that the total was really a summation ofseveral epidemics with different time intervals. To examine this, the deathswere taken out in the five broad groups-West, North, Central, East and South,as defined by the Registrar-General, for the nineteenth-century epidemics.The surmise proved to be incorrect since there is no apparent difference in thecommencement of the outbreak in the various divisions, but the interestingdiscovery was made that the course of the epidemic varied from district todistrict. The time of the maxima differs, the most outstanding case is in 1871when the largest number of deaths in the East district occurred in February andin the North in June. The order in which the London districts reach theirmaximum varies, but for the years examined the North tends to occupy thelast position. In 1863 the maximum for London as a whole, the North andSouth was in the 19th-22nd weeks of the year, whilst the largest number ofdeaths in the West, Central and East occurred in the preceding four-weekperiod. The outbreak in 1870-2 attained a maximum for London as a whole andthe South in the 17th-20th weeks of 1871, whilst that for the West, Central andEast was in the 5th-8th weeks. The climax of the epidemic in the Northdistrict was reached in the 21st-24th weeks. During the epidemic of 1876-8,which was bimodal, the first maximum for London as a whole and the Southwas reached in the first four-week period of 1877, in the East in February andin the North in March. The Central and West districts did not have a definitemaximum. The second peak for London as a whole, North and East appearedin the 13th-16th weeks of 1878 and that for the South and West in the pre-ceding and succeeding months respectively. In 1881 the peaks of the distribu-tion for London as a whole, West and Central districts were simultaneous,17th-20th weeks, that for the East, South and North in the 13th-16th,21st-24th and 25th-28th weeks respectively. The distribution of deaths forthe epidemic of 1884-5 was trimodal, and although the maxima of the fivedistricts were not so clearly defined as in the preceding outbreaks, they showthe same time irregularities as did the previous epidemics. The majority ofthe deaths in the 1901-2 outbreak occurred in M.A.B. ships and hospitals at

Dartford, so that a similar analysis in districts for this epidemic is of littlevalue. It is not possible to analyse the eighteenth-century data in subdivisions.The notifications for the years 1901-2 and 1928-31 were also examined in thefive districts. These two distributions show varying maxima and irregularitiessimilar to those displayed by the deaths for the nineteenth century. The

16

maximum number of cases of smallpox for London as a whole, North andSouth, for the epidemic of 1901-2, was reached in the 9th-12th weeks of 1902,the maximum of the Central, West and East districts had occurred earlier inthe 49th-52nd weeks of 1901, lst-4th and 5th-8th weeks of 1902 respectively.The distribution of cases of smallpox during 1928-31 is irregular. The largestnumber of cases for London as a whole, North and Central occurred during the13th-16th weeks, the East in the 9th-12th weeks of 1930, whilst in the Southdistrict the peak of the epidemic was in the 45th-48th weeks of 1929. TheWest district had very few cases. A Type VII, a symmetrical curve, smoothesthe 1901-2 notifications. The observed and smoothed values and the propor-tion of the total deaths occurring in each four-weekly period, for each of theLondon epidemics, are given in Tables II, III and IV.

This analysis makes it clear that no precise formal distinction differentiatesthe London epidemics of the eighteenth century from those of the late nine-teenth and early twentieth centuries. There is perhaps a tendency for themodern epidemics to be more concentrated in time. This can be most easilyseen by an examination of the proportional distribution as given in Table IV.If we now take the three four-weekly periods of which that containing themaximum is central, then in the three eighteenth-century epidemics these threeperiods include 34*4, 29-3 and 30*6 per cent. of the total epidemic deaths; thesix nineteenth- and twentieth-century epidemics have respectively 34 7, 39.5,34.9, 41.4, 40*4 and 48*1 per cent. of their totals within the defined limits.There is a suggestion that as we approach more nearly the modern epoch thisconcentration increases. Graphs I and II illustrate the comparison. Thisphenomenon could be explained in many ways, but, for the reasons alreadypointed out, e.g. the non-uniformity of distribution throughout the area ofLondon, we have no adequate means of verifying any speculations. All thatcan fairly be said is that, in respect of evolution in time, no sharp distinctionbetween epidemic smallpox in pre- and post-vaccination times can be estab-lished.

TOWNS OTHER THAN LONDON.It is of some interest to compare the forms of the London epidemics with

those of other cities. Glasgow is the only British city for which suitable datawere available and a Type II curve graduated the 1863-4 experience.

The 1870 epidemic was common to most European countries and makes com-parison possible between different cities. The distributions are given by Prinzingin his Epidemics resulting from Wars. A normal curve describes the Hamburgepidemic of 1870-2; Leipzig 1871 yields a Type IV, the fall being more rapidthan the rise; Breslau (cases) 1871-2 gives a Type I with steeper decline thanrise. The distributions for Danzig and suburbs and Berlin are two-peaked, anda fourth order parabola fails to describe them. The curve for Paris 1869-71 canbe regarded as two curves, the first a Type I, with a slower rise than fall, andthe second a Type II. The epidemic was apparently declining when it received

Jourm. of Hyg. xxxiv 2

W. J. MARTIN 17

Epidemic Curve of Smallpox

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Epidemic Curve of Smallpox

Table III. Unclassified distributions. London.

1722-4

4-week No. ofperiods deaths13-1617-2021-2425-2829-3233-3637-4041-4445-4849-521-45-89-1213-1617-2021-2425-2829-3233-3637-4041-4445-4849-521-45-8

6193

1231321532192543083974144253441981911862582432692422312051481128160

1795-7

4-week No. ofperiods deaths37-4041-4445-4849-521-45-89-1213-1617-2021-2425-2829-3233-3637-4041-4445-4849-521-45-8

891382102181691921661722163463744003254062652072028342

1884-5.-A

4-week No. ofperiods deaths1-45-89-1213-1617-2021-2425-2829-3233-3637-4041-4445-4849-5253-34-78-1112-1516-1920-2324-2728-3132-3536-3940-43

1511234685

1551859761519716420822621912215224020612763353512

Table IV. London epidemics. Proportion of deaths in each month.

Month 1716-17 1751-31st 0-026 0-0172nd 0 037 0-0263rd 0-057 0-0264th 0 090 00455th 0-114 0-0516th 0-124 0 0597th 0-106 0-0718th 0-103 0 0689th 0-085 0-079

10th 0-064 0 09111th 0-063 0-11312th 0-056 0-08913th 0.044 0-06914th 0-032 0 04915th 0 03916th 0-03517th 0-02818th 0*02519th 0-018

1780-20-0190-0250-0250-0320-0310-0300 0350 0400-0640-07600940-1030-1140.0890-0630-06800430 0340-016

1862-30-0270-0560 0490-0620-0810-1220-1260 0990-0850-0820-0580-0520 0390 0350027

1870-100400 0750-1140-1060-1260-1400-1290-1020 05900430-0360-031

1876-700070-00800090-0180-0250 0700 0990-1270-1230-1190-0980097008100480 0350-0200-016

1877-80-0290-0630 0750 0940-1150-1180-1550-1410-0960-0560 0400-018

1880-1 1901-20-024 0-0010-033 0-0100-067 0-0130-089 0-0170-087 0-0470-125 0-0590-143 0-1050-136 0-1440-113 0-1880-064 0-1490 053 0-1080 035 0-0840-031 0 044

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20

W. J. MARTIN 21

LONDON 1751-53Type II

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Epidemic Curve of Smallpox

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22

W. J. MARTIN

1899 1900

Four-week periods

Graph III.

23

0z-.4

24 Epidemic Curve of Smallpoxa stimulus from the arrival, in Paris, during September 1870 of newlyorganised troops, consisting of young men who had not been vaccinated orrevaccinated. The constants, equations, observed and smoothed values for thesecities are given in Tables V, VI and VII.

INDIA.Finally the deaths from smallpox in British India have been analysed, and

curves fitted to the data of Bombay and Calcutta. The data as a whole for the49 years 1882-1930 give a remarkably smooth curve, with a steady increase toa maximum in the spring and a steady decrease to a minimum in the autumn.The peaks of the curve occur as follows:

Maxima Minima10 in March 3 in September22 in April 40 in October16 in May 6 in NovemberI in June

Totals 49 49

There is a suggestion of periodicity in the severity of the epidemics in theIndian data, the maxima following a wave-like trend. The more severeepidemics (and to a less extent the minor epidemics) tend to occur in pairs, infollowing years. The actual maximum and minimum values for this period aregiven in Table VIII.

Curves have been fitted to the deaths, summed in four-week periods, for theepidemics in Calcutta, during the years 1905-6 and 1908-9, and for Bombay for1899-1900 and 1904-5. In each case a Type IV curve gives a good description.In these Indian cities, the concentration around the period of maximum is fargreater than in the London experience. Thus in Bombay, 1899-1900, 75-8 percent. of the deaths fall to the three four-weekly periods in which the maximumis central (see Graph III). In 1904-5, the percentage was 70 5. In the twoCalcutta epidemics the proportions were 60x8 and 78x9 per cent. The constants,equations, observed and smoothed values are given in Tables IX and X.

CONCLUSIONS.It is not possible, on the basis taken, to differentiate between the course of

the epidemic before and after the introduction of vaccination. The shape of thedistribution of monthly totals of the outbreaks of 1870 for various Europeancities are dissimilar. The majority of the fitted curves are symmetrical oralmost so; the most skew distributions, of London, examined are those of1716-17 and 1780-2. It does not seem practicable, from these data, to makethe London epidemics a function of time only. The shorter period for whichthe epidemic rages in India, the greater severity, the striking seasonal variation-the spring maxima and autumn minima-probably account for the fourIndian curves being of the same type and giving a better description of theobserved distribution than most of the other curves.

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Table VII. Unclassified epidemics.

Berlin 1870-2

Month DeathsNov. 9Dec. 22Jan. 48Feb. 80March 176April 349May 430June 648July 532Aug. 528Sept. 490Oct. 600Nov. 660Dec. 671Jan. 445Feb. 256March 151April 117May 76June 33July 18Aug. 10

Danzig and suburbs 1870-2Month CasesOct. 4Nov. 13Dec. 34Jan. 123Feb. 129March 201April 365May 459June 442July 182Aug. 130Sept. 111Oct. 124Nov. 136Dec. 135Jan. 245Feb. 222March 153April 89May 34June 19July 13Aug. 8

Table VIII. India.Monthly deaths

Maxima Minima11,612 3,01338,023 4,28467,516 5,13912,620 2,2187,043 1,9138,789 2,82913,277 3,03520,803 2,63118,409 3,40712,676 4,13914,418 3,5379,305 2,1686,305 1,4686,691 1,603

22,673 3,59932,521 2,7549,493 1,8226,270 2,79812,024 3,92712,238 3,88918,348 2,93413,354 2,4577,859 1,8528,540 3,26014,916 3,488

Year190719081909191019111912191319141915191619171918191919201921192219231924192519261927192819291930

Monthly deaths

Maxima Minima13,918 3,32728,773 2,87919,347 2,2747,317 1,9317,895 2,474

11,726 3,10215,239 2,16410,958 2,25613,837 1,9638,876 2,1127,284 3,45312,966 3,33820,728 3,44016,625 1,8496,542 9924,783 2,4975,320 1,9187,568 1,91813,439 2,63217,648 2,68018,986 2,30616,109 2,54411,927 1,68311,775 1,238

26

Year1882188318841885188618871888188918901891189218931894189518961897189818991900190119021903190419051906

W. J. MARTIN 27

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Table X.

Bombay 1899-1900Deaths

f

4-week Observed Smoothedperiod values values

41-4445-4849-521-45-89-1213-1617-2021-2425-2829-3233-3637-40

164815165498479235612044191443

639

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Bombay 1904-5Deaths

4-week Observed Smoothedperiod values values41-44 6 345-48 16 1749-52 78 801-4 238 2545-8 519 5089-12 649 61313-16 426 45417-20 210 22221-24 81 7825-28 33 2229-32 4 5

Totals 2260 2256

Totals 3205 3202

Calcutta 1905-6Deaths

4-week Observed Smoothedperiod values values41-44 4 345-48 12 1649-52 82 781-4 273 2655-8 575 5769-12 707 77213-16 741 64617-20 331 36021-24 131 14625-28 41 4729-32 26 1333-36 3 4

Totals 2926 2926

4-weekperiod42-4546-4950-531-45-89-1213-1617-2021-2425-2829-32

Calcutta 1908-9Deaths

Observed Smoothedvalues values

5 124 883 50

329 267851 9311360 1449780 835278 21062 3417 52 1

Totals 3791 3791

A theoretical objection to taking Type I, which occurs four times, as an

epidemic curve is the fact that its range is limited in both directions. The same

objection can be raised against Type II, which describes the majority of Londonoutbreaks, since it is a special case of Type I, a symmetrical form. The normalcurve and Types VII and IV fulfil the theoretical condition of unlimited range

in both directions, i.e. the epidemic never absolutely disappears although itbecomes negligible. Type VII is a particular case of Type IV arising from a

symmetrical form. The normal curve is the transition point between a Type IIand Type VII. Type IV, which has been found to fit the majority of otherepidemics, is leptokurtic, i.e. a narrow top curve, whilst most of the observeddistributions are platykurtic, hence this curve is unsuitable for describing themajority of the smallpox epidemics.

28

W. J. MARTIN 29

REFERENCES.BROWNLEE, J. (1915, 1916). On the curve of the epidemic. Brit. Med. J. 8. v. 15 and

29. vii. 16.(1915). Historical note on Farr's theory of epidemics. Ibid. 14. viii. 15.

EvAs, G. H. (1866-76). Some arithmetical considerations of the progress of epidemics.Trans. Epidem. Soc. of London, 3, 551.

FARR, W. (1840). Second Annual Report, Registrar-General. Dr Farr's Appendix, p. 91.GREENWOOD, M. (1913). The factors that determine the rise and spread and degree of severity

of epidemic diseases. The 17th International Congress ofMedicine (Hygiene and PreventiveMedicine), Section xvm, London, pp. 49-80.(1930). The vaccination problem. J. Roy. Stat. Soc. 93, Part ii, pp. 233-57.

Guy, W. A. (1882). On two hundred and fifty years of smallpox in London. Ibid. 45,pp. 399-437.

(MS. received for publication 19. x. 1933.-Ed.)