16
AMEBICAN JOUBNAL or EPIDEMIOLOGY Vol. 98, No. 5 Copyright O 1973 by The John* Hopkins University Printed in U£.A. RECURRENT OUTBREAKS OF MEASLES, CHICKENPOX AND MUMPS I. SEASONAL VARIATION IN CONTACT RATES 1 WAYNE P. LONDON-" AND JAMES A. YORKE* (Received for publication April 19,1973) London, W. P. (Mathematical Research Branch, National Institute of Arthritis, Metabolism, and Digestive Diseases, Bethesda, Md. 20014) and J. A. Yorke. Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates. Am J Epidemiol 98:453-468, 1973. —Recurrent outbreaks of measles, chickenpox and mumps in cities are studied with a mathematical model of ordinary differential delay equa- tions. For each calendar month a mean contact rate (fraction of suscepti- bles contacted per day by an infective) is estimated from the monthly re- ported cases over a 30- to 35-year period. For each disease the mean monthly contact rate is 1.7 to 2 times higher in the winter months than in the summer months; the seasonal variation is attributed primarily to the gathering of children in school. Computer simulations that use the seasonally varying contact rates reproduce the observed pattern of un- damped recurrent outbreaks: annual outbreaks of chickenpox and mumps and biennial outbreaks of measles. The two-year period of measles outbreaks is the signature of an endemic infectious disease that would exhaust itself and become nonendemic if there were a minor in- crease in infectivity or a decrease in the length of the incubation period. For populations in which most members are vaccinated, simulations show that the persistence of the biennial pattern of measles outbreaks implies that the vaccine is not being used uniformly throughout the population. chickenpox; communicable diseases; disease outbreaks; epidemiologic methods; measles; models, theoretical; mumps; varicella INTRODUCTION rent outbreaks in large populations (2). Outbreaks of infectious diseases have The seasonal variation in the reported cases been studied frequently by mathematical of measles, for example, has been long models (1-9). Although useful in describ- recognized (3, 10) but whether or not there ing single outbreaks of a few months' dura- is seasonal variation in the contact rate has tion in small populations, deterministic not been investigated, models have not predicted undamped recur- The contact rate of a disease in a given ... , population is the fraction of the suscepti- 'This research was partially supported under ,, , , . , ,. ... National Science Foundation Grant GP-313S6X1. ™™ that an average infective successfully •Mathematical Research Branch, National In- exposes per day. In this paper contact rates stitute of Arthritis, Metabolism, and Digestive for measles, chickenpox and mumps are Diseases, Bethesda, Md. 20014. estimated for each month of a 30- or 35- * Institute for Fluid Dynamics and Applied . . , , , , , , Mathematics, University of Maryland. College Year-period from the monthly reports of Park. Md. 20740. cases of the three diseases in New York 453 at Stanford Medical Center on September 15, 2013 http://aje.oxfordjournals.org/ Downloaded from

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Page 1: RECURRENT OUTBREAKS OF MEASLES, CHICKENPOX AND …

AMEBICAN JOUBNAL or EPIDEMIOLOGY Vol. 98, No. 5

Copyright O 1973 by The John* Hopkins University Printed in U£.A.

RECURRENT OUTBREAKS OF MEASLES,CHICKENPOX AND MUMPS

I. SEASONAL VARIATION IN CONTACT RATES1

WAYNE P. LONDON-" AND JAMES A. YORKE*

(Received for publication April 19,1973)

London, W. P. (Mathematical Research Branch, National Institute ofArthritis, Metabolism, and Digestive Diseases, Bethesda, Md. 20014) andJ. A. Yorke. Recurrent outbreaks of measles, chickenpox and mumps. I.Seasonal variation in contact rates. Am J Epidemiol 98:453-468, 1973.—Recurrent outbreaks of measles, chickenpox and mumps in cities arestudied with a mathematical model of ordinary differential delay equa-tions. For each calendar month a mean contact rate (fraction of suscepti-bles contacted per day by an infective) is estimated from the monthly re-ported cases over a 30- to 35-year period. For each disease the meanmonthly contact rate is 1.7 to 2 times higher in the winter months than inthe summer months; the seasonal variation is attributed primarily to thegathering of children in school. Computer simulations that use theseasonally varying contact rates reproduce the observed pattern of un-damped recurrent outbreaks: annual outbreaks of chickenpox andmumps and biennial outbreaks of measles. The two-year period ofmeasles outbreaks is the signature of an endemic infectious disease thatwould exhaust itself and become nonendemic if there were a minor in-crease in infectivity or a decrease in the length of the incubation period.For populations in which most members are vaccinated, simulations showthat the persistence of the biennial pattern of measles outbreaks impliesthat the vaccine is not being used uniformly throughout the population.

chickenpox; communicable diseases; disease outbreaks; epidemiologicmethods; measles; models, theoretical; mumps; varicella

INTRODUCTION rent outbreaks in large populations (2).Outbreaks of infectious diseases have The seasonal variation in the reported cases

been studied frequently by mathematical o f measles, for example, has been longmodels (1-9). Although useful in describ- recognized (3, 10) but whether or not thereing single outbreaks of a few months' dura- is seasonal variation in the contact rate hastion in small populations, deterministic not been investigated,models have not predicted undamped recur- The contact rate of a disease in a given

. . . , population is the fraction of the suscepti-'This research was partially supported under , , , , . , ,. . . .

National Science Foundation Grant GP-313S6X1. ™™ that an average infective successfully•Mathematical Research Branch, National In- exposes per day. In this paper contact rates

stitute of Arthritis, Metabolism, and Digestive for measles, chickenpox and mumps areDiseases, Bethesda, Md. 20014. estimated for each month of a 30- or 35-

* Institute for Fluid Dynamics and Applied . . , , , , , ,Mathematics, University of Maryland. College Year-period from the monthly reports ofPark. Md. 20740. cases of the three diseases in New York

453

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454 LONDON AND YOBKE

City and measles in Baltimore. For eachcalendar month the 30 or 35 monthly con-tact rates are averaged to obtain a meanmonthly contact rate for that month. Theyear-to-year variation in the contact ratefor any month is small relative to theseasonal variation: for the three diseasesthe mean monthly contact rates are 1.7 to 2times higher during the autumn and wintermonths than during the summer months.This seasonal variation is apparently causedby the close contacts made by children,particularly during the coldest months,when school is in session. Simulations thatuse the seasonally varying contact ratesshow the pattern of the outbreaks of mea-sles, chickenpox and mumps: undampedrecurrent outbreaks that peak in the springmonths. The large seasonal variation in thecontact rates appears to be an essentialfeature of any realistic model of recurrentoutbreaks of these diseases in cities.

Measles with its biennial pattern of re-current outbreaks is shown to be in anarrow border region between "highly effi-cient" nonendemic diseases and "less effi-cient" diseases such as chickenpox andmumps that are endemic with regular one-year outbreaks. Our simulations reproducethe annual outbreaks of chickenpox andmumps and the biennial outbreaks of mea-sles in which the observed ratio of cases inthe high year vs. the low year is 5:1. Thesimulations show further that if the incuba-tion period of measles were longer than 12—13 days or if the infectivity were slightlylower the outbreaks of measles would occurannually. If the incubation period were asshort as 10 days or if the infectivity wereslightly higher, the disease would die out, atleast locally, and no regular pattern ofoutbreaks would be observed.

Recurrent outbreaks in a population inwhich many members are vaccinated aremore difficult to model accurately becausethe numbers and social characteristics ofthose vaccinated usually are not known.Simulations show, however, that the contin-uing biennial pattern of measles outbreaks

implies that the present use of the vaccineis strongly non-uniform and that in spite ofthe reduced numbers of cases, in somegroups in society the disease is as prevalentas ever.

In a subsequent paper (11) the estimatedcontact rates are used to study the spreadof the three infections in society and sto-chastic effects of populations of differentsizes are analyzed. A general formulation ofthe model will appear elsewhere (12).

THE DATA

The monthly number of reported cases ofmeasles, chickenpox and mumps in NewYork City and measles in Baltimore isshown in figure 1. In New York City, from1945 until widespread use of the vaccine inthe early 1960's, outbreaks of measles oc-curred every other year in the even-num-bered years. Prior to 1945 outbreaksoccurred essentially every two years withextra high years in 1931, 1936, and 1944;two consecutive low years occurred in 1939and 1940, followed by an exceedingly highyear in 1941. From 1929 to 1963 the aver-age number of reported annual cases inNew York City was about 18,000. In Balti-more from 1928-1959 outbreaks of measlesoccurred essentially even' second or thirdyear with no apparent pattern to the bien-nial or triennial recurrences; the averagenumber of reported annual cases was about5000. In both New York City and Balti-more the dramatic effect of extensive vacci-nation is seen after 1966. In both cities thelargest number of cases of measles occurredin the spring: in the high years in March,April or May and in the low years in April,May or June. The minimum number ofcases occurred in August or September.

In New York City outbreaks of chicken-pox and mumps that peaked in the springmonths occurred annually. The average an-nual number of reported cases was about9800 for chickenpox and about 6500 formumps.

The fraction of cases of each disease thatare reported can be estimated from the

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6000-

4000-

2000-

2000-

1000

New' iM City

20001-

1000-

2000|-

YEAR

FIGURE 1. Monthly number of reported cases of measles, chickenpox and mumps in NewYork City and Mensles in Baltimore.

birth rates (13, 14) if the changes in sus-ceptibles due to immigration and emigra-tion are neglected and if it is assumed thatby age 20 nearly all children acquire mea-sles, 68 per cent acquire chickenpox and 50per cent acquire mumps (15). On this basis,the rate of reporting of each disease is 1 in8 cases of measles in New York City, 1 in 3to 4 cases of measles in Baltimore and 1 in10 to 12 cases of chickenpox and mumps inNew York City. (Since at least 25 per cent

of infections of mumps are subclinical, thereported fraction of infections of mumpswould be correspondingly smaller.)

THE MODEL

A contact or an exposure of a susceptibleby an infective is defined as an encounter inwhich the infection is transmitted. Thecontact rate is denned as the fraction ofsusceptibles in a given population contactedper infective per day. The contact rate

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reflects the social behavior of the membersof society and the ease with which thedisease is transmitted; both factors mayvary during the year. Suppose that, at timet, E(t) is the number of exposures per day,S{t) is the number of susceptibles and I{t)is the number of infectives. Then the frac-tion of susceptibles exposed per day by allinfectives is E(t)/S{t) and the contact ratefi(t) is given by

The number of exposures per day is givenby

E{t) = (1)

(Alternatively if v is the total number ofcontacts of both susceptibles and immunesmade by an infective per day and N is thetotal population, then the number of expo-sures per day is given by E(t) = v(S(t)/N)I(t) and the contact rate is given by /?= v/N; if v varies during the year, so does/S-)

We assume that the net rate of entry ofsusceptibles into the population is constant.This net rate is the sum of the rates ofentry of susceptibles from births and fromimmigration minus the rate of loss of sus-ceptibles from emigration and the rate ofthe loss of individuals who do not acquirethe disease by, for example, age 20, andhence leave the school-aged susceptiblepopulation. (In 1935-1936, for example, byage 20, 5 per cent of an urban populationhad no history of measles, 32 per cent nohistory of chickenpox, and 50 per cent nohistory of mumps (15).) The assumption ofa constant input of susceptibles is, ofcourse, an approximation. The birth rate inNew York and Baltimore, for example,decreased by about 30 per cent during the1930's and rose again during the next twodecades (13, 14). Significant immigrationand emigration also occurred in both thesecities, but it is virtually impossible to meas-ure these migrations or to know the fractionof the immigrant or emigrant populationthat was susceptible.

Since we are interested in diseases thatcan be acquired only once, the rate ofchange of susceptibles, dS/dt, equals theconstant net rate of entry of susceptibles yminus the rate of exposure E(t), that is

dS/dt = y - E{t) (2)

This equation implies that after being con-tacted by an infective a susceptible imme-diately leaves the susceptible population; itis shown later that multiple contacts of asusceptible can be neglected.

We assume that all individuals exposedat time t incubate the disease for time Tx,are infectious for time T% and then cease tobe infectious and remain permanently im-mune to reinfection. The rate of change ofinfectives dl/dt equals the rate of appear-ance of infectives, which is the exposurerate time 7\ ago, minus the rate of disap-pearance of infectives, which is the expo-sure rate time Ti + T2 ago. Thus

dl/dl = E(t - Ti) - E(l - T1 - T^.

Integration of this equation yields

= / E(s) ds. (3)

This equation states that the number ofinfectives at any time equals the sum of theexposures made in the previous Ti 4- T2 toTi days. (In the more general formulationof the problem (12) the definition of I(t)differs from the definition here by the mul-tiplicative factor T->.)

The basic equations 1, 2 and 3 follownaturally from the assumptions that therate of exposure is proportional to both thenumber of susceptibles and the number ofinfectives, that the disease confers perma-nent immunity and that there exists anincubation and an infectious period. Thesame equations appear, for example, in thework of Wilson and Burke (16).

The two delays—T1? the time from expo-sure to infectivity and T2, the duration ofinfectivity—require interpretation. We areinterested in the spread of disease in so-ciety, and not among siblings in a house-

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hold (the cases of which are less likely toaffect the spread of disease in society andless likely to be reported (17)); therefore,we assume that an infective ceases to trans-mit the disease when he is confined at homeby the severity or the characteristic fea-tures of the disease (e.g., the rash). Theinfective is assumed to manifest constantinfectivity for a period of time equal to T2

before being withdrawn from society. 7\ isthen the time from exposure to the begin-ning of infectivity; although called theincubation period, it is perhaps a few daysshorter than the usual incubation period,which is defined as the time from exposureto the onset of symptoms. The choices ofthe length of the delays, Tx and T2 arebased on the epidemiology of the individualdiseases (18, 19).

For purposes of computation we use afixed time interval A, usually one day, letthe nth time interval be tn = nA, andapproximate equations 1-3 by the differencedelay equations (A — 1),

E(tn) = fKQKQSiL) (4)

S(tn+l) = S(tn) ~ E(tn) + y (5)

/(WO = E E(k-i + 1) (6). - 7 - !

A distribution of incubatio?i periods andmodels without delays. A broad distributionof incubation periods similar to that re-ported by Sartwell (19) can be incorpo-rated into the delay equation model byassuming, for example, that 1/8 of theexposed individuals incubate the disease for9 days, 1/8 for 10 days, . .. , 1/8 for 16days. Equation 3 is replaced by

7(0 = (1/8) [ [ /3(u)S(u)T(u)duds.•I i-Tj J$-n

(A distribution of incubation periods inwhich half the exposed individuals incubatethe infection for 12 days and half for 13days is denoted by T± - 12 to 13.)

The incubation and infectious periods canbe also modeled by ordinary differentialequations without delays (see appendix 1).

METHOD

The data used to calculate the monthlycontact rates are the notifications of casesof each infectious disease received by thecity health departments (usually by postalcard) during any month. (The monthlynotifications are given in the appendix ofthe following paper (11).) Since the delayfrom exposure to diagnosis is about twoweeks and the delay from diagnosis to thereceipt of notification in the health depart-ment is estimated to be about 10 days, themonthly totals for one month representmainly the exposures from the previousmonth. Thirty-five consecutive years ofdata (prior to the use of the vaccine) wereused to calculate the monthly contact ratesof measles in New York City and Baltimoreand 30 consecutive years of data for chick -enpox and mumps in New York City.

We define the disease year of, say 1950,as the 12 months from September 1, 1949,through August 31, 1950. For measles, ahigh year is a disease year in which manycases were reported (in New York City,greater than 21,000; in Baltimore, greaterthan 4900 cases) and a low year is a diseaseyear in which few cases were reported (inNew York City, less than 13,000; in Balti-more, less than 3800 cases).

In order to calculate a contact rate foreach month of the 30 or 35 disease years,the number of susceptibles at the beginningof each epidemic year was estimated. Theestimation of susceptibles is independent ofthe model or the choice of parameters of themodel {T-i, T2, y, or fi(t)). A mean contactrate for each calendar month was thenestimated from the data of monthly notifi-cations. This was done for each choice ofTj, the duration of the incubation period,and r2, the duration of the infectious pe-riod. The constant net input of susceptibles,7, was not required in the estimation of themonthly contact rates. Finally, for eachchoice of Tx and T2 the correspondingestimated mean monthly contact rates anda constant net input of susceptibles, y, was

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used in a simulation of the recurrent out-breaks.

Estimation of susceptibles for the calcu-lation of contact rates. The number ofsusceptibles at the beginning of each of the30 or 35 disease years was estimated byassuming that at the peak of each outbreakthe number of susceptibles equals a con-stant number Sp. Thus, at the beginning ofa particular September the number of sus-ceptibles equals Sp plus the number ofreported cases from that September to thepeak of the outbreak that year. Under thisassumption each observed annual outbreakof chickenpox is assumed to begin withessentially the same number of susceptiblesbecause the number of reported cases fromeach September to the peak of each out-break is roughly the same. The same is truefor the observed annual outbreaks ofmumps. For measles, a high year is as-sumed to begin with a large number ofsusceptibles (Sp plus the large number ofreported cases from that September to thepeak) and each low year is assumed tobegin with a correspondingly smaller num-ber of susceptibles. The more contagious thedisease the more contacts each infectivemakes and hence the smaller the number ofsusceptibles at the peak. That the numberof susceptibles should equal a constant atthe peak of an outbreak is intuitive since atthe peak the number of exposures per unittime is neither increasing nor decreasing.The idea may be justified in two ways.

The result was found empirically byHedrich (20), who estimated the number ofsusceptibles to measles for each month inBaltimore from 1900-1930 from census dataand the number of reported cases. From1900-1914 the average number of suscepti-bles at the peak of each outbreak was63,700 with a coefficient of variation(standard deviation divided by the mean)of less than 4 per cent; for 1921-1930 thecorresponding figures were 74,000 with acoefficient of variation of less than 6 percent. (The population of Baltimore rosesubstantially in 1918 (14).)

Second, a slight variation of the resultcan be proved for the model. We firstconsider a simple model without delays thatassumes no incubation period. (This modelis discussed in appendix 1.) Equation 3 isreplaced by dl/dt = p{t)S{t)I{t) - (1/S)I(t) where 8 is the mean length of theinfectious period. At tf, the time of the peakof the outbreak, dl/dt = 0 and S{tp) = Sp

= l/(S/3), where /S is the contact rate atthe time of the peak. For the more realisticdelay equation model that has an incuba-tion period: an infective infects fiS(t) sus-ceptibles per day so that during T2 days ofinfectivity fiS(t)T2 susceptibles are in-fected. At the peak of an outbreak eachinfective infects exactly one susceptible andso (within a minor correction for the de-lays) pS{tP)T2 = 1 or S(tp) = l/(T2f3).For Sp to be independent of the time of thepeak these arguments require that /3(£) doesnot change much during the months whenthe peaks occur.

The values of Sp and the average annualnumber of reported cases are given in table1. These values of Sp for New York Cityyield a total susceptible population that isclose to that calculated from census data(13) and age specific attack rates (15). Formeasles in Baltimore, the value of *SP yieldsa total susceptible population that agreeswith the findings of Hedrich (20). Thevalues of Sp for chickenpox and mumps canbe changed by at least 50 per cent without

TABLE 1

Values of Sp and average annual numberof reported canes of measles, chickenpox

and mumps in A'eio York City andmeasles in Baltimore

City and disease

New York CityMeaslesChickenpoxMumps

BaltimoreMeasles

sr

70,00085,00090,000

20,000

Averageannual

reportedcasej

18,0009,8006,600

5,000

Ratio

3.9

8.813.7

4.0

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altering the relative shape of the curve ofthe mean monthly contact rates or theresults of the simulations. For measles thechoice of Sf is quite critical. If Sp is de-creased by about 15 per cent too few sus-ceptibles are available at the end of thehigh year and the corresponding contactrates are systematically high; if Sp is de-creased by about 10 per cent the simula-tions yield biennial outbreaks but the ratioof cases between the high and low years isgreater than the observed ratio of 5:1. If Sj,is increased by about 10 per cent the simu-lations yield a ratio of cases that is lessthan the observed ratio; a 20 per centincrease in Sp yields simulations of annualoutbreaks (ratio of cases of 1:1).

Estimation of the mean monthly contactrate. A contact rate for each month of a 30-or 35-year-period was calculated from thedata of monthly notifications by the use ofequations 4-6 in the following way. Thenumber of susceptibles at the beginning ofeach of the 30 or 35 disease years wasspecified by the method described. A con-tact rate was found for each month, start-ing with September of the first year, suchthat the calculated number of exposuresequalled the reported number of exposuresfor that month. (The contact rate wasfound by a "shooting" technique: succes-sively smaller contact rates were tried untilthe calculated exposures equalled the re-ported exposures). To start the calculationsthe reported exposures for the precedingAugust were distributed equally throughoutthe month; thereafter, the pattern of expo-sures calculated for a month was used incalculating the contact rate for the nextmonth. The number of susceptibles at thebeginning of a month equalled the numberof susceptibles at the beginning of theprevious month minus the exposures for themonth. (In some calculations new suscepti-bles were added each day throughout theyear, but, in general, the constant net inputof susceptibles, y, was zero.) Each Septem-ber the number of susceptibles was speci-fied; susceptibles were not carried over

from year to year. The mean monthlycontact rate for each calendar month is theaverage of the 30 (or 35) contact rates forthat month. A mean monthly contact ratewas calculated for each choice of T^_ and T2.

The monthly contact rates calculated bythe above method showed a "see-saw" high-low pattern even after they were averagedfor all years. If the rate for one month wasexceptionally high, the rate for the nextmonth was unduly low. The raw monthlycontact rates (/3r) were smoothed accordingto the formula

/S(t) = 0.26j8,(i - 1)

+ 0.5 j8r(i) + 0.25 0,(i + 1)

where t = 1 , . . . , 12 (if i — 1, use 12 for i —1, etc.). The statements about the meanmonthly contact rates do not depend on thesmoothing.

Simulations. Equations 4-6 were used tosimulate the recurrent outbreaks. For eachchoice of J"i and T^ the corresponding curveof the 12 mean monthly contact rates esti-mated from the data of monthly notifica-tions was used. The constant net input ofsusceptibles y equalled the average annualnumber of reported cases of each disease. Inmost simulations it made little difference ifthe susceptibles were added equallythroughout the year, or, to mimic the gath-ering of children in school, added at thebeginning of the disease year. An arbitraryinitial estimate of susceptibles and infec-tives was needed to begin the iteration ofequations 4-6; after several years of simu-lated time, a pattern of stable, recurrent,undamped outbreaks that persist indefi-nitely was obtained. The estimation of sus-ceptibles that was used in the calculation ofmonthly contact rates was not required forthe simulations.

Most simulations were made to determineunder what conditions the equations wouldproduce biennial outbreaks similar to thoseof measles in New York City (figure 1). Inthese outbreaks the average number of re-ported cases was about 30,000 in the high

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year and about 6000 in the low years, aratio of 5:1. For measles the criterion of asuccessful simulation was a recurrent out-break every other year with a ratio of highyear to low year cases of about 5:1.

Multiple contacts of a susceptible. In thecalculation of the mean monthly contactrates and in the simulations, the differenceequations 4-6 were solved numerically on acomputer. The step size A was selected suchthat further reduction had negligible effecton the results; one iteration per day wassufficient.

Because the time interval of iteration wasone day or less, and not 14 days (3, 4), thechance of a susceptible being contacted bymore than one infective is very small. Atthe peak of the simulated outbreak ofmeasles in New York City there are about560,000 susceptibles and about 2400 expo-sures per day; the daily probability ofexposure for any susceptible is 2400/560,-000 or about .0043. The number of occur-rences of a susceptible being contacted bytwo infectives in one day is (.0043)2- 560,-000 = 10.4, which is negligible with respectto the 2400 exposures. Thus, the correctionfor multiple contacts, which is the distin-guishing feature of the Reed-Frost model(4), changes the exposure rate by at most0.43 per cent (10.4/2400). The Reed-Frostcorrection, usually employed when the iter-ation step size is 14 days, causes an error(and corresponding correction) 142 timeslarger than the correction here. That simi-lar mean monthly contact rates and similarsimulations are obtained with iteration stepsizes of one day or as small as 1/8 of a dayimplies that multiple contacts of a suscepti-ble are not important.

RESULTS

Mean monthly contact rates. The meanmonthly contact rates for measles, chicken-pox and mumps in New York City andmeasles in Baltimore are shown in figure 2.In both cities the measles contact rates inJune, July and August are low. The curvesrise sharply from August to October, re-

main high from November until March orApril and then fall steeply from May toJune. The ratio of the peak month to thelowest month is about 1.7 in New York; inBaltimore, 1.6. The contact rate for mumpsshows similar features as the curves formeasles. The curve for chickenpox, how-ever, peaks in October and has a sharpdecline from October to December; theratio of the peak month to the lowest monthis about 2. In a subsequent paper (11) thesespecial features of the seasonal variation inthe contact rate for chickenpox are relatedto the clustering of susceptibles with infec-tives and to the spread of chickenpoxthrough the population.

Figure 2 shows that for any given monththe variation in the monthly contact ratesfrom year to year for that month is smallcompared to the seasonal variation frommonth to month. The coefficients of varia-tion (standard deviation divided by themean) for measles, chickenpox, and mumpsin New York City are about 10, 7 and 5 percent, respectively, and about 18 per cent formeasles in Baltimore. Some months showsystematic differences in the mean monthlycontact rates for all three diseases betweenthe years with a large number of cases andyears with fewer cases and for chickenpoxbetween the early and late years of study.These systematic differences, which contrib-ute to the variation in the monthly contactrates from year to year, are discussed in asubsequent paper (11).

The curves of the mean monthly contactrates are not changed by any of the follow-ing alterations: 1) the length of the incuba-tion period is changed by 50 per cent or adistribution of incubation periods is as-sumed (for measles as long as 9 to 16days); 2) the infectious period is changed toone or three days; 3) the level of suscepti-bles (Sp) is changed by as much as 50 percent (for measles, if Sp is decreased by 15per cent, not enough susceptibles remain atthe end of a high year and erroneously highcontact rates are calculated for thesemonths); 4) susceptibles are added each

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1.4

1.2

1.0 -

0.8

0.6

1-2 h<i -oo

. 0.8<UJ

0.6

1.2

1.0 -

0 8 -

0.6

•1

MEASLES

CHICKENPOX

MUMPS

J J A S O N D J F M A M J J AFIGURE 2. Mean monthly contact rates for

measles, chickenpox and mumps in New YorkCity. The contact rates are normalized by #, theaverage of the 12 mean monthly contact rates.The bars show one standard deviation on eitherside of the mean. The dashed line in the top panelis the mean monthly contact rate for measles inBaltimore. Because of delays due to the incuba-tion period and in reporting, the notificationsfrom one month correspond to the contact rate ofthe previous month. For measles, T\ = 12 to 13days: in New York City (1929-1963), $ = 7.25 XlO-«; in Baltimore (1925-1959), 0 = 2.65 X HT8.

day throughout the year; 5) the contactrates are calculated using models of ordi-nary differential equations that assume anincubation period (see appendix 1). Meanmonthly contact rates calculated from datafrom individual boroughs in New York Cityshow the same features as those calculatedfor the entire city.

Simulations with a constant contact rate.The first result from the simulations is thatseasonal variation in the contact rate isnecessary to simulate undamped recurrentoutbreaks that peak in the spring months.If the contact rate is assumed constant andsusceptibles are added equally throughoutthe year, simulations yield damped wavesof outbreaks that approach a constant en-demic level of disease. This constant limit-ing solution is

where p is the constant contact rate, T2 theinfectious period, and y the constant netinput of susceptibles. If the contact rateis assumed constant but the susceptibles areadded only at the beginning of the diseaseyear (to model the gathering of children inschool), the simulations show outbreaks inwhich either the cases are distributed evenlyin all months of the year or if a well definedpeak occurs, the peak is not in the springmonths.

Simulations of measles epidemics: allow-able incubation periods. The following sim-ulations were done with the seasonallyvarying mean monthly contact rates calcu-lated from the data of monthly notifica-tions.

An important factor in simulating bien-nial outbreaks of measles that have theobserved ratio of high year to low yearcases is the duration of the incubationperiod. If the incubation period is assumed

For chickenpox in New York City (1931-19(30),7\ = 13 days and 0 = 6.07 X 10"«. For mumps inNew York City (1934-1963), 7\ = 16 days and £ =5.64 X 10"«. For all diseases Tt = 2 days.

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6000

4000

2000

1000

1000

MEASLES

MUMPS

S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A S O N D J F M A M J J A

MONTH

FIGURE 3. Simulations of the recurrent outbreaks of measles, ehickenpox and mumps inNew York City. The mean monthly contact rates and the values of 7\ and T2 from figure 2were used. The constant net input of susceptibles, y, which equals the average annual num-ber of reported cases, was measles—18,000; ehickenpox—9800; and mumps—6500. The dashedline is the simulation of the recurrent outbreaks of measles with the modified exposure rate,pU)SU)I(t) (l-c/(£)), where c = 0.00015.

to be 12 to 13 days, biennial outbreaks aresimulated. For the mean monthly contactrate from New York City the ratio of highyear to low year cases from the simulatedoutbreaks is 5:1 (figure 3); for the meanmonthly contact rate from Baltimore theratio from the simulated outbreaks is 6.5:1.If the incubation period is assumed to beshorter than 12 days, biennial outbreaks arenot simulated. Too many cases occur when

susceptibles are plentiful and in succeedingyears not enough susceptibles and infectivesare present to sustain the disease. (In theyear following the high year the suscepti-bles are replenished but the infectives arenow too low to sustain the disease. Intro-duction of additional infectives prevents thedisease from fading out in that secondyear.) If the duration of the incubationperiod is 14 days, too few cases occur when

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susceptibles are plentiful; biennial out-breaks are simulated but the ratio of highyear to low year cases is about 3:1. For anassumed incubation period of 16 days thesimulations show annual outbreaks (ratioof cases of 1:1).

As long as the duration of the incubationperiod is 12 to 13 days, assumptions aboutthe range of the distribution of incubationperiods (19) and the length of period ofinfectivity are not critical. For example, theassumption that half the individuals incu-bate the disease for 12 days and half for 13days gives the same simulation as the as-sumption that 1/8 of the individuals incu-bate the disease for 9 days, 1/8 for 10 days,. . . , and 1/8 for 16 days (the mean incuba-tion period being 13 days). It also makes nodifference if the period of infectivity isassumed to be one, two or three days.(There is a redundancy in the choice ofdifference values for r x and T2. For exam-ple, the assumption that half the individu-als incubate the infection for 12 days andhalf for 13 days and then are infectious forone day is the same as the assumption thatall individuals incubate the infection for 12days and then are infectious for two days.)

Simulations of measles outbreaks with amodified exposure rate. Mean monthly con-tact rates were also calculated with a modi-fied exposure rate of the formp(t)S(t)I(t) (1 - cl(t)). As discussed in asubsequent paper (11) the modification elim-inates a systematic difference in the meanmonthly contact rates between the high andlow years that occurs in the spring months.The maximum modification of the exposurerate is about 7 per cent at the peak of theoutbreak, and the parameter c can be cho-sen small enough so that the number ofinfectives in the low years is so small thatthe modification has negligible effect. Thismodification produces simulations thatmost faithfully reproduce the observationsin New York City. Biennial outbreaks withthe observed ratio of cases of 5:1 aresimulated (figure 3). Like the outbreaks inNew York City (figure 1) the peak of

simulated high year outbreak occurs earlier(March and April) than the peak of thesimulated outbreak in the low year (Apriland May); further, the exposures in thesummer after the low year are about 25 percent higher than the exposures in the sum-mer following the high year. These twoqualitative features are not present in thesimulation already mentioned in which theexposure rate is not modified.

The role of the incubation period andlevel of infectiousness in determining epi-demic patterns. Once the parameters aredetermined, the model allows us to deter-mine the role of the incubation period andthe role of the level of infectiousness inepidemic patterns. For example, if measleshad a shorter (or longer) incubation period,would it still be endemic in the New YorkCity area, and if endemic, would it stillhave a biennial pattern of outbreaks with aratio of high year cases to low year cases ofabout 5:1? Equivalently, we may ask, sup-pose a virus disease appeared with the samelevel of infectiousness as measles, that is,with the same contact rate, but having adifferent incubation period, what would bethe pattern of recurrent outbreaks? Usingthe contact rate obtained for measles (cal-culated using Tt = 12 to 13 and T2 = 2days), we simulate these situations by as-suming incubation periods of differentlengths (T2 always equals two days). If theincubation period T\ is assumed to be threedays, the peak of the outbreak occurs inDecember or January. The disease dies outafter the peak, is no longer endemic, andthe pattern of outbreaks appears similar tothat of influenza (18). If the incubationperiod is increased slightly, the ratio ofhigh year cases to low year cases becomessmaller. Finally, if Ti is greater than orequal to 16 days, annual outbreaks (ratioof cases is 1:1) that peak in the spring aresimulated.

We may similarly ask how the level ofinfectivity affects the pattern of measlesoutbreaks. If the infectivity is increased by15 per cent (modeled by a 15 per cent

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increase in the contact rate for all months)biennial outbreaks are simulated but theratio of high year to low year cases is about8:1; with a 20 per cent increase in inf ectiv-ity too many cases occur when susceptiblesare plentiful and the outbreak is not sus-tained in succeeding years. If the infectiv-ity is decreased by 10 per cent (modeled bya 10 per cent decrease in the contact rate forall months) biennial outbreaks are simu-lated but the ratio of cases is only 3:1; an18 per cent decrease in infectivity yieldssimulations of annual outbreaks.

Siviulations of outbreak of chickenpoxand mumps. Simulations with meanmonthly contact rates for chickenpox (fig-ure 2) reproduce the pattern of outbreaks inNew York City, annual outbreaks thatpeak in February and March (figure 3).Simulations with the mean monthly contactrate for mumps (figure 2) show annualoutbreaks that peak in March (figure 3). Ifthe mean monthly contact rate for mumps(calculated with Tx = 16) is used in asimulation with Tx = 12, annual outbreaksare still simulated; this shows that a short-ened incubation period is.not sufficient to.simulate biennial outbreaks of mumps.

DISCUSSION

The mean monthly contact rates for mea-sles, chickenpox and mumps that are esti-mated from reported monthly cases showsubstantial seasonal variation (figure 2).This cyclic variation is large relative to thevariation from year to year. The shape ofthe curve of mean monthly contact ratesdoes not change significantly when the con-tact rates are computed from a wide choiceof values for the incubation period, theinfectious period or the level of susceptibles.The curve of the mean monthly contactrate for measles in Baltimore is almostidentical to the curve for measles in XewYork City. The increased contact rate inthe autumn and winter months for measles,chickenpox and mumps suggests that this isan essential feature of any realistic modelof recurrent outbreaks of these diseases in

cities. Comparison of the contact rate ofpolio (that has a relatively short incubationperiod) with the contact rate of infectioushepatitis (that has a relatively long incuba-tion period) would be useful in understand-ing the seasonal variation of these diseases.

Because the absolute values of the meanmonthly contact rates are inversely propor-tional to the level of susceptibles, which isdifficult to determine, the absolute values ofthe mean contact rates are not a reliablemeasure of the actual number of contactsmade by an infective per day or of theinfectiousness of the three diseases. A validmeasure of infectivity is discussed in asubsequent paper (11).

The cause of the seasonal variation in thecontact rates. What accounts for the sea-sonal variation in the contact rate? Thecontact rate is affected by two classes offactors: first, climatic factors that mightenhance the transmission of infectious di-seases, such as cold weather, decreased in-door relative humidity, or possiblydecreased resistance to infectious diseasesduring colder months, and second, the socialbehavior of children aged 4—15, who pre-sumably make more contacts when they arein school.

The sharp rise and fall in the graph ofthe mean monthly contacts for all threediseases (figure 2) coincide with the open-ing and closing of school. The meanmonthly contact rate for measles that isestimated from the reported cases from thelow years (figure 1 in reference 11)) isapproximately constant during the schoolyear. These features suggest that the in-creased contacts made by children in schoolare the main cause of the seasonal variationin the monthly contact rates. In addition,the contact rate is high during the coldermonths when, both in school and in thehome, children spend more time indoorswith each other. Data of weekly or bi-weekly notifications might show drops inthe contact rate due to holiday vacationsfrom school, but the monthly totals ana-lyzed here did not. For the years and cities

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for which we have data, apparently therewere no instances when, except during thesummer, school was closed for more than amonth. In England and Russia, however,disruption in the usual pattern of schoolattendance during World War II, alteredthe seasonal and biennial patterns of theoutbreaks of measles (21-23). The Russianworkers (23) present other evidence includ-ing different seasonal incidences of measlesbetween urban and rural populations, toshow that the critical factors are social andnot climatic. Finally, Hope Simpson (24)measured the infectiousness or contagious-ness of measles, chickenpox and mumps inhouseholds during all months of the year.(In that study the infectiousness was de-fined as the proportion of exposures ofsusceptibles in the home leading to trans-mission of the disease.) In the home theinfectiousness of the three diseases showedno seasonal variation. For these reasons, theseasonal variation in the contact rate isattributed to social factors, particularly thegathering of children in school.

The necessity of seasonal variation in thecontact rate. For the delay equation modelstudied here, if a simulation is to giveundamped recurrent outbreaks that peak inthe spring, the contact rate £(£) must haveseasonal variation. Computer simulations inwhich the contact rate is constant andsusceptibles are added equally throughoutthe year show waves of outbreaks of everdecreasing amplitude and the disease ap-proaches a constant endemic level in so-ciety. Simulations in which the contact rateis constant but susceptibles are added onlyat the beginning of the disease year do notshow cyclic outbreaks that peak in thespring. Although time delays in ordinarydifferential equations often introduce oscil-latory or even periodic solutions, if thecontact rate is constant, the delays of atmost 20 days are insufficient to producecyclic annual or biennial outbreaks. Theidea, for example, that undamped biennialoutbreaks can occur merely by depletingsusceptibles in a high year and then replen-

ishing susceptibles in the succeeding lowyear is not sufficient to explain the biennialpattern.

Biennial outbreaks. A second question iswhy the annual variation in the contactrate produces annual outbreaks of chicken-pox and mumps but biennial outbreaks ofmeasles. As suggested by the simulations,biennial outbreaks of measles occur becausethe disease is sufficiently contagious and hasa brief enough incubation period that suffi-ciently many cases occur when susceptiblesare plentiful to deplete substantially thepopulation of susceptibles. In comparisonwith chickenpox and mumps, the relativelysmall population of susceptibles to measlesreflects the contagiousness of the disease.Indeed, an urban resident had almost nochance of escaping measles by age 20, butabout one chance in three of not contractingchickenpox and one chance in two of notcontracting mumps (15). In the simulationsof measles outbreaks, the susceptible popu-lation is depleted in a high year by 40 percent; in a low year by 7 per cent. Incontrast, the population susceptible tochickenpox and mumps is depleted annuallyby less than 10 per cent. If in the calcula-tion of the monthly contact rate the numberof susceptibles at the peak of the outbreak(Sp) is increased by 20 per cent, the suscep-tible population is too large, the depletionduring a high year is less and only annualoutbreaks appear in the simulations. Con-ceivably, a three-year pattern could beestablished (and has been observed in Bal-timore and elsewhere (25)) in which twoyears are needed to replenish the suscepti-bles; if, however, too many cases occur in ahigh year the susceptibles and then theinfectives subsequently become depleted sothat, without the introduction of extra in-fectives, the disease dies out. Contrary tofindings based on another model (26), thatsmall changes in infectivity do not affectperiodicity, 5 per cent changes in the infec-tivity are enough to change substantiallythe ratio of cases in the high and low years,and an 18 per cent decrease in the infectiv-

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ity yields simulations of annual outbreaks.Although the infectivity of each disease isdifficult to determine, chickenpox is appar-ently 35-65 per cent and mumps 19-̂ 42 percent as infectious as measles in society(11). Finally, to simulate biennial out-breaks in which the ratio of high year tolow year cases is the observed ratio of 5:1,the mean duration of the incubation periodmust be 12 to 13 days. A longer incubationperiod (e.g., 16 days) yields simulations ofannual outbreaks, and with a shorter incu-bation period too many cases occur whensusceptibles are plentiful and the diseasedies out.

The simulations show, therefore, thatmeasles is in a narrow border region be-tween "highly efficient" and "less efficient"infectiou3 diseases. The hypothetical"highly efficient" diseases are highly infec-tious, have brief incubation periods (fewerthan 12 days), deplete substantially thesusceptible population during an outbreakand thus are not endemic in cities. Becausea disease that is not endemic in cities wouldprobably fail to perpetuate itself, thereappears to be no example of a hypothetical"highly efficient" disease. An infectious dis-ease that has a brief incubation period buta low contact rate would not be "highlyefficient" and could be endemic in cities.The "less efficient" diseases, such as chick-enpox and mumps, are less infectious, haverelatively long incubation periods, do notdeplete substantially the susceptible popu-lation during an outbreak and are endemicwith a pattern of recurrent annual out-breaks.

Outbreaks that peak in the spring. In-cubation periods of at least 12 days alsoexplain why these outbreaks peak in thespring months even though the contact raterises sharply in the autumn months. With a"generation time" of about two weeks,seven or eight months are needed to buildup the level of infectives and then depletethe susceptibles until the outbreak can nolonger be sustained. Simulations of a dis-ease that has the same contact rate as

measles but that has an incubation periodof three days show that the peak of theoutbreak occurs in the early winter, afterwhich the outbreak dies out. The peak ofthe outbreaks of influenza, which has anincubation period of at most three days,occurs in the late autumn or winter (18).

Non-uniform use of the measles vaccine.The model can be used to simulate wide-spread use of, for example, the measlesvaccine, by introducing fewer susceptiblesannually into the population. To modeluniform use of the vaccine, the originalmonthly contact rate is used (because thetotal population is unchanged); the simula-tions show annual outbreaks. To modelnonuniform use of the vaccine, that is, asubpopulation that has not received thevaccine, the monthly contact rate is in-creased proportionally (because the totalpopulation is smaller); the simulationsshow biennial outbreaks. The persistence ofthe biennial pattern of measles outbreaks inspite of widespread use of the vaccine, atleast in Chicago (27) and New York City(figure 1) suggests nonuniform use of thevaccine. The simulations suggest the exist-ence of a subpopulation that is not receiv-ing the vaccine and that is depletedsubstantially after an outbreak. Indeed, thepreschool population interacts socially withitself and represents a larger percentage ofthe cases than was previously true (27, 28).The replacement of the biennial pattern byannual outbreaks suggests uniform use ofthe vaccine. With nonuniform use of thevaccine, immunes should not be counted inthe total population, and, as noted by oth-ers (8), the concept of herd immunity doesnot apply to the pockets of susceptibleswithin an otherwise totally immune urbanpopulation (29).

REFERENCES

1. Serfling RE: Historical review of epidemictheory. Hum Biol 24:145-166,1952

2. Bailey NTF: The mathematical theory ofepidemics. New York, Hafner, 1957

3. Soper HE: The interpretation of periodicity indisease prevalence. J R Stat Soc 92:34-73, 1929

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4. Abbey H: An examination of the Reed-Frosttheory of epidemics. Hum Biol 24:201-233,1952

5. Bnrtlett MS: Deterministic and stochasticmodels for recurrent epidemics. Proceedings ofthe Third Berkeley Symposium on Mathemati-cal Statistics and Probability. Berkeley andLos Angeles, University of California Press,1956, Vol 4, pp Sl-109

6. Wilson LO: AJI epidemic model involving athreshold. Math Biosci 15:109-121, 1972

7. Ewy W, Ackerman E, Gatewood LC, et al: Ageneralized stochastic model for simulation ofepidemics in a heterogeneous population(model VI). Comput Biol Med 2:45-58, 1972

8. Fox JP, Elveback L, Scott W, et al: Herdimmunity: basic concept and relevance topublic immunization practices. Am J Epide-miol 94:179-189, 1971

9. Hoppensteadt F, Waltman P: A problem inthe theory of epidemics. II. Math Biosci12:133-146, 1971

10. Bliss CI, Blevins DL: The analysis of seasonalvariation in measles. Am J Hyg 70:328-334,1959

11. Yorke JA, London WP: Recurrent outbreaksof measles, chickenpox and mumps. II. Sys-tematic differences in contact rates and sto-chnstic effects, Am J Epidemiol 98:469-482,1973

12. Yorke JA, London WP: (in preparation)13. Summary of Vital Statistics 1968 The City of

New York. Published by the Department ofHealth, The City of New York, 1968

14. Baltimore Health News 48:10. Published byBaltimore City Health Department, 1971

15. Collins SD, Wheeler RE, Shannon RD: Theoccurrence of whooping cough, chicken pox,mumps, measles and German measles in 200,000surveyed families in 28 large cities. SpecialStudy Series, No 1, Division of Public HealthMethods, NIH, USPHS, Washington DC, 1942

16. Wilson EB, Burke M: The epidemic curve.Proc Natl Acad Sci USA 28:361-367, 1942

17. Chope HD: A study of factors that influencereporting of measles. Virus and RickettsialDiseases. A Symposium held at the HarvardSchool of Public Health, June 12-June 17,1939. Cambridge, Harvard University Press,1940, pp 283-308

18. Debre R, Celers, J (editors): Clinical Virology(The Evaluation and Management of HumanViral Infections). Philadelphia, WB SaundersCompany, 1970

19. Sartwell PE: The distribution of incubationperiods of infectious disease. Am J Hyg51:310-318, 1950

20. Hedrich AW: Monthly estimates of the child

population "susceptible" to measles, 1900-1931,Baltimore, Md. Am J Hyg 17:613-636, 1933

21. Gunn W: Measles. Modern Practices in Infec-tious Fevers. Vol 2. Edited by HS Banks. NewYork, PB Hoeber, 1951, pp 499-520

22. Butler W: Whooping cough and measles, anepidemiological concurrence and contrast. ProcR Soc Med 40:384-398, 1947

23. Guslits SV: Measles. A Course in Epidemiol-ogy. Edited by II Elkin. New York, PergamonPress, 1961, pp 353-362

24. Hope Simpson RE: Infectiousness of commu-nicable diseases in the household (measles,chicken pox, and mumps). Lancet 2:549-554,1952

25. Emerson H: Measles and whooping cough. AmJ Public Health (Suppl) 27:1-153, 1937

26. Bartlett MS: The critical community size formeasles in the United States. J R Stat SocSeries A 123 :37^4, 1960

27. Hardy CE, Kassanoff I, Orbach HG, et al:The failure of a school immunization campaignto terminate an urban epidemic of measles.Am J Epidemiol 91:2S6-293, 1970

28. Landrigan PJ, Conrad JL: Current status ofmeasles in the United States. J Infect Disease124:620-622, 1971

29. Scott HD: The elusiveness of measles eradica-tion: insights gained from three years ofintensive surveillance in Rhode Island. Am JEpidemiol 94:37-42, 1971

A P P E N D I X 1

Models based on ordinary differentialequations. The incubation and infectiousperiods can also be modeled by ordinarydifferential equations that have no delays.In the following three models the averageduration of the infectious state is 5 days. Allthree models implicitly assume that the prob-ability of an infective ceasing to be infec-tious by time t is independent of the durationof his infectivity. Likewise, models two andthree assume that the probability of an indi-vidual ceasing to incubate the disease bytime t is independent of how long he hasincubated the disease. The first model as-sumes no incubation period, that is, an ex-posed individual immediately becomes infec-tious.

The equations of the first model are

dS/dt = -p(l)S(t)I(f) + y (la)

dl/dt = p(t)S(t)I(t) - (l/5)/(0 (2a)

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Equation la is the same as Equation 2 ofthe delay equation model.

The second model assumes an incubationstate W that has mean life of 6 days. Theequations are la and

dW/dt = 0(05(0/(0 ~ O./0)W(t) (3a)

cll/dl = (1/0)1^(0 - (1/5)7(0 (4a)

A more complicated approximation to thedelay of the incubation period is to assumea sequence of incubation states W\, . . . , Wr

each with a mean life of one day. The equa-tions of model three are la and

dWJdt = p(l)S(t)I(t) - Wx{t) (5a)

dWi/dt = H',_i(0 - Wi(t) (6a)

i = 2, . . . r

dl/dt = WT(t) - (1/8)1 (0 (7a)

For measles, in all three models 5 = 2days, in model two, 6 = 12 days, and inmodel three, r = 12.

The graph of the mean monthly contactrates for measles that is calculated usingmodel one is irregular, has large unsystematicdifferences between the contact rates calcu-

lated irom the high and low years, and showsno consistent seasonal variation. The meanmonthly contact rates for measles, calcu-lated using models two and three, are verysimilar to the contact rates calculated fromthe delay equation model (figure 2). Formodels two and three the simulations of themeasles outbreaks that use the seasonallyvarying contact rates show biennial out-breaks but the ratio of high year to low yearcases is too low. For model two the ratio isabout 2.5; for model three the ratio is about3.3.

In comparing the delay equation modeland the ordinary differential equationmodels, the model that assumes no incuba-tion state (model one) is too simplistic toallow the calculation of realistic contactrates. The models that assume an incubationstate (models two and three) yield realisticcontact rates but the simulations are notsatisfactory. Clearly, ordinary differentialequation models without delays that aremore complicated than model three canapproximate the incubation period, but thesimplest and most satisfactory model is thedelay equation model.

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