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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) 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 o f 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 recurThe 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 Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 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 equations. For each calendar month a mean contact rate (fraction of susceptibles contacted per day by an infective) is estimated from the monthly reported 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 undamped recurrent outbreaks: annual outbreaks of chickenpox and mumps and biennial outbreaks of measles. T h e 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 increase 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. 454 LONDON AND YOBKE implies that the present use of the vaccine is strongly non-uniform and that in spite of the reduced numbers of cases, in some groups in society the disease is as prevalent as ever. In a subsequent paper (11) the estimated contact rates are used to study the spread of the three infections in society and stochastic effects of populations of different sizes are analyzed. A general formulation of the model will appear elsewhere (12). THE DATA The monthly number of reported cases of measles, chickenpox and mumps in New York City and measles in Baltimore is shown in figure 1. In New York City, from 1945 until widespread use of the vaccine in the early 1960's, outbreaks of measles occurred every other year in the even-numbered years. Prior to 1945 outbreaks occurred essentially every two years with extra high years in 1931, 1936, and 1944; two consecutive low years occurred in 1939 and 1940, followed by an exceedingly high year in 1941. From 1929 to 1963 the average number of reported annual cases in New York City was about 18,000. In Baltimore from 1928-1959 outbreaks of measles occurred essentially even' second or third year with no apparent pattern to the biennial or triennial recurrences; the average number of reported annual cases was about 5000. In both New York City and Baltimore the dramatic effect of extensive vaccination is seen after 1966. In both cities the largest number of cases of measles occurred in the spring: in the high years in March, April or May and in the low years in April, May or June. The minimum number of cases occurred in August or September. In New York City outbreaks of chickenpox and mumps that peaked in the spring months occurred annually. The average annual number of reported cases was about 9800 for chickenpox and about 6500 for mumps. The fraction of cases of each disease that are reported can be estimated from the Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 City and measles in Baltimore. For each calendar month the 30 or 35 monthly contact rates are averaged to obtain a mean monthly contact rate for that month. The year-to-year variation in the contact rate for any month is small relative to the seasonal variation: for the three diseases the mean monthly contact rates are 1.7 to 2 times higher during the autumn and winter months than during the summer months. This seasonal variation is apparently caused by the close contacts made by children, particularly during the coldest months, when school is in session. Simulations that use the seasonally varying contact rates show the pattern of the outbreaks of measles, chickenpox and mumps: undamped recurrent outbreaks that peak in the spring months. The large seasonal variation in the contact rates appears to be an essential feature of any realistic model of recurrent outbreaks of these diseases in cities. Measles with its biennial pattern of recurrent outbreaks is shown to be in a narrow border region between "highly efficient" nonendemic diseases and "less efficient" diseases such as chickenpox and mumps that are endemic with regular oneyear outbreaks. Our simulations reproduce the annual outbreaks of chickenpox and mumps and the biennial outbreaks of measles in which the observed ratio of cases in the high year vs. the low year is 5:1. The simulations show further that if the incubation period of measles were longer than 12— 13 days or if the infectivity were slightly lower the outbreaks of measles would occur annually. If the incubation period were as short as 10 days or if the infectivity were slightly higher, the disease would die out, at least locally, and no regular pattern of outbreaks would be observed. Recurrent outbreaks in a population in which many members are vaccinated are more difficult to model accurately because the numbers and social characteristics of those vaccinated usually are not known. Simulations show, however, that the continuing biennial pattern of measles outbreaks 455 OUTBREAKS OP MEASLES, CHICKENPOX AND MUMPS. I. 6000- 4000- 2000- 1000 N e w ' i M City 20001- 1000- 2000|- YEAR FIGURE 1. Monthly number of reported cases of measles, chickenpox and mumps in New York City and Mensles in Baltimore. birth rates (13, 14) if the changes in susceptibles due to immigration and emigration are neglected and if it is assumed that by age 20 nearly all children acquire measles, 68 per cent acquire chickenpox and 50 per cent acquire mumps (15). On this basis, the rate of reporting of each disease is 1 in 8 cases of measles in New York City, 1 in 3 to 4 cases of measles in Baltimore and 1 in 10 to 12 cases of chickenpox and mumps in New York City. (Since at least 25 per cent of infections of mumps are subclinical, the reported fraction of infections of mumps would be correspondingly smaller.) THE MODEL A contact or an exposure of a susceptible by an infective is defined as an encounter in which the infection is transmitted. The contact rate is denned as the fraction of susceptibles in a given population contacted per infective per day. The contact rate Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 2000- 456 LONDON AXD YORKB Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 reflects the social behavior of the members Since we are interested in diseases that of society and the ease with which the can be acquired only once, the rate of disease is transmitted; both factors may change of susceptibles, dS/dt, equals the vary during the year. Suppose that, at time constant net rate of entry of susceptibles y t, E(t) is the number of exposures per day, minus the rate of exposure E(t), that is S{t) is the number of susceptibles and I{t) dS/dt = y - E{t) (2) is the number of infectives. Then the fraction of susceptibles exposed per day by all This equation implies that after being coninfectives is E(t)/S{t) and the contact rate tacted by an infective a susceptible immefi(t) is given by diately leaves the susceptible population; it is shown later that multiple contacts of a susceptible can be neglected. The number of exposures per day is given We assume that all individuals exposed by at time t incubate the disease for time Tx, are infectious for time T% and then cease to E{t) = (1) be infectious and remain permanently im(Alternatively if v is the total number of mune to reinfection. The rate of change of contacts of both susceptibles and immunes infectives dl/dt equals the rate of appearmade by an infective per day and N is the ance of infectives, which is the exposure total population, then the number of expo- rate time 7\ ago, minus the rate of disapsures per day is given by E(t) = v(S(t)/ pearance of infectives, which is the expoN)I(t) and the contact rate is given by /? sure rate time Ti + T2 ago. Thus = v/N; if v varies during the year, so does dl/dl = E(t - Ti) - E(l - T1 - T^. /S-) We assume that the net rate of entry of Integration of this equation yields susceptibles into the population is constant. This net rate is the sum of the rates of = / E(s) ds. (3) entry of susceptibles from births and from immigration minus the rate of loss of sus- This equation states that the number of ceptibles from emigration and the rate of infectives at any time equals the sum of the the loss of individuals who do not acquire exposures made in the previous Ti 4- T2 to the disease by, for example, age 20, and Ti days. (In the more general formulation hence leave the school-aged susceptible of the problem (12) the definition of I(t) population. (In 1935-1936, for example, by differs from the definition here by the mulage 20, 5 per cent of an urban population tiplicative factor T->.) had no history of measles, 32 per cent no The basic equations 1, 2 and 3 follow history of chickenpox, and 50 per cent no naturally from the assumptions that the history of mumps (15).) The assumption of rate of exposure is proportional to both the a constant input of susceptibles is, of number of susceptibles and the number of course, an approximation. The birth rate in infectives, that the disease confers permaNew York and Baltimore, for example, nent immunity and that there exists an decreased by about 30 per cent during the incubation and an infectious period. The 1930's and rose again during the next two same equations appear, for example, in the decades (13, 14). Significant immigration work of Wilson and Burke (16). and emigration also occurred in both these The two delays—T1? the time from expocities, but it is virtually impossible to meas- sure to infectivity and T2, the duration of ure these migrations or to know the fraction infectivity—require interpretation. We are of the immigrant or emigrant population interested in the spread of disease in sothat was susceptible. ciety, and not among siblings in a house- OUTBREAKS OF MEASLES, CHICKENPOX AND MUMPS. I. 457 Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 METHOD hold (the cases of which are less likely to affect the spread of disease in society and The data used to calculate the monthly less likely to be reported (17)); therefore, contact rates are the notifications of cases we assume that an infective ceases to transof each infectious disease received by the mit the disease when he is confined at home city health departments (usually by postal by the severity or the characteristic feacard) during any month. (The monthly tures of the disease (e.g., the rash). The notifications are given in the appendix of infective is assumed to manifest constant the following paper (11).) Since the delay infectivity for a period of time equal to T2 from exposure to diagnosis is about two before being withdrawn from society. 7\ is weeks and the delay from diagnosis to the then the time from exposure to the beginreceipt of notification in the health departning of infectivity; although called the ment is estimated to be about 10 days, the incubation period, it is perhaps a few days monthly totals for one month represent shorter than the usual incubation period, mainly the exposures from the previous which is defined as the time from exposure month. Thirty-five consecutive years of to the onset of symptoms. The choices of data (prior to the use of the vaccine) were the length of the delays, Tx and T2 are used to calculate the monthly contact rates based on the epidemiology of the individual of measles in New York City and Baltimore diseases (18, 19). and 30 consecutive years of data for chick For purposes of computation we use a enpox and mumps in New York City. fixed time interval A, usually one day, let We define the disease year of, say 1950, the nth time interval be tn = nA, and as the 12 months from September 1, 1949, approximate equations 1-3 by the difference through August 31, 1950. For measles, a delay equations (A — 1), high year is a disease year in which many E(tn) = fKQKQSiL) (4) cases were reported (in New York City, greater than 21,000; in Baltimore, greater S(tn+l) = S(tn) ~ E(tn) + y (5) than 4900 cases) and a low year is a disease year in which few cases were reported (in /(WO = E E(k-i + 1) (6) New York City, less than 13,000; in Balti.-7-! A distribution of incubatio?i periods and more, less than 3800 cases). In order to calculate a contact rate for models without delays. A broad distribution each month of the 30 or 35 disease years, of incubation periods similar to that rethe number of susceptibles at the beginning ported by Sartwell (19) can be incorpoof each epidemic year was estimated. The rated into the delay equation model by estimation of susceptibles is independent of assuming, for example, that 1/8 of the the model or the choice of parameters of the exposed individuals incubate the disease for model {T-i, T , y, or fi(t)). A mean contact 2 9 days, 1/8 for 10 days, . .. , 1/8 for 16 rate for each calendar month was then days. Equation 3 is replaced by estimated from the data of monthly notifi7(0 = (1/8) [ [ /3(u)S(u)T(u)duds. cations. This was done for each choice of Tj, the duration of the incubation period, •I i-Tj J$-n (A distribution of incubation periods in and r 2 , the duration of the infectious pewhich half the exposed individuals incubate riod. The constant net input of susceptibles, the infection for 12 days and half for 13 7, was not required in the estimation of the monthly contact rates. Finally, for each days is denoted by T± - 12 to 13.) The incubation and infectious periods can choice of Tx and T2 the corresponding be also modeled by ordinary differential estimated mean monthly contact rates and equations without delays (see appendix 1). a constant net input of susceptibles, y, was 458 LONDON AND YOEKE Second, a slight variation of the result can be proved for the model. We first consider a simple model without delays that assumes no incubation period. (This model is discussed in appendix 1.) Equation 3 is replaced by dl/dt = p{t)S{t)I{t) - (1/S) I(t) where 8 is the mean length of the infectious period. At tf, the time of the peak of the outbreak, dl/dt = 0 and S{tp) = Sp = l/(S/3), where /S is the contact rate at the time of the peak. For the more realistic delay equation model that has an incubation period: an infective infects fiS(t) susceptibles per day so that during T2 days of infectivity fiS(t)T2 susceptibles are infected. At the peak of an outbreak each infective infects exactly one susceptible and so (within a minor correction for the delays) pS{tP)T2 = 1 or S(tp) = l/(T2f3). For Sp to be independent of the time of the peak these arguments require that /3(£) does not change much during the months when the peaks occur. The values of Sp and the average annual number of reported cases are given in table 1. These values of Sp for New York City yield a total susceptible population that is close to that calculated from census data (13) and age specific attack rates (15). For measles in Baltimore, the value of *SP yields a total susceptible population that agrees with the findings of Hedrich (20). The values of Sp for chickenpox and mumps can be changed by at least 50 per cent without TABLE 1 Values of Sp and average annual number of reported canes of measles, chickenpox and mumps in A'eio York City and measles in Baltimore City and disease New York City Measles Chickenpox Mumps Baltimore Measles sr Average annual reported casej 70,000 85,000 90,000 18,000 9,800 6,600 13.7 20,000 5,000 4.0 Ratio 3.9 8.8 Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 used in a simulation of the recurrent outbreaks. Estimation of susceptibles for the calculation of contact rates. The number of susceptibles at the beginning of each of the 30 or 35 disease years was estimated by assuming that at the peak of each outbreak the number of susceptibles equals a constant number Sp. Thus, at the beginning of a particular September the number of susceptibles equals Sp plus the number of reported cases from that September to the peak of the outbreak that year. Under this assumption each observed annual outbreak of chickenpox is assumed to begin with essentially the same number of susceptibles because the number of reported cases from each September to the peak of each outbreak is roughly the same. The same is true for the observed annual outbreaks of mumps. For measles, a high year is assumed to begin with a large number of susceptibles (Sp plus the large number of reported cases from that September to the peak) and each low year is assumed to begin with a correspondingly smaller number of susceptibles. The more contagious the disease the more contacts each infective makes and hence the smaller the number of susceptibles at the peak. That the number of susceptibles should equal a constant at the peak of an outbreak is intuitive since at the peak the number of exposures per unit time is neither increasing nor decreasing. The idea may be justified in two ways. The result was found empirically by Hedrich (20), who estimated the number of susceptibles to measles for each month in Baltimore from 1900-1930 from census data and the number of reported cases. From 1900-1914 the average number of susceptibles at the peak of each outbreak was 63,700 with a coefficient of variation (standard deviation divided by the mean) of less than 4 per cent; for 1921-1930 the corresponding figures were 74,000 with a coefficient of variation of less than 6 per cent. (The population of Baltimore rose substantially in 1918 (14).) OUTBREAKS OF MEASLES, CHICKENPOX AND MUMPS. I. from year to year. The mean monthly contact rate for each calendar month is the average of the 30 (or 35) contact rates for that month. A mean monthly contact rate was calculated for each choice of T^_ and T2. The monthly contact rates calculated by the above method showed a "see-saw" highlow pattern even after they were averaged for all years. If the rate for one month was exceptionally high, the rate for the next month was unduly low. The raw monthly contact rates (/3r) were smoothed according to 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 mean monthly contact rates do not depend on the smoothing. Simulations. Equations 4-6 were used to simulate the recurrent outbreaks. For each choice of J"i and T^ the corresponding curve of the 12 mean monthly contact rates estimated from the data of monthly notifications was used. The constant net input of susceptibles y equalled the average annual number of reported cases of each disease. In most simulations it made little difference if the susceptibles were added equally throughout the year, or, to mimic the gathering of children in school, added at the beginning of the disease year. An arbitrary initial estimate of susceptibles and infectives was needed to begin the iteration of equations 4-6; after several years of simulated time, a pattern of stable, recurrent, undamped outbreaks that persist indefinitely was obtained. The estimation of susceptibles that was used in the calculation of monthly contact rates was not required for the simulations. Most simulations were made to determine under what conditions the equations would produce biennial outbreaks similar to those of measles in New York City (figure 1). In these outbreaks the average number of reported cases was about 30,000 in the high Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 altering the relative shape of the curve of the mean monthly contact rates or the results of the simulations. For measles the choice of Sf is quite critical. If Sp is decreased by about 15 per cent too few susceptibles are available at the end of the high year and the corresponding contact rates are systematically high; if Sp is decreased by about 10 per cent the simulations yield biennial outbreaks but the ratio of cases between the high and low years is greater than the observed ratio of 5:1. If Sj, is increased by about 10 per cent the simulations yield a ratio of cases that is less than the observed ratio; a 20 per cent increase in Sp yields simulations of annual outbreaks (ratio of cases of 1:1). Estimation of the mean monthly contact rate. A contact rate for each month of a 30or 35-year-period was calculated from the data of monthly notifications by the use of equations 4-6 in the following way. The number of susceptibles at the beginning of each of the 30 or 35 disease years was specified by the method described. A contact rate was found for each month, starting with September of the first year, such that the calculated number of exposures equalled the reported number of exposures for that month. (The contact rate was found by a "shooting" technique: successively smaller contact rates were tried until the calculated exposures equalled the reported exposures). To start the calculations the reported exposures for the preceding August were distributed equally throughout the month; thereafter, the pattern of exposures calculated for a month was used in calculating the contact rate for the next month. The number of susceptibles at the beginning of a month equalled the number of susceptibles at the beginning of the previous month minus the exposures for the month. (In some calculations new susceptibles were added each day throughout the year, but, in general, the constant net input of susceptibles, y, was zero.) Each September the number of susceptibles was specified; susceptibles were not carried over 459 460 LONDON AND YOBKE RESULTS Mean monthly contact rates. The mean monthly contact rates for measles, chickenpox and mumps in New York City and measles in Baltimore are shown in figure 2. In both cities the measles contact rates in June, July and August are low. The curves rise sharply from August to October, re- main high from November until March or April and then fall steeply from May to June. The ratio of the peak month to the lowest month is about 1.7 in New York; in Baltimore, 1.6. The contact rate for mumps shows similar features as the curves for measles. The curve for chickenpox, however, peaks in October and has a sharp decline from October to December; the ratio of the peak month to the lowest month is about 2. In a subsequent paper (11) these special features of the seasonal variation in the contact rate for chickenpox are related to the clustering of susceptibles with infectives and to the spread of chickenpox through the population. Figure 2 shows that for any given month the variation in the monthly contact rates from year to year for that month is small compared to the seasonal variation from month to month. The coefficients of variation (standard deviation divided by the mean) for measles, chickenpox, and mumps in New York City are about 10, 7 and 5 per cent, respectively, and about 18 per cent for measles in Baltimore. Some months show systematic differences in the mean monthly contact rates for all three diseases between the years with a large number of cases and years with fewer cases and for chickenpox between the early and late years of study. These systematic differences, which contribute to the variation in the monthly contact rates from year to year, are discussed in a subsequent paper (11). The curves of the mean monthly contact rates are not changed by any of the following alterations: 1) the length of the incubation period is changed by 50 per cent or a distribution of incubation periods is assumed (for measles as long as 9 to 16 days); 2) the infectious period is changed to one or three days; 3) the level of susceptibles (Sp) is changed by as much as 50 per cent (for measles, if Sp is decreased by 15 per cent, not enough susceptibles remain at the end of a high year and erroneously high contact rates are calculated for these months); 4) susceptibles are added each Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 year and about 6000 in the low years, a ratio of 5:1. For measles the criterion of a successful simulation was a recurrent outbreak every other year with a ratio of high year to low year cases of about 5:1. Multiple contacts of a susceptible. In the calculation of the mean monthly contact rates and in the simulations, the difference equations 4-6 were solved numerically on a computer. The step size A was selected such that further reduction had negligible effect on the results; one iteration per day was sufficient. Because the time interval of iteration was one day or less, and not 14 days (3, 4), the chance of a susceptible being contacted by more than one infective is very small. At the peak of the simulated outbreak of measles in New York City there are about 560,000 susceptibles and about 2400 exposures per day; the daily probability of exposure for any susceptible is 2400/560,000 or about .0043. The number of occurrences of a susceptible being contacted by two infectives in one day is (.0043)2- 560,000 = 10.4, which is negligible with respect to the 2400 exposures. Thus, the correction for multiple contacts, which is the distinguishing feature of the Reed-Frost model (4), changes the exposure rate by at most 0.43 per cent (10.4/2400). The Reed-Frost correction, usually employed when the iteration step size is 14 days, causes an error (and corresponding correction) 142 times larger than the correction here. That similar mean monthly contact rates and similar simulations are obtained with iteration step sizes of one day or as small as 1/8 of a day implies that multiple contacts of a susceptible are not important. OUTBREAKS OF MEASLES, CHICKENPOX AND MUMPS. I. 1.4 MEASLES 1.2 1.0 - •1 0.8 CHICKENPOX 1-2 h < i- o o day throughout the year; 5) the contact rates are calculated using models of ordinary differential equations that assume an incubation period (see appendix 1). Mean monthly contact rates calculated from data from individual boroughs in New York City show the same features as those calculated for the entire city. Simulations with a constant contact rate. The first result from the simulations is that seasonal variation in the contact rate is necessary to simulate undamped recurrent outbreaks that peak in the spring months. If the contact rate is assumed constant and susceptibles are added equally throughout the year, simulations yield damped waves of outbreaks that approach a constant endemic level of disease. This constant limiting solution is . 0.8 < UJ 0.6 MUMPS 1.2 1.0 - 0 8- 0.6 JJASONDJFMAMJJA FIGURE 2. Mean monthly contact rates for measles, chickenpox and mumps in New York City. The contact rates are normalized by #, the average of the 12 mean monthly contact rates. The bars show one standard deviation on either side of the mean. The dashed line in the top panel is the mean monthly contact rate for measles in Baltimore. Because of delays due to the incubation period and in reporting, the notifications from one month correspond to the contact rate of the previous month. For measles, T\ = 12 to 13 days: in New York City (1929-1963), $ = 7.25 X lO-«; in Baltimore (1925-1959), 0 = 2.65 X HT8. where p is the constant contact rate, T2 the infectious period, and y the constant net input of susceptibles. If the contact rate is assumed constant but the susceptibles are added only at the beginning of the disease year (to model the gathering of children in school), the simulations show outbreaks in which either the cases are distributed evenly in all months of the year or if a well defined peak occurs, the peak is not in the spring months. Simulations of measles epidemics: allowable incubation periods. The following simulations were done with the seasonally varying mean monthly contact rates calculated from the data of monthly notifications. An important factor in simulating biennial outbreaks of measles that have the observed ratio of high year to low year cases is the duration of the incubation period. 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 in New York City (1934-1963), 7\ = 16 days and £ = 5.64 X 10"«. For all diseases Tt = 2 days. Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 0.6 461 462 LONDON AND YORKE MEASLES 6000 4000 1000 MUMPS 1000 SONDJFMAMJJASONDJFMAMJJASONDJFMAMJJASONDJFMAMJJA MONTH FIGURE 3. Simulations of the recurrent outbreaks of measles, ehickenpox and mumps in New York City. The mean monthly contact rates and the values of 7\ and T2 from figure 2 were used. The constant net input of susceptibles, y, which equals the average annual number of reported cases, was measles—18,000; ehickenpox—9800; and mumps—6500. The dashed line 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 are simulated. For the mean monthly contact rate from New York City the ratio of high year to low year cases from the simulated outbreaks is 5:1 (figure 3); for the mean monthly contact rate from Baltimore the ratio from the simulated outbreaks is 6.5:1. If the incubation period is assumed to be shorter than 12 days, biennial outbreaks are not simulated. Too many cases occur when susceptibles are plentiful and in succeeding years not enough susceptibles and infectives are present to sustain the disease. (In the year following the high year the susceptibles are replenished but the infectives are now too low to sustain the disease. Introduction of additional infectives prevents the disease from fading out in that second year.) If the duration of the incubation period is 14 days, too few cases occur when Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 2000 OUTBREAKS OF MEASLES, CHICKENPOX AND MUMPS. I. simulated high year outbreak occurs earlier (March and April) than the peak of the simulated outbreak in the low year (April and May); further, the exposures in the summer after the low year are about 25 per cent higher than the exposures in the summer following the high year. These two qualitative features are not present in the simulation already mentioned in which the exposure rate is not modified. The role of the incubation period and level of infectiousness in determining epidemic patterns. Once the parameters are determined, the model allows us to determine the role of the incubation period and the role of the level of infectiousness in epidemic patterns. For example, if measles had a shorter (or longer) incubation period, would it still be endemic in the New York City area, and if endemic, would it still have a biennial pattern of outbreaks with a ratio of high year cases to low year cases of about 5:1? Equivalently, we may ask, suppose a virus disease appeared with the same level of infectiousness as measles, that is, with the same contact rate, but having a different incubation period, what would be the pattern of recurrent outbreaks? Using the contact rate obtained for measles (calculated using Tt = 12 to 13 and T2 = 2 days), we simulate these situations by assuming incubation periods of different lengths (T2 always equals two days). If the incubation period T\ is assumed to be three days, the peak of the outbreak occurs in December or January. The disease dies out after the peak, is no longer endemic, and the pattern of outbreaks appears similar to that of influenza (18). If the incubation period is increased slightly, the ratio of high year cases to low year cases becomes smaller. Finally, if Ti is greater than or equal to 16 days, annual outbreaks (ratio of cases is 1:1) that peak in the spring are simulated. We may similarly ask how the level of infectivity affects the pattern of measles outbreaks. If the infectivity is increased by 15 per cent (modeled by a 15 per cent Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 susceptibles are plentiful; biennial outbreaks are simulated but the ratio of high year to low year cases is about 3:1. For an assumed incubation period of 16 days the simulations show annual outbreaks (ratio of cases of 1:1). As long as the duration of the incubation period is 12 to 13 days, assumptions about the range of the distribution of incubation periods (19) and the length of period of infectivity are not critical. For example, the assumption that half the individuals incubate the disease for 12 days and half for 13 days gives the same simulation as the assumption that 1/8 of the individuals incubate the disease for 9 days, 1/8 for 10 days, . . . , and 1/8 for 16 days (the mean incubation period being 13 days). It also makes no difference if the period of infectivity is assumed to be one, two or three days. (There is a redundancy in the choice of difference values for r x and T2. For example, the assumption that half the individuals incubate the infection for 12 days and half for 13 days and then are infectious for one day is the same as the assumption that all individuals incubate the infection for 12 days and then are infectious for two days.) Simulations of measles outbreaks with a modified exposure rate. Mean monthly contact rates were also calculated with a modified exposure rate of the form p(t)S(t)I(t) (1 - cl(t)). As discussed in a subsequent paper (11) the modification eliminates a systematic difference in the mean monthly contact rates between the high and low years that occurs in the spring months. The maximum modification of the exposure rate is about 7 per cent at the peak of the outbreak, and the parameter c can be chosen small enough so that the number of infectives in the low years is so small that the modification has negligible effect. This modification produces simulations that most faithfully reproduce the observations in New York City. Biennial outbreaks with the observed ratio of cases of 5:1 are simulated (figure 3). Like the outbreaks in New York City (figure 1) the peak of 463 464 LONDON AND YOBKE DISCUSSION The mean monthly contact rates for measles, chickenpox and mumps that are estimated from reported monthly cases show substantial seasonal variation (figure 2). This cyclic variation is large relative to the variation from year to year. The shape of the curve of mean monthly contact rates does not change significantly when the contact rates are computed from a wide choice of values for the incubation period, the infectious period or the level of susceptibles. The curve of the mean monthly contact rate for measles in Baltimore is almost identical to the curve for measles in Xew York City. The increased contact rate in the autumn and winter months for measles, chickenpox and mumps suggests that this is an essential feature of any realistic model of recurrent outbreaks of these diseases in cities. Comparison of the contact rate of polio (that has a relatively short incubation period) with the contact rate of infectious hepatitis (that has a relatively long incubation period) would be useful in understanding the seasonal variation of these diseases. Because the absolute values of the mean monthly contact rates are inversely proportional to the level of susceptibles, which is difficult to determine, the absolute values of the mean contact rates are not a reliable measure of the actual number of contacts made by an infective per day or of the infectiousness of the three diseases. A valid measure of infectivity is discussed in a subsequent paper (11). The cause of the seasonal variation in the contact rates. What accounts for the seasonal variation in the contact rate? The contact rate is affected by two classes of factors: first, climatic factors that might enhance the transmission of infectious diseases, such as cold weather, decreased indoor relative humidity, or possibly decreased resistance to infectious diseases during colder months, and second, the social behavior of children aged 4—15, who presumably make more contacts when they are in school. The sharp rise and fall in the graph of the mean monthly contacts for all three diseases (figure 2) coincide with the opening and closing of school. The mean monthly contact rate for measles that is estimated from the reported cases from the low years (figure 1 in reference 11)) is approximately constant during the school year. These features suggest that the increased contacts made by children in school are the main cause of the seasonal variation in the monthly contact rates. In addition, the contact rate is high during the colder months when, both in school and in the home, children spend more time indoors with each other. Data of weekly or biweekly notifications might show drops in the contact rate due to holiday vacations from school, but the monthly totals analyzed here did not. For the years and cities Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 increase in the contact rate for all months) biennial outbreaks are simulated but the ratio of high year to low year cases is about 8:1; with a 20 per cent increase in inf ectivity too many cases occur when susceptibles are plentiful and the outbreak is not sustained in succeeding years. If the infectivity is decreased by 10 per cent (modeled by a 10 per cent decrease in the contact rate for all months) biennial outbreaks are simulated but the ratio of cases is only 3:1; an 18 per cent decrease in infectivity yields simulations of annual outbreaks. Siviulations of outbreak of chickenpox and mumps. Simulations with mean monthly contact rates for chickenpox (figure 2) reproduce the pattern of outbreaks in New York City, annual outbreaks that peak in February and March (figure 3). Simulations with the mean monthly contact rate for mumps (figure 2) show annual outbreaks that peak in March (figure 3). If the mean monthly contact rate for mumps (calculated with Tx = 16) is used in a simulation with Tx = 12, annual outbreaks are still simulated; this shows that a shortened incubation period is.not sufficient to .simulate biennial outbreaks of mumps. OUTBREAKS OF MEASLES, CHICKENPOX AND MUMPS. I. ishing susceptibles in the succeeding low year is not sufficient to explain the biennial pattern. Biennial outbreaks. A second question is why the annual variation in the contact rate produces annual outbreaks of chickenpox and mumps but biennial outbreaks of measles. As suggested by the simulations, biennial outbreaks of measles occur because the disease is sufficiently contagious and has a brief enough incubation period that sufficiently many cases occur when susceptibles are plentiful to deplete substantially the population of susceptibles. In comparison with chickenpox and mumps, the relatively small population of susceptibles to measles reflects the contagiousness of the disease. Indeed, an urban resident had almost no chance of escaping measles by age 20, but about one chance in three of not contracting chickenpox and one chance in two of not contracting mumps (15). In the simulations of measles outbreaks, the susceptible population is depleted in a high year by 40 per cent; in a low year by 7 per cent. In contrast, the population susceptible to chickenpox and mumps is depleted annually by less than 10 per cent. If in the calculation of the monthly contact rate the number of susceptibles at the peak of the outbreak (Sp) is increased by 20 per cent, the susceptible population is too large, the depletion during a high year is less and only annual outbreaks appear in the simulations. Conceivably, a three-year pattern could be established (and has been observed in Baltimore and elsewhere (25)) in which two years are needed to replenish the susceptibles; if, however, too many cases occur in a high year the susceptibles and then the infectives subsequently become depleted so that, without the introduction of extra infectives, the disease dies out. Contrary to findings based on another model (26), that small changes in infectivity do not affect periodicity, 5 per cent changes in the infectivity are enough to change substantially the ratio of cases in the high and low years, and an 18 per cent decrease in the infectiv- Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 for which we have data, apparently there were no instances when, except during the summer, school was closed for more than a month. In England and Russia, however, disruption in the usual pattern of school attendance during World War II, altered the seasonal and biennial patterns of the outbreaks of measles (21-23). The Russian workers (23) present other evidence including different seasonal incidences of measles between urban and rural populations, to show that the critical factors are social and not climatic. Finally, Hope Simpson (24) measured the infectiousness or contagiousness of measles, chickenpox and mumps in households during all months of the year. (In that study the infectiousness was defined as the proportion of exposures of susceptibles in the home leading to transmission of the disease.) In the home the infectiousness of the three diseases showed no seasonal variation. For these reasons, the seasonal variation in the contact rate is attributed to social factors, particularly the gathering of children in school. The necessity of seasonal variation in the contact rate. For the delay equation model studied here, if a simulation is to give undamped recurrent outbreaks that peak in the spring, the contact rate £(£) must have seasonal variation. Computer simulations in which the contact rate is constant and susceptibles are added equally throughout the year show waves of outbreaks of ever decreasing amplitude and the disease approaches a constant endemic level in society. Simulations in which the contact rate is constant but susceptibles are added only at the beginning of the disease year do not show cyclic outbreaks that peak in the spring. Although time delays in ordinary differential equations often introduce oscillatory or even periodic solutions, if the contact rate is constant, the delays of at most 20 days are insufficient to produce cyclic annual or biennial outbreaks. The idea, for example, that undamped biennial outbreaks can occur merely by depleting susceptibles in a high year and then replen- 465 466 LONDON AXD YOHKE measles but that has an incubation period of three days show that the peak of the outbreak occurs in the early winter, after which the outbreak dies out. The peak of the outbreaks of influenza, which has an incubation 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 widespread use of, for example, the measles vaccine, by introducing fewer susceptibles annually into the population. To model uniform use of the vaccine, the original monthly contact rate is used (because the total population is unchanged); the simulations show annual outbreaks. To model nonuniform use of the vaccine, that is, a subpopulation that has not received the vaccine, the monthly contact rate is increased proportionally (because the total population is smaller); the simulations show biennial outbreaks. The persistence of the biennial pattern of measles outbreaks in spite of widespread use of the vaccine, at least in Chicago (27) and New York City (figure 1) suggests nonuniform use of the vaccine. The simulations suggest the existence of a subpopulation that is not receiving the vaccine and that is depleted substantially after an outbreak. Indeed, the preschool population interacts socially with itself and represents a larger percentage of the cases than was previously true (27, 28). The replacement of the biennial pattern by annual outbreaks suggests uniform use of the vaccine. With nonuniform use of the vaccine, immunes should not be counted in the total population, and, as noted by others (8), the concept of herd immunity does not apply to the pockets of susceptibles within an otherwise totally immune urban population (29). REFERENCES 1. Serfling RE: Historical review of epidemic theory. Hum Biol 24:145-166,1952 2. Bailey NTF: The mathematical theory of epidemics. New York, Hafner, 1957 3. Soper HE: The interpretation of periodicity in disease prevalence. J R Stat Soc 92:34-73, 1929 Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 ity yields simulations of annual outbreaks. Although the infectivity of each disease is difficult to determine, chickenpox is apparently 35-65 per cent and mumps 19-^42 per cent as infectious as measles in society (11). Finally, to simulate biennial outbreaks in which the ratio of high year to low year cases is the observed ratio of 5:1, the mean duration of the incubation period must be 12 to 13 days. A longer incubation period (e.g., 16 days) yields simulations of annual outbreaks, and with a shorter incubation period too many cases occur when susceptibles are plentiful and the disease dies out. The simulations show, therefore, that measles is in a narrow border region between "highly efficient" and "less efficient" infectiou3 diseases. The hypothetical "highly efficient" diseases are highly infectious, have brief incubation periods (fewer than 12 days), deplete substantially the susceptible population during an outbreak and thus are not endemic in cities. Because a disease that is not endemic in cities would probably fail to perpetuate itself, there appears to be no example of a hypothetical "highly efficient" disease. An infectious disease that has a brief incubation period but a low contact rate would not be "highly efficient" and could be endemic in cities. The "less efficient" diseases, such as chickenpox and mumps, are less infectious, have relatively long incubation periods, do not deplete substantially the susceptible population during an outbreak and are endemic with a pattern of recurrent annual outbreaks. Outbreaks that peak in the spring. Incubation periods of at least 12 days also explain why these outbreaks peak in the spring months even though the contact rate rises sharply in the autumn months. With a "generation time" of about two weeks, seven or eight months are needed to build up the level of infectives and then deplete the susceptibles until the outbreak can no longer be sustained. Simulations of a disease that has the same contact rate as 467 OUTBREAKS OF MEASLES, CHICKENPOX AND MUMPS. I. 21. 22. 23. 24. 25. 26. 27. 28. 29. population "susceptible" to measles, 1900-1931, Baltimore, Md. Am J Hyg 17:613-636, 1933 Gunn W: Measles. Modern Practices in Infectious Fevers. Vol 2. Edited by HS Banks. New York, PB Hoeber, 1951, pp 499-520 Butler W: Whooping cough and measles, an epidemiological concurrence and contrast. Proc R Soc Med 40:384-398, 1947 Guslits SV: Measles. A Course in Epidemiology. Edited by II Elkin. New York, Pergamon Press, 1961, pp 353-362 Hope Simpson RE: Infectiousness of communicable diseases in the household (measles, chicken pox, and mumps). Lancet 2:549-554, 1952 Emerson H : Measles and whooping cough. Am J Public Health (Suppl) 27:1-153, 1937 Bartlett MS: The critical community size for measles in the United States. J R Stat Soc Series A 123 :37^4, 1960 Hardy CE, Kassanoff I, Orbach HG, et al: The failure of a school immunization campaign to terminate an urban epidemic of measles. Am J Epidemiol 91:2S6-293, 1970 Landrigan PJ, Conrad JL: Current status of measles in the United States. J Infect Disease 124:620-622, 1971 Scott H D : The elusiveness of measles eradication: insights gained from three years of intensive surveillance in Rhode Island. Am J Epidemiol 94:37-42, 1971 APPENDIX Models based on 1 ordinary differential equations. The incubation and infectious periods can also be modeled by ordinary differential equations that have no delays. In the following three models the average duration of the infectious state is 5 days. All three models implicitly assume that the probability of an infective ceasing to be infectious by time t is independent of the duration of his infectivity. Likewise, models two and three assume that the probability of an individual ceasing to incubate the disease by time t is independent of how long he has incubated the disease. The first model assumes no incubation period, that is, an exposed individual immediately becomes infectious. The equations of the first model are dS/dt = -p(l)S(t)I(f) + y dl/dt = p(t)S(t)I(t) - (l/5)/(0 (la) (2a) Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 4. Abbey H : An examination of the Reed-Frost theory of epidemics. Hum Biol 24:201-233, 1952 5. Bnrtlett MS: Deterministic and stochastic models for recurrent epidemics. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Berkeley and Los Angeles, University of California Press, 1956, Vol 4, pp Sl-109 6. Wilson LO: AJI epidemic model involving a threshold. Math Biosci 15:109-121, 1972 7. Ewy W, Ackerman E, Gatewood LC, et al: A generalized stochastic model for simulation of epidemics in a heterogeneous population (model VI). Comput Biol Med 2:45-58, 1972 8. Fox JP, Elveback L, Scott W, et al: Herd immunity: basic concept and relevance to public immunization practices. Am J Epidemiol 94:179-189, 1971 9. Hoppensteadt F, Waltman P : A problem in the theory of epidemics. II. Math Biosci 12:133-146, 1971 10. Bliss CI, Blevins DL: The analysis of seasonal variation in measles. Am J Hyg 70:328-334, 1959 11. Yorke JA, London WP: Recurrent outbreaks of measles, chickenpox and mumps. II. Systematic differences in contact rates and stochnstic 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 of Health, The City of New York, 1968 14. Baltimore Health News 48:10. Published by Baltimore City Health Department, 1971 15. Collins SD, Wheeler RE, Shannon RD: The occurrence of whooping cough, chicken pox, mumps, measles and German measles in 200,000 surveyed families in 28 large cities. Special Study Series, No 1, Division of Public Health Methods, NIH, USPHS, Washington DC, 1942 16. Wilson EB, Burke M: The epidemic curve. Proc Natl Acad Sci USA 28:361-367, 1942 17. Chope H D : A study of factors that influence reporting of measles. Virus and Rickettsial Diseases. A Symposium held at the Harvard School 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 Human Viral Infections). Philadelphia, WB Saunders Company, 1970 19. Sartwell P E : The distribution of incubation periods of infectious disease. Am J Hyg 51:310-318, 1950 20. Hedrich AW: Monthly estimates of the child 46S LOXDOX AXD YORKE Equation la is the same as Equation 2 of the delay equation model. The second model assumes an incubation state W that has mean life of 6 days. The equations are la and dW/dt = 0(05(0/(0 ~ O./0)W(t) (3a) cll/dl (4a) = (1/0)1^(0 - (1/5)7(0 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 = 2 days, in model two, 6 = 12 days, and in model three, r = 12. The graph of the mean monthly contact rates for measles that is calculated using model one is irregular, has large unsystematic differences between the contact rates calcu- Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013 A more complicated approximation to the delay of the incubation period is to assume a sequence of incubation states W\, . . . , Wr each with a mean life of one day. The equations of model three are la and lated irom the high and low years, and shows no consistent seasonal variation. The mean monthly contact rates for measles, calculated using models two and three, are very similar to the contact rates calculated from the delay equation model (figure 2). For models two and three the simulations of the measles outbreaks that use the seasonally varying contact rates show biennial outbreaks but the ratio of high year to low year cases is too low. For model two the ratio is about 2.5; for model three the ratio is about 3.3. In comparing the delay equation model and the ordinary differential equation models, the model that assumes no incubation state (model one) is too simplistic to allow the calculation of realistic contact rates. The models that assume an incubation state (models two and three) yield realistic contact rates but the simulations are not satisfactory. Clearly, ordinary differential equation models without delays that are more complicated than model three can approximate the incubation period, but the simplest and most satisfactory model is the delay equation model.