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 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. 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 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.
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
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454 LONDON AND YOBKE
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
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
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OUTBREAKS OP MEASLES, CHICKENPOX AND MUMPS. I. 455
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 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
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456 LONDON AXD YORKB
reflects the social behavior of the members
of society and the ease with which the
disease is transmitted; both factors may
vary during the year. Suppose that, at time
t, 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 fraction
of susceptibles exposed per day by all
infectives is E(t)/S{t) and the contact rate
fi(t) is given by
The number of exposures per day is given
by
E{t) = (1)
(Alternatively if v is the total number of
contacts of both susceptibles and immunes
made by an infective per day and N is the
total population, then the number of exposures
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 of
susceptibles into the population is constant.
This net rate is the sum of the rates of
entry of susceptibles from births and from
immigration minus the rate of loss of susceptibles
from emigration and the rate of
the loss of individuals who do not acquire
the disease by, for example, age 20, and
hence leave the school-aged susceptible
population. (In 1935-1936, for example, by
age 20, 5 per cent of an urban population
had no history of measles, 32 per cent no
history of chickenpox, and 50 per cent no
history of mumps (15).) The assumption of
a constant input of susceptibles is, of
course, an approximation. The birth rate in
New York and Baltimore, for example,
decreased by about 30 per cent during the
1930's and rose again during the next two
decades (13, 14). Significant immigration
and emigration also occurred in both these
cities, but it is virtually impossible to measure
these migrations or to know the fraction
of the immigrant or emigrant population
that was susceptible.
Since we are interested in diseases that
can be acquired only once, the rate of
change of susceptibles, dS/dt, equals the
constant net rate of entry of susceptibles y
minus the rate of exposure E(t), that is
dS/dt = y - E{t) (2)
This equation implies that after being contacted
by an infective a susceptible immediately
leaves the susceptible population; it
is shown later that multiple contacts of a
susceptible can be neglected.
We assume that all individuals exposed
at time t incubate the disease for time Tx,
are infectious for time T% and then cease to
be infectious and remain permanently immune
to reinfection. The rate of change of
infectives dl/dt equals the rate of appearance
of infectives, which is the exposure
rate time 7\ ago, minus the rate of disappearance
of infectives, which is the exposure
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 of
infectives at any time equals the sum of the
exposures made in the previous Ti 4- T2 to
Ti days. (In the more general formulation
of the problem (12) the definition of I(t)
differs from the definition here by the multiplicative
factor T->.)
The basic equations 1, 2 and 3 follow
naturally from the assumptions that the
rate of exposure is proportional to both the
number of susceptibles and the number of
infectives, that the disease confers permanent
immunity and that there exists an
incubation and an infectious period. The
same equations appear, for example, in the
work of Wilson and Burke (16).
The two delays—T1? the time from exposure
to infectivity and T2, the duration of
infectivity—require interpretation. We are
interested in the spread of disease in society,
and not among siblings in a house-
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OUTBREAKS OF MEASLES, CHICKENPOX AND MUMPS. I. 457
hold (the cases of which are less likely to
affect the spread of disease in society and
less likely to be reported (17)); therefore,
we assume that an infective ceases to transmit
the disease when he is confined at home
by the severity or the characteristic features
of the disease (e.g., the rash). The
infective is assumed to manifest constant
infectivity for a period of time equal to T2
before being withdrawn from society. 7\ is
then the time from exposure to the beginning
of infectivity; although called the
incubation period, it is perhaps a few days
shorter than the usual incubation period,
which is defined as the time from exposure
to the onset of symptoms. The choices of
the length of the delays, Tx and T2 are
based on the epidemiology of the individual
diseases (18, 19).
For purposes of computation we use a
fixed time interval A, usually one day, let
the nth time interval be tn = nA, and
approximate equations 1-3 by the difference
delay 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 and
models without delays. A broad distribution
of incubation periods similar to that reported
by Sartwell (19) can be incorporated
into the delay equation model by
assuming, for example, that 1/8 of the
exposed individuals incubate the disease for
9 days, 1/8 for 10 days, . .. , 1/8 for 16
days. 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 in
which half the exposed individuals incubate
the infection for 12 days and half for 13
days is denoted by T± - 12 to 13.)
The incubation and infectious periods can
be also modeled by ordinary differential
equations without delays (see appendix 1).
METHOD
The data used to calculate the monthly
contact rates are the notifications of cases
of each infectious disease received by the
city health departments (usually by postal
card) during any month. (The monthly
notifications are given in the appendix of
the following paper (11).) Since the delay
from exposure to diagnosis is about two
weeks and the delay from diagnosis to the
receipt of notification in the health department
is estimated to be about 10 days, the
monthly totals for one month represent
mainly the exposures from the previous
month. Thirty-five consecutive years of
data (prior to the use of the vaccine) were
used to calculate the monthly contact rates
of measles in New York City and Baltimore
and 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, a
high year is a disease year in which many
cases were reported (in New York City,
greater than 21,000; in Baltimore, greater
than 4900 cases) and a low year is a disease
year in which few cases were reported (in
New York City, less than 13,000; in Baltimore,
less than 3800 cases).
In order to calculate a contact rate for
each month of the 30 or 35 disease years,
the number of susceptibles at the beginning
of each epidemic year was estimated. The
estimation of susceptibles is independent of
the model or the choice of parameters of the
model {T-i, T2, y, or fi(t)). A mean contact
rate for each calendar month was then
estimated from the data of monthly notifications.
This was done for each choice of
Tj, the duration of the incubation period,
and r2, the duration of the infectious period.
The constant net input of susceptibles,
7, was not required in the estimation of the
monthly contact rates. Finally, for each
choice of Tx and T2 the corresponding
estimated mean monthly contact rates and
a constant net input of susceptibles, y, was
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458 LONDON AND YOEKE
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).)
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
70,000
85,000
90,000
20,000
Average
annual
reported
casej
18,000
9,800
6,600
5,000
Ratio
3.9
8.8
13.7
4.0
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OUTBREAKS OF MEASLES, CHICKENPOX AND MUMPS. I. 459
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 30-
or 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
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
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