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

Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013

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

Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013

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

Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013

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-

Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013

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
Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013

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

Downloaded from http://aje.oxfordjournals.org/ at Stanford Medical Center on September 15, 2013

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

Downloaded from

## No comments:

## Post a Comment