Herd Immunity: History, Theory, Practice
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Epidemiologic Reviews Vol. 15, No. 2
Copyright © 1993 by The Johns Hopkins University School of Hygiene and Public Health Printed in U.S.A.
All rights reserved
Herd Immunity: History, Theory, Practice
Paul E. M. Fine
INTRODUCTION Along with the growth of interest in herd
immunity, there has been a proliferation of
Herd immunity has to do with the pro- views of what it means or even of whether
tection of populations from infection which it exists at all. Several authors have written
is brought about by the presence of immune of data on measles which "challenge" the
individuals. The concept has a special aura, principle of herd immunity (3-5) and others
in its implication of an extension of the pro- cite widely divergent estimates (from 70 to
tection imparted by an immunization pro- 95 percent) of the magnitude of the herd im-
gram beyond vaccinated to unvaccinated in- munity threshold required for measles eradi-
dividuals and in its apparent provision of a cation (6-8). Still other authors have com-
means to eliminate totally some infectious mented on the failure or "absence" of herd
diseases. It is a recurrent theme in the medi- immunity against rubella (9, 10) and diph-
cal literature and has been discussed fre- theria (11). Authorities continue to argue
quently during the past decade. This new over the extent to which different types of
popularity comes as a consequence of sev- polio vaccine can, let alone do, induce herd
eral recent major achievements of vaccina- immunity (12-14). Given such differences
tion programs, i.e.: the historic success of of opinion, there is need for clarification.
the global smallpox eradication program; Many authors have based their discus-
dramatic increases in vaccination coverage sions of herd immunity on an influential pa-
stimulated by national programs and by the per published in 1971 by Fox et al. titled
Expanded Programme on Immunization; the "Herd immunity: basic concept and rel-
commitment of several countries to eradi- evance to public health immunization prac-
cate measles; and international dedication to tices" (15). This paper took as its starting
eliminate neonatal tetanus and to eradicate point a medical dictionary's definition of
poliomyelitis from the world by the year herd immunity as "the resistance of a group
2000.' to attack by a disease to which a large pro-
portion of the members are immune, thus
lessening the likelihood of a patient with a
Received for publication January 27, 1993, and in
final form July 29, 1993. disease coming into contact with a suscep-
From the Communicable Disease Epidemiology Unit, tible individual" (16). While useful, even
London School of Hygiene and Tropical Medicine, this definition lends itself to different inter-
Keppel Street, London WC1, England. (Reprint re-
quests to Dr. Paul E. M. Fine at this address.)
pretations; these may be either quantitative
(herd immunity as partial resistance, re-
1
Though the words "eradicate" and "eliminate" have flected in reductions in frequency of disease
been used interchangeably by some authors in the past,
current usage of eradication implies reduction of both
due to reductions in numbers of source cases
infection and disease to zero whereas elimination im- and of susceptibles) or qualitative (herd im-
plies either regional eradication, or reduction of disease munity as total resistance, implying a
incidence to some tolerably low level, or else reduction
of disease to zero without total removal of the infectious
threshold number or percentage of immunes
agent (1). Thus the 42nd World Health Assembly rec- above which an infection cannot persist).
ommended "elimination of neonatal tetanus by 1995 Each of these interpretations has its place,
and global eradication of poliomyelitis by the year 2000" but they are sometimes confused in debates
(2).
265266 Fine
on the subject. A given population may ex- Wilson introduced the term in the follow-
hibit one (partial, quantitative) without the ing manner: "Consideration of the results
other (total, qualitative) form of herd im- obtained during the past five years . . . led
munity. It will be found that such definitions us to believe that the question of immunity
do not easily fit situations in which vaccine- as an attribute of a herd should be studied
derived immunity is transferred either di- as a separate problem, closely related to,
rectly (as in the case of maternal antibodies but in many ways distinct from, the prob-
against tetanus) or indirectly (as in the case lem of the immunity of an individual host"
of secondary spread of oral polio vaccines) (38, p. 243). After describing experiments
between members of a population, or in showing that immunized mice had lower
which vaccines impart different levels of mortality rates from, and were less likely
protection against infection, disease, or to transmit, Bacillus enteritidis, the au-
transmission (as in diphtheria, pertussis, and thors concluded by posing an " . . . obvious
perhaps malaria). problem to be solved.... Assuming a
The paper of Fox et al. (15) is also of im- given total quantity of resistance against a
portance because of its method and the na- specific bacterial parasite to be available
ture of the conclusions which were dictated among a considerable population, in what
by that approach. Sufficient years have now way should that resistance be distributed
elapsed for both the method and the con- among the individuals at risk, so as best to
clusions to be reviewed in perspective. ensure against the spread of the disease, of
Interest in applying the "magic" of herd which the parasite is the causal agent?"
immunity in disease control has encouraged (38, pp. 248-9). Wilson later recalled that
mathematical research exploring the theo- he had first heard the phrase "herd immu-
retical implications of the subject (6-8, 17— nity" in the course of a conversation with
37). Though much of this work has been Major Greenwood (G. S. Wilson, London
published in journals and in language unfa- School of Hygiene and Tropical Medicine,
miliar to the medical and public health com- personal communication, 1981); and
munities, its isolation has been reduced in Greenwood employed it in his 1936 text-
recent years largely through the publications book Epidemics and Crowd Diseases (40).
of Anderson and May and their colleagues Although these authors did not distinguish
(8, 17, 20, 21, 23, 24, 28, 29, 31, 33, 36). clearly between direct and indirect protec-
It is the intent of this review to bring to- tion stemming from vaccine-derived im-
gether the literature on the history, theory, munity, later authors picked up the phrase
and practical experience of herd immunity, and applied it in particular to the indirect
to consider the variety of issues raised by the protection afforded to nonimmune indi-
application of the concept to different dis- viduals by the presence and proximity of
eases, and to consider how well current others who are immune.
theory and practice correspond with each
another. That the presence of immune individuals
could provide indirect protection to others
was itself recognized at least as far back as
HISTORY
the 19th century. Farr had noted in 1840 that
The first published use of the term "herd "The smallpox would be disturbed, and
immunity" appears to have been in a paper sometimes arrested, by vaccination, which
published in 1923 by Topley and Wilson protected a part of the population . . ." (41).
titled "The spread of bacterial infection: Such observations, that epidemics often
the problem of herd immunity" (38). This came to an end prior to the involvement of
was one of a classic series of studies by all susceptibles, led in turn to a major epi-
these authors on epidemics of various in- demiologic controversy in the early years of
fections in closely monitored populations this century. This controversy was between
of laboratory mice (39). Topley and those who believed that epidemics termi-Herd Immunity 267
nated because of changes in the properties of mathematical epidemiology relating to
the infectious agent (e.g., loss of "virulence" vector-borne diseases has been repeatedly a
resulting from serial passage) (42) and those source of important insights for the field of
who argued that it reflected the dynamics of vaccination and herd immunity.
the interaction between susceptible, in-
fected, and immune segments of the popu- THEORY
lation (43). Each argument was supported
by observations and by mathematical rea- Three separate theoretical perspectives
soning (44). It was the latter explanation that have been used to derive measures of herd
won the day; and its simple mathematical immunity. Over recent years, these perspec-
formulation, the "mass action principle," tives have converged into a general theory.
which has become a cornerstone of epide-
miologic theory, provides one of the sim- The mass-action principle
plest logical arguments for indirect protec-
tion by herd immunity. The theoretical basis of herd immunity
The concept of herd immunity is often in- was introduced by Hamer (43) in 1906 in the
voked in the context of discussions of dis- context of a discussion of the dynamics of
ease eradication programs based on vacci- measles. Hamer argued that the number of
nation. It is significant that both Jenner (45) transmissions (he called it the "ability to in-
and Pasteur (46), key figures in the early fect") per measles case was a function of the
development of vaccines, recognized the number of susceptibles in the population.
potential of vaccines to eradicate specific We can paraphrase his argument as:
diseases, but neither appears to have con-
sidered the practical issues closely enough C, + JC, varies with S,, (1)
to have touched on herd effects. Further- where S, and C, are numbers of susceptibles
more, the major focus of eradication think- and cases, respectively, in some time period
ing in the first half of this century did not t, C, +, is the number of cases in the suc-
involve vaccines or vaccine-preventable ceeding time period, and Cl+l/C, is, thus,
diseases at all, but concerned vector-borne the number of successful transmissions per
diseases, malaria in particular. This current case (see figure 1). The time period
stemmed from the writings of Ross (47) used in this formulation is the average in-
who, in work on the dynamics of malaria, terval between successive cases in a chain of
had deduced that it was not necessary to transmission, sometimes called the "serial
eliminate mosquitoes totally in order to interval" (50), which is approximately 2
eradicate the disease. Ross's so-called weeks for infections such as measles and
"mosquito theorem" was the first recogni- pertussis (see table 1). This relation can be
tion of a quantitative threshold which could expressed:
serve as a target for a disease elimination
program. So powerful was the argument,
and so influential was the tradition of quan-
titative thinking which it engendered, that where r is a transmission parameter, or
the World Health Organization attempted "contact rate," in effect the proportion of all
global eradication of malaria before that of possible contacts between susceptible and
any other disease (48).2 This tradition of infectious individuals which lead to new in-
fections. In order to simulate successive
changes over time, the number of suscep-
1955 World Health Assembly recommended
that the World Health Organization take the initiative in tibles is recalculated for each new time
"a programme having as its ultimate objective the world- period as
wide eradication of malaria." It was not until 1965 that
the Assembly first declared "the worldwide eradication
of smallpox to be one of the major objectives of the
organization" (49). where 5, + , is the number of susceptibles in268 Fine
TIME- TABLE 1. Approximate serial intervals, basic
reproduction rates (in developed countries) and
NEXT implied crude herd immunity thresholds
TIME
PERIOD
(H, calculated as 1 - 1/ff0) for common
potentially-vaccine-preventable diseases. Data
from Anderson and May (8), Mcdonald (54), and
Benenson (135). It must be emphasized that the
SUSCEPTIBLES
values given in this table are approximate,
and do not properly reflect the tremendous
range and diversity between populations.
CASES They nonetheless give an appreciation of
order-of-magnitude comparability
Serial interval
Infection (%)
IMMUNES (range)
Diphthenat 2->:30 days 6-7 85
Influenza}: 1-10 days ? ?
DEATHS Malaria§ >20 days 5-100 80-99
FIGURE 1. Relation between susceptibles (S), infec- Measles|| 7-16 days 12-18 83-94
tious cases (C), and immunes (/) in successive time Mumps 8-32 days 4-7 75-86
intervals (t, t + 1) in the simple discrete time mass action
Pertussisl 5-35 days 12-17 92-94
or Reed-Frost models. In each time period some
(Ci + i) susceptibles become cases and the others re- Polio# 2-45 days 5-7 80-86
main susceptible. Each case is assumed to remain in- Rubella 7-28 days 6-7 83-85
fectious for no more than a single time period (= serial Smallpox 9-45 days 5-7 80-85
interval). B, individuals may enter as susceptible births Tetanus NA* NA NA
during each time period (e.g., equation 3). Note that Tuberculosis** Months-years ? ?
neither the simple mass action (equations 2 and 3) nor
Reed-Frost (equation 9) equations include an explicit * flo, basic case reproduction rate; H, herd immunity thresh-
term for immunes. By implication, deaths prior to infec- old defined as the minimum proportion to be immunized in a
population for elimination of infection; NA, not applicable.
tion are not considered in these simplest models and the t Long-term infectious carriers of Corynebacterium diphthe-
total population is assumed constant (i.e., in each pe- riae occur. See the text for a discussion of the definition of im-
riod the same number of immunes die as susceptibles munity.
are born into the population). t Ro of influenza viruses probably varies greatly between
subtypes.
§ All these variables differ also between Plasmodium spe-
cies The serial interval may extend to several years. See the
text for a discussion of implications of genetic subtypes.
the next time period and B, is the number of || See the text for a discussion and variation in estimates of
Ro in table 5.
new susceptibles added (e.g., born into) to H See the text for a discussion relating to the definition of
the population per time period. immunity in pertussis.
# Distinct properties of different polio vaccines need to be
The relation in equation 2, that future in- considered in interpreting the herd immunity thresholds.
cidence is a function of the product of cur- " f l o has been declining in developed countries; protective
immunity is not well defined.
rent prevalence times current number sus-
ceptible, has become known as the
theoretical work on the dynamics of infec-
epidemiologic "law of mass action" by anal-
tions in populations (23, 52).
ogy with the physical chemical principle
that the rate or velocity of a chemical reac- Figure 2 illustrates what happens when
tion is a function of the product of the initial equations 2 and 3 are iterated and serves to
concentrations of the reagents.3 Often ex- illustrate several fundamental principles of
pressed as a differential (continuous time) the epidemiology of those acute immunizing
rather than a difference (discrete time) equa- infections (such as measles, mumps, rubella,
tion, as here, this relation underlies most chickenpox, poliomyelitis, pertussis, etc.)
which affect a high proportion of individuals
in unvaccinated communities.
3
This analogy was apparently first made by Soper First, the model predicts cycles of infec-
(51). The inspiration from physical chemistry is of more
than passing interest in that it reflects a tradition among tion incidence, such as are well recognized
biomedical theorists to strive for the simplicity and el- for many of the ubiquitous childhood infec-
egance of the physical sciences. Not only mass action, tions (figure 3). The incidence of infection
but also the concepts of catalysis and of critical mass
have close analogies in the behavior of infections, as cycles above and below the "birth" rate, or
mentioned below. rate of influx of new susceptibles.Herd Immunity 269
14
Susceptibles (S )
0 4
(0
CD
1 2
Cases (C{) Births (B
0 10 20 30 40 50 60 70 80 90 100
Time Periods (Serial Intervals)
FIGURE 2. Mass action model. Results obtained on reiteration of equations 2 and 3. The illustrated simulation was
based on 12,000 susceptibles and 100 cases at the start, r = 0.0001 and 300 births per time period. Note that the
incidence of cases cycles around the birth rate and that the number of susceptibles cycles around the epidemic
threshold: S a = 1/r= 10,000.
Second, the number of susceptibles also and the relation between the interepidemic
cycles, but around a number which is some- interval and the time required for the number
times described as the "epidemic threshold," of susceptibles to reach the epidemic thresh-
Se. Simple rearrangement of equation 1 to old (23, 43, 51, 52). Though it was not em-
Ct+l/C, = S, r reveals that this threshold is phasized explicitly by the earlier authors,
numerically equivalent to the reciprocal of who dealt in numbers or "density," rather
the transmission parameter r; as incidence than proportions, of susceptibles, the epi-
increases (i.e., C, + , > C,) when, and only demic threshold provides a simple numeri-
when, S, > 1/r; and, thus, Se = 1/r. This cal measure of a herd immunity criterion. If
important relation is implicit in Hamer's the proportion immune is so high that the
original paper (43), and was formalized as a number of susceptibles is below the epi-
"threshold theorem" in 1927 by Kermack demic threshold, then incidence will de-
and McKendrick (53). The principle may be crease. We can express this algebraically as:
illustrated by analogy with the physical con-
cept of a "critical mass"—the epidemic H = 1 - SJT = 1 - 1//T (4)
threshold represents a critical mass (density where T is the total population size, Se is the
per some area) of susceptibles, which, if ex- epidemic threshold number of susceptibles
ceeded, will produce an explosive increase for the population, and H is the herd immu-
in incidence of an introduced infection. The nity threshold, i.e., the proportion of im-
correspondence between the case and sus- munes which must be exceeded if incidence
ceptible lines in figure 2 illustrates this re- is to decrease.
lation. Figure 4 presents another way of illustrat-
Hamer and his successors used this logic ing the herd immunity threshold, i.e., in
to explain several aspects of the dynamics of terms of the relation between the proportion
measles and other childhood infections, immunized at birth and the ratio of the cu-
such as cyclical epidemics, the persistence mulative incidence during the postvaccina-
of susceptibles at the end of an epidemic, tion period to that during the prevaccination270 Fine
Measles: England and Wales B Pertussis: England and Wales
Measles notifications
S 8
Year Year
Measles: USA Pertussis: USA
Year Year
FIGURE 3. Reported incidence of common childhood vaccine-preventable diseases. Measles showed a tendency
to biennial epidemics in England and Wales prior to vaccination (A). This pattern was less dramatic in data for the
entire United States (C) because of the size and heterogeneity of the population (not all areas were in phase with
one another). All areas showed a strong seasonal oscillation in addition to the biennial cycle. Pertussis shows a 3-4
year cycle with little obvious seasonality in the United Kingdom (B). This cycling is also seen in national data for the
United States prior to 1970 (D). Notification efficiency was approximately 60% for measles in England and Wales
prior to vaccination (55) but was considerably lower for pertussis and for both diseases in the United States.
period, either among those not immunized at vent the number of susceptibles from reach-
birth (figure 4A), or in the entire population ing the epidemic threshold, then incidence
(figure 4B). Insofar as the immunization of should continue to decline, ultimately to ex-
individuals removes both susceptibles and tinction. Hamer's original principle implied
potential sources of infection from the com- the simplistic assumption of an homoge-
munity, it will lead to a reduction in inci- neous, randomly mixing population, like
dence rates and, hence, in cumulative inci- that of molecules in the ideal gasses for
dence. If the proportion immunized at birth which the mass action principle was most
is maintained at or above the threshold, H, appropriate. However, given the power of
then the cumulative incidence is reduced to the analogy, elaboration of the theory was
zero, indicating that the infection has been only a matter of time.
eliminated from the population.
It was only many years after Hamer that Case reproduction rates
the wide use of vaccines meant that these
epidemic and herd immunity thresholds If an infection is to persist, each infected
could be considered as targets for interven- individual must, on average, transmit that
tion. If appropriate vaccination could pre- infection to at least one other individual. IfHerd Immunity 271
IF NO INDIRECT PROTECTION
1.0
. A
X IF INDIRECT
PROTECTION
\
\
OCCURS \
0% 50% 100%
% IMMUNIZED AT BIRTH
1.0 n
O ui
PROPORTION NOT
IMMUNE BUT W/A
D 2
STILL ESCAPE V///<
2 V INFECTION
§ B
^ i
z 5 \ ,
^t IF NO
UJ U. ^ INDIRECT
O O ^ PROTECTION
o z
z o IF INDIRECT 3 ^ /
PROTECTION —' X^>/
2
S
5
s. 0 H
0% 50% 100%
% IMMUNIZED AT BIRTH
FIGURE 4. Cumulative incidence (e.g., per lifetime) of infection after a vaccination program as a proportion of prior
cumulative incidence among individuals not immunized by the vaccine (A) and among the total population (B). In
each diagram the dotted line refers to an infection for which the vaccine offers no indirect protection (e.g., tetanus
vaccination of males) and the solid line refers to an infection for which the vaccine does impart indirect protection
(e.g., measles). The vertical distance between the two lines reflects the nonimmunized individuals who escape
infection as a proportion of all nonimmunized individuals (A) or of the total population (B).
this does not occur, the infection will dis- tistic is one which was formulated originally
appear progressively from the population. by Macdonald (54), in the context of malaria
This average number of actual infection studies, as the average number of secondary
transmissions per case is an extremely pow- cases who contract an infection from a
erful concept, and has thus been discussed single primary case introduced into a totally
by many researchers. The fundamental sta- susceptible population. He called this num-272 Fine
ber the "basic case reproduction rate", by rate Rn should be equivalent to the basic case
analogy with the demographic concept of reproduction rate Ro times the proportion
the intrinsic reproduction rate, the average susceptible in the population:
number of potential progeny per individual
if there were no constraints to fertility (26). = R0S,/T. (6)
This definition can be translated directly
This has interesting implications. If an en-
into the mass action equation (equation 2) by
demic infection persists in a population of
letting C, = 1 and 5, = T, to represent the
constant size, then Rn should, on average,
single case introduced into a fully suscep-
over a long period of time, be equivalent to
tible population. The number of secondary
unity (i.e., each case leads on average to a
cases, Cl+i, is then equivalent, by defini-
single subsequent case). Therefore, "on av-
tion, to the basic case reproduction rate (Ro):
erage" from equation 6:
R0 = Tr. (5)
Ro = Tlaverage S, = T/Se. (7)
On reflection, we appreciate that this basic
case reproduction rate describes the spread- In words, for endemic infections, the basic
ing potential of an infection in a population, case reproduction rate should be equivalent
and that it will be a function both of the to the reciprocal of the "average" proportion
biologic mechanism of transmission and of susceptible in the population. That the av-
the rate of contact or interaction between erage number of susceptibles is equivalent
members of the host population. Analogous to Se should be evident from figure 2. An
or identical statistics have been defined by important implication of this relation is the
several authors, and given different names prediction that the average proportion sus-
such as "expected number of contacts" (15), ceptible should remain constant in a popu-
"contact number" (25), or "basic reproduc- lation, even in the face of extensive and ef-
tion number" (26).4 Examples of numerical fective vaccination, as long as the infection
values of this statistic, applicable to differ- remains endemic (and as long as the popu-
ent infections and derived by methods de- lation remains of constant size). Analysis of
scribed below, are shown in table 1. A data on measles has confirmed this relation
simple way of illustrating the concept is pre- (55).
sented in figure 5A. Combination of equations 4 and 7 pro-
Of course, in the real world there are con- vides us with an expression for the herd im-
straints to unlimited infection transmission. munity threshold in terms of Ro:
For example, some of the "contacts" of an H=1-VRO = (Ro - (8)
infected person may be individuals who are
already infected or immune. As a result, the This is illustrated graphically in figure 6
average number of actual infection trans- which shows the implications for persis-
missions per case, in a real population, will tence or eradication of infections depending
be less than the basic case reproduction on the proportion of immunes in the popu-
rate, and has been defined, again first by lation.5
Macdonald (54), as the "net reproduction
rate" /?„. Other authors have called this the The Reed-Frost heterogeneous
"actual" or "effective" reproduction rate population simulation approach
(23). This is illustrated in figure 5B. It is
clear from figure 5 that the net reproduction The paper by Fox et al. (15) cited in the
introduction has been one of the most fre-
"•Different symbols have been used for the statistic by
5
different authors. The original work by Macdonald (54) This important relation was published explicitly first
employed ZQ for the basic reproduction rate. Several by Dietz (18), in 1975, though it is implicit in some earlier
authors have noted that the statistic is not a proper rate, work, in particular a graph published by Smith (56) in
but that term is now imbedded in the literature (26). 1970.Herd Immunity 273 FIGURE 5. Cartoon illustrating implications of a basic reproduction rate Ro = 4. In each successive time (serial) interval, each individual has effective contact with four other individuals. If the population is entirely susceptible (A), incidence increases exponentially, fourfold each generation (until the accumulation of immunes slows the process). If 75% of the population is immune (B), then only S/T= 25% of the contacts lead to successful transmissions, and the net reproductive rate Rn = Ro (S/T) = 1. quently cited references on herd immunity. cines. By 1971, the initial successes and fail- This paper is of historical interest, and also ures of these programs were on record (e.g., of interest because of its theoretical argu- figure 3C), and Fox et al. set out to explain ment and conclusions. them. The appearance of the Fox et al. paper in They based their theoretical argument not 1971 was significant. Four years before, in on the mass action arguments outlined 1967, the World Health Organization had above, but on an alternative approach, declared its intention of eradicating small- rooted in the Johns Hopkins University pox from the world within 10 years, and the School of Hygiene and Public Health (58). United States Public Health Service had de- This model, named the Reed-Frost for clared its intention of eradicating measles its developers Lowell Reed and Wade from the United States within 1 year (57). Hampton Frost, assumes the same discrete Both of these tasks were to be achieved by time schema illustrated in figure 1 but pro- the induction of herd immunity with vac- poses an alternative to the mass action equa-
274 Fine
,100 for births (B, in equation 3), the authors
could only address questions relating to epi-
demics in closed populations.
£ 75
Their first step was to explore these equa-
H = (R0-1)/RQ tions for simple randomly mixing popula-
50
tions. Table 2 presents a portion of the initial
results, on the basis of which the authors
25 concluded " . . . application of the Reed-
Frost model... demonstrates that, over a
CD
wide range of variations, the number of sus-
10 20 30 40 50 ceptibles and the rate of contact between
Basic Reproduction Rate (R-. ) them determine epidemic potentials in ran-
domly mixing populations. If these are held
FIGURE 6. Relation between herd immunity threshold constant, changes in population size and,
(H) and basic reproduction rate Ro, as in equation 8: H
= 1 - MR0. therefore, in the proportion immune do not
influence the probability of spread" (15, p.
182). The emphasis in this conclusion on
numbers and probability of spread deserves
tion (equation 2 above) as: comment. The perspective reflects the pa-
C f + 1 = S,{l - ( 1 - p)Q} (9) per's focus on epidemic potential in closed
populations rather than on infection persis-
where p equals the "probability of effective tence in open populations. Though the au-
contact," or the probability that any two in- thors calculated statistics analogous to basic
dividuals in the population have, in one time and net reproduction rates (see table 2), they
period (serial interval), the sort of contact neither used that terminology nor derived
necessary for transmission of the infection thresholds. Indeed, on the surface, their con-
in question (58). The logic of this equation clusion implies there is no threshold ("the
is such that the risk of infection among sus-
proportion immune do not influence the
ceptibles is equal to the probability of hav-
probability of spread"), though this is a con-
ing effective contact with at least one in-
sequence of the assumption that "numbers
fectious case.6 This model had traditionally
of susceptibles and the rate of contact" are
been applied to simulate epidemics in closed
held constant. But, given the definition of
populations (with no births or influx of sus-
ceptibles). Fox et al. continued this tradition, the Reed-Frost contact rate as the probabil-
and thus calculated susceptibles for succes- ity that any two individuals have effective
sive time periods as contact in one time period, it is unreasonable
to consider alteration of population size
S = 9 —
:,+ ,. (io) without accepting its implications for some
This is important, as, by omitting any term consequent change in contact probabilities.
(For example, the probability for any two
6
people chosen at random in a small com-
lf the same value is substituted for r in equation 2 munity to meet, by chance, in 1 week, may
and p in equation 9, the mass action predicts a higher
number of successive cases than does the Reed-Frost be 0.1, but this probability will surely be
for any given S, and C,. This is because the mass action smaller if they live in a very large popula-
equation does not correct for the fact that multiple in-
fections on a single susceptible can lead to only a single
tion). Viewed from this perspective, the au-
subsequent case. It can be shown by the binomial ex- thors' first conclusion, as quoted above, ap-
pansion that the Reed-Frost model approximates the pears almost spurious.
mass action if p is small, in which case the Reed-Frost
p and the mass action r become the same statistic The paper then took a crucially important
(59). This is reasonable in that as p is reduced, the prob- step. The authors explored an alternative to
ability of a susceptible contacting more than one case
per serial interval (e.g., p 2 is the probability of contacting the basic assumption of homogeneous ran-
two cases, etc.) becomes vanishingly small. dom mixing, which had been implicit in allHerd Immunity 275
TABLE 2. Extract from a table published by Fox et al. (15) to illustrate the behavior of infections in a
randomly mixing population, as predicted by the Reed-Frost model
Expected number of effective
Initial population composition "Probability Probability
contacts by case in first interval
of effective of no
contact" With spread
Susceptibies Cases Immune S Total Total
(P) susceptibies
(S) (C) (W)
10 1 0 11 0.2 2 2 0.11
10 1 5 16 0.2 2 3 0.11
10 1 5 16 0.133 1.3 2 0.23
• Analogous to the net reproduction rate, Rn.
t Analogous to the basic reproduction rate, fl0-
t The probability that all 10 susceptibies fail to have contact with the single index case.
modeling arguments to that time. They set susceptibies. The optimum immunization
up a structured community in which 1,000 program is one which will reduce the supply
individuals were separately assigned family, of susceptibies in all subgroups. No matter
school, and social groupings, each of which how large the proportion of immunes in the
had a different internal contact probability. total population, if some pockets of the com-
By using Monte Carlo techniques, they munity, such as low economic neighbor-
simulated the consequences of introducing hoods, contain a large enough number of
infections into such populations with and susceptibies among whom contacts are fre-
without opportunities for special mixing quent, the epidemic potential in these
within and between the social groups. Table neighborhoods will remain high. Success
3 presents a portion of the results of these of a systematic immunization program re-
simulations, which led the authors to con- quires knowledge of the age and subgroup
clude: "Free living populations of commu- distribution of the susceptibies and maxi-
nities are made up of multiple and interlock- mum effort to reduce their concentration
ing mixing groups, defined in such terms as throughout the community, rather than
families, family clusters, neighborhoods, aiming to reach any specified overall pro-
playgroups, schools, places of work, ethnic portion of the population" (15, p. 186).
and socioeconomic subgroups. These mix- While the argument that social structure is
ing groups are characterized by different important in determining patterns of infec-
contact rates and by differing numbers of tion is compelling, two points in this con-
TABLE 3. Relative frequency distributions of epidemic sizes predicted by the Reed-Frost model,
assuming different structures to a population of 1,000 persons. Data are based on 100 stochastic
simulations under each set of conditions, as published by Fox et al. (15)
Within Total number of cases per epidemic (%) Mean
Mixing group
contact epidemic
groups
(p value) 5-9 10-19 20-29 30-39 40-59 60-79 size
Total community 0.002 82* 15 2 1 1.2t
Total community 0.002 22 18 34 8 17 3.3
Families, [62]$ 0.5
Total community 0.002
Families, [62] 0.5 11 6 26 23 23 5.6
Playgroups [24] 0.1
Total community 0.002
Families, [62] 0.5
Playgroups [24] 0.1 23 4 28 45 45.0
Nursery school 0.1
* Thus, 82 of the 100 epidemics simulated under these conditions (in this case a randomly mixing community with probability of
effective contact, p = 0.002), terminated after a single case.
t The average total number of cases in all 100 simulated epidemics was 1.2.
t The numbers in brackets reflect the numbers of families, playgroups, and nursery schools in the simulated populations.276 Fine
elusion are less clear. First, the statement linking of the mass action and basic case
that it is important to reduce the supply of reproduction rate theories. The crucial in-
susceptibles in all subgroups is not strictly sight appeared in a 1975 paper by Dietz (18)
supported in the paper's theoretical results; which demonstrated that, if one assumes a
indeed, it is intuitively reasonable, and stable population in which the mortality
was later demonstrated in theory (see be- rates and the incidence rates of infection are
low), that targeting vaccination to groups both independent of age, then
with high contact probabilities can be
more efficient (in the sense of minimizing Ro = T/Se = 1 + L/A, (11)
the total number of vaccinations required)
where L is defined as the average expecta-
in reducing disease than is uniform cover-
age of an entire population. Second, the tion of life and A is the average age at in-
emphasis on curbing epidemic spread re- fection.7 Mathematical proofs of this rela-
mains. Although Fox et al. considered tion have been presented by several authors
their approach " . . . relevant to programs (18, 23, 25, 27). The derivations assume an
of systematic immunization .. . which exponential distribution of the population by
have as their ultimate goal elimination of age and age-independent incidence rates of
the causative agent from the country" (15, infection (figure 7A).8 The relation can take
p. 186), it was most relevant to epidemics an even simpler form if the population is
in closed populations, as it had no provi- assumed to have a rectangular age distribu-
sion for examining the implications of a tion (figure 7B), in which case
constant influx of susceptibles into the
population, as by birth. Ro = L/A. (12)
The Fox et al. paper deserves its consid- This latter relation can be illustrated neatly
erable influence. Its break from the tradition if we recall that Ro is equivalent to the re-
of random mixing populations was a cru- ciprocal of the proportion susceptible at
cially important development. Its theory equilibrium ((Ro = T/Se = l/s e ), and as-
was born of practical experience and disap- sume that everyone is infected at exactly age
pointment with progress in measles control A, the average age at infection, and dies at
in the United States, and its tone was pes- exactly age L, the average expectation of life
simistic and practical, compared with most (figure 7B). Assuming this rectangular age
of the past (and subsequent) literature on structure, the proportion susceptible isA/L;
herd immunity, which has trended to em- thus Ro = L/A. On this basis, we might
phasize simple thresholds. As we shall see, conclude that the higher crude estimates of
the paper still proves to be wise counsel. Ro implicit in equation 11 should in general
be more appropriate for developing coun-
Recent theoretical developments tries, with pyramidal or exponential age dis-
tributions (figures 7A and C), and the lower
The credibility of the simple formulations estimates of equation 12 for developed
of herd immunity thresholds is weakened by countries (figures 7B and D).
the fact that the logic and formulae are based
on obviously simplistic assumptions. In par- 7
ticular, the basic mass action models as- This insight represents another contribution stem-
ming from the traditions of the mathematics of vector-
sumed that populations are homogeneous, borne diseases (Dietz's paper (18) was on arthropod-
with no differences by age, social group, or borne viruses) and of physical chemistry (the
season, and that they mix at random. Math- assumption of an age-independent incidence rate is the
basis of the so-called "catalytic models" (60)).
ematically inclined workers have taken 8
ln brief, if u is the death rate and A is the force (person-
these failings as a challenge to adapt the time incidence rate) of infection, then the average du-
theory to more realistic assumptions. ration of life is 1/u = L and the average duration of sus-
ceptible life is 1/(A + u). As f?0 = M(proportion
The estimation of Ro. The centerpiece of susceptible), fl0 = (A + u)/u = 1 + A/p. If p is small
research on herd immunity has been the compared to A, then this expression is close to 1 + L/A.Herd Immunity 277
"Rectangular" population
"Exponential" population
A100 B 100 n
C
'5
50H
Immune
c
O
ID
Q.
O-i
0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 80 90
Age (years) Age (years)
Malawi' Population by Age England & Wales: Population by Age
1987 1991
c
„ 1,600-L.
4-
•8 1,400-
§1,200-
E 1 ,ooo I3" in
g 600-
75
6
400
200-
0
i'0~'r n
. •J.Uj_!.i_L!
7". V !.*• v ;-, •
1 -
o-L . •.? •;• v ^,;~ s !o •> g
Age (years) Age (years)
FIGURE 7. Schematic diagrams of exponential (A) or rectangular (B) age distributions compared with current
population distributions in Malawi (C) and England and Wales (D). The exponential model (A) assumes infection
and constant death rates at all ages. The average age at infection and average expectation of life are A and L years,
respectively. In the rectangular model, all individuals are assumed to become infected at age A and to die at
age L.
Equations 11 and 12 may be combined quires compartmentalization of the popula-
with the basic herd immunity expression tion by age groups as well as by infection
(equation 8) to give relations between crude status (i.e., with maternal immunity, or sus-
basic reproduction rates, herd immunity ceptible, or latent, or infectious, or with ac-
thresholds, and average age at infection, as tive immunity). Assumptions must then be
shown in figures 8A-8D. The availability of made as to how the risk of infection, within
such expressions has made it a straightfor- each age group in each time period, is a
ward matter to estimate crude basic repro- function of the prevalence of infectious
duction rates and herd immunity thresholds cases in the same and other age groups at
for a variety of diseases of childhood (see that time. A general scheme for this ap-
table 1). Beyond that, they have opened the proach is presented in figure 9. Several in-
way to explorations of more realistic (and vestigators have tackled the problem and
complicated) sets of assumptions. have thus been able explore the effects of
Age-related effects. The simple mass ac- different age-specific contact patterns, and
tion and Reed-Frost models make no pro- vaccination strategies, within simulated
vision for the fact that individuals pass populations (7,19,23,36). Not surprisingly,
through periods of different infection risk as the simple elegance of the basic mass action
they age. The inclusion of this factor re- model has been lost, and the results have278 Fine
"Exponential" Population Rectangular" Population
0 10 20
Average Age at Infection (A)
Average Age at Infection (A) years Average Age at Infection (A) years
FIGURE 8. Relation between fi0 (basic case reproduction rate), H (herd immunity threshold), A (average age at
infection), and L (average expectation of life), based on exponential (A and B) or rectangular (C and D) age dis-
tribution assumptions, derived from equations 8, 11, and 12.
become more complex, and less easily gen- from birth, for example,
eralized, as the number of variables has in-
Ro = 1 + (L - M)I(A - M). (13)
creased. On the other hand, several prin-
ciples have emerged. Another use of this approach has been
Inclusion of maternal immunity (trans- to explore the implications of vaccinating
placentally-acquired immunoglobulin G) in at different ages. Selection of the optimal
the models serves to increase slightly the age for vaccination is dependent on sev-
estimates of basic reproduction rates and eral factors, including the duration of in-
herd immunity thresholds calculated from terfering maternally-acquired antibodies,
equations 11 and 12 (23). This is intuitively logistic requirements of the health ser-
reasonable in that, as far as an infectious vices, and the need to protect children
agent is concerned, an individual does not prior to exposure to risk. The issue is com-
really enter the population until he or she has plicated further insofar as vaccination it-
lost maternal antibody protection (and, thus, self may reduce infection risks, and,
iheA andL parameters in equations 11 and hence, expand the "window" period prior
12 are, in effect, overestimates). The basic to any given level of cumulative incidence.
equations can thus be adapted to adjust ages On the other hand, age at vaccination is re-
as though they were calculated from the av- lated inversely to the reduction of suscep-
erage age of losing maternal immunity, M tibles in the population, and, hence, affects
(on the order of 0.5 years for measles but estimates of herd immunity thresholds.
less for many other infections), rather than This is easily described in terms of theHerd Immunity 279
100i
•O
N
I 80 PH=(L-A)/(L-V)
e
70: assumes L = 70
TIME
II 60^
FIGURE 9. Schema for age-structured model, based
on addition of age axes to figure 1. Simulation requires
accounting susceptible (S a ,,), case (Ca, /), and immune 50
(/a, ,) individuals over successive time periods. Such
models generally include births, latent infections, and 0 1 2 3
deaths (23).
Age at immunization (V yrs)
— A = 3 —A = 5 -*-A = 10
rectangular age distribution (figure 7B).
By seeking the proportion PH of a popula- FIGURE 10. Relation between PH (proportion of in-
tion which must be vaccinated at age V, in fants which must be immunized in order to attain herd
immunity threshold), A (average age at infection), and
order to produce an overall proportion of l/(age at immunization), assuming rectangular age dis-
immunes in the population equivalent to tribution (equation 14). Illustrated solutions assume L =
(L - A)/L (see figure 7B), we find directly 70.
(23, 28):
H = (L- A)I{L - V). (14)
certain age groups are at special risk for
This relation (figure 10) is unrealistic inso- childhood infections, and it is intuitively
far as it implies 100 percent vaccine effi- reasonable that this should be so considering
cacy and it neglects that the efficacy of the implications of aggregation in schools in
many vaccines is age-dependent (for ex- particular. Figure 11 shows annual risks of
ample, not reaching a maximum until age reported measles by age in England and
15 months for measles). On the other Wales prior to introduction of vaccination,
hand, it nicely illustrates an important showing the dramatic effect of the aggre-
point, that simple crude estimates of im- gation of children in primary schools from
munity thresholds, which implicitly as- the age of 5 years. Very few children made
sume vaccines to be given at birth or as it to their eighth birthday without having
soon as maternal immunity wanes, (and to contracted infection with the measles virus!
be 100 percent effective) will be optimisti- The actual risks of infection in any age
cally low; and that much higher coverage group (a) are a consequence of "contact" not
levels are required because, inter alia, of only within that group, but also between that
the inevitable delays in providing vaccines age group and each of the other age groups
to some members of the community. in the community. The simple mass action
The assumption of variations in infection formulation can be generalized to define the
risk by age has even more complicated and incidence of infection in age group a as the
important effects on herd immunity thresh- sum of infections acquired from contact
old estimates. It is common knowledge that within age group a, and between that and280 Fine
* —• 1950 cohort
limited numbers of age groups (in effect the
0.7-
| --« 1955 cohort ra*t parameters of equation 15). An example
05
UJ 0.6-
o- — - o 1960 cohort of such a matrix is shown in figure 12.
Analysis of these structures has revealed
IBL
/
£ 0.5- that, under different circumstances, age-
UJ
o /1 dependent contact rates can lead to either an
0.4- /
t 1
increase or a decrease in the estimates of Ro
1 and H compared with those derived from the
Z 0.3-
a // simple global mass action assumptions
o
z 0.2- above (36). In general, crude estimates of Ro
^ - ^ ^ . (e.g., from equations 11 or 12) will be too
0.1 - high if age-specific contact rates are highest
K ' ^C among the young and fall with age. This is
0 '1 1
2 ' 3 '4 ' 5 '6 ' 7°" ' 8 ' 9o '10 '11 1 12 I 13 I 14 I reasonable as older susceptibles will be rela-
AGE IN YEARS tively less relevant insofar as they are less
FIGURE 11. Age-specific risks of notified measles in likely to have the sort of contact necessary
three birth cohorts in England and Wales prior to the for transmission. In contrast, crude esti-
introduction of measles vaccination in 1968. Denomi-
nators are the numbers of individuals presumed sus- mates of #0 will be too low if contact rates
ceptible (not yet immunized or infected) in each age rise with age.
group (55). Note the steep increase at age 5 years on
entry to primary school. Low risk after age 6 years in the Season and other periodic changes.
1960 cohort reflects reduced transmission after intro- Most of the common vaccine-preventable
duction of vaccination. diseases are seasonal. The most obvious ex-
ample of this is the seasonal increase in
measles which follows the annual opening
each of the other age groups (i = of primary schools in many countries (61).
1,2,3- • . .a. . .n) to be considered: It was recognized long ago that this had im-
plications for the mass action theory as it
a, i + 2J (15)
AGE OF SOURCES OF INFECTION
;= i
Here, the a subscripts refer to separate age y
groups and ra», stands for the contact or (5-15)
transmission parameter between age groups
a and /. Reiteration is based on recalculation r r r
of numbers of susceptibles and cases in each UJ ( x.x y.x z.x
age group at each successive time period, m
taking into account transitions from one age o_
UJ
group to the next. O y r r
3 (5-15) x.y y.y rz.y
Exploration of the effects of this addi-
tional structure is hampered by the difficulty
(perhaps impossibility) of obtaining appro-
priate data defining the contact parameters r
x.z ry.z rz.z
within and between different age groups in
any population (let alone that any such pa-
rameters would vary between different FIGURE 12. "WAIFW" (Who Acquires Infection From
Whom) matrix of transmission parameters within and
populations and change over time). The between three different age groups, preschool, school-
theoretical implications of such age struc- age, and adult. Under most conditions such a matrix
ture were thus explored by Anderson and would be symmetric along the xx-yy axis, ( r ^ = r^),
though this need not necessarily be the case (e.g., the
May (36) in the context of simplified hygiene habits of younger children may be different
"WAIFW" ("Who Acquires Infection From making them particularly efficient at transmitting some
Whom") matrices defining contact between infections, in which case, for example, r^ > fyx).Herd Immunity 281
meant that there must be seasonal changes in mine a 'maximum initial infection reproduc-
the transmission parameter r (and in the ba- tion rate,' Rmax, which quantity must be used
sic reproduction rate) (51). Some early au- in defining conditions of herd immunity.. . .
thors tried to mimic these changes by at- As a consequence the present model implies
taching trigonometric functions to the herd immunity against measles with sub-
contact rates in their models (51, 62), but stantially lower immunization rates than are
more recent authors have taken more prag- predicted from global mass action theory.
matic approaches. Here the calculated critical immunization
Yorke et al. (63) discussed the implica- coverage would be 76 per cent if protection
tions of seasonally for eradication strategy by vaccination could be achieved in new-
employing the simple mass action approach. borns" (7, pp. 187-8). The extent to which
Though these authors did not argue in terms Schenzle's surprisingly low estimate of
of herd immunity thresholds or basic case measles herd immunity might have been at-
reproduction rates per se, they noted that tributable to his assumptions of annual
transmission is most tenuous (i.e., Ro is changes in transmission (low Ro values dur-
minimal) just before, or during, seasons of ing the summer months), in addition to the
lowest incidence, and that it should be easi- assumed age structure and age-dependent
est to break transmission at these times. contact rates, is unclear.
(Though they did not so express it, the im- Timing of interventions. The Schenzle
plication was that the herd immunity thresh- paper cited above, and work by others (64)
old is lowest during such periods, and, thus, have shown that the predicted impact of an
that a vaccine coverage level which is not intervention can also vary according to the
high enough to "interrupt transmission" in timing of its introduction into a population.
peak seasons may nonetheless be sufficient Though it has been proposed that certain
to do so during the annual low.) situations can lead to "chaotic" results (65),
The implications of periodic aggregation it is unclear to what extent such effects are
of children in schools was explored by relevant to actual programs, given that real
Schenzle (7) who constructed a compart- life includes many structured perturbations
mental model for measles simulation which (such as school year calendar variation and
included both age structure and appropriate holiday-dependent delays in notification)
changes in the transmission parameters to beyond the scope of the assumptions of
mimic the periodic aggregation of succes- simple mathematical models. On the other
sive cohorts of children in schools. His re- hand, such work lends another perspective
sults are of particular interest in that they to the interpretation of irregular incidence
provide a closer approximation to observed patterns.
measles trends and the impact of vaccination Social and geographic clustering. The
(in England and in Germany) than has been disparity between the homogeneous mixing
achieved by any other published model. As assumption of basic models and the hetero-
with the other models incorporating age geneity in structure and mixing of real hu-
structure and a declining contact rate with man populations is obvious. The importance
age, Schenzle's simulations suggested a of social aggregations such as families, play
herd immunity threshold for measles which groups, neighborhoods, and schools, and
was appreciably lower than that predicted by geographic distinctions between towns and
the simple homogeneous mixing model. In
urban and rural areas, mean that human
his own words: "The quantity [/?„ =
populations are partitioned in a complex set
T/Se] has no meaning at all in the presence
of interlocking patterns with inevitable im-
of age-dependent contact rates, where infec-
plications for the transmission of infections.
tives of differing ages are assigned different
Fox et al. (15) showed great insight in tack-
infectious potentials. These have to be
weighted appropriately in order to deter- ling this problem in their original paper on
herd immunity. Since then, though several282 Fine
subsequent investigators have attempted to transmission characteristics. They found
build models with social or geographic that eradication could be achieved with
structure, few useful generalizations have fewer overall vaccinations if they were dis-
arisen (7,20,22,23,29). In one sense, social tributed primarily to the high contact rate
and geographic partitioning of populations groups (e.g., cities) than if they were dis-
just represents an extension of the sort of tributed uniformly to the overall population
partitioning represented by age. All indi- (but see also (22)). Beyond this intuitively
viduals belong to many different subgroups sensible qualitative result, that it may be ad-
in society, and the transitions from one sub- vantageous to target interventions at high
group to another (by aging, migration, etc.), risk groups, we are left with the conclusion
as well as the contact rates within and be- of Fox et al. (e.g., table 3) that social struc-
tween all subgroups, will vary according to ture can have profound effects on the like-
many different factors, many of which will, lihood and patterns of infection transmission
in turn, be confounded with one another (so- and, hence, upon herd immunity thresholds.
cioeconomic status, political, social, and Overall implications of additional vari-
historical context, behavior, hygiene level, ables. Implications of the various supple-
crowding, season, mode of infection trans- mental assumptions which have been ex-
mission, etc.). In an effort to describe just plored in recent theoretical work on herd
the most superficial level of such complex- immunity are summarized in table 4. The
ity, May and Anderson (29) formulated a set difficulty of making precise estimates of
of general equations describing populations herd immunity thresholds in any particular
broken into several groups with two differ- context is evident for each of the various
ent within and between group (high and low) influences even without considering the in-
TABLE 4. Implications of different assumptions for theoretical estimates of the herd immunity
threshold (H), with reference to simple global estimates as obtained by equation 8 , 1 1 , and 12
Implications
Variable + assumption for herd References
immunity
Maternal immunity If vaccines not effective until maternal immunity wanes, (23)
crude H estimates will be too low; this may be corrected by
considering that a child is not born until maternal immunity
disappears (equation 13)
Variation in age at vaccination Herd immunity effect greatest (H threshold lowest) when (8, 28)
vaccination occurs at earliest possible age; delayed vaccin-
ation implies threshold coverage level will be higher than
simple estimates
Age differences in "contact" Implications vary with relation between age and contact (7, 36)
rates or infection risk rate; falling contact rate with age implies true H may be
lower than simple global estimate
Seasonal changes in contact Seasonality may imply lower true herd immunity threshold if (7, 63)
rates seasonal change is marked, and fade out can occur during
low transmission period
Geographic heterogeneity In theory, geographic differences in contact rates may (20)
permit elimination with lower overall vaccine coverage than
that implied by H based on total population by targeting
high risk groups
Social structure (nonrandom Social structure can have complicated implications as it (15)
mixing) implies group differences in vaccination uptake and/or
infection risk; existence of vaccine-neglecting high contact
groups means true H will be higher than simple estimatesYou can also read
