Epidemiologic ReviewsCopyright © 1993 by The Johns Hopkins University School of Hygiene and Public HealthAll rights reserved

Vol. 15, No. 2Printed in U.S.A.

Herd Immunity: History, Theory, Practice

Paul E. M. Fine

INTRODUCTION

Herd immunity has to do with the pro-tection of populations from infection whichis brought about by the presence of immuneindividuals. The concept has a special aura,in its implication of an extension of the pro-tection imparted by an immunization pro-gram beyond vaccinated to unvaccinated in-dividuals and in its apparent provision of ameans to eliminate totally some infectiousdiseases. It is a recurrent theme in the medi-cal literature and has been discussed fre-quently during the past decade. This newpopularity comes as a consequence of sev-eral recent major achievements of vaccina-tion programs, i.e.: the historic success ofthe global smallpox eradication program;dramatic increases in vaccination coveragestimulated by national programs and by theExpanded Programme on Immunization; thecommitment of several countries to eradi-cate measles; and international dedication toeliminate neonatal tetanus and to eradicatepoliomyelitis from the world by the year2000.'

Received for publication January 27, 1993, and infinal form July 29, 1993.

From the Communicable Disease Epidemiology Unit,London School of Hygiene and Tropical Medicine,Keppel Street, London WC1, England. (Reprint re-quests to Dr. Paul E. M. Fine at this address.)

1Though the words "eradicate" and "eliminate" havebeen used interchangeably by some authors in the past,current usage of eradication implies reduction of bothinfection and disease to zero whereas elimination im-plies either regional eradication, or reduction of diseaseincidence to some tolerably low level, or else reductionof disease to zero without total removal of the infectiousagent (1). Thus the 42nd World Health Assembly rec-ommended "elimination of neonatal tetanus by 1995and global eradication of poliomyelitis by the year 2000"(2).

Along with the growth of interest in herdimmunity, there has been a proliferation ofviews of what it means or even of whetherit exists at all. Several authors have writtenof data on measles which "challenge" theprinciple of herd immunity (3-5) and otherscite widely divergent estimates (from 70 to95 percent) of the magnitude of the herd im-munity threshold required for measles eradi-cation (6-8). Still other authors have com-mented on the failure or "absence" of herdimmunity against rubella (9, 10) and diph-theria (11). Authorities continue to argueover the extent to which different types ofpolio vaccine can, let alone do, induce herdimmunity (12-14). Given such differencesof opinion, there is need for clarification.

Many authors have based their discus-sions of herd immunity on an influential pa-per published in 1971 by Fox et al. titled"Herd immunity: basic concept and rel-evance to public health immunization prac-tices" (15). This paper took as its startingpoint a medical dictionary's definition ofherd immunity as "the resistance of a groupto attack by a disease to which a large pro-portion of the members are immune, thuslessening the likelihood of a patient with adisease coming into contact with a suscep-tible individual" (16). While useful, eventhis definition lends itself to different inter-pretations; these may be either quantitative(herd immunity as partial resistance, re-flected in reductions in frequency of diseasedue to reductions in numbers of source casesand of susceptibles) or qualitative (herd im-munity as total resistance, implying athreshold number or percentage of immunesabove which an infection cannot persist).Each of these interpretations has its place,but they are sometimes confused in debates

265

266 Fine

on the subject. A given population may ex-hibit one (partial, quantitative) without theother (total, qualitative) form of herd im-munity. It will be found that such definitionsdo not easily fit situations in which vaccine-derived immunity is transferred either di-rectly (as in the case of maternal antibodiesagainst tetanus) or indirectly (as in the caseof secondary spread of oral polio vaccines)between members of a population, or inwhich vaccines impart different levels ofprotection against infection, disease, ortransmission (as in diphtheria, pertussis, andperhaps malaria).

The paper of Fox et al. (15) is also of im-portance because of its method and the na-ture of the conclusions which were dictatedby that approach. Sufficient years have nowelapsed for both the method and the con-clusions to be reviewed in perspective.

Interest in applying the "magic" of herdimmunity in disease control has encouragedmathematical research exploring the theo-retical implications of the subject (6-8, 17—37). Though much of this work has beenpublished in journals and in language unfa-miliar to the medical and public health com-munities, its isolation has been reduced inrecent years largely through the publicationsof Anderson and May and their colleagues(8, 17, 20, 21, 23, 24, 28, 29, 31, 33, 36).

It is the intent of this review to bring to-gether the literature on the history, theory,and practical experience of herd immunity,to consider the variety of issues raised by theapplication of the concept to different dis-eases, and to consider how well currenttheory and practice correspond with eachanother.

HISTORY

The first published use of the term "herdimmunity" appears to have been in a paperpublished in 1923 by Topley and Wilsontitled "The spread of bacterial infection:the problem of herd immunity" (38). Thiswas one of a classic series of studies bythese authors on epidemics of various in-fections in closely monitored populationsof laboratory mice (39). Topley and

Wilson introduced the term in the follow-ing manner: "Consideration of the resultsobtained during the past five years . . . ledus to believe that the question of immunityas an attribute of a herd should be studiedas a separate problem, closely related to,but in many ways distinct from, the prob-lem of the immunity of an individual host"(38, p. 243). After describing experimentsshowing that immunized mice had lowermortality rates from, and were less likelyto transmit, Bacillus enteritidis, the au-thors concluded by posing an " . . . obviousproblem to be solved.... Assuming agiven total quantity of resistance against aspecific bacterial parasite to be availableamong a considerable population, in whatway should that resistance be distributedamong the individuals at risk, so as best toensure against the spread of the disease, ofwhich the parasite is the causal agent?"(38, pp. 248-9). Wilson later recalled thathe had first heard the phrase "herd immu-nity" in the course of a conversation withMajor Greenwood (G. S. Wilson, LondonSchool of Hygiene and Tropical Medicine,personal communication, 1981); andGreenwood employed it in his 1936 text-book Epidemics and Crowd Diseases (40).Although these authors did not distinguishclearly between direct and indirect protec-tion stemming from vaccine-derived im-munity, later authors picked up the phraseand applied it in particular to the indirectprotection afforded to nonimmune indi-viduals by the presence and proximity ofothers who are immune.

That the presence of immune individualscould provide indirect protection to otherswas itself recognized at least as far back asthe 19th century. Farr had noted in 1840 that"The smallpox would be disturbed, andsometimes arrested, by vaccination, whichprotected a part of the population . . ." (41).Such observations, that epidemics oftencame to an end prior to the involvement ofall susceptibles, led in turn to a major epi-demiologic controversy in the early years ofthis century. This controversy was betweenthose who believed that epidemics termi-

Herd Immunity 267

nated because of changes in the properties ofthe infectious agent (e.g., loss of "virulence"resulting from serial passage) (42) and thosewho argued that it reflected the dynamics ofthe interaction between susceptible, in-fected, and immune segments of the popu-lation (43). Each argument was supportedby observations and by mathematical rea-soning (44). It was the latter explanation thatwon the day; and its simple mathematicalformulation, the "mass action principle,"which has become a cornerstone of epide-miologic theory, provides one of the sim-plest logical arguments for indirect protec-tion by herd immunity.

The concept of herd immunity is often in-voked in the context of discussions of dis-ease eradication programs based on vacci-nation. It is significant that both Jenner (45)and Pasteur (46), key figures in the earlydevelopment of vaccines, recognized thepotential of vaccines to eradicate specificdiseases, but neither appears to have con-sidered the practical issues closely enoughto have touched on herd effects. Further-more, the major focus of eradication think-ing in the first half of this century did notinvolve vaccines or vaccine-preventablediseases at all, but concerned vector-bornediseases, malaria in particular. Thisstemmed from the writings of Ross (47)who, in work on the dynamics of malaria,had deduced that it was not necessary toeliminate mosquitoes totally in order toeradicate the disease. Ross's so-called"mosquito theorem" was the first recogni-tion of a quantitative threshold which couldserve as a target for a disease eliminationprogram. So powerful was the argument,and so influential was the tradition of quan-titative thinking which it engendered, thatthe World Health Organization attemptedglobal eradication of malaria before that ofany other disease (48).2 This tradition of

1955 World Health Assembly recommendedthat the World Health Organization take the initiative in"a programme having as its ultimate objective the world-wide eradication of malaria." It was not until 1965 thatthe Assembly first declared "the worldwide eradicationof smallpox to be one of the major objectives of theorganization" (49).

mathematical epidemiology relating tovector-borne diseases has been repeatedly asource of important insights for the field ofvaccination and herd immunity.

THEORY

Three separate theoretical perspectiveshave been used to derive measures of herdimmunity. Over recent years, these perspec-tives have converged into a general theory.

The mass-action principle

The theoretical basis of herd immunitywas introduced by Hamer (43) in 1906 in thecontext of a discussion of the dynamics ofmeasles. Hamer argued that the number oftransmissions (he called it the "ability to in-fect") per measles case was a function of thenumber of susceptibles in the population.We can paraphrase his argument as:

C, + JC, varies with S,, (1)

where S, and C, are numbers of susceptiblesand cases, respectively, in some time periodt, C, +, is the number of cases in the suc-ceeding time period, and Cl+l/C, is, thus,the number of successful transmissions percurrent case (see figure 1). The time periodused in this formulation is the average in-terval between successive cases in a chain oftransmission, sometimes called the "serialinterval" (50), which is approximately 2weeks for infections such as measles andpertussis (see table 1). This relation can beexpressed:

where r is a transmission parameter, or"contact rate," in effect the proportion of allpossible contacts between susceptible andinfectious individuals which lead to new in-fections. In order to simulate successivechanges over time, the number of suscep-tibles is recalculated for each new timeperiod as

where 5, + , is the number of susceptibles in

268 Fine

TIME-

NEXTTIME

PERIOD

SUSCEPTIBLES

CASES

IMMUNES

DEATHSFIGURE 1 . Relation between susceptibles (S), infec-tious cases (C), and immunes (/) in successive timeintervals (t, t + 1) in the simple discrete time mass actionor Reed-Frost models. In each time period some(Ci + i) susceptibles become cases and the others re-main susceptible. Each case is assumed to remain in-fectious for no more than a single time period (= serialinterval). B, individuals may enter as susceptible birthsduring each time period (e.g., equation 3). Note thatneither the simple mass action (equations 2 and 3) norReed-Frost (equation 9) equations include an explicitterm for immunes. By implication, deaths prior to infec-tion are not considered in these simplest models and thetotal population is assumed constant (i.e., in each pe-riod the same number of immunes die as susceptiblesare born into the population).

the next time period and B, is the number ofnew susceptibles added (e.g., born into) tothe population per time period.

The relation in equation 2, that future in-cidence is a function of the product of cur-rent prevalence times current number sus-ceptible, has become known as theepidemiologic "law of mass action" by anal-ogy with the physical chemical principlethat the rate or velocity of a chemical reac-tion is a function of the product of the initialconcentrations of the reagents.3 Often ex-pressed as a differential (continuous time)rather than a difference (discrete time) equa-tion, as here, this relation underlies most

3This analogy was apparently first made by Soper(51). The inspiration from physical chemistry is of morethan passing interest in that it reflects a tradition amongbiomedical theorists to strive for the simplicity and el-egance of the physical sciences. Not only mass action,but also the concepts of catalysis and of critical masshave close analogies in the behavior of infections, asmentioned below.

TABLE 1. Approximate serial intervals, basicreproduction rates (in developed countries) andimplied crude herd immunity thresholds(H, calculated as 1 - 1/ff0) for commonpotentially-vaccine-preventable diseases. Datafrom Anderson and May (8), Mcdonald (54), andBenenson (135). It must be emphasized that thevalues given in this table are approximate,and do not properly reflect the tremendousrange and diversity between populations.They nonetheless give an appreciation oforder-of-magnitude comparability

InfectionSerial interval

(range)(%)

DiphthenatInfluenza}:Malaria§Measles||MumpsPertussislPolio#RubellaSmallpoxTetanusTuberculosis**

2->:30 days1-10 days>20 days7-16 days8-32 days5-35 days2-45 days7-28 days9-45 days

NA*Months-years

6-7?

5-10012-184-7

12-175-76-75-7NA?

85?

80-9983-9475-8692-9480-8683-8580-85

NA?

* flo, basic case reproduction rate; H, herd immunity thresh-old defined as the minimum proportion to be immunized in apopulation for elimination of infection; NA, not applicable.

t Long-term infectious carriers of Corynebacterium diphthe-riae occur. See the text for a discussion of the definition of im-munity.

t Ro of influenza viruses probably varies greatly betweensubtypes.

§ All these variables differ also between Plasmodium spe-cies The serial interval may extend to several years. See thetext for a discussion of implications of genetic subtypes.

|| See the text for a discussion and variation in estimates ofRo in table 5.

H See the text for a discussion relating to the definition ofimmunity in pertussis.

# Distinct properties of different polio vaccines need to beconsidered in interpreting the herd immunity thresholds.

" f l o has been declining in developed countries; protectiveimmunity is not well defined.

theoretical work on the dynamics of infec-tions in populations (23, 52).

Figure 2 illustrates what happens whenequations 2 and 3 are iterated and serves toillustrate several fundamental principles ofthe epidemiology of those acute immunizinginfections (such as measles, mumps, rubella,chickenpox, poliomyelitis, pertussis, etc.)which affect a high proportion of individualsin unvaccinated communities.

First, the model predicts cycles of infec-tion incidence, such as are well recognizedfor many of the ubiquitous childhood infec-tions (figure 3). The incidence of infectioncycles above and below the "birth" rate, orrate of influx of new susceptibles.

Herd Immunity 269

14

0 4

(0CD

1 2

Susceptibles (S )

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 wasbased on 12,000 susceptibles and 100 cases at the start, r = 0.0001 and 300 births per time period. Note that theincidence of cases cycles around the birth rate and that the number of susceptibles cycles around the epidemicthreshold: Sa = 1/r= 10,000.

Second, the number of susceptibles alsocycles, but around a number which is some-times described as the "epidemic threshold,"Se. Simple rearrangement of equation 1 toCt+l/C, = S, r reveals that this threshold isnumerically equivalent to the reciprocal ofthe transmission parameter r; as incidenceincreases (i.e., C, + , > C,) when, and onlywhen, S, > 1/r; and, thus, Se = 1/r. Thisimportant relation is implicit in Hamer'soriginal paper (43), and was formalized as a"threshold theorem" in 1927 by Kermackand McKendrick (53). The principle may beillustrated by analogy with the physical con-cept of a "critical mass"—the epidemicthreshold represents a critical mass (densityper some area) of susceptibles, which, if ex-ceeded, will produce an explosive increasein incidence of an introduced infection. Thecorrespondence between the case and sus-ceptible lines in figure 2 illustrates this re-lation.

Hamer and his successors used this logicto explain several aspects of the dynamics ofmeasles and other childhood infections,such as cyclical epidemics, the persistenceof susceptibles at the end of an epidemic,

and the relation between the interepidemicinterval and the time required for the numberof susceptibles to reach the epidemic thresh-old (23, 43, 51, 52). Though it was not em-phasized explicitly by the earlier authors,who dealt in numbers or "density," ratherthan proportions, of susceptibles, the epi-demic threshold provides a simple numeri-cal measure of a herd immunity criterion. Ifthe proportion immune is so high that thenumber of susceptibles is below the epi-demic threshold, then incidence will de-crease. We can express this algebraically as:

H = 1 - SJT = 1 - 1//T (4)

where T is the total population size, Se is theepidemic threshold number of susceptiblesfor the population, and H is the herd immu-nity threshold, i.e., the proportion of im-munes which must be exceeded if incidenceis to decrease.

Figure 4 presents another way of illustrat-ing the herd immunity threshold, i.e., interms of the relation between the proportionimmunized at birth and the ratio of the cu-mulative incidence during the postvaccina-tion period to that during the prevaccination

270 Fine

Measles: England and Wales

Measles notifications

S 8

Year

Measles: USA

B Pertussis: England and Wales

Year

Pertussis: USA

Year YearFIGURE 3. Reported incidence of common childhood vaccine-preventable diseases. Measles showed a tendencyto biennial epidemics in England and Wales prior to vaccination (A). This pattern was less dramatic in data for theentire United States (C) because of the size and heterogeneity of the population (not all areas were in phase withone another). All areas showed a strong seasonal oscillation in addition to the biennial cycle. Pertussis shows a 3-4year cycle with little obvious seasonality in the United Kingdom (B). This cycling is also seen in national data for theUnited States prior to 1970 (D). Notification efficiency was approximately 60% for measles in England and Walesprior to vaccination (55) but was considerably lower for pertussis and for both diseases in the United States.

period, either among those not immunized atbirth (figure 4A), or in the entire population(figure 4B). Insofar as the immunization ofindividuals removes both susceptibles andpotential sources of infection from the com-munity, it will lead to a reduction in inci-dence rates and, hence, in cumulative inci-dence. If the proportion immunized at birthis maintained at or above the threshold, H,then the cumulative incidence is reduced tozero, indicating that the infection has beeneliminated from the population.

It was only many years after Hamer thatthe wide use of vaccines meant that theseepidemic and herd immunity thresholdscould be considered as targets for interven-tion. If appropriate vaccination could pre-

vent the number of susceptibles from reach-ing the epidemic threshold, then incidenceshould continue to decline, ultimately to ex-tinction. Hamer's original principle impliedthe simplistic assumption of an homoge-neous, randomly mixing population, likethat of molecules in the ideal gasses forwhich the mass action principle was mostappropriate. However, given the power ofthe analogy, elaboration of the theory wasonly a matter of time.

Case reproduction rates

If an infection is to persist, each infectedindividual must, on average, transmit thatinfection to at least one other individual. If

Herd Immunity 271

1.0IF NO INDIRECT PROTECTION

. A XIF INDIRECT \PROTECTION \OCCURS \

0% 50%

% IMMUNIZED AT BIRTH

100%

1.0 nO ui

D 2

2 §

^ iz 5UJ U.

O Oo zz o

5

2S

s. 0

B

PROPORTION NOTIMMUNE BUT W/ASTILL ESCAPE V///<

V INFECTION

\ ,^ t IF NO

^ INDIRECT^<t> PROTECTION

IF INDIRECT 3 ^ /PROTECTION —' X ^ > /

0% 50% H 100%% IMMUNIZED AT BIRTH

FIGURE 4. Cumulative incidence (e.g., per lifetime) of infection after a vaccination program as a proportion of priorcumulative incidence among individuals not immunized by the vaccine (A) and among the total population (B). Ineach diagram the dotted line refers to an infection for which the vaccine offers no indirect protection (e.g., tetanusvaccination 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 escapeinfection as a proportion of all nonimmunized individuals (A) or of the total population (B).

this does not occur, the infection will dis-appear progressively from the population.This average number of actual infectiontransmissions per case is an extremely pow-erful concept, and has thus been discussedby many researchers. The fundamental sta-

tistic is one which was formulated originallyby Macdonald (54), in the context of malariastudies, as the average number of secondarycases who contract an infection from asingle primary case introduced into a totallysusceptible population. He called this num-

272 Fine

ber the "basic case reproduction rate", byanalogy with the demographic concept ofthe intrinsic reproduction rate, the averagenumber of potential progeny per individualif there were no constraints to fertility (26).This definition can be translated directlyinto the mass action equation (equation 2) byletting C, = 1 and 5, = T, to represent thesingle case introduced into a fully suscep-tible population. The number of secondarycases, Cl+i, is then equivalent, by defini-tion, to the basic case reproduction rate (Ro):

R0 = Tr. (5)

On reflection, we appreciate that this basiccase reproduction rate describes the spread-ing potential of an infection in a population,and that it will be a function both of thebiologic mechanism of transmission and ofthe rate of contact or interaction betweenmembers of the host population. Analogousor identical statistics have been defined byseveral authors, and given different namessuch as "expected number of contacts" (15),"contact number" (25), or "basic reproduc-tion number" (26).4 Examples of numericalvalues of this statistic, applicable to differ-ent infections and derived by methods de-scribed below, are shown in table 1. Asimple way of illustrating the concept is pre-sented in figure 5A.

Of course, in the real world there are con-straints to unlimited infection transmission.For example, some of the "contacts" of aninfected person may be individuals who arealready infected or immune. As a result, theaverage number of actual infection trans-missions per case, in a real population, willbe less than the basic case reproductionrate, and has been defined, again first byMacdonald (54), as the "net reproductionrate" /?„. Other authors have called this the"actual" or "effective" reproduction rate(23). This is illustrated in figure 5B. It isclear from figure 5 that the net reproduction

"•Different symbols have been used for the statistic bydifferent authors. The original work by Macdonald (54)employed ZQ for the basic reproduction rate. Severalauthors have noted that the statistic is not a proper rate,but that term is now imbedded in the literature (26).

rate Rn should be equivalent to the basic casereproduction rate Ro times the proportionsusceptible in the population:

= R0S,/T. (6)

This has interesting implications. If an en-demic infection persists in a population ofconstant size, then Rn should, on average,over a long period of time, be equivalent tounity (i.e., each case leads on average to asingle subsequent case). Therefore, "on av-erage" from equation 6:

Ro = Tlaverage S, = T/Se. (7)

In words, for endemic infections, the basiccase reproduction rate should be equivalentto the reciprocal of the "average" proportionsusceptible in the population. That the av-erage number of susceptibles is equivalentto Se should be evident from figure 2. Animportant implication of this relation is theprediction that the average proportion sus-ceptible should remain constant in a popu-lation, even in the face of extensive and ef-fective vaccination, as long as the infectionremains endemic (and as long as the popu-lation remains of constant size). Analysis ofdata on measles has confirmed this relation(55).

Combination of equations 4 and 7 pro-vides us with an expression for the herd im-munity threshold in terms of Ro:

H=1-VRO = (Ro - (8)

This is illustrated graphically in figure 6which shows the implications for persis-tence or eradication of infections dependingon the proportion of immunes in the popu-lation.5

The Reed-Frost heterogeneouspopulation simulation approach

The paper by Fox et al. (15) cited in theintroduction has been one of the most fre-

5This important relation was published explicitly firstby Dietz (18), in 1975, though it is implicit in some earlierwork, in particular a graph published by Smith (56) in1970.

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, andthe net reproductive rate Rn = Ro (S/T) = 1.

quently cited references on herd immunity.This paper is of historical interest, and alsoof interest because of its theoretical argu-ment and conclusions.

The appearance of the Fox et al. paper in1971 was significant. Four years before, in1967, the World Health Organization haddeclared its intention of eradicating small-pox from the world within 10 years, and theUnited States Public Health Service had de-clared its intention of eradicating measlesfrom the United States within 1 year (57).Both of these tasks were to be achieved bythe induction of herd immunity with vac-

cines. By 1971, the initial successes and fail-ures of these programs were on record (e.g.,figure 3C), and Fox et al. set out to explainthem.

They based their theoretical argument noton the mass action arguments outlinedabove, but on an alternative approach,rooted in the Johns Hopkins UniversitySchool of Hygiene and Public Health (58).This model, named the Reed-Frost forits developers Lowell Reed and WadeHampton Frost, assumes the same discretetime schema illustrated in figure 1 but pro-poses an alternative to the mass action equa-

274 Fine

,100

£ 75

50

CD

25

H = (R0-1)/RQ

10 20 30 40

Basic Reproduction Rate (R-. )

50

FIGURE 6. Relation between herd immunity threshold(H) and basic reproduction rate Ro, as in equation 8: H= 1 - MR0.

tion (equation 2 above) as:

Cf+1 = S,{l - ( 1 - p)Q} (9)

where p equals the "probability of effectivecontact," or the probability that any two in-dividuals in the population have, in one timeperiod (serial interval), the sort of contactnecessary for transmission of the infectionin question (58). The logic of this equationis such that the risk of infection among sus-ceptibles is equal to the probability of hav-ing effective contact with at least one in-fectious case.6 This model had traditionallybeen applied to simulate epidemics in closedpopulations (with no births or influx of sus-ceptibles). Fox et al. continued this tradition,and thus calculated susceptibles for succes-sive time periods as

S = 9 — :,+ ,. (io)

This is important, as, by omitting any term

6lf the same value is substituted for r in equation 2and p in equation 9, the mass action predicts a highernumber of successive cases than does the Reed-Frostfor any given S, and C,. This is because the mass actionequation does not correct for the fact that multiple in-fections on a single susceptible can lead to only a singlesubsequent case. It can be shown by the binomial ex-pansion that the Reed-Frost model approximates themass action if p is small, in which case the Reed-Frostp and the mass action r become the same statistic(59). This is reasonable in that as p is reduced, the prob-ability of a susceptible contacting more than one caseper serial interval (e.g., p2 is the probability of contactingtwo cases, etc.) becomes vanishingly small.

for births (B, in equation 3), the authorscould only address questions relating to epi-demics in closed populations.

Their first step was to explore these equa-tions for simple randomly mixing popula-tions. Table 2 presents a portion of the initialresults, on the basis of which the authorsconcluded " . . . application of the Reed-Frost model. . . demonstrates that, over awide range of variations, the number of sus-ceptibles and the rate of contact betweenthem determine epidemic potentials in ran-domly mixing populations. If these are heldconstant, changes in population size and,therefore, in the proportion immune do notinfluence the probability of spread" (15, p.182). The emphasis in this conclusion onnumbers and probability of spread deservescomment. The perspective reflects the pa-per's focus on epidemic potential in closedpopulations rather than on infection persis-tence in open populations. Though the au-thors calculated statistics analogous to basicand net reproduction rates (see table 2), theyneither used that terminology nor derivedthresholds. Indeed, on the surface, their con-clusion implies there is no threshold ("theproportion immune do not influence theprobability of spread"), though this is a con-sequence of the assumption that "numbersof susceptibles and the rate of contact" areheld constant. But, given the definition ofthe Reed-Frost contact rate as the probabil-ity that any two individuals have effectivecontact in one time period, it is unreasonableto consider alteration of population sizewithout accepting its implications for someconsequent change in contact probabilities.(For example, the probability for any twopeople chosen at random in a small com-munity to meet, by chance, in 1 week, maybe 0.1, but this probability will surely besmaller if they live in a very large popula-tion). Viewed from this perspective, the au-thors' first conclusion, as quoted above, ap-pears almost spurious.

The paper then took a crucially importantstep. The authors explored an alternative tothe basic assumption of homogeneous ran-dom mixing, which had been implicit in all

Herd Immunity 275

TABLE 2. Extract from a table published by Fox et al. (15) to illustrate the behavior of infections in arandomly mixing population, as predicted by the Reed-Frost model

Initial population composition

Susceptibies(S)

Cases(C)

Immune S Total(W)

"Probabilityof effective

contact"(P)

Expected number of effectivecontacts by case in first interval

Withsusceptibies

Total

• 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.

Probabilityof no

spread

101010

111

055

111616

0.20.20.133

221.3

232

0.110.110.23

modeling arguments to that time. They setup a structured community in which 1,000individuals were separately assigned family,school, and social groupings, each of whichhad a different internal contact probability.By using Monte Carlo techniques, theysimulated the consequences of introducinginfections into such populations with andwithout opportunities for special mixingwithin and between the social groups. Table3 presents a portion of the results of thesesimulations, which led the authors to con-clude: "Free living populations of commu-nities are made up of multiple and interlock-ing mixing groups, defined in such terms asfamilies, family clusters, neighborhoods,playgroups, schools, places of work, ethnicand socioeconomic subgroups. These mix-ing groups are characterized by differentcontact rates and by differing numbers of

susceptibies. The optimum immunizationprogram is one which will reduce the supplyof susceptibies in all subgroups. No matterhow large the proportion of immunes in thetotal population, if some pockets of the com-munity, such as low economic neighbor-hoods, contain a large enough number ofsusceptibies among whom contacts are fre-quent, the epidemic potential in theseneighborhoods will remain high. Successof a systematic immunization program re-quires knowledge of the age and subgroupdistribution of the susceptibies and maxi-mum effort to reduce their concentrationthroughout the community, rather thanaiming to reach any specified overall pro-portion of the population" (15, p. 186).While the argument that social structure isimportant in determining patterns of infec-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 stochasticsimulations under each set of conditions, as published by Fox et al. (15)

Mixinggroups

Withingroup

contact(p value)

Total number of cases per epidemic (%)

5-9 10-19 20-29 30-39 40-59

Meanepidemic

60-79 size

Total communityTotal community

Families, [62]$Total community

Families, [62]Playgroups [24]

Total communityFamilies, [62]Playgroups [24]Nursery school

0.0020.0020.50.0020.50.10.0020.50.10.1

82*22

11

23

15 2 118 34 8

6 26 23

4

17

23

1.2t3.3

5.6

28 45 45.0

* Thus, 82 of the 100 epidemics simulated under these conditions (in this case a randomly mixing community with probability ofeffective 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 statementthat it is important to reduce the supply ofsusceptibles in all subgroups is not strictlysupported in the paper's theoretical results;indeed, it is intuitively reasonable, andwas later demonstrated in theory (see be-low), that targeting vaccination to groupswith high contact probabilities can bemore efficient (in the sense of minimizingthe total number of vaccinations required)in reducing disease than is uniform cover-age of an entire population. Second, theemphasis on curbing epidemic spread re-mains. Although Fox et al. consideredtheir approach " . . . relevant to programsof systematic immunization . . . whichhave as their ultimate goal elimination ofthe causative agent from the country" (15,p. 186), it was most relevant to epidemicsin closed populations, as it had no provi-sion for examining the implications of aconstant influx of susceptibles into thepopulation, as by birth.

The Fox et al. paper deserves its consid-erable influence. Its break from the traditionof random mixing populations was a cru-cially important development. Its theorywas born of practical experience and disap-pointment with progress in measles controlin the United States, and its tone was pes-simistic and practical, compared with mostof the past (and subsequent) literature onherd immunity, which has trended to em-phasize simple thresholds. As we shall see,the paper still proves to be wise counsel.

Recent theoretical developments

The credibility of the simple formulationsof herd immunity thresholds is weakened bythe fact that the logic and formulae are basedon obviously simplistic assumptions. In par-ticular, the basic mass action models as-sumed that populations are homogeneous,with no differences by age, social group, orseason, and that they mix at random. Math-ematically inclined workers have takenthese failings as a challenge to adapt thetheory to more realistic assumptions.

The estimation of Ro. The centerpiece ofresearch on herd immunity has been the

linking of the mass action and basic casereproduction rate theories. The crucial in-sight appeared in a 1975 paper by Dietz (18)which demonstrated that, if one assumes astable population in which the mortalityrates and the incidence rates of infection areboth independent of age, then

Ro = T/Se = 1 + L/A, (11)

where L is defined as the average expecta-tion of life and A is the average age at in-fection.7 Mathematical proofs of this rela-tion have been presented by several authors(18, 23, 25, 27). The derivations assume anexponential distribution of the population byage and age-independent incidence rates ofinfection (figure 7A).8 The relation can takean even simpler form if the population isassumed to have a rectangular age distribu-tion (figure 7B), in which case

Ro = L/A. (12)

This latter relation can be illustrated neatlyif we recall that Ro is equivalent to the re-ciprocal of the proportion susceptible atequilibrium ((Ro = T/Se = l/se), and as-sume that everyone is infected at exactly ageA, the average age at infection, and dies atexactly age L, the average expectation of life(figure 7B). Assuming this rectangular agestructure, the proportion susceptible isA/L;thus Ro = L/A. On this basis, we mightconclude that the higher crude estimates ofRo implicit in equation 11 should in generalbe more appropriate for developing coun-tries, with pyramidal or exponential age dis-tributions (figures 7A and C), and the lowerestimates of equation 12 for developedcountries (figures 7B and D).

7This insight represents another contribution stem-ming from the traditions of the mathematics of vector-borne diseases (Dietz's paper (18) was on arthropod-borne viruses) and of physical chemistry (theassumption of an age-independent incidence rate is thebasis of the so-called "catalytic models" (60)).

8ln brief, if u is the death rate and A is the force (person-time incidence rate) of infection, then the average du-ration of life is 1/u = L and the average duration of sus-ceptible life is 1/(A + u). As f?0 = M(proportionsusceptible), fl0 = (A + u)/u = 1 + A/p. If p is smallcompared to A, then this expression is close to 1 + L/A.

Herd Immunity 277

"Exponential" population

A 1 0 0

0 10 20 30 40 50 60 70

Age (years)

Malawi' Population by Age1987

c„ 1,600-L.

•8 1,400-§1,200-E 1 ,ooog 600-75 4006 200-

0i '0~'rn. •J.Uj_!.i_L!

7". V !.*• v ;-, •

"Rectangular" population

B100nC'5

50HcO

IDQ .

O - i

Immune

0 10 20 30 40 50 60 70 80 90

Age (years)

England & Wales: Population by Age1991

4-

I3"1 -

o-L

in. •.? •;• v ^,;~ s !o •><•• i- S> g

Age (years) Age (years)

FIGURE 7. Schematic diagrams of exponential (A) or rectangular (B) age distributions compared with currentpopulation distributions in Malawi (C) and England and Wales (D). The exponential model (A) assumes infectionand 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 atage L.

Equations 11 and 12 may be combinedwith the basic herd immunity expression(equation 8) to give relations between crudebasic reproduction rates, herd immunitythresholds, and average age at infection, asshown in figures 8A-8D. The availability ofsuch expressions has made it a straightfor-ward matter to estimate crude basic repro-duction rates and herd immunity thresholdsfor a variety of diseases of childhood (seetable 1). Beyond that, they have opened theway to explorations of more realistic (andcomplicated) sets of assumptions.

Age-related effects. The simple mass ac-tion and Reed-Frost models make no pro-vision for the fact that individuals passthrough periods of different infection risk asthey age. The inclusion of this factor re-

quires compartmentalization of the popula-tion by age groups as well as by infectionstatus (i.e., with maternal immunity, or sus-ceptible, or latent, or infectious, or with ac-tive immunity). Assumptions must then bemade as to how the risk of infection, withineach age group in each time period, is afunction of the prevalence of infectiouscases in the same and other age groups atthat time. A general scheme for this ap-proach is presented in figure 9. Several in-vestigators have tackled the problem andhave thus been able explore the effects ofdifferent age-specific contact patterns, andvaccination strategies, within simulatedpopulations (7,19,23,36). Not surprisingly,the simple elegance of the basic mass actionmodel has been lost, and the results have

278 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 atinfection), 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-eralized, as the number of variables has in-creased. On the other hand, several prin-ciples have emerged.

Inclusion of maternal immunity (trans-placentally-acquired immunoglobulin G) inthe models serves to increase slightly theestimates of basic reproduction rates andherd immunity thresholds calculated fromequations 11 and 12 (23). This is intuitivelyreasonable in that, as far as an infectiousagent is concerned, an individual does notreally enter the population until he or she haslost maternal antibody protection (and, thus,iheA andL parameters in equations 11 and12 are, in effect, overestimates). The basicequations can thus be adapted to adjust agesas though they were calculated from the av-erage age of losing maternal immunity, M(on the order of 0.5 years for measles butless for many other infections), rather than

from birth, for example,

Ro = 1 + (L - M)I(A - M). (13)

Another use of this approach has beento explore the implications of vaccinatingat different ages. Selection of the optimalage for vaccination is dependent on sev-eral factors, including the duration of in-terfering maternally-acquired antibodies,logistic requirements of the health ser-vices, and the need to protect childrenprior to exposure to risk. The issue is com-plicated further insofar as vaccination it-self may reduce infection risks, and,hence, expand the "window" period priorto any given level of cumulative incidence.On the other hand, age at vaccination is re-lated inversely to the reduction of suscep-tibles in the population, and, hence, affectsestimates of herd immunity thresholds.This is easily described in terms of the

Herd Immunity 279

100i

TIME

FIGURE 9. Schema for age-structured model, basedon addition of age axes to figure 1. Simulation requiresaccounting susceptible (Sa,,), case (Ca, /), and immune(/a, ,) individuals over successive time periods. Suchmodels generally include births, latent infections, anddeaths (23).

rectangular age distribution (figure 7B).By seeking the proportion PH of a popula-tion which must be vaccinated at age V, inorder to produce an overall proportion ofimmunes in the population equivalent to(L - A)/L (see figure 7B), we find directly(23, 28):

H = (L- A)I{L - V). (14)

This relation (figure 10) is unrealistic inso-far as it implies 100 percent vaccine effi-cacy and it neglects that the efficacy ofmany vaccines is age-dependent (for ex-ample, not reaching a maximum until age15 months for measles). On the otherhand, it nicely illustrates an importantpoint, that simple crude estimates of im-munity thresholds, which implicitly as-sume vaccines to be given at birth or assoon as maternal immunity wanes, (and tobe 100 percent effective) will be optimisti-cally low; and that much higher coveragelevels are required because, inter alia, ofthe inevitable delays in providing vaccinesto some members of the community.

The assumption of variations in infectionrisk by age has even more complicated andimportant effects on herd immunity thresh-old estimates. It is common knowledge that

•O

N

I 80e

70:

II 60

50

PH=(L-A)/(L-V)

assumes L = 70

0 1 2 3

Age at immunization (V yrs)

— A = 3 —A = 5 -*-A = 10

FIGURE 10. Relation between PH (proportion of in-fants which must be immunized in order to attain herdimmunity threshold), A (average age at infection), andl/(age at immunization), assuming rectangular age dis-tribution (equation 14). Illustrated solutions assume L =70.

certain age groups are at special risk forchildhood infections, and it is intuitivelyreasonable that this should be so consideringthe implications of aggregation in schools inparticular. Figure 11 shows annual risks ofreported measles by age in England andWales prior to introduction of vaccination,showing the dramatic effect of the aggre-gation of children in primary schools fromthe age of 5 years. Very few children madeit to their eighth birthday without havingcontracted infection with the measles virus!The actual risks of infection in any agegroup (a) are a consequence of "contact" notonly within that group, but also between thatage group and each of the other age groupsin the community. The simple mass actionformulation can be generalized to define theincidence of infection in age group a as thesum of infections acquired from contactwithin age group a, and between that and

280 Fine

05UJ

IBL

£UJo

Z

aoz

0.7-

0.6-

0.5-

0.4-

0.3-

0.2-

0.1 -K

0 ' 1

|

/

/ 1/t1

//

1 2 ' 3 '4 ' 5 '6

*

o- —

1

' C°" o' 7 '8 ' 9 '10

—• 1950 cohort--« 1955 cohort- o 1960 cohort

^ - ^ ^ .

'11 112I13I14I

AGE IN YEARS

FIGURE 11. Age-specific risks of notified measles inthree birth cohorts in England and Wales prior to theintroduction of measles vaccination in 1968. Denomi-nators are the numbers of individuals presumed sus-ceptible (not yet immunized or infected) in each agegroup (55). Note the steep increase at age 5 years onentry to primary school. Low risk after age 6 years in the1960 cohort reflects reduced transmission after intro-duction of vaccination.

each of the other age groups (i =1,2,3- • . .a. . .n) to be considered:

a, i + 2J (15);= i

Here, the a subscripts refer to separate agegroups and ra», stands for the contact ortransmission parameter between age groupsa and /. Reiteration is based on recalculationof numbers of susceptibles and cases in eachage group at each successive time period,taking into account transitions from one agegroup to the next.

Exploration of the effects of this addi-tional structure is hampered by the difficulty(perhaps impossibility) of obtaining appro-priate data defining the contact parameterswithin and between different age groups inany population (let alone that any such pa-rameters would vary between differentpopulations and change over time). Thetheoretical implications of such age struc-ture were thus explored by Anderson andMay (36) in the context of simplified"WAIFW" ("Who Acquires Infection FromWhom") matrices defining contact between

limited numbers of age groups (in effect thera*t parameters of equation 15). An exampleof such a matrix is shown in figure 12.Analysis of these structures has revealedthat, under different circumstances, age-dependent contact rates can lead to either anincrease or a decrease in the estimates of Ro

and H compared with those derived from thesimple global mass action assumptionsabove (36). In general, crude estimates of Ro

(e.g., from equations 11 or 12) will be toohigh if age-specific contact rates are highestamong the young and fall with age. This isreasonable as older susceptibles will be rela-tively less relevant insofar as they are lesslikely to have the sort of contact necessaryfor transmission. In contrast, crude esti-mates of #0 will be too low if contact ratesrise with age.

Season and other periodic changes.Most of the common vaccine-preventablediseases are seasonal. The most obvious ex-ample of this is the seasonal increase inmeasles which follows the annual openingof primary schools in many countries (61).It was recognized long ago that this had im-plications for the mass action theory as it

AGE OF SOURCES OF INFECTION

y(5-15)

UJ (m

o_UJO y3 (5-15)

rx.x

rx.y

rx.z

ry.x

ry.y

ry.z

rz.x

rz.y

rz.z

FIGURE 12. "WAIFW" (Who Acquires Infection FromWhom) matrix of transmission parameters within andbetween three different age groups, preschool, school-age, and adult. Under most conditions such a matrixwould be symmetric along the xx-yy axis, ( r ^ = r^),though this need not necessarily be the case (e.g., thehygiene habits of younger children may be differentmaking them particularly efficient at transmitting someinfections, in which case, for example, r^ > fyx).

Herd Immunity 281

meant that there must be seasonal changes inthe transmission parameter r (and in the ba-sic reproduction rate) (51). Some early au-thors tried to mimic these changes by at-taching trigonometric functions to thecontact rates in their models (51, 62), butmore recent authors have taken more prag-matic approaches.

Yorke et al. (63) discussed the implica-tions of seasonally for eradication strategyemploying the simple mass action approach.Though these authors did not argue in termsof herd immunity thresholds or basic casereproduction rates per se, they noted thattransmission is most tenuous (i.e., Ro isminimal) just before, or during, seasons oflowest incidence, and that it should be easi-est to break transmission at these times.(Though they did not so express it, the im-plication was that the herd immunity thresh-old is lowest during such periods, and, thus,that a vaccine coverage level which is nothigh enough to "interrupt transmission" inpeak seasons may nonetheless be sufficientto do so during the annual low.)

The implications of periodic aggregationof children in schools was explored bySchenzle (7) who constructed a compart-mental model for measles simulation whichincluded both age structure and appropriatechanges in the transmission parameters tomimic the periodic aggregation of succes-sive cohorts of children in schools. His re-sults are of particular interest in that theyprovide a closer approximation to observedmeasles trends and the impact of vaccination(in England and in Germany) than has beenachieved by any other published model. Aswith the other models incorporating agestructure and a declining contact rate withage, Schenzle's simulations suggested aherd immunity threshold for measles whichwas appreciably lower than that predicted bythe simple homogeneous mixing model. Inhis own words: "The quantity [/?„ =T/Se] has no meaning at all in the presence

of age-dependent contact rates, where infec-tives of differing ages are assigned differentinfectious potentials. These have to beweighted appropriately in order to deter-

mine a 'maximum initial infection reproduc-tion rate,' Rmax, which quantity must be usedin defining conditions of herd immunity.. . .As a consequence the present model impliesherd immunity against measles with sub-stantially lower immunization rates than arepredicted from global mass action theory.Here the calculated critical immunizationcoverage would be 76 per cent if protectionby vaccination could be achieved in new-borns" (7, pp. 187-8). The extent to whichSchenzle's surprisingly low estimate ofmeasles herd immunity might have been at-tributable to his assumptions of annualchanges in transmission (low Ro values dur-ing the summer months), in addition to theassumed age structure and age-dependentcontact rates, is unclear.

Timing of interventions. The Schenzlepaper cited above, and work by others (64)have shown that the predicted impact of anintervention can also vary according to thetiming of its introduction into a population.Though it has been proposed that certainsituations can lead to "chaotic" results (65),it is unclear to what extent such effects arerelevant to actual programs, given that reallife includes many structured perturbations(such as school year calendar variation andholiday-dependent delays in notification)beyond the scope of the assumptions ofsimple mathematical models. On the otherhand, such work lends another perspectiveto the interpretation of irregular incidencepatterns.

Social and geographic clustering. Thedisparity between the homogeneous mixingassumption of basic models and the hetero-geneity in structure and mixing of real hu-man populations is obvious. The importanceof social aggregations such as families, playgroups, neighborhoods, and schools, andgeographic distinctions between towns andurban and rural areas, mean that humanpopulations are partitioned in a complex setof interlocking patterns with inevitable im-plications for the transmission of infections.Fox et al. (15) showed great insight in tack-ling this problem in their original paper onherd immunity. Since then, though several

282 Fine

subsequent investigators have attempted tobuild models with social or geographicstructure, few useful generalizations havearisen (7,20,22,23,29). In one sense, socialand geographic partitioning of populationsjust represents an extension of the sort ofpartitioning represented by age. All indi-viduals belong to many different subgroupsin society, and the transitions from one sub-group to another (by aging, migration, etc.),as well as the contact rates within and be-tween all subgroups, will vary according tomany different factors, many of which will,in turn, be confounded with one another (so-cioeconomic status, political, social, andhistorical context, behavior, hygiene level,crowding, season, mode of infection trans-mission, etc.). In an effort to describe justthe most superficial level of such complex-ity, May and Anderson (29) formulated a setof general equations describing populationsbroken into several groups with two differ-ent within and between group (high and low)

transmission characteristics. They foundthat eradication could be achieved withfewer overall vaccinations if they were dis-tributed primarily to the high contact rategroups (e.g., cities) than if they were dis-tributed uniformly to the overall population(but see also (22)). Beyond this intuitivelysensible qualitative result, that it may be ad-vantageous to target interventions at highrisk groups, we are left with the conclusionof Fox et al. (e.g., table 3) that social struc-ture can have profound effects on the like-lihood and patterns of infection transmissionand, hence, upon herd immunity thresholds.

Overall implications of additional vari-ables. Implications of the various supple-mental assumptions which have been ex-plored in recent theoretical work on herdimmunity are summarized in table 4. Thedifficulty of making precise estimates ofherd immunity thresholds in any particularcontext is evident for each of the variousinfluences even without considering the in-

TABLE 4. Implications of different assumptions for theoretical estimates of the herd immunitythreshold (H), with reference to simple global estimates as obtained by equation 8,11, and 12

Variable + assumptionImplications

for herdimmunity

References

Maternal immunity

Variation in age at vaccination

Age differences in "contact"rates or infection risk

Seasonal changes in contactrates

Geographic heterogeneity

Social structure (nonrandommixing)

If vaccines not effective until maternal immunity wanes, (23)crude H estimates will be too low; this may be corrected byconsidering that a child is not born until maternal immunitydisappears (equation 13)

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 thansimple estimates

Implications vary with relation between age and contact (7, 36)rate; falling contact rate with age implies true H may belower than simple global estimate

Seasonality may imply lower true herd immunity threshold if (7, 63)seasonal change is marked, and fade out can occur duringlow transmission period

In theory, geographic differences in contact rates may (20)permit elimination with lower overall vaccine coverage thanthat implied by H based on total population by targetinghigh risk groups

Social structure can have complicated implications as it (15)implies group differences in vaccination uptake and/orinfection risk; existence of vaccine-neglecting high contactgroups means true H will be higher than simple estimates

Herd Immunity 283

evitable interactions between them (i.e., dif-ferent age groups have different social struc-tures and seasonal patterns of aggregation).

PRACTICE

This section examines the relation be-tween theory and experience of herd immu-nity with reference to particular vaccine-preventable diseases.

Smallpox

The historic elimination of smallpox wasone of the important stimuli behind the re-cent interest in herd immunity. The initialWorld Health Organization encouragementtoward global eradication of smallpox camein a resolution passed by the 12th WorldHealth Assembly in 1959, which stated that" . . . eradication of smallpox from an en-demic area can be accomplished by success-fully vaccinating or revaccinating 80 per-cent of the population within a period of fourto five years, as has been demonstrated inseveral countries" (66). The wording is ofinterest in its explicit stipulation of a herdimmunity threshold and also in its implica-tion that waning vaccine-derived immunitymight pose an obstacle to achieving thethreshold (thus the call for revaccination).

The disappearance of smallpox frommany regions despite the continued pres-ence of large numbers of unvaccinated sus-ceptibles was evident from the historical re-cord (as had been noted by Farr (41) morethan a century ago). This is consistent withrelatively low estimates of household sec-ondary attack rates, basic reproduction ratesand, hence, herd immunity thresholds forsmallpox (table 1) (67). It is notable that the1959 World Health Organization recom-mendation implied an RQ of 5. Though thisis consistent with more recent theory-derived estimates, it was based originallyupon experience alone, having been madeprior to the development of the elegant herdimmunity theory discussed above. On theother hand, the validation of such estimates,however derived, remains difficult. In prac-tice, the severity of smallpox, in particular

variola major, was such that outbreaks gen-erally led to active intervention, in effect todifferent forms of quarantine and ring vac-cination, and, hence, it is not always clear towhat extent the disappearance of the diseasefrom different populations was due to thegeneral or to the selective vaccination.

Arita et al. (68) assembled data on crudepopulation densities and smallpox vaccina-tion coverage in African and Asian coun-tries during the late 1960s and early 1970s.Despite inevitable problems of nonuniformdistributions of populations and of vacci-nations, let alone the inaccuracy of vacci-nation statistics themselves, these data in-dicate that smallpox disappeared earlyfrom countries in which the crude densityof susceptibles (unvaccinated individuals)fell below 10 persons per km2 (corre-sponding to 80 percent coverage in popu-lations with crude population density lessthan 50 persons per km2. The infectionpersisted in more densely populated re-gions, however, in particular Nigeria (54persons per km2), Pakistan (83 persons perkm2), India (175 persons per km2), andBangladesh (502 persons per km2).Whether or not continued reliance uponpopulation-wide vaccination programsmight ultimately have been sufficient toeliminate smallpox from the more denselypopulated nations of Africa and Asia isnow a moot point. If the 10 susceptiblesper km2 threshold is a guide, then 98 per-cent vaccination coverage would havebeen necessary for Bangladesh, and suchcoverages were impracticable. However, itwas recognized by 1970 that variola viruscould be eliminated from populationsmore effectively by a policy of active casedetection, contact tracing, and the breakingof individual chains of transmission byquarantine and ring vaccination than by re-lying entirely upon herd immunity frommass vaccination programs (69). In effect,the focus of prevention activity shiftedfrom the population back to the individual.The success of this policy is now a matterof record (67).

284 Fine

Among the major lessons from the small-pox program was the inadequacy of relyingtoo heavily upon reported vaccine uptakestatistics and herd immunity predictions fordisease eradication. Many experiences illus-trated that reported data could be extremelyunreliable, and that implicit assumptions ofuniform or random coverage with vaccineswere misleading. High coverage statisticsoften obscured the fact that important seg-ments of a population were inadequatelyvaccinated and could serve to maintain andtransport the infection for long periods anddistances (67, 68).9 The disappearance ofsmallpox from many populations prior to theintensive campaigns of the final eliminationprogram are consistent with herd immunityand indirect protection of unvaccinated sus-ceptibles having contributed importantly tothe overall decline of this disease. Beyondthat, the persistence of the disease in denselypopulated third world countries despite ap-parent vaccination coverages far in excess ofthe World Health Organization's recom-mended 80 percent herd immunity thresholdprobably reflects two important factors: 1)that Ro varies importantly between popula-tions and is a function of population density,and 2) that it varies importantly within popu-lations as a consequence of complex socialpatterns.

The smallpox experience is thus salutaryin demonstrating both the validity and thelimits of herd immunity in practice. It shouldalso be appreciated that several features ofthe natural history of smallpox favored theshift in strategy away from the emphasisupon herd immunity, in particular the highcase-to-infection ratio and characteristic pa-thology (which facilitated detection ofcases) and the relatively low transmissibility(see table 1) (which facilitated control byidentification of contacts and ring vaccina-tion). Without these characteristics, much

9lt was such experiences which led to the naming ofthe World Health Organization's Expanded Programmeon Immunization, the intent being to increase immuni-zations, not just vaccinations (R. H. Henderson, WorldHealth Organization, Geneva, Switzerland, personalcommunication, 1993).

greater emphasis would have had to beplaced on raising general herd immunitylevels in order to achieve eradication of thisdisease.

Measles

No disease has been studied more in-tensely with reference to herd immunitythan has measles (3, 4, 6, 7, 27, 28, 43, 51,55, 57, 61, 70). There are two reasons forthis: 1) measles has long been a favorite sub-ject for theoretical modeling, because of itsfrequency, its regular behavior, and the highquality of available data, and 2) there hasbeen serious discussion ever since 1967 ofthe possibility of eliminating measles bothnationally and internationally (57, 71-74).These discussions have relied heavily onperceived estimates and implications of herdimmunity.

Table 5 lists published estimates of herdimmunity thresholds for measles, withnotes commenting on the assumptionsupon which each was based. The earliestcited estimate, explicit in the publisheddeclaration that measles would be eradi-cated from the United States during 1967,was derived from a combination of in-tuition, epidemiologic experience, andbold interpretation of a classic paper byHedrich (75). Hedrich had analyzedmeasles notifications in Baltimore, Mary-land, between the years 1900 and 1931and showed, by cumulating age-specificnotifications, that measles epidemics ap-peared when the proportion immuneamong children (under 15 years of age)fell below 55 percent (76). The 1967 USPublic Health Service prediction ofmeasles elimination was based upon thisfigure as an estimate of the herd immunitythreshold, neglecting the population over15 years of age because in unvaccinatedpopulations such older age groups werethen not involved in measles transmission.In retrospect, we see two problems withthis threshold estimate. First, as soon asvaccination is introduced, transmission isreduced, and the mean age of cases in-creases, and given that all age groups are

Herd Immunity 285

TABLE 5. Measles herd immunity thresholds H* as predicted in the published literature

Situation(assumptions) References

55 Based upon Hedrich's (75) analysis of Baltimore, Maryland, (57)data indicating that epidemics began when less than 55%of children under 15 years were immune; invalid becauseolder individuals were neglected

70 Compartmental model, assumptions not clear from publica- (6)tion but may have included inappropriate parameter values(23)

76 Compartmental mass action model with age and season; (7)data from England and West Germany

85 Stochastic simulation of a West Africa situation; measles (136)elimination predicted if 85% of susceptibles immunizedevery year

94-96 Compartmental mass action model with age, but no season (8)

95 Simple discrete time mass action with season but no age (63)

Not specified Reed-Frost model simulation of population with social struc- (4, 15)ture but no consideration of age, season, or introduction ofsusceptibles

* H, herd immunity threshold defined as the minimum proportion to be immunized in a population for elimination of infection.

potentially able to participate in measlesvirus transmission, the total populationshould be included in the denominator.Indeed, if everyone aged greater than 15years were immune, then the estimate of55 percent immunes among those agedless than 15 years corresponds roughly to90 percent immunes among the total popu-lation, and is thus consistent with thetheory discussed above and the simple es-timates of Ro and H shown in table 1. Thesecond problem is the implicit assumptionof homogeneous mixing.

The 1967 US Public Health Service pre-diction has been discussed by Langmuir inseveral lectures and publications (3, 77).These discussions are of particular interestin that they reflect the influence of earlymodeling theory upon the formulation ofpublic health policy. Langmuir states that hewas influenced strongly by his exposure tothe Reed-Frost model while at the JohnsHopkins University School of Hygiene andPublic Health during the 1940s, and that thiswas important in encouraging the 1967 pre-diction. It is thus ironic that it was, in part,the failure of this prediction (see figure 3C)which led to the work of Fox et al. in ap-plying the Reed-Frost model explicitly to

the problem of herd immunity (15). Fox etal.'s conclusion differed from Langmuir's,but was no less dogmatic. Twelve years afterthe original publication, Fox (4) reiteratedhis views with direct reference to measles,and in effect argued that herd immunity didnot apply because of heterogeneity of con-tact within populations.

Though Fox was reticent (perhaps be-cause of his experience) or unable (becauseof the modeling approach he used) to give aprecise estimate of the proportion immunerequired to stem transmission of the measlesvirus, his pessimism was not shared by sev-eral modelers who subsequently publishedpredictions based on variations of the massaction model approach (table 5). The rangeof these estimates, from 70 to 96 percent, isitself instructive in showing the implicationsof different sets of assumptions. Indeed, therange is such that those responsible for set-ting vaccination strategy may find that Fox'sconclusion, though less precise (he providedno threshold estimates) and less apparentlyrigorous in its mathematical base, is themost useful of them all! In general, simpletheoretical approaches provide crude esti-mates of ./?o in excess of 10 for measles indeveloped countries (except for some rural

286 Fine

area populations), and, hence, imply herdimmunity thresholds in excess of 90 percent(table 1). The extremely low estimate pro-vided by Cvjetanovic et al. (6) was basedupon simulations that may have been logi-cally flawed (23).

The comparison of theory with experi-ence is complicated by the nature of theavailable data. Measles elimination has beendeclared policy in several countries, e.g.,Canada, Czechoslovakia, Sweden, and theUnited States, and more recently the Euro-pean and Caribbean regions of the WorldHealth Organization. The strategy in eachcountry is different, in terms of the numberand timing of vaccine doses, and haschanged over time. The United States ex-perience is informative in its complexity be-cause of the size of the population and theaggressiveness with which the eliminationgoal has been pursued. Given that globaleradication is still impracticable and theconsequent inevitability of measles impor-tations, the United States has phrased itsmeasles elimination target pragmatically, asa level of population immunity and of pro-gram capacity such that indigenous trans-mission of measles virus does not persist andthat no more than two generations of trans-mission occur subsequent to any importa-tion (74).

It is difficult to describe the immunityprofile of a large nation such as the UnitedStates, because of several factors: 1) un-derreporting of measles cases (this haslessened in recent years, but was consider-able during the 1960s), 2) the fact thatmeasles cases were not reported by preciseyear of age until 1982, 3) the absence ofprecise age-year-specific vaccination up-take data, 4) variations in the estimates ofmeasles vaccine efficacy, 5) absence ofrepresentative serologic data, and 6) con-troversies over the interpretation of differ-ent serologic assays (78), let alone thesheer size and heterogeneity of the popula-tion. It is evident that the incidence ofmeasles in the United States has fallen byapproximately 99 percent since the intro-duction of vaccination in 1963, even ac-

cepting the resurgence which began in1989, despite the fact that a smaller per-centage of individuals have been immu-nized. (Though approximately 98 percentof children in the United States have beenvaccinated by school entry in recent years,an appreciable proportion escape vaccina-tion until they approach school age, and itis known that only some 95 percent ofvaccinations succeed in immunizing therecipients; thus, the proportion of the pre-school population effectively immunizedis probably less than 90 percent.) This initself is indicative of a certain degree ofindirect protection of nonimmunes by thepresence of immunes and, hence, a formof herd immunity. However, despite thedecline, measles transmission persists inthe United States. Analyses of surveillancedata suggest that transmission has beencontinuous in several large urban popula-tions, in particular those with large poorinner city populations (New York, NewYork, Los Angeles, California, etc.) andonly sporadic through the remainder of thecountry (5). It is likely that current immu-nity levels are high enough to prohibitcontinued transmission throughout most ofthe country but are insufficient in these ur-ban areas, where special initiatives will berequired to attain the high coverage requi-site for interruption of transmission. Un-fortunately, these urban centers present anextremely difficult challenge to publichealth providers, as the social conditionsare least conducive to high vaccine uptakein the very areas where the highest uptakeis required. Given the extent of populationmovement in such a nation, it is not sur-prising that the measles virus repeatedlyescapes from urban centers into schoolsand communities throughout the land.

Faced with this situation, the AdvisoryCommittee on Immunization Practice to theUS Public Health Service recommended in1989 that all American children receive twodoses of measles vaccine, at 15 months ofage and at school entry (79). It is hoped toincrease overall coverage and to reduce thenumber of primary and secondary vaccine

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failures from approximately 5 percent to lessthan 1 percent by this procedure.

Sporadic outbreaks of measles in highlyvaccinated populations have raised anotherproblem for herd immunity. Some authorshave implied that such events challenge theconcept of population protection by a highprevalence of immunes (3-5). This is toopessimistic an appraisal. The fact that indi-rect protection fails to occur in some com-munities or small populations (perhaps be-cause of a chance aggregation of vaccinefailures or an exceptionally high exposureintensity) does not invalidate that it gener-ally does occur, just as the failure of a vac-cine in one individual does not refute its ef-fectiveness in most. That said, experiencedoes suggest that most theoretically-derivedestimates of vaccination uptake and herd im-munity thresholds have been optimisticallylow because they do not cater for importantheterogeneity within real populations.

Rubella

Though the basic transmission dynamicsof rubella are similar to those of measles, itraises different questions relating to herdimmunity. Public health concern with ru-bella is concentrated on the congenital ru-bella syndrome and, thus, upon infectionsoccurring in women in their reproductiveyears (30). Control can in theory be broughtabout in two ways, either by reducing theproportion susceptible among women or byreducing their risk of infection. Differentvaccination strategies have emphasizedthese two approaches to different degrees.Vaccination of adolescent girls, as practicedin the United Kingdom between 1971 and1988, emphasized the reduction of suscep-tibles by ensuring a maximum percentage offemales would acquire either natural orvaccine-derived immunity prior to their re-productive years. On the other hand, vacci-nation of boys and girls in their second yearof life, as practiced in the United States since1971 and in the United Kingdom since 1988,also leads to reduction of circulation of ru-bella virus and, hence, to the reduction ofrisk of infection for any remaining suscep-

tibles in the adult female population. Theherd immunity implications of these twopolicies are paradoxical as this is a situationin which low coverage vaccination (a littleinduced herd immunity) can be "worse"than none at all. Low vaccination coverageof young children of both sexes can, intheory, have a detrimental effect by reduc-ing the transmission of rubella virus to sucha degree that the proportion of women ofreproductive age still susceptible to the vi-rus, and the number of consequent cases ofcongenital rubella syndrome, actually in-crease. Several investigations have con-cluded that the threshold vaccination cov-erage which must be achieved andmaintained in young children of both sexes,in order for the incidence of congenital ru-bella syndrome to decrease in the long term,is in the region of 50 to 80 percent (25, 30-32). The higher the initial intensity of trans-mission in the population, the higher thethreshold of vaccination coverage requiredamong young children in order to avoid in-creasing the incidence of congenital rubellasyndrome. Given that vaccination uptakerates in the early 1970s in the United Statesand the United Kingdom were on the orderof 90 and 50 percent, respectively, each na-tion's strategy was probably appropriate un-der the circumstances. As incidence rates ofrubella infection are extremely high in somethird world countries (e.g., The Gambia,which is one of the few populations withappropriate data), it would be unwise forthem to introduce measles-mumps-rubellavaccine until they can confidently ensureand maintain coverage levels over 90 per-cent (23).

According to current estimates, rubella isless transmissible than is measles, and, thus,a lower herd immunity threshold should berequired for its elimination (table 1). Giventhat measles and rubella vaccines are com-monly combined in a single preparation, thestrategy and success of the measles eradi-cation efforts will have interesting implica-tions for herd immunity to rubella and, thus,for herd immunity theory in general. It maybe that rubella will disappear as a conse-

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quence of measles elimination activitieswith no special additional efforts such asoutbreak containment (e.g., school exclu-sion as is practiced as part of measles controlin the United States). Such a disappearancewould confirm the theoretical predictions ofrubella Ro levels and would demonstrate thepower of herd immunity alone to dictateeradication of an infection.

Mumps

Mumps is similar to measles (both areparamyxoviruses maintained by respiratoryspread) but is less transmissible in house-hold settings and has a lower crude Ro, and,hence, a lower herd immunity threshold(table 1) (33). Mumps vaccine was licensedin the United States in 1968 but not recom-mended by the Advisory Committee on Im-munization Practice until 1977. In theUnited Kingdom, mumps vaccine was notintroduced until 1988. Routine mumps sur-veillance began in the United States in 1968.These data indicate that the incidence of thedisease fell sharply over the 9 years betweenlicensure and universal use of the vaccine inchildren. Mumps notifications have nowfallen by more than 95 percent since the in-troduction of vaccination. Given that vac-cine uptake has only recently reached thatlevel among school entrants, that uptakeamong preschoolers is far below that level,and that mumps vaccine efficacy is probablybelow 90 percent (73, 80, 81), this declinein incidence is appreciably greater thanwould be predicted by direct protectionalone. Assuming that the decline in reportedcases reflects incidence and not a decline innotification efficiency, then this is evidencefor indirect protection of susceptibles byherd immunity.

Only Sweden has thus far declared an in-tent to eradicate mumps (73). However, theroutine administration of mumps along withmeasles antigens, coupled with the lowerherd immunity threshold of mumps, indi-cates that it may disappear from severalcountries as a consequence of efforts di-rected at measles elimination. Indeed, if thisdoes not occur, it will be of interest as an

indication of special population heterogene-ity relevant to mumps virus transmission.

Pertussis

Whooping cough is an ubiquitous diseaseof childhood. Responsible for considerablemorbidity and mortality in the past, it hasbeen a target for routine vaccination pro-grams in many countries since the 1940s.These programs have been successful in re-ducing the burden of disease due to pertus-sis, and it is probable that herd immunity, inthe sense of indirect protection, has playeda role in this effect. For example, the pro-tection of older children by vaccination hasprobably reduced the risk of infection foryounger siblings who are at highest risk ofsevere complications of whooping cough.On the other hand, there has been little se-rious discussion of eradicating Bordetellapertussis (82). There is good reason for thisreticence (83).

The cyclical pattern of pertussis providesa classic example of mass action dynamics(compare figures 2 and 3B) (34, 84). Thecrude basic reproduction rate of B. pertussishas been estimated to be approximately 15for developed countries in recent decades(table 1). This is similar to measles and im-plies a crude herd immunity threshold of 93percent. Consideration of age-dependenttransmission has suggested a slightly lowerestimate, 88 percent, assuming no waning ofimmunity (34). Given that these herd im-munity estimates are higher than most esti-mates of the protective efficacy of a com-plete course of pertussis vaccine (85), andthat there is evidence of waning vaccine-derived protection (85, 86), it appears thateradication of this infection is not currentlypossible by childhood vaccination alone.

Immunity to pertussis is extremely diffi-cult to define, either in individuals or inpopulations. There is as yet no good sero-logic or other immunologic correlate forprotective immunity (85, 87); history of dis-ease is neither highly sensitive nor highlyspecific as an indicator of past infection and,hence, natural immunity; and there is con-siderable controversy over the efficacy of

Herd Immunity 289

available pertussis vaccines (85, 87). In ad-dition, there is evidence that pertussis vac-cines provide greater protection against per-tussis disease than they do against infectionwith B. pertussis, and that adults may par-ticipate in transmission of the infectionwithout manifesting characteristic signs ofthe disease (37, 84, 85, 87). Given all theseunknowns, we are not able to make convinc-ing predictions of the global herd immunitythresholds for this disease.

Diphtheria

Diphtheria is one of the success stories ofpublic health. Though vaccine-induced herdimmunity probably played a role in this suc-cess, the role was not straightforward andserves to illustrate additional complexitiesof herd immunity processes.

Diphtheria was a major cause of morbid-ity and mortality in Europe and NorthAmerica during the last century. Incidencefell from the early years of this century butthe decline accelerated along with introduc-tion of widespread toxoid vaccination in theUnited States and the United Kingdom dur-ing the 1940s. As vaccination of less than 90percent of children has led to more than99.99 percent fall in disease, it appears thatthe herd immunity threshold against diph-theria was achieved in these populations.But what was the threshold, and how did itwork?

One of the earliest published estimates ofa herd immunity threshold for any diseasewas by Godfrey (88) who, in 1933, predictedthat vaccination (three doses of diphtheriatoxoid) of 30 percent of infants and children0-4 years old and 50 percent of children5-14 years old would be sufficient to elimi-nate diphtheria. Later authors proposedhigher figures, on the order of 70-90 percent(89, 90) based on experience in developedcountries, and application of simple theorygives an estimate of approximately 85 per-cent (17). Estimates aside, the actual pro-portion of diphtheria immunes in today'spopulations is an elusive quantity. Vaccineuptake is difficult to define as at least threedoses are recommended, though one or two

doses provide some protection (91). Theprotection imparted by diphtheria toxoidvaccines has never been evaluated in formaltrials, but observational studies provide es-timates ranging from 55 to 90 percent (11,91, 92). Serologic studies have shown thatvaccine-induced antitoxin titers decline withtime or age (93), but may in some popula-tions be lower among individuals born inrecent decades, perhaps because they havenot been boosted by exposure to natural in-fections (90). Surveys carried out in devel-oped countries have shown a wide range inprevalence of "protective" antitoxin levelsamong adults, from 50 to 80 percent, leadingto recommendations that adults should re-ceive booster doses of diphtheria vaccine(90).

An even more fundamental issue relatesto the nature of immunity induced by diph-theria toxoid vaccines and how it may differfrom infection-attributable immunity. In thesense that herd immunity implies indirectprotection, it requires immunity against in-fection. However, given that diphtheriatoxin is not a normal constituent of Cory-nebacterium diphtheriae, the immunity in-duced by toxoid vaccination may not pro-vide protection against infection at all. Thisview has been expressed by numerous au-thors; e.g., ". . . immunization with diphthe-ria toxoid is protective only against the phage-mediated toxin, and not against infection bythe C. diphtheriae organism" (94, p. 1396).Some studies which have attempted tomeasure these two different types of immu-nity have found results consistent with thisprediction (11, 92). However, if diphtheriatoxoid vaccines did not impart any protec-tion against infection, then one might pre-dict that there should have been no changein the incidence of C. diphtheriae infectionin the community and no change in the riskof disease in unvaccinated individuals. Ineffect there should be no evidence of herdimmunity, a prediction which is inconsistentwith the extremely low rates of diphtheria inrecent years. Resolution of this paradox isprobably related to the fact that transmissionof the diphtheria bacillus is much more ef-

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ficient from clinical cases than from sub-clinical carriers (95); thus, the vaccines pro-tect against infection transmission, not (ormore than) against infection receipt! Reso-lution of the implications of the variousforms of immunity to diphtheria would re-quire a major research effort. It is unlikelythat this will ever be accomplished, giventhat the disease is no longer a major publichealth problem.

Tetanus

Tetanus is not directly communicable be-tween hosts, and, thus, vaccination cannotlead to indirect protection in the source-reduction sense implied in many definitionsof herd immunity. Strict adherence to thedefinition quoted by Fox et al. (15) (seeabove) would mean that herd immunity isnot relevant to tetanus at all. Certainly thereis no threshold proportion of immunes, be-low 100 percent, which can ensure total ab-sence of tetanus from a community.

There is little doubt that the introductionof routine tetanus toxoid vaccination in the1940s had an impact upon trends and pat-terns of the disease. However, the fact thatthe incidence of tetanus was decliningprior to widespread vaccination, becauseof decreasing exposure (fewer people incontact with soil and animal feces whichare the main reservoirs of the tetanus ba-cillus) and the widespread use of tetanustoxoid in wound management make it dif-ficult to assess the precise extent to whichthe prophylactic vaccination has contrib-uted to the decline in tetanus morbidity.

Despite the noncommunicability of teta-nus, vaccination of certain individuals doesimpart indirect protection to others in thecommunity. Antitetanus immunity of moth-ers is transmitted across the placenta, andtwo doses of toxoid during pregnancy canprotect a woman's offspring against neona-tal disease (96). This is extremely importantin that the public health importance of teta-nus on a global scale is attributable largelyto neonatal disease. In 1989, the WorldHealth Assembly declared an initiative toeliminate neonatal tetanus by 1995 (2).

Though the intervention will include effortsto improve birth practices, it will be basedlargely upon provision of tetanus toxoidvaccine to girls and women (97). If "elimi-nation" were to be interpreted as reductionto zero, then this initiative requires 100 per-cent effective vaccination coverage of 100percent of the target population.

Poliomyelitis

The issue of herd immunity in polio hasbeen debated for more than 3 decades. Thedebate has been notable for its partisan fer-vor and confusing for its shifting focus toand from different types of herd immunityinduction by different types of polio vac-cines (12-14,98). The ecology and herd im-munity characteristics of polioviruses areheavily dependent upon levels of hygiene.Serologic surveys carried out in the pastamong unvaccinated populations revealedthat the average age at infection ranged fromless than 2 years in nonhygienic environ-ments in developing countries to more than10 years in developed countries (99-101).Interpretation of such values in the contextof equations 8, 12, and 13 suggests that thebasic reproduction rate ranges from 5 to 30,and that the herd immunity threshold rangesfrom 80 to 97 percent depending on the levelof hygiene.

The polio herd immunity controversy hasbeen part of a broader argument concerningthe relative advantages of killed, inactivatedpolio vaccine versus live oral polio vaccine.Among the arguments favoring the live vac-cines has been the claim that they providemuch greater herd immunity than do inac-tivated polio vaccines (12, 14). Two pointsare embedded in this claim. The first is thatlive vaccines impart greater intestinal (local,immunoglobulin A-mediated) immunity,and, hence, impart greater protection againstinfection than do the killed vaccines (whichinduce protection more directly against tis-sue invasion and disease). To the extent thatthis is so, then recipients of killed vaccinesmay be protected effectively against diseasebut still be susceptible to enteric wild po-liovirus infection, and thus provide little or

Herd Immunity 291

no indirect protection to their unvaccinatedneighbors. If this were so in the extreme,then herd immunity thresholds would be in-valid for such vaccines, and only 100 per-cent inactivated polio vaccine coverage of apopulation would suffice to protect it fromdisease. However, this argument has some-times been overstated. Though there is evi-dence that prior live oral polio vaccine re-cipients excrete less virus in their feces thando prior recipients of inactivated polio vac-cine, after subsequent challenge with livepolio vaccine virus strains, it has also beendemonstrated that fecal and oropharyngealvirus excretion is reduced among prior in-activated polio vaccine recipients comparedwith unvaccinated individuals (102, 103).Thus, inactivated polio vaccines do providesome protection against infection transmis-sion. The greater propensity of inactivatedpolio vaccine to reduce oropharyngeal ex-cretion of virus might be particularly im-portant in populations with high levels ofsanitation, in which respiratory transmissionof poliovirus is more important than in areaswith poor sanitation conditions, wheretransmission is overwhelmingly by thefecal-oral route (14).

The second argument for greater herd im-munity induction by live oral rather than in-activated polio vaccine is based upon thefact that live polio vaccine virus is excretedin the feces and the oropharynx in sufficientquantities for it to be transmitted to contacts.This unique attribute of live oral polio vac-cine provides a special mechanism for in-direct protection of nonvaccinees, in effectby vaccinating them surreptitiously. Thefrequency of such live oral polio vaccinespread is dependent on hygiene behaviorand intimacy of contact, and varies greatlybetween populations. Studies carried out inthe 1950s in Louisiana and in the Seattle,Washington, virus watch program, showedthat oral polio vaccine virus was transmittedto 35-80 percent of child contacts of liveoral polio vaccine recipients within low so-cioeconomic group households, though lessfrequently within better-off households, andthat considerable transmission also occurred

beyond the confines of households (104,105). This means that the proportion immu-nized in a population receiving live oral po-lio vaccine is a function of three factors: vac-cine uptake, vaccine efficacy, and vaccinevirus transmission. The advantage inherentin this unique attribute of the live polio vac-cines is tempered only by the fact that thelive oral polio vaccine virus may rarely un-dergo reversion to virulence, and, hence, asmall proportion of the contacts of vaccinevirus may actually contract paralytic disease(this risk has been estimated to be of theorder of one such case per million vaccinedoses administered) (106).

It appears that wild polio viruses ceased tocirculate in most of the United States by1970, at which time only some 65 percent ofchildren were receiving a complete courseof live oral polio vaccine (14). However,given the complex history of previous in-activated polio vaccine and then live oralpolio vaccine programs in the country, andthe propensity of live oral polio vaccine vi-ruses to circulate in the community, theoverall level of immunity in the populationis unknown. Given the evidence for disap-pearance of wild polio viruses from theUnited States (107), it is probable that theprevalence (or subpopulation-specific preva-lences) of immunity was (were) consider-ably above whatever herd immunity thresh-old^) might have been in force.

In addition to the virologic evidence forreduced fecal excretion of virus in inacti-vated polio vaccine recipients, there is epi-demiologic evidence for indirect protectionby killed polio virus vaccines. An analysisof surveillance data from the United Statessuggested that polio incidence fell by agreater degree during the years 1955-1961(when only killed vaccines were in use) thancould be explained by the direct protectionof vaccinees alone (108, but see also 14).More convincingly, countries which haveused only killed vaccines (e.g., Sweden, Fin-land, and the Netherlands) have experiencedvirtual elimination of circulating wild polioviruses for long periods of time (109, 110).An outbreak of 10 cases in Finland in 1984-

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1985 was attributed to a type 3 virus dif-ferent from that included in the vaccine(110). Outbreaks in the Netherlands havebeen restricted almost entirely to a religiouscommunity which refuses vaccination alto-gether, with no evidence of transmission inthe population at large despite the presenceof at least 400,000 individuals who havenever been vaccinated at all (98, 111).

Current efforts at global eradication of po-lio highlight the importance of herd immu-nity. Given the low case-to-infection ratio ofpolio (probably less than 1 percent of in-fections are recognizable clinically) and thepotential of poliovirus to spread throughsewage, water, and foodstuffs, case findingand outbreak containment will be less effi-cient in controlling poliovirus spread thanthey were against smallpox and might beagainst measles. There will thus be a greaterreliance upon high levels of herd immunityin the strategy for eradication of polio thanfor these other diseases. This has been rec-ognized in the use of mass live oral poliovaccine campaigns, first in Cuba, then inBrazil, and more recently throughout LatinAmerica (112). By providing live oral poliovaccine to large segments of the population(e.g., all children under 5 years of age) si-multaneously, this approach ensures flood-ing of the environment with live oral poliovaccine virus to such a degree that very fewindividuals escape direct or indirect vacci-nation (figure 13). Though the approach hasbeen manifestly successful in eliminatingwild polio virus from Latin America, ques-tions remain over its applicability in Africaand Asia because of greater logistic diffi-culties and evidence that the efficacy of liveoral polio vaccine may be lower in theseareas than in other parts of the world (98,113). It is possible that the lower efficacycould also indicate lower indirect transmis-sion of live oral polio vaccine in some en-vironments, as both may be impeded by thehigh prevalence of other enteric virus infec-tions. This and related concerns have led tocontinued debates over inclusion of com-bined inactivated-live oral polio vaccineregimens in the strategy. The population im-

plications of all these environmental, vac-cine type, and vaccination strategy variablesare complicated. Given the pace of the glo-bal program in the face of its year 2000 tar-get, it is unlikely that there will be sufficienttime and research to comprehend fully theherd immunity mechanisms of polio control,unless the strategies fail and resources arediverted back from operations to research.

Influenza

Type A influenza viruses present yet an-other set of herd immunity problems. Giventhe genetic lability of these viruses, as mani-fested in frequent major (shift) and minor(drift) antigenic changes of their hemagglu-tinin (H) and neuraminidase (N) antigens,and their persistence in many different ver-tebrate species, there is no prospect of theirtotal eradication. On the other hand, herdimmunity has frequently been invoked in theliterature as an explanation for the changingprofile of influenza viruses in human popu-lations and the successive disappearance ofspecific antigenic subtypes (35, 114). Theargument is that increasing proportions im-mune to each individual influenza subtype,and varying degrees of cross protection pro-vided between subtypes, should provide aselective pressure favoring the spread ofnew antigenic variants. Though such amechanism appears to fit the available evi-dence, it does not lend itself to precise nu-merical description, given the complicatedimmunologic relations between virus sub-types, the possibility that immunity to in-fluenza may be less durable than immunityto many other viruses, and the unpredictablenature of the antigenic changes of theseviruses.

The hypothesis that herd immunity to in-fluenza viruses has been a driving force inthe selection of new predominant strains inthe human population has another interest-ing feature. One of the peculiarities of in-fluenza epidemiology is the observationthat, although prior to 1977 only a singlemajor virus (shift) subtype was found cir-culating in the human population worldwideat any time, more recent years have wit-

Herd Immunity 293

NUMBER OF CASES600

NVD-OPV

unumuiiuuiunui

i l l — r n — i — i i i i—i i i i i i—i—rn—i i i i i—TT—l—i i i i

1946 1950 1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1992

YEAR

SOURCE: COUNTRY REPORTS TO PAHO

CASES500

National Vaccination Days: J. J June and Augustevery year

u u

1976 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992

SOURCE: PAHOFIGURE 13. Notified cases of polio in Cuba (A) and Brazil (B) in recent decades showing the effect of nationalvaccination days (arrows). PAHO, Pan American Health Organization.

nessed the cocirculation of different sub-types (e.g., HJNJ and H3N2) simultaneouslyin the same populations (115). Why thisshould have occurred in unclear. If the ob-servation is correct, and does not reflectchanges in virologic surveillance, then the

recent appearance of cocirculating virusesmay indicate one of two possibilities: 1) ei-ther the viruses are now different, perhapsproviding less cross-subtype (e.g., H3N2

versus H1N1) protection than in the past, or2) the human population has changed, per-

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haps by increasing in total number, in num-ber of new susceptibles added per year,and/or in worldwide communication, tosuch extent that individual virus subtypesmay reduce susceptibles to below thresholdlevels in some populations but still persist inothers for long enough to allow sufficientaccumulation of susceptibles in the firstgroup to again support transmission. If thisis so, and the recent appearance of multiplecocirculating influenza viruses does reflectsuch changes in the human population, thenthis could have implications for the world-wide control of other infectious agents.

Though eradication of type A influenzaviruses is not possible, their control by im-munization is an important public healthactivity in all the wealthier countries.There has been much discussion of influ-enza vaccination strategies, given thechanging antigenic nature of the viruses,their rapid spread, explosive epidemics,and serious impact in terms of sickness ab-sences among the employed and mortalityamong the elderly. One proposal has beento reduce community spread by concen-trating on vaccination of schoolchildren, astransmission within crowded classroomsleads to rapid dispersal throughout thecommunity, and into the homes where sus-ceptible adults reside. It is of interest thatFox et al. (15), whose seminal paper onherd immunity was discussed above, wereparticularly interested in influenza andused their heterogeneous-population simu-lation model to explore various strategiesof influenza control, including selectivevaccination of schoolchildren (116). Acomparison of influenza spread betweentwo communities in Michigan, one withand one without schoolwide vaccination,provided evidence of the effectiveness ofthis selective herd immunity approach(117), and the strategy has been nationalpolicy in Japan for many years (118). De-spite such theory and evidence, the na-tional policy for influenza control in theUnited States (and most other wealthycountries) has emphasized direct protec-tion of high risk individuals and not indi-

rect herd protection through reduced trans-mission (119).

Tuberculosis

Tuberculosis is included in this review inrecognition of the fact that immunologic in-tervention, in the form of BCG vaccination,remains an important element in the controlof this disease in most countries of theworld. More people alive today have re-ceived BCG than have received any othervaccine. In addition, it presents yet a dif-ferent perspective of the problem of herdimmunity, given that natural immunity totuberculosis is generally associated withpersistent, rather than self-limited, infec-tion. (In this sense, it may be compared withother persistent infections such as those as-sociated with the herpes viruses).

There has been little discussion of herdimmunity with reference to tuberculosis. Amajor reason for this silence is the rudimen-tary level of our understanding of the natureand implications of either natural or vaccine-derived immunity to this disease (120). Wedeal here with an infection whose majorsources of transmission in most communi-ties are due not to failures to acquire priorprotective immunity but to the losses of pro-tection in older, long infected, individuals.There is no evidence thus far that availablevaccines are able to prevent this loss of pro-tection. Indeed, despite the widespread useof BCG vaccines and the good evidence thatthey can impart appreciable protectionagainst pulmonary disease in some (but notall) populations (121), there is no convinc-ing evidence that the use of BCG vaccineshas reduced the risk of infection with thetubercle bacillus in any population (122). Inthe absence of greater basic understandingof the nature and implications of the immuneresponse to tuberculosis, it is of question-able utility to ponder its theoretical herd im-plications.

Malaria

Though malaria is not generally includedamong the vaccine-preventable diseases, itdeserves mention here because it illustrates

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yet another set of problems related to herdimmunity. Considerable effort is now beingdevoted to the development of malaria vac-cines which may ultimately provide meansfor manipulating the immunity of humanpopulations against these pathogens. Theconcepts of a basic reproduction rate, and ofan eradication threshold, were formulatedwith reference to malaria before being ap-plied to any other infection (54). The cal-culation of Ro is different with reference tovector-borne than to directly communicableinfections, as the contact parameter (r or pin the simple mass action or Reed-Frost for-mulations) is a function of the density, sur-vival rate, and feeding behaviors of the vec-tor populations (23, 54). Studies in variousregions of the world have provided esti-mates of RQ for malaria in the range from 5to 100, which would imply herd immunitythresholds from 80 to 99 percent. These re-flect tremendous regional variations in theepidemiology of malaria and have been in-terpreted as indicating that vaccines alonewill never be sufficient to eliminate malaria,in particular from the holoendemic regionsof Africa. However, recent studies on theantigenic diversity of Plasmodium falcipa-rum indicate that multiple genotypes of theparasite may cocirculate in endemic areas. Ifthese reflect independent populations, thenprevious estimates of Ro, which have im-plicitly assumed a single population of para-sites, may have been too high (S. Gupta etal., Imperial College London, manuscript inpreparation). These new results suggest thatindividual genotypic populations each havei?0 values on the order of 7, and, hence,might be amenable to elimination by highvaccine coverage (>87 percent), and raisenew questions about the genetic diversityand stability of the parasite population.

Another unusual feature of malaria relat-ing to herd immunity is the fact that severaldifferent types of malaria vaccines are underdevelopment, and these may have differentindividual as well as population actions. Thesimplest vaccines, based on sporozoites ormerozoites, would, in theory, provide pro-tection against infection in the recipient and,

hence, work like most conventional vac-cines. However, there are also vaccinesagainst the transmissible stages (gameto-cytes), which would, in theory, provide noprotection against initial phases of the in-fection or against disease in the individualrecipients, but only against the transmissi-bility of the infection (123, 124). We thushave the potential for a new possibility, vac-cines which protect against transmissibilitybut not against disease! Apart from the ethi-cal problems (is it acceptable to give a vac-cine which imparts no direct protection tothe recipient?), this raises new strategicquestions concerning the appropriate de-ployment of such reagents in order to opti-mize their impact.

DISCUSSION

This brief review of various infections re-veals numerous complexities to the mea-surement and interpretation of functionalherd immunity. The concept is simplest inthe context of the nontransmissible infec-tions, such as tetanus, in which it refers onlyto the direct protection of that proportion ofthe population actually immune (thoughcomplicated in this particular example bythe important passive transfer of maternalantibodies which can protect infants fromneonatal disease). It becomes more subtlewith reference to directly transmitted viralinfections such as measles, rubella, andmumps. Infection with these (wild orattenuated-vaccine) viruses leads to long-lasting immunity against subsequent infec-tion, and we can expect that the risk of in-fection in individuals still susceptible willvary in some inverse fashion with the pro-portion of such immunes in a population.Further complexities are introduced by thefact that both vaccination and contact be-havior have highly clustered distributions inreal populations, and these distributions willdetermine the net effect of the presence ofimmunes. For many other infections, exem-plified in this review by pertussis, diphthe-ria, poliomyelitis, tuberculosis, and malaria,the complexity is much greater yet, as a con-sequence of the fact that vaccines can pro-

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vide various sorts of immunity, i.e., whichmay act against infection, or against disease,or against transmission, or which may de-cline with time or age, or which may betransferred indirectly to unvaccinated indi-viduals. Our understanding of the implica-tions of population immunity in all these lat-ter infections is seriously hindered by ourincomplete understanding of the full impli-cations of immunity in individuals.

There is also a sense in which our under-standing of immunity in individuals is de-pendent upon that in populations. Measuresof vaccine efficacy—in effect the proportionwith protective immunity among those whohave been vaccinated—may be exaggeratedif vaccination coverage is not random in apopulation. If vaccinated individuals areclustered in community groups, then theybenefit both directly, from individual receiptof the vaccine, but also indirectly, from re-duced transmission in their neighborhoods(because of the herd immunity associatedwith the concentration of vaccinated indi-viduals). In such circumstances the vacci-nated and unvaccinated individuals are notequally exposed to infection, and derivedefficacy estimates will overestimate the im-munizing capacity of the vaccine among in-dividual recipients (85, 125). This is a par-ticular problem in observational studies ofvaccines, but may also affect trials in whichrandomization is by group and not by indi-vidual.

Much of the literature on herd immunityto various infections emphasizes the estima-tion of theoretical threshold proportions ofimmunes which, if reached and sustained(e.g., by vaccination), should supposedlylead to progressive elimination of the infec-tion from the population. Such estimatesprovide a rough ranking of the probable lev-els of natural and vaccine-derived immunityrequired for eradication of these infections.On the other hand, their validity should notbe accepted uncritically; for, as shown intable 5, they vary greatly dependent upontheir assumptions, and even the most elabo-rate derivations omit important features ofthe immune response and of the practical

logistics and nonuniformity of populationsand of vaccination programs. In addition,their relevance is mitigated by the fact thatmost public health programs aim at "con-trol," rather than elimination or eradicationof infections. Even if the goal is eradication,the practical approach will not be to just at-tain some threshold and sustain it, but to aimfor and sustain the highest possible cover-age, in theory 100 percent, as this will maxi-mize the rapidity of the disappearance of theinfection in question. Merely achieving aherd immunity threshold does not mean im-mediate disappearance of the infection, itonly starts a downward trend.

Such caveats are not to argue that herdimmunity is not a valid and a useful concept.That indirect protection occurs is obvious,both in logic and in observation. Preventionof a communicable infection in any indi-vidual reduces by one the potential sourcesof infection—and, hence, the potential risk(which is a probability, by definition) ofinfection—for that individual's peers. Thatis indirect protection and a form of herd im-munity. The observation of apparent excep-tions, small communities in which infec-tions appear to be transmitted despite veryhigh levels of vaccination coverage, do notrefute this principle, just as the failure of avaccination in some individual recipientsneed not refute an overall high efficacy ofthe vaccine.

The herd immunity threshold conceptprovides an important epidemiologic at-tribute with which to characterize and un-derstand any particular infection. Thoughprecision may not be possible, even crudeestimates are themselves of use in giving arough guideline for predicting the impact ofa vaccination program and at least the po-tential for eradication. As experience grows,we will come to appreciate better how thevarious subtleties of the epidemiology ofdifferent infections (e.g., those attributableto the nature of the immune response and tothe social structure of populations) implygreater or lesser biases in the estimates de-rived from simple models. Those authorswho would discuss fully the eradicability of

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any infection must deal with the herd im-munity issue. Finally, and perhaps most im-portant, the theory of herd immunity is use-ful in the context of teaching. It is part of thebasic science of infectious disease epidemi-ology, and, like all the basic sciences, pro-vides an essential background for under-standing the real world.

The emphasis upon elimination thresh-olds in the herd immunity literature distractsfrom important dynamic implications ofchanging patterns and levels of immunity inpopulations over time. A vaccination inter-vention entails a massive disruption of theprevious "natural" balance and can destabi-lize epidemiologic patterns for many years.For example, the introduction of an effectivevaccination program among children mayreduce infection incidence to such a degreethat a large number of susceptibles can ac-cumulate among those individuals born justtoo early to receive the vaccinations, andwho thus escape both the natural infectionand the benefits of vaccination. The accu-mulation of such susceptible groups maylead to changes in the age distribution ofcases in the future, as has been reported formeasles, mumps, and pertussis in recentyears (126-129). Discussion of suchchanges is sometimes confused by presen-tation in terms of proportions of cases indifferent age groups, as it is possible, forexample, for the proportion of measles casesamong adults to increase dramatically eventhough their absolute number decreases.Prediction of such effects requires simula-tion with models which take into accountdifferences in contact within and betweenage groups. Schenzle (7), and Anderson andMay (36) have made an important contri-bution to this subject in the exploration ofmatrices to describe different age-dependentpatterns of contact. The stipulation of cor-rect matrices, and the derivation of correcttransmission parameters, present majorlogical difficulties (23, 36). Many differentmatrices will be consistent with any givenage distribution of immunity. The only wayto derive convincing descriptions is by theaccumulation of detailed analyses of age

specific data over time, preferably beforeand after a vaccine intervention.

The growth of emphasis upon vaccinationprograms, and the recognition of the com-plexity of their implications, highlight theimportance of immunologic monitoring ofpopulations. Only by accumulating suchdata will we ultimately be able to understandthe dynamics of herd immunity and the fulleffects of vaccine interventions. In addition,such monitoring should enable detection ofaccumulating pockets of susceptibles and,hence, the prediction of delayed epidemicssuch as have been observed after a period ofvaccine-program-attributable low incidence(126, 127, 130). A further example of thelong-term implications of vaccine interven-tions is the recent evidence for lower levelsof passive immunity among children ofmothers who received measles vaccine com-pared with those whose mothers had expe-rienced measles infection (131). Recogni-tion of this trend may lead to lowering of therecommended age for vaccination.

Current measles and polio programs aredestined to enlarge greatly our understand-ing of herd immunity. The continued effortto eliminate measles in the United States hasled to repeated changes in policy: changes inthe recommended age for vaccination,changes in policy of revaccination, and theformulation of special recommendations fordealing with outbreaks and with inner citypopulations (5, 79). These changes have oc-curred in response to growing understandingof the subtleties of measles epidemiology,i.e., the recognition of long-duration mater-nal antibody, appreciation of the implica-tions of changes in vaccine formulation, evi-dence for extremely high potential trans-mission risks in high school populations, thedifficulty of achieving high vaccination cov-erage in inner city areas, and the possibilitythat vaccine-derived immunity to measlesmay wane with time (5, 132). Despite theinadequacy of the data at any point in time,the public health policy has had to be de-cisive. If measles is ultimately eliminatedfrom the United States, it will be unclearwhether two doses of vaccine were really

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necessary, whether intensive outbreak con-trol was really essential, or whether merelyshifting some of the resources to urban areaswould have been sufficient. In that sense, wewill never know just what part herd immu-nity played in the success. Ironically, wemay learn more about herd immunity by ob-serving what happens to mumps and rubella,as a consequence of the measles eliminationeffort, than by observing measles itself. Ifmumps or rubella do disappear, it will beattributable largely to the passive effects ofindirect protection, to herd immunity alone.

Even more aggressive attempts at mea-sles elimination are currently underway inthe Caribbean and in Latin America, basedon mass campaigns targeted at all childrenaged from 9 months to 15 years (133).Early impressive results indicate cessationof measles virus transmission over broadareas, but the long-term implications interms of preventing importations, and follow-up vaccination strategy, have yet to be de-fined. The issue of herd immunity thus ex-pands from the protection of individualsby vaccination of other individuals, to theprotection of populations through vaccina-tion of other populations.

The goal of polio eradication from theworld by the end of this decade raises ad-ditional herd immunity problems. It nowappears that wild polio viruses no longercirculate in the New World as the last con-firmed case attributed to continued trans-mission had onset in August 1991. Thissuccess was achieved by mass live oralpolio vaccine campaigns and was no doubtassisted by the spread of live oral poliovaccine strains within the populations in-volved. This advantage of live oral poliovaccine must be balanced against thelower efficacy of these vaccines, relativeto inactivated polio virus, as measured inAfrica and in Asia (113, 134). Insofar asone of the reasons for the low efficacy oflive oral polio vaccine may be the pres-ence of other enteric infections, there maybe a complicated relation between the effi-cacy of these vaccines in individual recipi-ents, and their tendency to spread in the

population. Prediction of the overall effectof a strategy will thus be difficult for anygiven population, and optimal strategiesmay require the combination of inactivatedand live oral polio vaccines. Whatever thestrategy may be, there will be a need tomaintain high levels of herd immunity inthe New World to prevent reintroductionof polio viruses until full global eradica-tion has been achieved.

This review has avoided emphasizing anysingle definition of herd immunity, rather,accepting the varied uses of the term by dif-ferent authors. This is in keeping with thefirst published use of the term which posedthe problem of herd immunity as the prob-lem of how to distribute any given amountof immunity (antibodies, vaccinations, etc.)so as best to protect a population from dis-ease (38). The mechanisms will be several:direct protection of vaccinees against dis-ease or transmissible infection and indirectprotection of nonrecipients by virtue of sur-reptitious vaccination, passive antibody, orjust reduced sources of transmission and,hence, risks of infection in the community.And the solutions will likewise depend onmany factors: the nature of the population,the infection, the vaccine, and the health ser-vices. The population and the infection aregenerally given, the vaccine we may try toimprove, but the distribution of that vaccineis up to the public health community. Howto optimize that distribution remains, in thebroadest sense, the problem of herd immu-nity.

ACKNOWLEDGMENTS

This review began during a period as visitingscientist in the Immunization Division at theCenters for Disease Control and Prevention inAtlanta, Georgia. The author is indebted to theLondon School of Hygiene and Tropical Medi-cine, London, England, and the Centers for Dis-ease Control and Prevention for having facili-tated that arrangement, and to colleagues in bothinstitutions for many hours of discussion of thematerial presented here. Susan Ashayer and JoelAlmeido helped greatly with the document.

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