american journal of...

AMERICAN

Journal of EpidemiologyFormerly AMERICAN JOURNAL OF HYGIENE

© 1979 by The Johns Hopkins University School of Hygiene and Public Health

VOL. 109 FEBRUARY, 1979 NO. 2

Reviews and Commentary

SEASONALITY AND THE REQUIREMENTS FOR PERPETUATION ANDERADICATION OF VIRUSES IN POPULATIONS

JAMES A. YORKE,1 NEAL NATHANSON,1 GIULIO PIANIGIANI1 J AND JOHN MARTIN*

Perpetuation and persistence. Persis-tence of a virus in a population is aphenomenon distinct from persistence ina cell culture or in an individual animal,and the term "perpetuation" is used inthis essay to maintain that distinction.Most viruses are capable of perpetuatingin populations, regardless of the durationof infection in individual hosts. The abil-ity of a virus to persist in an individualmay, however, be critical for its perpetua-tion in certain populations, as discussedbelow.

Eradication is the converse of perpetua-tion and represents the ultimate methodfor control of an infectious disease. To de-termine the potential for eradication, it is

1 Institute for Physical Science and Technologyand Department of Mathematics, University ofMaryland; Laboratory of Theoretical Biology, NCI,NIH. (Reprint requests to Dr. Yorke at IPST,University of Maryland, College Park, MD 20742.)

1 Division of Infectious Diseases, Department ofEpidemiology, School of Hygiene and Public Health,The Johns Hopkins University, Baltimore, MD21205.

3 University of Florence, Florence, Italy.A preliminary version of this paper was presented

at the ICN-UCLA Symposium on Persistent Viruses(1). The authors express their deep appreciation toDr. Lila Elveback, Professor of Biostatistics, MayoMedical School, for suggesting the collaborationwhich resulted in this paper. This work was sup-ported in part by USPHS grants NS 07077, NS11451, and AI 13233.

necessary first to understand the re-quirements for perpetuation. Thus thesubject is highly relevant to practicalgoals in public health and preventivemedicine.

While we emphasize viral infectionshere, the principles governing perpetua-tion also apply to other microbial dis-eases, such as gonorrhea (2). Further-more, eradication is a potential goal foragents such as tuberculosis, diphtheria,and pertussis (3).

Viruses in human populations: majorecologic patterns. The principal ecologicpatterns of viruses in host populations areset forth in table 1, with particular refer-ence to human infections. In the majorityof instances, human viruses persist inonly a single species and are transmitteddirectly from host to host. An infrequentvariant of this pattern involves an inter-mediate insect vector, as with dengue.The other major pattern is that of thezoonotic viral infections, where the agentis maintained in another species, but maybe transmitted to humans. In most suchinstances, man is a "dead end" host, al-though occasionally the agent may besubsequently maintained by human-to-human passage (urban yellow fever,perhaps influenza).

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104 YOHKE, NATHANSON, PIAN1GIANI AND MARTIN

In this brief review, we will limit atten-tion to the maintenance of a virus withina single naturally infected species. Animplicit premise in the analysis which fol-lows is that the long-lasting type-specificimmunity conferred by viral infections(with possible odd exceptions) means thatimmune individuals can be consideredpermanently withdrawn from the suscep-tible population, as either potential casesor as potential new links in the infectionchain.

Determinants of perpetuation. Descrip-tive epidemiologic studies, reviewed byMatumoto (4), have indicated a number ofparameters which are important deter-minants of virus perpetuation. Discreteparameters which can be readily definedare listed in table 2. These parametersmay be grouped into two composite fac-tors which most directly affect transmis-sion dynamics and perpetuation.

1) Population turnover per generationperiod. The number of new susceptiblesintroduced per generation period definesthe theoretical limiting requirement forperpetuation. In a small population, ifless than one new susceptible is intro-duced per generation period, a virus willalmost certainly not perpetuate.

2) Transmissibility: The fraction of thesusceptibles infected by each existing in-fection. This critical variable (5, 6) de-pends on biological, social, and physicalfactors which are generally impossible tosort out with precision. Although difficultto quantify its individual components,transmissibility clearly differs amongviruses, as documented by the classicalhousehold studies of Hope Simpson (7).The variables, transmissibility and gen-eration time will determine the rate ofspread of an agent through a population,for a given set of population parameters.Paradoxically, an agent which spreadsrapidly may exhaust susceptibles anddisappear more quickly from a smallpopulation than a virus which spreads in-dolently. Among viruses endemic in a

given population, those with highertransmissibility will have smaller pools ofsusceptibles.

Seasonal variation in transmissibility.Human virus infections differ widely inthe extent of seasonal fluctuation in inci-dence, which is very dramatic for measlesand poliomyelitis, but relatively modestfor hepatitis B. Where seasonality ismarked, it can play a major role in thesize of the population required for per-petuation, as illustrated subsequently formeasles. Among viruses which showgreat seasonal fluctuation, peak incidencemay occur at different times of year.Thus, in the United States, measles peaksin early spring while poliomyelitis peakedin early fall. The mechanisms underlying

TABLE 1

Ecologic patterns of viruses in human populations

1) Infection confined to a single vertebrate host:most human viruses

2) Vector-borne with a single vertebrate host:dengue, urban yellow fever

3) Multiple vertebrate hosts (with or without avector): influenza?, jungle yellow fever

4) Human population not involved in virusperpetuation: arboviruses, arenaviruses,rabies

TABLE 2

Parameters which influence virus perpetuation

1) Size of population

2) Rate of population turnover: Losses arepartially immune and are replaced bysusceptibles.

3) Density of population: Contacts per unit timemay increase with density.

4) Immunity of population: Primary infectionconfers long-lasting immunity.

5) Transmissibility or infectiousness: Theprobability per unit time that an infectedindividual will transmit to a contact.

6) Duration of infectiousness

7) Generation period: The average intervalbetween acquisition and transmission ofinfection.




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PERPETUATION AND ERADICATION OF VIRUSES 105

seasonality still remain essentially un-explained.

Core population. Populations are in-homogeneous in many respects, includingthe transmission of infectious agents. Forsome viruses, such as those which aresexually transmitted, the source of mostinfections can be attributed to a "core"group which is responsible for perpetua-tion. The same principle probably oper-ates for other viruses, although the corepopulation is less clearcut. Thus, infantsmay not have sufficient extrafamilial con-tacts to act as important transmitters,even though they are highly susceptibleto many agents. In general, most viruseswill depend for perpetuation on a coreconsisting of one or more subgroups of ef-ficient transmitters (5, 6).

PERPETUATION IN SMALL POPULATIONS

Observations on the perpetuation ofviruses in populations are scatteredthrough the epidemiologic literature (4).No attempt has been made to compilethese systematically. Instead, a fewselected examples will be considered to il-lustrate the impact of some of theparameters noted above. Particular at-tention will be devoted to population size,population turnover, immunity levels,seasonal cycles, and to a comparison ofviruses with differing patterns ofpathogenesis.

Isolated human populations

Primitive human populations which

have minimal contact with the outsideworld are occasionally available forserologic study. These populations pro-vide a unique opportunity to determinethe ability of viruses to perpetuate them-selves in their natural host. The studies ofBlack et al. (9, 10) and others (8, 11-14)have delineated two principal patterns(table 3) for agents which have no ex-trahuman reservoir. Viruses which areeliminated following primary infectiontend to cause abrupt outbreaks after in-troduction into the population and thendie out until reintroduced. Viruses whichare capable of persisting in individualhosts are able to perpetuate in small iso-lated groups. Epidemiologically, a dis-tinction must be made between two kindsof persistent infection: those, like the her-pesviruses, which are accompanied bylong-term (if intermittent) shedding, andthose where persistently infected indi-viduals are not infectious, such as inmeasles or rubella panencephalitis.

Figure 1 shows the age distribution ofneutralizing antibodies to poliovirustypes 1, 2, and 3 in the isolated Eskimovillage of Narssak (population 450) insouthern Greenland, from a study by Paf-fenbarger and Bodian (13). The profilesare striking and show that type 1 virushad infected essentially the entire popu-lation 25 years previously, while type 2virus had caused a similar wave of infec-tions 15 years previously and type 3 hadbeen present only one year prior to thecollection of sera in 1952. Also, it is clear

TABLE 3

Human viruses which persist in small populations or in individuals

Agents Persist inindividuals

Persist insmall populations

Herpes virusesHepatitis B virusKuruAdenovirusesPolyoma viruses

Measles virusRubella virusCreutzfeldt -Jacob




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106 YORKE, NATHANSON. PIANIGIANI AKD MARTIN

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TYPE 2

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100

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250 5 10 15 20AGE IN YEARS

FIGURE 1. Age distribution of poliomyelitis antibodies in an isolated Eskimo village, Narssak, Greenland.Adapted from: Paffenbarger and Bodian (13).

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10 20 30 40AGE IN YEARS

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60

FIGURE 2. Age distribution of hepatitis B surface antigen (HBsAg) and antibody to HBsAg (anti-HBs) inEskimos of southwest Greenland. Adapted from: Skinhoj (14).

that following its intrusion and exhaus-tion of susceptibles, each virus had essen-tially disappeared from the population.(The type 1 antibodies at 15-24 yearspresumably represent cross-reactions dueto infection with type 2 virus).

In contrast, hepatitis B (HB) produces a

considerable proportion of persistent in-fections, particularly in primitive popula-tions. Figure 2, from a study by Skinhpj(14) of Eskimos in southwest Greenland,shows that infections have occurred at allages, with a gradual rise in cumulativeincidence to 60 per cent by age 60. The




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presence of hepatitis B surface antigen(HBsAg) positives documents themechanism of perpetuation.

Small animal populationsAnimal populations differ radically

from human populations in their rela-tively rapid rate of turnover, and this is acritical variable in virus perpetuation.Depending upon the circumstances, meanlife expectancy in many animal popula-tions ranges from 6-12 months, comparedwith a range of 30-70 years for humans.In contrast to these enormous differences,the kinetics of infection are similarin an individual human or animal host.For instance, the median incubationperiod (to rash) is about 14 days insmallpox and 10 days in mousepox. Thus,a human population of 1000 might com-monly experience a turnover of one indi-vidual per smallpox generation period,while the same would be true formousepox in a mouse population of about35.

Differences in population turnover pergeneration period undoubtedly play animportant role in the perpetuation of cer-tain acute viral infections in wild animalpopulations. Thus, rabies can be main-tained in the rather sparse fox populationby moving back and forth between adja-cent geographic areas. Infections withlong incubation periods (such as progres-sive pneumonia and scrapie of sheep) maypersist unrecognized because most ani-mals are slaughtered prior to the onset offrank clinical signs.

Laboratory animal colonies have pro-vided an opportunity to document thequantitative impact of population turn-over. Observations in such colonies areof more than academic interest, since un-wanted virus infections present an impor-tant practical problem which frequentlyconfounds biomedical research (15—18).

Ectromelia. Ectromelia or mousepox

(16) closely resembles variola in man. In-fection is initiated either by entrythrough the skin or by inhalation, and inthe systemic infection which follows, theliver and spleen are critically involvedtarget organs and there is a generalizedpox-like rash of the skin.

As in smallpox, infections can be fatal(depending upon many variables, includ-ing virus strain, and age and genotype ofmouse), but recovered mice are solidlyimmune and have cleared virus fromtheir tissues. Virus persistence, if it oc-curs, is at most a rare phenomenon, in-adequate to perpetuate infection in apopulation (17). Ectromelia is a relativelyinfectious agent and a notorious cause ofacute and devastating epizootics (18).When a few infected mice are placed in alarger group of susceptible animals, thevirus spreads rapidly, and all mice are in-fected within 20 days or two incubationperiods (figure 3). Many mice die, and allthe survivors are immune.

In these circumstances, it would not beexpected that ectromelia could be per-petuated in a small mouse population.However, a series of classical studies byGreenwood and his colleagues (19) andsubsequently by Fenner (16, 17, 20-22),elegantly demonstrated that this viruswas readily perpetuated in mouse popula-tions of 100-200 animals.

Figure 4 summarizes one experiment,based on Greenwood's data, which havebeen interpreted and consolidated in thelight of Fenner's subsequent publications(16, 17, 20-22). A mouse colony was es-tablished with 25 normal and 20 infectedmice, three uninfected mice were addeddaily, and animals were only removed ifthey died. During the 30-month observa-tion period shown, the total colony sizegradually rose to a constant level of about230 animals. At equilibrium, about 40 percent of this population died and was re-placed each month and the mean lifetime




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108 YORKE, NATHANSON, PIANIGIANI AND MARTIN

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FIGURE 3. An acute "outbreak" of ectromelia in a closed mouse colony, to show survival curves. On day 0,eight mice were infected and placed in a cage with 50 uninfected animals. Daily records were kept of micewhich were healthy, ill, or dead, and immunity of survivors was confirmed by serologic test or challenge.Adapted from: Fenner (20).

of animals entering the colony was about75 days. The infection was maintainedthroughout the existence of the colonyand at equilibrium about one-fifth themice were not yet infected, one-fifth wereactively infected, and three-fifths wereimmune survivors. The mortality pro-duced by acute infection was 50-60 percent, typical for the Hampstead mousepassage strain. These observations indi-cate that, under the conditions of this ex-periment, turnover was about 25 mice pergeneration period, and this considerationmakes perpetuation less surprising. Al-though the minimal population size inwhich ectromelia could perpetuate wasnot determined, Fenner (17) observed onepersistently infected population whichstabilized at 70 animals.

Rat virus. Rat virus is a parvovirus ofrats which is mainly transmitted as anenteric infection. This agent was unwit-tingly maintained (23) in a juvenile col-

ony of about 1000 animals of which 250five-month-old animals were replacedmonthly with 250 susceptible weanlings.Rat virus spread at a rate such that 50 percent of animals were infected by age fivemonths and the overall proportion im-mune averaged about 30 per cent. If thepopulation had been permitted to remainfor a normal life expectancy, it is likelythat all animals would have converted toimmunes; under such conditions theagent might have failed to perpetuate.

Measles. An interesting study (24) ofthe ecology of measles virus in rhesusmonkeys indicated that animals in thewild were free of infection. Following cap-ture, monkeys rapidly acquired infectionduring holding and shipment, so that allanimals had been infected within eightweeks. The rapid turnover in the export-ers' compounds in India, where monkeyswere held for only a few days to a month,constantly introduced enough suscepti-




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FIGURB 4. Persistent ectromelia infection in a small mouse colony, to show population dynamics. Recon-struction from data of Greenwood et al. (19), interpreted in light of the findings of Fenner (16, 17, 20-22).7: the average days spent as uninfected, acutely infected, or immune after entry into colony, fj: the 50 percent survival time in days.

bles to perpetuate the virus. In contrast,measles infection could not be maintainedin a stable rhesus colony.

PERPETUATION IN LARGE POPULATIONS:MEASLES

Descriptive epidemiologic data. To fol-low the pattern of virus infection in largepopulations, it is necessary to rely on re-portable diseases which are clinically dis-tinctive, and for which a high proportionof infections are clinically apparent.Measles is one of the few virus diseaseswhich meets these criteria and its be-havior in populations has been a recur-rent subject of interest to epidemiologists(25-30). Even measles has its limitations

as a subject for study, since only 10-55per cent of cases are reported.

Although measles is an ubiquitous andhighly infectious virus, its ability to per-petuate itself in human populations issurprisingly fragile. Bartlett, in his clas-sical studies (25, 26), found that in urbanpopulations of less than 500,000, measleswould periodically disappear. As table 4shows, in five cities of 200,000-300,000,the reported disease faded out for at leastone month in 56 of the 100 years recorded.Black (27) compiled data on measles inisland populations, which led to a similarestimate of critical population size.

Since a population of 300,000 might beexpected to experience about 6000 cases of




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City

TABLE 4

Reported cases of measles in cities of North America, 1921 -1940*

PopulationAverage

lreports

Years witha negative

month

Estimated% reported

New YorkChicagoPhiladelphiaDetroitLos AngelesMontrealClevelandBaltimoreBostonTorontoWashingtonPittsburghMilwaukeeBuffaloMinneapolisVancouverRochesterDallasAkronWinnipeg

7,500,0003,400,0001,900,0001,600,0001,500,0001,000,000

900,000900,000800,000700,000700,000700,000600,000600,000500,000300,000300,000300,000200,000200,000

21,00011,000

8,0008,0003,5003,5004,0005,5004,0003,9002,3003,8005,3002,2002,800

9001,2001,400

9002,100

000000100000000

203

1887

1719252014213339373622355624351624292455

* Adapted from: Bartlet (26).

measles in an average year, it is at firstsurprising that this would be insufficientto maintain the infection. The explana-tion becomes more apparent when theseasonal cycle of measles is examined (fig-ure 5). Data from a large US city (Balti-more, Maryland) show that the seasonalcycle is very marked so that only about 1per cent of the annual total occurs in thefive low months (August-December). Inthe lowest month (August), less than1/3000 of the annual total occurs per12-day generation period. This is abouttwo cases in a population of 300,000, andthere is considerable statistical variationaround this average. From this viewpoint,it is not implausible that periodic fade-outs would occur in populations below500,000 (table 5).

It has been assumed in the foregoingthat measles infects close to 100 per centof the population, or that the average an-nual number of infections is equal to an-

nual births. Figure 6 indicates that this isessentially correct, and that in one urbanarea (Baltimore, Maryland, 1900-1931)the average age of infection (t) was 5.7years. Data of the kind in figure 6 suggestthat the average size of the susceptiblepopulation (S) may be conveniently de-fined in terms of the equivalent number ofannual birth cohorts. S is slightly lessthan t, since infants are not susceptibleuntil they have lost maternal antibody,assumed here to occur at 0.7 years of age.In this example, the mean time a personis susceptible is 5.0 years and the averagevalue of S is 5.0 annual birth cohorts.

Estimates of the size of the susceptiblepopulation, as a proportion of the total,are important because they determineboth the maximum reduction whichmight be achieved by immunization, andthe minimum number required for per-petuation. In turn, these estimates areimportant elements in projecting the po-




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TABLE 5

The minimum population requiredfor perpetuation of measles

Population

Average annual cases(birth rate 20/1000)

Cases in trough month (0.1%)

Cases in trough generation(12 days)

Actual

500,000

10,000

10

3

Reported(30%)

3000

2-3

1

tential impact of control programs. Thisapproach is further developed in the con-cluding section.

Seasonality and persistence. The forego-ing observations suggest that the persis-tence of a short cycle infection in a largepopulation is critically dependent uponthe relationship between the generationtime and incidence during the seasonaltrough. Seasonality is clearly a key to un-derstanding the requirements for per-petuation in large populations. Therefore,we have attempted to quantify theparameters which determine the seasonalcycle. To this end, we have developed amodified and simplified method for cal-culating seasonal variation in transmis-sibility. Application of this analysis tomeasles illustrates the method and shows




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FIGURE 6. Age-specific susceptibility to measles.Baltimore, Maryland, 1900—1931, based on age dis-tribution of cases. Adapted from: Hedrich (29).United States, 1975, is a composite reconstructionbased on measles immunization surveys andmeasles age distribution. Adapted from: CDC (46).The average age of acquisition of immunity (by in-fection or by immunization) is denoted by T. The sizeof the susceptible population, expressed as the equiv-alent number of annual birth cohorts, is denoted asS. See text for details.

entering the population. If all personseventually acquire measles, then the av-erage daily number of cases (33,600 intwo years or 46 daily) is also the dailygain in susceptibles. (Infant mortality isignored.)

This procedure generates a periodiccurve of the fluctuation in susceptiblesaround an average level (figure 7, middlepanel). This average is independently es-timated as described subsequently. Thenumber of susceptibles (S) is expressed interms of the equivalent number of annualbirth cohorts, as described above. S mayalso be expressed as a per cent of the totalpopulation, if an annual birth rate is es-timated. In figure 7, 3.6 years multipliedby two births per 100 person-years yields7.2 per cent susceptible.

3) Transmissibility of measles variesby season, and is calculated as follows.The number of persons infected on day tcan be taken to be C, + 12, that is the caseswith onset on day (t + 12), assuming a12-day incubation period. Then

how the model may be used to examinethe role of various parameters in the per-petuation of an infectious agent.

Computer accounting. To construct anestimate of seasonal variation in measlestransmissibility, the problem is attackedin steps.

1) Between 1948 and 1964, measles inNew York City (28, 30) exhibited a regu-lar biennial cycle, alternating betweenyears of high and low incidence. Usingthese data, an average two-year cycle ofmeasles incidence is constructed. Monthlycases are spread by individual days togive a smooth curve (figure 7, top panel).

2) The number of susceptibles is tabu-lated on a daily basis, by a bookkeepingprocedure which accounts for both lossesand gains. Losses on a particular day areequivalent to the cases with onset one in-cubation period (taken as 12 days) later.Gains are the number of new susceptibles

S,

is the fraction of susceptibles infected onday t. If C, is the number of infectious in-dividuals on day t, assuming infectivity isconfined to the day of onset of illness, thenthe fraction of the susceptibles infectedper infectious individual is

where Tt is transmissibility on day t (31).From the daily tabulations of C and S,values of T can be computed and plotted(figure 7, bottom panel). (Transmissiontimes are actually spread around the dayof onset, but a correction for this spreadhas no significant influence on the plotsin figure 7).

4) The average of susceptibles is set bytrial-and-error, in which the two-yearcurve is generated using different aver-age values of S. The seasonal curves of T




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FIGURE 7. Computer accounting of seasonal fluctuation in measles. The daily number of cases (top panel)has been used to generate the curves showing seasonal variation in number of susceptibles and in transmis-sion rate. The high and low years are based on reported cases for New York City (28, 30), and it is assumedthat 16,800 susceptibles enter the population annually, equivalent to a birthrate of 20 per 1000 in apopulation of 840,000. See text for details.




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114 YORKE, NATHANSON, PIANIGIAN1 AND MARTIN

for high and low years are then superim-posed, and a mean value of S taken asthat which minimizes the high-low differ-ences in the seasonal configuration of T.In the accounting shown in figure 7, themean number of annual cohorts of suscep-tibles (S) is 3.6, less than Hedrich's (29)estimate of 5.0 years (figure 6). This isconsistent with the concept of a "core"group of susceptibles who are potentiallyefficient transmitters, excluding infantsunder age two.

Observations on the model. Figure 7sets forth the major parameters in themodel over a biennial period, which thenrepeats indefinitely. Several points arenoteworthy.

(a) Transmissibility (T) shows a rathersmall seasonal variation (about twofold),with somewhat greater variation duringtransitory seasonal extremes.

(b) Transmissibility is on the wholesimilar in the low and high years. (Closeexamination suggests that T is actuallyslightly lower in the high year, an unex-pected finding.) Thus, the model suggeststhat the larger number of cases in highincidence years is due to the accumula-tion of susceptibles at the beginning of theseasonal rise.

(c) The seasonal drop in low years oc-curs without a decrease in susceptiblesand is ascribable to a reduction in trans-missibility. In high years the case ratepeaks earlier, following a drop in suscep-tibles. Subsequently, the early summerdrop in transmissibility occurs, and thefalling incidence begins to plummet.Thus, cyclic changes in S and T interplayin producing the familiar biennial cycle ofendemic measles.

(d) An outbreak of measles can be con-structed. At first, susceptibles are numer-ous and the number of cases is low butincreasing. It may take a number of gen-erations of growth in the case rate beforesusceptibles are affected substantially.Then susceptibles drop to some critical

level where the case rate peaks. Tem-porarily, each infective is only replacinghimself. The continuing drop in the levelof susceptibles makes it less and lesslikely that an infective will transmit theinfection, so the level of cases drops moreand more steeply.

If the initial level of susceptibles wasfar above the critical level needed for oneinfective to infect one susceptible, then atthe end of the outbreak the number ofsusceptibles will be far below that level.After a considerable interval, susceptibleswill climb back to the critical level andwill continue to climb since it will nowtake a long time before the level of infec-tives grows high enough to initiate a de-cline in susceptibles.

This picture ignores seasonal variation.Soper (32) incorrectly believed that thisnatural pattern of oscillation was suffi-cient to account for sustained fluctuationin incidence. Without a computer he couldnot tell what happens over a long time.Had he carried out his calculations overseveral cycles, he would have found theamplitude of oscillation slowly dampingdown to a constant endemic level (28, 30).We may think of the level of susceptiblesas similar to a pendulum swinging backand forth past equilibrium. Seasonal vari-ation gives the pendulum a shove everyyear and these regular shoves are re-quired to keep the pendulum in motion.

(e) Bartlett (25, 26) based his theoreti-cal analysis of measles fadeout onstochastic effects, excluding seasonal vari-ation from consideration. We believe thatstochastic effects play a role, but onewhich is secondary to seasonal variationin transmissibility. Construction of sea-sonal waves based on large populations,such as New York City (figure 7),minimizes stochastic effects and docu-ments the reality of seasonality as amajor underlying cycle. During the sea-sonal trough generated by this cycle,small numbers of cases occur and stochas-




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tic effects can come into play, resultingin irregular and unpredictable fadeouts.

It should be kept in mind that theforegoing analysis treats of the macropatterns seen in large populations.Enormous variations from these averagesare seen in small subpopulations.

Level of susceptibles and perpetuation. Ifisolated large populations have a highlevel of susceptibles, a major epidemicmay occur when infection is introduced,which reduces susceptibles to a very lowlevel. During the extended period beforesusceptibles can accumulate, a fadeout ispossible even in populations larger thanBartlett's critical 300,000. Hence, a situa-tion of isolation is conducive to fadeout ofan infectious agent (27).

The simulation model may be manipu-lated to examine the influence of differentvariables on perpetuation. An example isshown in figure 8, in which a single infec-tion is introduced during the period ofpeak transmissibility (mid-January). Ifsusceptibles are at a level of 6 per cent orhigher, the virus persists and begins to

move into the biennial cycle. With a rela-tively small reduction in susceptibles (toabout 5 per cent), fadeout occurs duringthe first seasonal trough. This simulationis conducted with a number of casesroughly equivalent to those seen in apopulation of 840,000, and is, therefore,relevant to what might be expected in acity of moderate size, which has experi-enced a fadeout followed by reintroduc-tion of measles. It indicates the difficultyin preventing measles above a criticallevel of susceptibles and the potential forcontrol if immunization has reduced sus-ceptibles below that level.

Generation period and perpetuation.The extent of seasonality for a particularagent is determined not only by cycles intransmissibility but also by the genera-tion period. A given cycle in transmissi-bility will generate increasing degrees ofseasonality as the generation period is re-duced. Thus, if the transmissibility ofmeasles drops markedly over a 60-day in-terval the effect is cumulated over fivegeneration periods, that is, by the power

to

sinUJ

to

ui-o

I CASE

IOOOO

1000

100

10

S• XPOPULATION SUSCEPTIBLETO MEASLES

^AOEOUT

M MMONTH

FICUEE 8. Computer simulation of measles, following introduction of one case at seasonal peak (January),to show effect of variation in the per cent of the population which is susceptible to measles at beginning ofsimulation. The simulation is based on a population of 840,000 with an annual birthrate of 20 per 1000, anduses the seasonally variable transmission rates shown in the prior figure. See text for details.




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of 5. Applied to influenza, using an aver-age generation period of three days, thesame effect would be raised to a power of20. If, for example, incidence drops by afactor of 0.7 per generation over a two-month period of low transmissibility, theresulting decrease in measles is 0.75

(about sixfold) and in influenza is 0.720

(over 1000-fold). For this reason, in-fluenza might be unable to persist inany human population in the absence ofantigenic variation.

ERADICATION: PUBLIC HEALTHIMPLICATIONS OF VIRUS PERPETUATION

The recent dramatic successes in globalsmallpox control have demonstrated thateradication is an attainable objective forselected viral infections of man. The sa-lient epidemiologic features of smallpox(3, 33) which made eradication possiblewere: 1) The relatively long incubationperiod (about 14 days) and relatively lowinfectiousness (requiring close direct con-tact) cause variola to spread slowly withinor between rural communities. This madeit practical to place emphasis on asearch-and-containment strategy, inwhich local outbreaks were identified andaborted by intensive immunizationaround each focus. This strategy consti-tuted a critical addition to ongoing massvaccination programs. 2) The markedseasonality of smallpox meant that, evenin epidemic areas, the number of caseswas relatively low during the troughperiod. Exploitation of this phenomenonplayed a key role in achieving eradicationin India and Bangladesh. 3) The highratio of cases to infections (approachingunity) and the ready recognition ofsmallpox were also of great importance.

In light of this experience, it is of in-terest to examine the current status ofseveral important viral infections of man,and the prospects for eradication at a con-tinental if not a global level. Considera-tion is limited to the United States wherecopious data are available.

Poliomyelitis. The annual incidence ofpoliomyelitis in the United States isshown in figure 9, for 1951-1977 (34).Paralytic cases have been reduced from alevel of 10,000-20,000 to 10-20, or about1000-fold. The epidemiologic classifica-tion of cases for 1969-1977 is set forth infigure 10 (34, 35). Putting aside vaccine-associated cases and imported cases, itcan be seen that the last domestic out-break of poliomyelitis occurred in 1972.During the five years 1973-1977, therehave been a total of only seven "endemic"cases (i.e., non-epidemic, non-imported)with no identified vaccine association,and the scattering of these seven in spaceand time makes it possible that they wererelated to the vaccine virus. Apparentlyindigenous natural poliomyelitis has beeneradicated from the United States.

In searching for an explanation of thefadeout of wild poliovirus, an attempt hasbeen made to estimate the reduction inthe pool of susceptibles attained by oralpoliovirus vaccination. Figure 11 setsforth the age-specific distribution of sus-ceptibles in 1955, at the threshold of massimmunization. The 1955 estimates arebased on the age-distribution of paralyticpoliomyelitis (36) and involve several as-sumptions: (a) that most paralytic cases

10000

M I O O°

<oS IOO10

1955 INACTIVATEDPOLIOVIRUS VACCINE1

1963 ORALPOLIOVIRUS VACCINE

1950 1955 I960 1965 1970 1975

FIGURE 9. Annual reported cases of paralyticpoliomyelitis, United States, 1951-1977. Adaptedfrom: CDC (34, 35).




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EPIDEMIC

ENDEMICNO VACCINE ASSOCIATION

VACCINE-ASSOCIATED

1969 1971 1973 1975 1977

FipuRE 10. Paralytic poliomyelitis by epi-demiologic category, United States, 1969-1977.Adapted from: CDC (34, 35).

were caused by type 1 poliovirus; (b) thatthe cumulative lifetime infection rate was100 per cent; (c) that the age distributionof cases reflects the age distribution of in-fections (this may slightly overestimatethe number of susceptibles); and (d) thatmaternal antibody conferred protection toage eight months (0.7 years). The 1955data suggest that the susceptible pool (S)was about 11.2 annual birth cohorts or22.4 per cent of the population.

The size of the susceptible population in1977 is much harder to estimate, becauseof the lack of systematic serosurveys inthe United States in recent years. The re-sults of annual United States Immuniza-tion Surveys over the period 1965-1976(34, 37) indicate a gradual increase in theproportion of children ages 0-14 whohave not been immunized (5 per cent in1965 and 8.8 per cent in 1976). Thissuggests that serosurveys throughout thisperiod may be used to estimate the size ofthe susceptible population. Such surveys(38-41) have been based on small num-bers and have not employed stratifiedrandom samples. Not surprisingly, the re-sults vary considerably. One of the most

100 r

UJ

5 75

CO 50

UJ

UJQ_

S« 11.2 YEARS

7*11.9 YEARS

10 15 20

AGE IN YEARS

25 30

FIGURE 11. Estimated population susceptible to poliomyelitis (S), based on age distribution forUnited States in 1955. Adapted from: Hall (36).




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recent and larger studies, conducted inEngland, is shown in table 6. A compari-son of these surveys suggests an overallestimate, for children ages 0-14, of only70-80 per cent with neutralizing anti-body against type 1 poliovirus. In turn,this estimate implies a maximum reduc-tion in susceptibles, between 1955 and1976, of no more than threefold.

An apparent threefold reduction in sus-ceptibles has led to a reduction inpoliomyelitis of 1000-fold and brought theUnited States over the threshold of eradi-cation. This remarkable achievement in-dicates that the pool of susceptibles hasdropped below the level required for per-petuation of wild poliovirus. Such an un-predicted development requires explana-tion. An examination of the seasonal cycleof poliomyelitis suggests a possiblemechanism.

Figure 12 shows the seasonal cycle forpoliomyelitis in the United States, basedon pooled data for 1953-1955 (42).Poliomyelitis showed a marked seasonal-ity, such that incidence during the sea-sonal trough was 30-fold below peak inci-dence. From this figure it can also be seenthat the seasonality is due almost entirelyto fluctuations in transmission rate (T)since the pool of susceptibles (S) is solarge that it shows little change duringthe course of the year.

During the period 1955-1970, of de-clining poliomyelitis incidence, the infec-tion retained its seasonality. The last

TABLE 6

Poliomyelitis neutralizing antibodies (titer *1:4)from non-random sera collected during 1976-1977

in England*

1 - 45-19

20-2930-39

40+

No

67255171120403

PercentTypel

6465697673

withType

7076748573

antibody to2 TypeS

5149607359

Adapted from: Codd and White (38).

year with enough natural cases foranalysis was 1970 (figure 13). Althoughthere were only 32 cases in 1970 (43), sea-sonality was still apparent. There wasonly one case during the periodJanuary-March, and again incidencedropped to zero in December. Sinceparalytic cases represent about 1 per centof all infections, the 1970 data do notprove that the virus was unable to over-winter. However, when total infectionsfor the entire United States drop to theorder of 100 per month, it is plausible thatthe chain of infection might be entirelyinterrupted in major sections of the coun-try.

The eradication of wild poliovirus fromthe United States has certain practicalimplications which deserve brief mention:(a) Importation becomes an important po-tential source of future poliomyelitis, anda requirement for adequate immunizationof all entrants into the continental UnitedStates seems logical. Since Mexico is'themajor source of importations, all possibleassistance should be provided to imple-ment Mexican efforts at control, (b) It isimportant to maintain immunizationlevels to minimize the possibility thatwild poliovirus could be re-established fol-lowing the foreign introductions whichare now occurring (figure 10).

Measles. Measles represents an infec-tion which has been markedly reduced inthe United States by immunization, butwhich is far from eradicated. Prior to livemeasles virus vaccine, about 450,000cases were reported annually or 15 percent of an estimated annual 3,000,000cases. Following introduction of livemeasles vaccine in 1963, reported caseshave decreased to 5-10 per cent of thepre-vaccine level. If, as seems likely, re-porting of measles has improved paripassu with reduction in incidence, thenmeasles may be well below 5 per cent ofthe pre-immunization level. However,during the last 10 years, the number ofcases reported each year has remained at




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9000x

sa 3000<ra.

900

300

POLIOMYELITIS(REPORTED. CASES)

-—9600

'-—300

J F M A M J J A S O N D J F M A

orOOoicc

- I Dif)

XL.

o<rUJCD

s

11.4 -

11.2

Z. 11.0 -

SUSCEPTIBLES(S t OR PER CENT)

> / 11.2 YEARS

1 1 1 1 1

^-v^— MEAN x

\

VMEAN/1.02

1 1 1

1.02

/

22.4% /

- 22.8

22.4

22.0 9

J F M A M J J A S O N O J F M A

14

.12

.10

0 8

TRANSMISSION. ( T )

1 1 1 1

RATE

\ / ' •

1 1 1 1 1 1 1 1 1 I I

X

- 1.50

- 1.25

- 1.00

- 0.75

J F M A M J J A S O N D J F M AMONTH

FIGURE 12. Computer simulation of seasonal fluctuation in poliomyelitis, based on an average of totalreported cases for United States, 1953 through 1955. Derived from data of CDC (42). The three panels showcases, susceptibles OS) and transmission rate (D- An incubation period of 10 days was used and S wasestimated from figure 11 (36). See figure 7 and text for details.




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2 l0

8 8oo 6

<or<o- 0 L

FIGURE 13. Seasonal distribution of paralyticpoliomyelitis (32 cases), United States, 1970.Adapted from: CDC (43).

a plateau level, and the absence of adownward trend has markedly dampened(44, 45) earlier enthusiastic predictions ofeventual eradication.

It is interesting to examine the possibil-ity of measles eradication from the view-point of virus perpetuation. As a firststep, it is necessary to estimate the size ofthe susceptible population prior to andafter mass measles immunization. Figure6 compares the age-specific distribution ofsusceptibles for a typical pre-vaccineperiod (Baltimore, 1900-1931) with es-timates for the United States for 1975. Toderive the 1975 curve, data (46) from theUnited States Immunization Survey wasused to estimate the proportion of infantsimmunized at age 12-15 months; the agedistribution of measles cases (46) in theunvaccinated was then applied to the re-sidual susceptibles to complete the curve.The susceptible population is estimated atabout 10 per cent prior to immunizationcompared to about 4.4 per cent currently.

If these estimates are applied to thepopulation size requirements for measlesperpetuation (table 7), it can be estimatedthat close to 50,000 susceptibles are re-quired to maintain measles in a NorthAmerican city. If the size of the suscepti-ble pool is reduced by 50-60 per centthrough immunization, the absolutenumber of susceptibles in a large city,such as New York, remains above thethreshold for perpetuation. However, acity of moderate size, such as Baltimore,drops below the estimated threshold re-quirement for measles perpetuation. It is

interesting that measles reports (table 8)indicate that measles continues to be con-stantly present in New York, as it was inBaltimore before mass measles immuni-zation. However, since 1965 (data for1968-1971 are shown) measles has beensubject to periodic fadeouts in Baltimore.Parallel observations for Rhode Islandwere published by Scott (44), and similarpredictions were published by Griffiths(45).

Thus the United States may be closer tomeasles eradication than is generally ap-preciated. If a renewed effort to eradicatemeasles were undertaken, this reviewsuggests several components whichshould be included in such a plan: 1) Thevigorous enforcement of measles immuni-zation as a requirement for attendance inschools and in preschool day care centers

TABLE 7

Measles: hypothetical number of susceptiblesbefore and after immunization program*

Population

9,000,000900,000500,000300,000

Susceptiblesbefore

immunization

900,00090,00050,00030,000t

Susceptiblesafter

immunization

400,00040,000t22,0OOt13,000t

* Estimates of the number of susceptibles arebased on data in figure 6.

t Predicted fadeouts.

TABLE 8

Perpetuation of measles before and aftermeasles immunization program*-t

Year

1958195919601961

1968196919701971

No. of reportsNew York

479743

123

11393139

in low monthBaltimore

14221119

0000

* Measles vaccine has been widely used in UnitedStates since 1963.

t Adapted from: Yorke and London (28).




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and the like. This would reduce the poten-tial "core" population to a minimum. 2)The application of a containment strategywhere each school is considered a unit.The continued persistence of measles inspite of present day levels of immunity isprobably explicable by the high transmis-sion rates within such units. Therefore,when measles involves a school, the beststrategy may be the identification oftangential schools at high risk of second-ary spread and the vigorous immuniza-tion of non-immunes within these adja-cent units. 3) The seasonal trough shouldbe exploited in two ways. Since only about10 per cent of annual measles reportsoccur during the period August throughNovember (46), this low season (plus thetwo preceding months) is the best time toquickly identify and cope with all newoutbreaks. Furthermore, transmissibilityis at a seasonal low during the sameperiod, and this will favor a vigorous in-tervention program. 4) Required measlesimmunization for all entrants into theUnited States would complement otherelements in any containment program(44).

Future prospects for eradication. Theprinciples discussed above can be used tospeculate about the possible eradicationof other human viral infections (table 9).

Rubella and mumps are two infectionsfor which vaccines exist. For rubella, it ishighly questionable whether an attemptto eradicate is preferable to a more fo-cussed approach directed at women ofchild-bearing age. For mumps, it can bedebated whether the cost of eradication

TABLE 9

A speculative list of human viral infectionswhich are potentially eradicable

Infection

SmallpoxPoliomyelitisMeaslesHepatitis AHepatitis B

Eradicablein USA

194619721982?2000?2050?

Eradicableworldwide

1977??1

?

would be justified by the potential bene-fits.

Of the viral infections for which vac-cines will probably become available inthe future, the most logical candidate foreradication (in the United States) ishepatitis A. Since the epidemiology ofhepatitis A closely resembles that ofpoliomyelitis, eradication seems a realis-tic goal.

Hepatitis B also deserves consideration,since its ability to persist is offset by itsvery limited communicability. Three ob-servations are relevant: 1) Personalhabits and public sanitation have alreadyhad a marked impact on hepatitis B, asindicated by the much lower prevalence(47) of anti-HBs in developed than in de-veloping countries (less than 10 per centcontrasted with up to 70 per cent). 2) Inthe United States, transmission ofhepatitis B from a carrier to contacts isvery restricted, since only about 30 percent of intimate (household) contacts areinfected (48). 3) Most (probably 90 percent) hepatitis B infections are acquiredafter age six (49) when children are read-ily accessible for immunization programs.These considerations suggest that, if 90per cent of infections could be preventedby school entry immunization, thecumulative reduction over several gener-ations would be significant. However,further elucidation of the pathogenesis,immunology, and mode of transmission ofhepatitis B is a necessary prerequisite forserious evaluation of the feasibility oferadication.

SUMMARY

Perpetuation of a virus in a populationis distinct from the ability to persist in acell culture or individual host. Parame-ters which determine perpetuation in-clude: 1) the size of the population; 2) theturnover of the population; 3) the propor-tion of immunes in the population; 4) thetransmissibility of the infection; and5) the generation time between sequen-




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tial infections. These parameters maybe grouped into two composite factorswhich most directly affect transmissiondynamics and perpetuation: (a) popula-tion turnover per generation period, and(b) transmissibility or the fraction of sus-ceptibles infected per existing infection.

Perpetuation in small populations usu-ally requires either the ability to persistin individuals or rapid population turn-over. Conversely, human viruses whichinitiate only acute infections requirelarger populations to persist. Seasonalvariation in transmissibility can greatlyincrease the minimum population size inwhich persistence is possible, and weargue that the population size of 500,000for measles persistence (described byBartlett) is primarily a consequence ofseasonal variation. Computer modellingcan be used to examine the effect ofchanges in parameters which determinethe seasonal cycle of virus perpetuationand fadeout.

Finally, human infections are reviewedto indicate those which have been eradi-cated (smallpox), are on the threshold oferadication (poliomyelitis), are possiblyeradicable (measles), or could be candi-dates for future efforts (hepatitis A andhepatitis B). In developing a strategy foreradication two points are of great poten-tial utility: first, the seasonal troughshould be exploited as a time for effectiveintervention; and, second, containmentefforts should be directed at epidemiologi-cally important population groupingssuch as schools.

REFERENCES

1. Nathanson N, Yorke JA, Pianigiani G, et al:Requirements for perpetuation and eradicationof viruses in populations. In Persistent Viruses.Edited by J Stevens, GJ Todaro, FL Fox. NewYork, Academic Press, in press

2. Yorke JA, Hethcote HW, Nold A: Dynamics andcontrol of the transmission of gonorrhea. Jour-nal of Sexually Transmitted Diseases 5:51-56,1978

3. Henderson DA: The eradication of smallpox. SciAm 235:25-33, 1976

4. Matumoto M: Mechanism of perpetuation ofanimal viruses in nature. Bacteriol Rev33:404-418, 1969

5. Fox JP, Elveback LR: Herd immunity-changingconcepts. In Viral Immunology and Im-munopathology. Edited by AL Notkins. NewYork, Academic Press, 1975, pp 273-290

6. Fox JP, Elveback L, Scott W, et al: Herd immu-nity: basic concept and relevance to public healthimmunization practices. Am J Epidemiol94:179-189, 1971

7. Hope Simpson RE: Infectiousness of communica-ble diseases in the household (measles, chicken-pox, and mumps). Lancet 2:649-554, 1952

8. Hope Simpson RE: Studies on shingles. Is thevirus ordinary chickenpox virus? Lancet2:1299-1302, 1954

9. Black FL, Hierholzer WJ, Pinheiro FD, et al:Evidence for persistence of infectious agents inisolated human populations. Am J Epidemiol100:230-250, 1974

10. Black FL: Infectious diseases in primitivesocieties. Science 187:515-518, 1975

11. Panum PL: Observations made during theepidemic of measles on the Faroe Islands in theyear 1846. New York, American Publishing As-sociation, 1940

12. Paul JH, Freese HL: An epidemiological andbacteriological study of "common cold" in an iso-lated Arctic community (Spitzbergen). Am JHyg 17:517-535, 1933

13. Paffenbarger RS, Bodian D: Poliomyelitis im-mune status in ecologically diverse populations,in relation to virus spread, clinical incidenceand virus disappearance. Am J Hyg 74:311-325, 1961

14. Skinhoj P: Hepatitis and hepatitis B-antigen inGreenland. II. Occurrence and interrelation ofhepatitis B associated surface, core and "e"antigen-antibody systems in a highly endemicarea. Am J Epidemiol 105:99-106, 1977

15. Holdenried R (Editor): Viruses of laboratory ro-dents. NCI Monograph No 20, Washington DC,US GPO, 1966

16. Fenner F: Mouse pox (infectious ectromelia ofmice): A review. J Immunol 63:341-373, 1949

17. Fenner F: The epizootic behavior of mousepox(infectious ectromelia of mice). II. The course ofevents in long-continued epidemics. J Hyg46:383-393, 1948

18. Briody B: Response of mice to ectromelia andvaccinia viruses. Bacteriol Rev 23:61-95, 1959

19. Greenwood M, Bradford Hill A, Topley WWC, etal: Experimental Epidemiology, Special ReportSeries, No 209, Medical Research Council,HMSO, London, 1936

20. Fenner F: Studies in mousepox (infectious ec-tromelia of mice). VII. The effect of the age ofthe host upon the response to infection. Aust JExp Biol Med Sci 27:45-53, 1949

21. Fenner F: The epizootic behavior of mouse-pox(infectious ectromelia). Br J Exp Pathol29:69-91, 1948

22. Fenner F, Fenner EMB: Studies in mousepox(infectious ectromelia of mice). V. Closed




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epidemics in herds of normal and vaccinatedmice. Aust J Exp Biol Med Sci 27:19-30, 1949

23. Robinson GW, Nathanson N, Hodous J: Sero-epidemiological study of rat virus infection in aclosed laboratory colony. Am J Epidemiol94:91-100, 1971

24. Meyer HM Jr, Brooks BE, Douglas RD, et al:Ecology of measles in monkeys. Am J Dis Child103:307-313, 1962

25. Bartlett MS: Measles periodicity and commun-ity size. J Roy Stat Soc Ser A 120:48-70, 1957

26. Bartlett MS: The critical community size formeasles in the United States. J Roy Stat Soc SerA 123:37-44, 1960

27. Black FL: Measles endemicity in insular popu-lations. Critical community size and itsevolutionary implication. J Theor Biol 11:207-211, 1966

28. Yorke JA, London WP: Recurrent outbreaks ofmeasles, chickenpox and mumps. II. Systematicdifferences in contact rates and stochastic ef-fects. Am J Epidemiol 98:469-482, 1973

29. Hedrich AW: Monthly estimates of the childpopulation "susceptible" to measles,1900-1931, Baltimore, Md. Am J Hyg17:613-636, 1933

30. London WP, Yorke JA: Recurrent outbreaks ofmeasles, chickenpox and mumps. I. Seasonalvariation in contact rates. Am J Epidemiol98:453-468, 1973

31. Frost WH: Some conceptions of epidemics ingeneral. Am J Epidemiol 103:141-151, 1976

32. Soper HE: The interpretation of periodicity indisease prevalence. J Roy Stat Soc Ser A92:34-73, 1929

33. Fenner F: The eradication of smallpox. ProgMed Virol 23:1-21, 1977

34. Center for Disease Control: Poliomyelitis sur-veillance summary 1974-1976, Atlanta, 1977

35. Center for Disease Control: Morbidity and Mor-

tality Weekly Report 26:424, 197736. Hall WJ, Nathanson N, Langmuir AD: The age

distribution of poliomyelitis in the UnitedStates in 1955. Am J Hyg 66:214-234, 1957

37. Center for Disease Control: US ImmunizationSurvey: 1976. HEW Publ No (CDC) 78-8221

38. Codd AA, White E: Protection againstpoliomyelitis. Lancet 2:1078, 1977

39. Oberhofer TR, Brown GC, Monto AS: Seroim-munity to poliomyelitis in an American com-munity. Am J Epidemiol 101:333-339, 1975

40. Poland JD, Plexico K, Flynt JW, et al: Poliovirusneutralizing antibody levels among preschoolchildren. Public Health Rep 83:507-512, 1968

41. Melnick JL, Burkhardt M, Taber LH, et al: De-veloping gap in immunity to poliomyelitis in anurban area. JAMA 209:1181-1185, 1969

42. Center for Disease Control: Poliomyelitis Sur-veillance Report No 56. Atlanta, 1956

43. Center for Disease Control: Poliomyelitis Sur-veillance Summary 1970. Atlanta, 1971

44. Scott HD: The elusiveness of measles eradica-tion: Insights gained from three years of inten-sive surveillance in Rhode Island. Am JEpidemiol 94:37-42, 1971

45. Griffiths DA: The effect of measles vaccinationon the incidence of measles in the community. JRoy Stat Soc A 136:441-449, 1973

46. Center for Disease Control: Measles Surveil-lance Report No 10, 1973-1976. Atlanta, 1977

47. Szmuness W, Dienstag JI, Purcell RH, et al: Theprevalence of antibody to hepatitis A antigen invarious parts of the world: a pilot study. Am JEpidemiol 106:392-398, 1977

48. Szmuness W, Harley EJ, Prince AM: Intrafa-milial spread of a symptomatic hepatitis B. AmJ Med Sci 270:293-304, 1975

49. Center for Disease Control: Hepatitis surveil-lance report No 39. Atlanta, 1977




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