Proc. Natl. Acad. Sci. USAVol. 93, pp. 7231-7235, July 1996Medical Sciences

Mass vaccination to control chickenpox: The influence of zoster(population dynamics/infectious diseases/mathematical modeling/varicella/immunization)

NEIL M. FERGUSON, RoY M. ANDERSON, AND GEOFF P. GARNETT*Wellcome Centre for the Epidemiology of Infectious Disease, Department of Zoology, Oxford University, South Parks Road, Oxford OX1 3PS, United Kingdom

Communicated by Robert May, University of Oxford, Oxford, United Kingdom, February 20, 1996 (received for review November 17, 1995)

ABSTRACT The impact of transmission events from pa-tients with shingles (zoster) on the epidemiology of varicellais examined before and after the introduction of mass immu-nization by using a stochastic mathematical model of trans-mission dynamics. Reactivation of the virus is shown to dampstochastic fluctuations and move the dynamics toward simpleannual oscillations. The force of infection due to zoster casesis estimated by comparison of simulated and observed inci-dence time series. The presence of infectious zoster casesreduces the tendency for mass immunization to increasevaricella incidence at older ages when disease severity istypically greater.

allow detailed investigation of the effects of different vacci-nation policies, where age structure and mixing patterns playa critical role (13). We therefore adapted a stochastic realisticage-structured (RAS) model design that has been appliedsuccessfully to investigate the pre- and postvaccination epide-miology of the measles virus (14-16) and used it to explore thepotential impact of varicella vaccination (3, 4). The RASmodel incorporates discrete age structure in the form of 20yearly cohorts and one group representing those over 20 yearsof age.The age-specific force of infection Aj is the rate at which

susceptible individuals in age cohort i acquire infection and isdefined as

Approval for the use of a varicella (chickenpox) vaccine wasgiven by the U.S. Food and Drug Administration in March1995 (1, 2). The predictions of mathematical models of vari-cella transmission dynamics played a part in the decision tolicense the vaccine, and they will have a role in the design ofany mass vaccination programs (3, 4). To date, however, allsuch models of varicella transmission have been based upon asomewhat simplified view of the natural history of varicellainfection. Following the primary disease stage of varicella-zoster virus (VZV) infection, the infectious agent becomesdormant in the dorsal root ganglia (5, 6), from where it canreactivate many years later, causing herpes-zoster (shingles).The first indication that the two diseases might be connectedwas the 1888 observation, by von Bokay, of varicella casesacquired through contact with zoster. However, beyond ac-knowledging its role in the continued presence ofVZV in smallisolated populations (7, 8), the importance of infections ac-quired from zoster has not been considered in studies of thetransmission dynamics of the virus. This is largely due toignorance of the relative infectiousness of zoster comparedwith varicella.By using a stochastic version of a simple, seasonally forced,

compartmental [SEIR-susceptible (X), exposed (H), infec-tious (Y) recovered (Z)] transmission model, Rand and Wilson(9) demonstrated that chaotic repellors can be stabilized bystochastic fluctuations. Stochastic dynamics with a positiveLyapunov characteristic exponent created large deviationsfrom the deterministic orbit. Fluctuations of such magnitudehad not been observed in the similar stochastic models ofOlsen and colleagues (10-12). Their models produced inci-dence time-series dominated by an annual cycle in goodagreement with data on varicella incidence from Bornholmand Copenhagen in Denmark, as a result of using an immi-gration rate of infectives (21 per year for N = 106, and 2 peryear for N = 50,000), much larger than that used in the Randand Wilson model. The effect of this immigration term is todestroy the chaotic repellor, and consequently reduce thefluctuations that would otherwise be seen.While SEIR models may be able to replicate the observed

incidence patterns of varicella quite accurately, they do not

A = lfijyj+4' [1]

Here 3ij is the transmission coefficient from cohortj to cohorti, and Yj is the number of infectives in cohort j (see Appendixfor model details). 4' represents a constant "background"force of infection, generated by the additional varicella infec-tions caused by herpes-zoster cases. The limited data that existon zoster incidence indicates that it has a white noise spectrum,with no discernible dynamical structure (17, 18), justifying ourassumption that the force of infection due to zoster can bepresumed to be constant (i.e., nonseasonal). However, sincezoster incidence data tell us little about the magnitude and agestructure of its contribution to VZV transmission, it is neces-sary to explore the effect of a wide range of 4' values on modeldynamics.

In general, the introduction of a constant rate of infectiveimmigration into the SEIR or RAS models has a profound(and often unremarked upon) effect on their dynamics (19-21), effectively imposing a minimum bound on Y, and therebydestroying all attractors with orbits dropping below that bound.In the case of measles models, where immigration is used as acrude representation of spatial effects (coupling a populationto an external reservoir of infection), the result is the destruc-tion of those attractors that produce the large amplitudechaotic behavior seen in the SEIR model with large seasonalforcing. For chickenpox, non-negligible immigration destroysthe chaotic repellor seen in the SEIR model (9), therebystabilizing the dynamics and moving them toward the deter-ministic annual limit cycle.

Seasonality is introduced by reducing the value of thetransmission coefficient 132 (see Appendix, Eq. 4), describingmixing within primary school children, outside school termtime. The magnitude of seasonal forcing is calculated as thedifference, AP3, between the in-school and out-of-school mean,B values (see Appendix). The deterministic version of thismodel generates a annual limit cycle for the range of values ofAP3 and 4' relevant to varicella in developed countries. Themagnitude of the oscillations increased with AP3, but varying

Abbreviations: VZV, varicella-zoster virus; SEIR model, susceptible,exposed, infectious, recovery model; RAS model, realistic age-structured model; FOI, force of infection; WAIFW, who acquiresinfection from whom matrix.*To whom reprint requests should be addressed.

7231

The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertisement" inaccordance with 18 U.S.C. §1734 solely to indicate this fact.

Dow

nloa

ded

by g

uest

on

Feb

ruar

y 3,

202

0

7232 Medical Sciences: Ferguson et al.

the 4' coefficients within the range 0 ' A' c 0.03 had littleeffect. Because the stochastic and deterministic models gavenearly identical results for N < 2 x 107, we used the deter-ministic model to estimate a realistic value for Aj3 by adjustingAP3 to match the oscillation amplitude of the model with thatfrom the time-averaged values recorded in a survey by theRoyal College of General Practitioners in the United Kingdompresented in Fig. la. Using a least-squares fitted (minimizingthe deviation between observed and simulated amplitudes ofthe annual oscillations in incidence) value of AP3 = 5.85 x 10-6,the fractional forcing magnitude, A(3/(3, is 0.32 (9-12). How-ever, the results presented below are relatively insensitive tothe precise value of A13, since the RAS model produces annuallimit cycles for all A13 in the range 0 ' A13 s f3.

It should be noted that there is no evidence that the RASmodel with the parameters used here possesses a chaoticrepellor of the type that so dramatically changes the dynamicsof the Rand and Wilson SEIR model of varicella transmission.Rather, the large dynamical fluctuations seen for small pop-ulations are a result of demographic stochasticity and extinc-tion effects. Unlike the SEIR model, the RAS model con-verges to the deterministic annual limit cycle asN is increased.We calculated the power ratio, r, a measure of stochastic

fluctuations in a time-series (22), which, because varicella hasa primary annual cycle, we defined to be the ratio of thetriennial to annual spectral power. As stochastically drivenfluctuations perturb the annual cycle, the triennial contribu-tion to the power spectrum increases, thereby increasing r. Thecontribution of these two periodicities to the power spectrumwas computed by calculating the area under the relevantsections of the spectral curve-in this case between frequen-cies 0.2 and 0.8 (which includes the 0.33 and 0.66 harmonics)and between 0.8 and 1.2. However, the precise definition usedmakes little difference with very similar results for a powerratio defined as the ratio of the triennial to annual spectral

(a)60

v

X 40

20

0

peak heights. The power ratio curves obtained by varying A4 fordifferent values of N (Fig. lb) encompass three differentdynamical regimes. For a low force of infection from zoster (A4< 10-5), when the population is small (N < 2 x 106) all thecurves converge to r 0.575, corresponding to long periods ofdisease extinction, interspersed by large epidemics (see Fig. 2ffor typical spectrum). Around the critical population size ofN 2 x 106, the dynamics becomes much less stable (reflectedby the larger error bars on the curve), intermittently switchingbetween cycles of different periodicities (1-4 years) andamplitudes (Fig. lc). For large populations N . 5,000,000,extinctions become very rare and the dynamics show a strongannual oscillatory component for the entire range of zosterforces of infection. An intermediate zoster force of infection(4' = 10-3) is sufficient to prevent frequent extinctions, andthe dynamics are then dominated by large-scale triennial andbiennial stochastic fluctuations (see Fig. 2e for typical spec-trum). As A4 increases further, stochastic fluctuations areincreasingly damped and annual oscillations dominate thedynamics. Increasing A4 is directly equivalent to increasing Nand the dynamics move rapidly toward the deterministic limit.For large populations (N < 107), the power ratio curve flattensoff and r << 1 across the entire range of 4' values.

Spatial heterogeneity is not included in the model presentedhere because spatial effects are unlikely to be as significant inthe transmission dynamics of varicella-zoster as in those ofmeasles. In latter case, spatial heterogeneity is key to explain-ing disease persistence (22), while this paper shows thatvaricella persists in even the smallest populations due to theexistence of the reservoir of infection provided by zoster cases.By comparison, the magnitude of any spatial coupling toneighboring regions is likely to be of secondary impor-tance-at least until high levels of vaccination coverage areachieved. Indeed, for this reason the transmission dynamics of

(c) 2000-

w 1500-a

In 1000-0

500

01 I 1 21 31 41 51

Week0 25 50 75

Yearloo 125 150

(b) Log(Zoster FOI) -43 / \LLog'(Population size)

0

Log(Power ratio) j 4

-2~~~~~~-

FIG. 1. (a) Weekly varicella incidence throughout an average year. Royal College of General Practitioners figures (solid line, ±1 SD) areaveraged over data for the period 1976-1987 and are scaled for 62% under-reporting (estimated by assuming that everyone is infected with VZVbefore their death). The Royal College of General Practitioners reports for varicella and zoster are collected voluntarily by a set of 200 generalpractitioners spread over the United Kingdom and are presented in refs. 17 and 18. Model figures (dashed line) are for deterministic run, at thebest-fit value of AP3 = 5.85 x 10-6 (AO - 0.03 Vi, and N = 5 x 107). (b) Spectral power ratio, r, plotted against the Zoster force of infection (FOI),A4, for population sizes between N = 25,000 and N = 107 (vertical lines indicate ± 1 SD), using logarithmic scales. Each point is the average ofseven runs of the stochastic RAS model. Every run was for 350 years, the spectra being calculated from the last 200 years of data. (c) A 150-yearexample incidence time series from stochastic RAS model (parameters used given in Appendix) with N = 106 and A4 = 0.0001, illustratingintermittency of dynamics.

,,,AAWII1 JU

Proc. Natl. Acad. Sci. USA 93 (1996)

Dow

nloa

ded

by g

uest

on

Feb

ruar

y 3,

202

0

Proc. Natl. Acad. Sci. USA 93 (1996) 7233

varicella are distinctively different from most other childhooddiseases.We adjusted the magnitude of the A4 (with 4' = k' Vij) to

achieve the best fit to incidence data from Copenhagen(population 600,000,t Bornholm (population 50,000), and theFaroe Islands (population 25,000) (Fig. 2). With increasing A4the magnitude of fluctuations away from annual cycles and thefrequency of disease extinction are both dramatically reduced.This is reflected in a reduction in the relative prominence ofthe 3-year compared with the 1-year cycle peak in the powerspectrum. For small 4', and N between 5 x 105 and 2 x 106,this model exhibits considerable intermittency, switching reg-ularly between periods of 1-, 2-, and 3-year cycles (see Fig. lc).Such behavior has been noted in RAS models of measles withsimilar population sizes (15), and may be indicative of thepresence of multiple coexisting attractors (21, 23) or a chaoticrepellor (9).With N = 25,000 and 4' = 0.03, the model appeared to

provide a good description of varicella incidence in the FaroeIslands, producing a time series with large fluctuations but fewextinctions. As spectra for simulations varied greatly, theestimate of 4' is clearly a crude approximation. However, thespectra obtained from model runs at n = 6 x 105 and 4' = 0.03were remarkably stable, and agreed closely with the spectrumcalculated from the Copenhagen incidence data. For smallpopulations such as the Faroe Islands, demographic stochas-ticity would be expected to generate a high extinction proba-bility for the disease, giving rise to incidence time series similarto that seen for measles on the islands (Fig. 3a), namely longperiods of disease extinction with infrequent epidemics. In-

tCopenhagen city is part of a larger metropolitan area with population1 million, but reported case notifications are only for the city proper.

deed, as the measles data for the Faroe Islands does not suggestthat a high immigration rate of infectives pertains, we rule outthis possibility as a further reservoir of infection for varicella.Simulated time series with a low zoster force of infection A4 =0.0001 (Fig. 2 c and f) highlight the need to include the forceof infection due to zoster in any model of varicella dynamics.This key result is relatively robust against perturbations in anyof the key parameters of the model, being essentially the onlyrealistic explanation of the difference in the extinction behav-iors of varicella and measles.To quantify the transmission coefficient of zoster for vari-

cella infections we assume A4 = 0.03 Vi, and use

A= ,byb [2]

where yb is the mean number of zoster cases, and fb is thetransmission coefficient of zoster. Royal College of GeneralPractitioners figures for zoster give a mean weekly incidence inEngland and Wales (N = 5 x 107) of 6000. Using an estimateof 4 weeks for the duration of zoster (17), we get an estimateof /b = 1.25 x 10-6, -7% of the mean value of ,B for varicellaof 1.82 x 10-5.We proceed to analyze the influence of vaccination on the

transmission of varicella comparing results with and withoutthe FOI term arising from zoster cases (Fig. 3 b-f). In itsabsence a pattern typical of other childhood infections isobserved, with a fall in the number of cases over a periodsimilar to the prevaccination average age of infection. This isfollowed by periodic epidemics, which become more pro-nounced for coverage levels similar to those observed in otherU.S. childhood immunization programs (25). Associated withthese changed dynamics is a rise in the mean age of infection(Fig. 3e), which is of concern due to the severe disease (Fig. 3f)

(d) 2-&-0 1.5-

_ 1 -

t 0.5 -

co o -

(e) 2.5-w 2 -04 1.5-1-

L 0.5q

() 1.5-¢ 1.2-; 0.9-3 0.6-w 0.3 -

(A 0

10 20 30Year

0 1Frequency (1/year)

15

10

5

0

10

5

0

2 0 1Frequency (1/year)

FIG. 2. (a) Monthly reported incidence figures for varicella in the Faroe Islands (population 25,000, Left) and Copenhagen (population 600,000,Right), in the periods 1944-1969 and 1938-1969, respectively. (b and c) Representative incidence time-series from stochastic RAS model(parameters used given in text) forN = 25,000 (Left) and N = 600,000 (Right), with A' = 0.03 (b) and A' = 0.0001 (c). (d) Spectra calculated fromreported Varicella incidence time-series in the Faroe Islands (population 25,000, Left) and Copenhagen (population 600,000, Right), for the periods1944-1969 and 1938-1969, respectively. (e) Representative spectra generated by stochastic RAS model (parameters used given in text) for N =

25,000 (25-year series used, Left) andN = 600,000 (30-year series used, Right), with A4 = 0.03. The level of stochastic variability in the N = 25,000spectrum indicates that the probability of the Faroe Island spectrum being "representative" of the dynamics is small. Time-series comparisonindicates =50% under-reporting in the Faroe Island data. (f) Representative spectra generated by stochastic RAS model for N = 25,000 (Left)and N = 600,000 (Right), with A4 = 0.0001 (spectra calculated using 150-year time-series). Tukey window of 150 used for all spectra. It appearsthat under-reporting in Copenhagen was >70%, calculating on the basis that a city of 600,000 needs to have an mean incidence rate of 660 permonth for everyone to have been infected by the time they die.

(a)

(b) 150 -

6800iI

'C 50

~ 00 -0

(c) 800-

6200 -

00a 20

O ;0

800 -

400 -

0-1500 -

1000 -

500 -

0-12000

8000 -

4000 -

I %0

0

N=600000

N=600000

11A

N=25000

10 20Year

N=25000

2

Medical Sciences: Ferguson et al.

Dow

nloa

ded

by g

uest

on

Feb

ruar

y 3,

202

0

7234 Medical Sciences: Ferguson et al.

750Faroes Measles

@ 600-a]

." 450 -

""300 -

= 150 -

1944 1954 1964Year

(b) 3000

250020001500

1000500

00 5025

Year

(c) 2000Q- 1500

10

r C 00

0 25Year

(d)100 -

,, 80-c

c 60

40t E 40-

m 0. 20 -

S O

Pre-vaccination

isHigh zoster FOI

Low zooler FOI

- M1M

0-5 5-10 10- 15- 20- 45- 65+15 20 45 65

Age class(e)

1 High zoster FOI (Pre-vacc)* Low zoster FOI (Pre-vacc)El High, flat zoster FOI (Post-vacc)E High zoster FOI for low ages (Post-vacc)* High zoster FOI for high ages (Post-vacc)1 Low, flat zoster FOI (Post-vacc)

(e)

1. 8(P0484) 0C*.4

;.t;--. w.- m,v E0 =

:. 041

Post-vaccination40-

35 4 High, flat zoster FOl

30 * High zoster FOI for low agesY 25 El High zoster FOI for high ages

220 Low, flat zosler FOI

1 5- FI10

57

0

0-5 5-10 10-15 15-20 20-45 45-65 65+

Age class

0.06 -

0.05 -

0.04 -

0.03 -

0.02 -

0.01

0

FIG. 3. (a) Monthly reported incidence figures for measles in the Faroe Islands (population 25,000) in the period 1944-1969. (b) Varicellaincidence in a population of 1 million with 60% vaccine coverage (90% efficacy), for low (Ab = 0.0003) and (c) high (Ab = 0.03) zoster FOI. Meanincidence of varicella by age (d) before vaccination with low and high zoster FOI, and (e) postvaccination, with low flat zoster FOI, high flat zosterFOI, zoster FOI concentrated at young ages, and zoster FOI concentrated at high ages. (f) Mean annual mortality due to varicella for the pre-and postvaccination choices of zoster FOI listed in d and e above, calculated from data given in ref. 24, and normalized so that high flat zosterFOI gives prevaccination mortality agreeing with mean deaths per year given in ref. 24.

often experienced by adults with varicella (24). Using a highzoster FOI, neither of these effects are observed. The build upof susceptibles is prevented by a steady low incidence ofvaricella generated by cases of zoster. The rise in the mean ageof infection is also inhibited, assuming the zoster force ofinfection is evenly distributed through age classes (agreeingwith the age mixing patterns assumed in most transmissionmodels, if zoster is seen predominantly in those older than 20years). This inhibition is more pronounced if the mixingbetween those with zoster (such as grandparents) and youngchildren is increased. However, if zoster infectiousness isconcentrated among young adult susceptibles, then the meanage of infection may still rise to a small degree.The model suggests (Fig. lb) that zoster has a less significant

role for large populations (N < 107), but this conclusion doesnot take into account the effect of spatial heterogeneities. Ashas been observed in the epidemiology of measles (22), it islikely that the effects of spatial decoupling on disease persis-tence and dynamics will become increasingly significant in thepostvaccination regime as incidence declines. Analyses takinginto account these spatial factors suggest that zoster will playa significant dynamical role, regardless of the size of popula-tion concerned. As the virus moves close to eradication as a

result of high vaccine uptake, the reservoir of infection pro-vided by zoster cases is likely to decline-assuming the vaccinevirus is significantly less likely to recrudesce than the wild virus(see discussion in ref. 18). We are currently examining theeffect of including such factors on the long-term dynamics ofvaricella-zoster transmission. However, over a period of many

decades before this state is achieved, as herd immunity rises as

a result of cohort immunization, our current analyses revealthe central role played by zoster in the transmission dynamicsand control of varicella virus.

Appendix

The deterministic RAS model is describedequations:

dIidt

dXidt

dHidt

dYidt

dZidt

by the following

= B - (d + pti)Ii

= dI, - (Ai + Ii)X,

= AiXi- (a + [ti)Hi

= o-Hi- (v + -i)Yi

= vYi-izi, [3]

where the subscripts denote a cohort-specific variable. I de-notes the number of individuals protected by maternal anti-bodies, X those susceptibles, H those infected but not infec-

(a)Low Zoster

-J50

Proc. Natl. Acad. Sci. USA 93 (1996)

Dow

nloa

ded

by g

uest

on

Feb

ruar

y 3,

202

0

Proc. Natl. Acad. Sci. USA 93 (1996) 7235

tious, Ythose infectious, and Z those who have recovered frominitial infection.

Children are born continuously, at rate B, into the firstcohort, but cohort members only move onto the next cohort atthe beginning of each (school) year. The age specific mortality,,i, is zero for the first 20 cohorts, and set to a constant rate PC(= B/N, for constant population N) in the last cohort.The equivalent stochastic Monte-Carlo model (as used for

the simulations presented here) is easily formulated; the termson the RHS of Eq. 3 represent rates for individual transmissionor birth/death events, and the time between successive events,At, is exponentially distributed, assuming At is sufficientlysmall (see, for example, ref. 13).

Parameter values used were as estimated in ref. 17-i.e.,mean duration of maternal immunity (d-1) of 4 months, meanincubation period (1/or) of 14 days, mean infectious period(1/ v) of 5 days, mean life span (20 + 1/,ut) of 75 years.The transmission coefficients ,Bij make up the WAIFW (who

acquires infection from whom) matrix (26). For simplicity,mixing between different age groups is in practice described interms of 4 coarser age classes, for ages 0-5 (preschool), 5-10(primary school), 10-20 (adolescents), and 20+ (adults). Interms of these classes, we used a WAIFW matrix with thefollowing block structure:

I3132 133 (4

H:(.33 13 313). [4]

04 04 4 P4

The 3i were calculated from the age-specific forces of infectionusing Eq. 1, where we used the equilibrium values of the Y1from the solution of the static RAS model. The Ai werecalculated (see ref. 27 for method) by using serological datafrom a range of surveys (see refs. 17 and 18 for details) withthe exact values used being 0.15, 0.24, 0.08, and 0.05 for thefour age classes described above. These differ slightly fromthose in ref. 17, due to the different age classes used here.The WAIFW matrix structure given by Eq. 4 is the form

most commonly used for modeling of childhood viral trans-mission (14-16), but the results presented above are relativelyinsensitive to the choice ofWAIFW matrix structure within therange discussed in ref. 13.The mean of (3, 3, is defined so that the non-age structured

SEIR model using a transmission coefficient of (3 wouldgenerate incident infections at the same rate as they aregenerated in the deterministic RAS model (Fig. la):

E jXjoi13Yi [5]

This gives (3 = 1.82 x 10-5, for the parameters used here. fi3Tand ,12ut, the in-school and out-of-school 32 values respectively,must therefore satisfy

Ap = qs((p in- pout) [6]

where q is the proportion of new cases that occur in theseasonally forced age class, s is the proportion of susceptiblesin that age class, and p is the proportion of the year thatchildren spend in school. Hence i3n and ,3°ut are given by:

gin =(1-p)A+(2u= (2+ qs

pA123q

[7]

[8]

where AP3 is the difference between the in-school and out-of-school values of ,B. AP3 and Ab have bounds determined by thecondition that (3i < 0 Vi.We used the school term definitions given in refs. 14 and 16,

but did not treat Sunday as an out-of-school period.

We thank Lars Olsen for generously providing us with varicella andmeasles data for Copenhagen, Bornholm, and the Faroe Islands;Bryan Grenfell for advice; and the Wellcome Trust for providing grantsupport.

1. Plotkin, S. A. (1986) Pediatrics 78, 705-7.2. VZV Research Foundation (1995) VZVFocus 3, 1-2.3. Halloran, M. E., Cochi, S. L., Lieu, T. A., Wharton, M. & Fehrs,

L. (1994) Am. J. Epidemiol. 140, 81-104.4. Lieu, T. A., Cochi, S. L., Black, S. B, Halloran, M. E., Shinefield,

H. R., Holmes, S. J., Wharton, M. W. & Washington, A. E.(1994) J. Am. Med. Assoc. 271, 375-381.

5. Gilden, D. H., Vafai, A., Shtram, Y., Pecker, Y., Berli, M. &Wellish, M. (1983) Nature (London) 306, 478-80.

6. Weller, T. H. (1983) N. Engl. J. Med. 309, 1362-1440.7. Hope Simpson, R. E. (1954) Lancet ii, 1299-1302.8. Black, F. L., Hierholzer, W. J., Pinheiro, F. P., Evans, A. S.,

Woodall, J. P., Opten, E. M., Emmons, J. E., West, B. S., Edsall,G., Downs, W. G. & Wallace, G. D. (1974)Am. J. Epidemiol. 100,230-250.

9. Rand, D. A. & Wilson, H. B. (1991) Proc. R. Soc. London B 246,179-184.

10. Olsen, L. F., Truty, G. L. & Schaffer, W. M. (1988) J. Theor.Popul. Biol. 33, 344-370.

11. Olsen, L. F. & Schaffer, W. M. (1990) Science 249, 499-504.12. Schaffer, W. M., Olsen, L. F., Truty, G. L. & Fulmer, S. L. (1990)

in The Ubiquity of Chaos, ed. Krasner, S. (Am. Assoc. for theAdvancement of Sci., Washington, DC),, pp. 136-166.

13. Anderson, R. M. & May, R. M. (1991) Infectious Diseases ofHumans: Dynamics and Control (Oxford Univ. Press, Oxford),2nd Ed.

14. Schenzle, D. (1984) IMA J. Math. Appl. Med. Biol. 1, 169-191.15. Bolker, B. M. & Grenfell, B. T. (1993) Proc. R. Soc. London B

251, 75-81.16. Babad, H. R., Nokes, D. J., Gay, N. J., Miller, E., Morgan-Caper,

P. & Anderson, R. M. (1995) Epidemiol. Inf 114, 319-344.17. Garnett, G. P. & Grenfell, B. T. (1992) Epidemiol. Inf: 108,

495-511.18. Garnett, G. P. & Grenfell, B. T. (1992) Epidemiol. Inf: 108,

513-528.19. Engbert, R. & Drepper, F. R. (1994) in Predicatability and

Nonlinear Modeling in Natural Sciences and Economics, eds.Wageningen, N. L. & Holden, A. V. (Wiley, New York).

20. Grenfell, B. T., Kleczkowski, A., Gilligan, C. A. & Bolker, B. M.(1995) Stat. Methods Med. Res. 4, 160-183.

21. Ferguson, N. M, Nokes, D. J. & Anderson, R. M. (1996) Math.Biosci., in press.

22. Bolker, B. M. & Grenfell, B. T. (1995) Proc. R. Soc. London B348, 308-320.

23. Schwartz, I. B. (1985) J. Math. Biol. 21, 347-61.24. Fairley, C. K. & Miller, E. (1995) J. Infect. Dis., in press.25. Centers for Disease Control and Prevention (1995) Morbid.

Mortal. Wkly.- Rep. 44, 613-623.26. Anderson, R. M. & May, R. M. (1985) J. Hyg. 94, 365-436.27. Grenfell, B. T. & Anderson, R. M. (1985) J. Hyg. 95, 419-436.

Medical Sciences: Ferguson et al.

Dow

nloa

ded

by g

uest

on

Feb

ruar

y 3,

202

0