Journal of Animal Persistence thresholds for phocine distemper virusEcology 0887\56\ 43Ð57 infection in harbour seal Phoca vitulina metapopulations

J[ SWINTON�\ J[ HARWOOD$\ B[ T[ GRENFELL% andC[ A[ GILLIGAN��Department of Plant Sciences\ University of Cambridge\ Downing Street\ Cambridge CB1 2EA\ UK^ $NERCSea Mammal Research Unit\ Gatty Marine Laboratory\ University of St Andrews\ St Andrews\ Fife\ KY05 7LB\UK^ and %Department of Zoology\ University of Cambridge\ Downing Street\ Cambridge CB1 2EJ\ UK

Summary

0[ This paper explores the concept of the critical community size for persistence ofinfection in wildlife populations[ We use as a case study the 0877 epidemic of phocinedistemper virus in the North Sea population of harbour seals\ Phoca vitulina[1[ We summarize the available data on this epidemic and use it to parameterize astochastic compartmental model for an infection spreading through a spatial arrayof patches coupled by nearest!neighbour mixing\ with replacement of susceptiblesoccurring as a discrete annual event[2[ A combination of analytical and simulation techniques is used to show that thehigh levels of transmission between di}erent seal subpopulations\ combined with thesmall annual birth cohort\ act to make persistence of infection impossible in thisharbour seal population at realistic population levels[ The well known mechanismsby which metapopulation structures may act to promote persistence can be seen tohave an e}ect only at weaker levels of spatial coupling\ and higher levels of hostrecruitment\ than those empirically observed[

Key!words] critical metapopulation distribution\ mathematical modelling\ morbilli!virus\ North Sea harbour seals\ persistence[

Journal of Animal Ecology "0887# 56\ 43Ð57

lett|s critical community size\ a scalar parameter\ to a0[ Introduction

critical metapopulation distribution of popula!In 0877\ phocine distemper virus "PDV# killed a large tion numbers across a particular metapopulationfraction of the North Sea population of harbour seals\ structure[Phoca vitulina L[ "Heide!Jo�rgenson et al[ 0881a#[ We ask which of the variables of this critical meta!Reviewing that mass mortality\ Hall "0884# com! population distribution are central in determining per!mented that {if PDV behaves like other morbilliviruses sistence] for example\ is it more important to knowit should have been eliminated from the North Sea\ patch population sizes\ the number of patches\ thebecause the surviving population of susceptible ani! level of interpatch mixing or the total population size<mals is too small to allow the disease to persist or to Metapopulation theory has been used to estimate thepermit the establishment of a new epidemic|[ This minimum amount of suitable habitat Ð MASH "Han!concept of a threshold susceptible population has been ski\ Moilanen + Gyllenberg 0885# Ð necessary for aa central one in ecological epidemiology since the population to persist[ On the other hand\ conservationwork of Bartlett on measles 39 years ago "Bartlett biologists have attempted to estimate the minimum

size necessary for a population to have a particular0845# and much theoretical work has been concernedto establish that persistence properties can be strongly probability of persisting for a certain length of time

"the minimum viable population\ MVP "Soule� 0876^dependent on mixing structure[ In this paper we usethe PDV example to study how the mixing that arises Bjo�rge\ Steen + Stenseth 0883##[ The fact that the

theoretical relationship between patch size and localfrom a patchy host population a}ects persistence[ Acommon theoretical approach to this problem is to extinction is well understood in an epidemiological

setting has enabled us to investigate the way in whichinterpret the host patches as habitat patches of apathogen metapopulation "Hanski + Gilpin 0880#[ local and total population size interact in determiningÞ 0887 British

Ecological Society Under this assumption\ we need to generalize Bart! population persistence[

43

44 PDV did not persist in harbour seals in 0877] this 259 × 092[ On the evidence of the Orkney study"Thompson + Harwood 0889#\ the stated ranges ofJ Swinton et al[ paper asks whether this should have been surprising\

or followed inevitably from the structure of the host uncertainty in these estimates are rather optimistic[population[ It has been noted previously "Grenfell0881^ Tidd et al[ 0882^ Grenfell et al[ 0883# that inter!

0[1 HARBOUR SEAL POPULATION STRUCTUREmediate levels of coupling promote persistence inmetapopulations] can the North Sea population be Population structuring\ and the mixing of di}erent

subpopulations\ is as important as total populationconsidered a metapopulation habitat for PDV and ifso was coupling too low or too high for persistence< size in determining epidemic behaviour[ Telemetry

studies "Thompson + Miller 0889^ Thompson et al[We also discuss a question of more general import!ance] under what conditions is it necessary to take 0880# suggest that adult harbour seals have a high

degree of site _delity\ but around 19) of pups haveaccount of the spatial structure of the population inorder to reach this conclusion< We begin by reviewing been observed to travel long distances "up to 499 km#

in short periods of time "Thompson\ Kovacs +the available data on harbour seals and PDV\ beforedescribing the mathematical model in detail and then McConnell 0883b#[ In the context of PDV epidemi!

ology\ {mixing| corresponds not to mating but to con!presenting the results of the numerical simulations[Finally\ we show how these results can be understood tact close enough to transmit the morbillivirus[

Although there is no empirical proof that this cannotanalytically and discuss the key features of the criticalcommunity size which emerge from this study[ happen in the water\ we take it as axiomatic here

that transmission only occurs between neighbouringanimals at haulouts[ The rapid spread of the virus in0877 as shown in Fig[ 0 demonstrates that there must

0[0 HARBOUR SEAL POPULATION SIZESbe signi_cant mixing between widely separated haul!outs if this axiom is correct and if there was no furtherHarbour\ or common seals Phoca vitulina are found

throughout much of the North Atlantic and North introduction of virus from elsewhere[ Haulouts do\nevertheless\ form a natural unit of population withinPaci_c "Bonner 0878#[ Like all marine animals\

reliable estimates of population size are di.cult to which to consider an epidemic taking place[ Such haul!outs are typically aggregated into regions of greatermake "Harwood et al[ 0878#[ However\ harbour seals

often spend time ashore on {haulouts|\ although the or lesser de_nition] recent genetic analyses based onmicrosatellite polymorphisms "Goodman 0886# sug!frequency with which they do so is variable and

depends strongly on age\ sex and time of year among gest six distinct population units] IrelandÐScotland\English east coast\ Wadden Sea\ Western Scan!other factors "Thompson 0878a\b#[ This behaviour

enables population estimates to be based on land dinavia\ East Baltic and Iceland[ This structuring intospatially distinct groups within each of which the epi!counts at the haulouts combined with estimates "of

varying degrees of sophistication# of the proportion demic fairly rapidly runs to completion "Heide!Jo�r!genson + Ha�rko�nen 0881# makes it natural to inter!of time that animals spend in the water[

How many harbour seals are there in northern pret haulouts and the regions they make up as unitsin a host metapopulation "Hanski 0880#[ Haulout sizesEuropean waters< Estimates of seal herd sizes

throughout Europe in 0878 are tabulated in Dietz\ vary substantially and can be strongly dependent onlocal topography\ but in the Wash haulout sizes areHeide!Jo�rgenson + Ha�rko�nen "0878#^ more recent

data which also covers the Baltic and Barents seas has in the range of 04Ð499 "SMRU\ unpublished data#[also been collected "Anonymous 0884#[ These esti!mates sum to about 49 × 092 animals\ but such a sumneeds to be treated with caution[ When Thompson +

0[2 HARBOUR SEAL FERTILITY ANDHarwood "0889# used haulout counts combined with

MORTALITYindividual behaviour from telemetry studies\ their esti!mate for the harbour seal population of Orkney was Harbour seals have an annual pupping season from

late May to early August with a peak in June "Thomp!approximately three times higher than earlier\ lesssophisticated estimates "Vaughan 0864^ McConnell son 0878b#[ Multiple births are rare\ and most females

do not ovulate until at least 2 years old[ Once mature\0874#[ They proposed that similar correction factorsshould be applied to other surveys in areas where 79Ð86) pup each year "Ha�rko�nen + Heide!Jo�rgenson

0889#[ This single annual birth per female provideshaulouts were predominantly rocky\ and suggested apre!epidemic population estimate in Britain of one major constraint on the population growth rate[

Helander + Bignert "0881# found the number of¼35 × 092\ rather than the previous 10 × 092[ As forworld populations\ Bigg "0870# estimated 399Ð pups born in the Swedish Baltic averaged 19Ð11) of

the maximum number of seals counted in the area^599 × 092 harbour seals in the world\ of which theÞ 0887 British

East Atlantic subspecies P[ vitulina vitulina made up Bigg "0858# observed 19) of a Canadian populationEcological Society29Ð099 × 092 animals[ Bonner "0878# claimed a world to be cubs[ Ha�rko�nen + Heide!Jo�rgenson "0889# usedJournal of Animal

Ecology\ 56\ 43Ð57 population of 299Ð399 × 092 and King "0872# stated animals found dead in 0877 in the Kattegat!Skagerrak

45 area as a cross!sectional population to calculate fer! 0[3 PHOCINE DISTEMPER VIRUS

PDV and seals tility and mortality data[ They recorded that 15) ofPhocine distemper virus "PDV# is a morbillivirus\

those that could be aged were pups[ Data from thatclosely related to canine distemper virus "CDV# and

study were used by Heide!Jo�rgenson\ Ha�rko�nen +one of a family of viruses including measles "Barrett

Aýberg "0881b# to construct a Leslie matrix populationet al[ 0884#[ A few studies of the pathology of PDV in

model[ In the asymptotic state of that model there areseals have been published "Visser et al[ 0878^ Harder

¼9=11 female pups per female[et al[ 0889\ 0881# and it appears similar to that of CDV

Mortality\ on the other hand\ acts throughout thein seals "Visser et al[ 0881#\ PDV in dogs "Osterhaus

year[ While seasonality in this factor is to be expected\et al[ 0877# and in other susceptible carnivores

there is no published quantitative analysis of this vari!"Blixenkrone!Mo�ller 0882 and references therein#[ In

ation\ which would in any case be obscured by sea!particular\ it takes several days for infection to become

sonality in haulout behaviour[ Accordingly\ in thispatent and infectiousness to develop[ Disease mor!

paper\ we take mortality to act at a constant ratetality is signi_cant "Harder et al[ 0881#\ although once

throughout the year[ The simplest modelling assump!recovered\ animals appear immune for life "Harder et

tion would be that this mortality rate is also inde!al[ 0889#[ Heide!Jo�rgenson + Ha�rko�nen "0881# esti!

pendent of age and sex] there are three sources of datamated the mean lengths of these latent and infectious

on this[ Boulva + McLaren "0868# found that catchperiods as 2 and 01 days\ respectively\ largely based

data from eastern Canada yielded an age pro_le con!on the studies mentioned above[ All the studies show

sistent with an age!independent mortality rate ofrecovery or mortality within 00Ð07 days\ so a total

9=077Ð9=082 year−0 in light hunting areas and 9=189mean generation time of 04 days is reasonable[ The

year−0 in a heavy hunting area[ Bigg "0858# publishedtime at which animals become infectious is harder to

a life table for a Canadian harbour seal populationestimate[ One approach is to be guided by analogy

subject to hunting pressure[ The log survivorships arewith CDV where dogs develop fever before they shed

approximately linear^ _ts to these data yields annualvirus\ which would suggest a latency period of rather

mortality rates for males of 9=18 year−0 and femalesmore than 2 days[ Another approach is to detect the

of 9=04 year−0\ and a combined mortality rate of 9=19presence of virus in leukocytes[ In two di}erent experi!

year−0[ Analysis of these data with a Cox proportionalments\ viraemia was not found until day 6 in one set

hazard approach "S!Plus routine coxph# yields an"Harder et al[ 0889# but before day 4 in another set

excess mortality of males over females of only 00)\"Harder et al[ 0881# in which the course of infection

and signi_cant only at the P � 9=95 level\ since thewas less similar to the wildtype pathology[ Accord!

additional mortality is largely apparent at larger age!ingly\ we take a latent period of ¼6 days and an

groups where sample size is small[ The _nal source ofinfectious period of ¼5 days as our informed guess

data on age!related mortality comes from the cross!at these parameters[ However\ as Heide!Jo�rgenson

sectional study of Ha�rko�nen + Heide!Jo�rgenson+ Ha�rko�nen point out\ the relative lengths of these

"0889# on those animals dying in 0877 in the Kattegat!periods is not particularly important in estimating

Skagerrak[ This population appeared to have beenoverall transmission during the period of infection

growing uniformly in the years prior to 0877 and"e}ectively\ R9\ the number of secondary cases pro!

would be expected to have an age!structure in whichduced by an initial primary case# which is estimated

cohort sizes decrease uniformly with age[ However\from the data and so the daily transmission rate

the age!distribution of those animals that died wasdepends in a simple way on the estimate of the infec!

markedly bimodal\ with a peak in both the _rst yeartious period[

class and the _fth year class\ which suggests that ani!mals under 4 were less susceptible to infection or mor!tality[ For animals over 4 years\ Ha�rko�nen + Heide!

0[4 THE 0877 PDV EPIDEMICJo�rgenson found that an age!independent annualmortality rate of 9=98 year−0 was consistent with the The epidemic of 0877 remains one of the best docu!

mented mass mortality events of wild sea mammals[data for males but not with the female data\ primarilybecause of the relatively large number of older females Much of the raw data was summarized by Dietz et al[

"0878# and reviewed by Heide!Jo�rgenson et al[found dead[ Nevertheless\ these three studies togetherdo suggest that an age!independent mortality rate in "0881a#^ the spatial spread is illustrated in Harwood

"0878# and Harwood "0889# and summarized in Fig[ 0[the absence of PDV is not an unreasonable modellingassumption\ particularly if the relatively high _rst! The timings of these reports provide a basis for assess!

ing the rate of spread of infection between hauloutsyear mortality is absorbed into the e}ective fertilityrate[ To compare this age!independent rate with the and regions[ Moreover\ the detailed local reporting in

some areas "Heide!Jo�rgenson + Ha�rko�nen 0881^ Hall\age!speci_c ones derived by Ha�rko�nen + Heide!Jo�rgenson\ we need to consider the asymptotic age! Pomeroy + Harwood 0881^ Thompson + Miller 0881#

Þ 0887 Britishdistribution of their Leslie population model\ which also enables assessment of the rate of spread withinEcological Societycorresponds "data not shown# to a population in which regions[Journal of Animal

Ecology\ 56\ 43Ð57 09=1) is removed annually by non!PDV mortality[ The initial source of the epidemic remains

46 unknown[ The most plausible hypothesis is that it was mission and restrict ourselves below to populations ofharbour seals only[J Swinton et al[ related to an invasion of Norwegian waters by harp

seals\ Phoca groenlandicus Erxleben 0666\ from popu!lations in the Barents and Greenland Seas where PDV 0[6 PERSISTENCE OF PDV IN HARBOUR SEAL

is believed to be endemic "Goodhart 0877^ Stuen et POPULATIONS

al[ 0883#[ In April 0877\ unusually large numbers ofEvidence for the continuing transmission of PDV inaborted foetuses were found on the island of Anholtharbour seals after 0877 remains equivocal[ Sero!"Heide!Jo�rgenson + Ha�rko�nen 0877#[ From there\positive samples were reported from Dutch seals borninfection appears to have spread both eastward\ to theafter 0877 "Visser et al[ 0882# but none from Germanysouthern Baltic and then the Swedish and Norwegian"Harder et al[ 0882#\ the Wash "Hughes et al[ 0881#coasts\ and westward to the Wadden Sea\ and then theand the Moray Firth "Thompson et al[ 0885#[ ThisWash\ Scotland and the Irish Sea "Fig[ 0#[ Estimates ofmay re~ect di}erences in transmission between areas\the total mortality from this epidemic rely on report!or may be accounted for by di}erences in protocoling of dead animals washed up at accessible sites[and interpretation[ There are no reports of continuedDietz et al[ "0878# summed all of these reports toexcess mortality at any of these sites\ so we assumeproduce a mortality estimate of 07 × 092 animals\ cor!below that transmission did not continue beyond earlyresponding to a case mortality rate of 25) if we0878\ or that if it did so\ it was of a strain of virus ofassume that every animal became infected and acceptreduced virulence[the estimate of 49 × 092 in Anonymous "0884# for the

pre!epidemic population size[Mortality rates appeared to vary around the North 1[ Mathematical formulation

Sea "Heide!Jo�rgenson et al[ 0881a#[ On the east coastWe now set up a mathematical model su.cient toof Scotland\ 09Ð19) of the population were estimatedreproduce the salient features of the 0877 epidemicto have died "Thompson + Miller 0881#\ while mor!using the biological knowledge described above[ Thetalities in the eastern North Sea may have reachedkey features of the model are represented in a typical59)[ It remains unclear "Thompson + Miller 0881#simulation output in Fig[ 1] it consists of discretewhether this re~ected decreased transmission orstochastic epidemics in a linear sequence of patches\decreased susceptibility to disease[ Only 41) "24:57#each weakly coupled to its nearest neighbours[of the Moray Firth animals were reported as sero!

positive "Thompson et al[ 0881# while 84) of females1[0 MODEL STRUCTUREin the Kattegat were believed infected based on the abor!

tion rate "Heide!Jo�rgenson et al[ 0881a#^ de Koeijer\ The population is divided into a number of di}erentDiekmann + Reijnders "0886# point out that in patches corresponding to seal populations at par!populations with the mass!action mixing de_ned ticular haulouts\ with Ni representing the populationbelow\ lower case mortality rates can themselves lead size for haulout i[ Within each patch there is a stan!to lower population prevalences of infection[ dard SEIR model\ for the numbers susceptible Si\

infected but not infectious Ei\ infectious Ii\ and re!covered Ri[

A deterministic representation of this model would0[5 PDV IN OTHER SEAL SPECIESbe as follows]

PDV is believed to be endemic in some other sealSþi � −liSi − mSi eqn 0species "Stuen et al[ 0883#[ In the North Sea the only

commonly found seals are the harbour seal and the Eþi � liSi − "m ¦ s#Ei eqn 1grey seal Halichoerus grypus Fabricius 0680[ The latter

Iþi � sEi − "m ¦ a ¦ g#Ii eqn 2is also thought to have been free of infection prior to0877 on the basis of serological evidence "Harwood et

Rþi � gIi − mRi eqn 3al[ 0878#[ At least 074 grey seals were found deadduring the 0877 epidemic with similar pathological together with an annual birth input[ Non!PDV mor!

tality is represented as an age!independent per!capitasymptoms to those in harbour seals\ so the infectionappears to have been present in the grey seal popu! rate m while infected animals leave the latent stage at

a rate s to become infectious[ While infectious theylation[ Serological evidence is that it was present atcomparable prevalences to that in the harbour seal su}er an additional mortality rate a or cease to be

infectious at a rate g "so that the fraction of animalspopulation "Harwood et al[ 0878#\ suggesting thatPDV mortality was a great deal lower in grey seals[ who die of the infection is m � a:"a ¦ g##[ Once re!

covered\ individuals remain immune for life[ TheThis lower mortality\ together with the fact that mostpopulations of grey seals are in remote areas\ means simulations use a stochastic version of the above

Þ 0887 Britishthat there are a great deal less data on the spread of model in which each per!capita transition rate is inter!Ecological Societythe epidemic within this species[ We assume in this preted as a hazard for transition occurring betweenJournal of Animal

Ecology\ 56\ 43Ð57 paper that there is no signi_cant cross!species trans! discrete population compartments\ so that the tran!

47

PDV and seals

Fig[ 0[ Timing of recorded mortality from the 0877 PDV epidemic[ Closed box] _rst recorded case in each location[ Triangle]centred on peak reporting time\ with width equal to length of period in which mid 89) of cases recorded and heightproportional to logarithm of total number of cases[ Open box marks last recorded case in those location where known "inother locations cases are those up to December 0877#[ Shaded period of month bar is approximate pupping season[ Data fromDietz et al[ "0878# and Harwood + Hall "unpublished data\ SMRU#[ The vertical ordering of sites is based partially on thatin Dietz et al[ "0878#\ and otherwise on geographical proximity[

sition Si : Si − 0 occurs in the interval "t\ t ¦ dt# with nitudes of transmission within and between patches[The resourceful work of Heide!Jo�rgenson + Ha�r!probability "m ¦ li# Sidt[ Using a _xed dt of 9=990

years\ the number of transition events of each type is ko�nen "0881# on the course of infection within haul!outs allowed an initial estimate of the mode and inten!calculated by treating it as the outcome of a sequence

of independent binomial trials of all possible tran! sity of transmission[ This can be described in terms ofthe basic reproductive number R9 of the infection] thesitions\ or an approximating Normal or Poisson vari!

ate when appropriate "Evans\ Hastings + Peacock number of secondary cases produced by an initialprimary case[ They showed that in populations of size0882^ Ripley 0876#[

The only element of this local dynamics which "N# varying from 79 to 0499\ R9 was approximatelyconstant[ In a terminology introduced by de Jong\depends on the rest of the system outside the patch is

the force of infection l\ which contains information Diekmann + Heesterbeck "0884#\ Diekmann et al["0885a# and de Koeijer et al[ "0886# pointed out thaton the mixing structure of the system] there is assumed

to be no movement of seals between patches[ There is this is more consistent with {mass action| mixing than{pseudo mass action mixing|[ "In the notation de_nedno age!structure in this model[

Births are assumed to occur once a year in a simple above\ {mass action| corresponds to a force of infec!tion of the form l � bI:N and pseudo mass action todensity!dependent manner[ Each patch is assigned a

carrying capacity K\ and the number of births in a l � bI[# Heide!Jo�rgenson + Ha�rko�nen "0881# gaveestimates for a quantity they called p¼N\ which can bepatch of size N is taken to be the minimum of KÐN

and rN where r is the maximum fecundity rate[ multiplied by the infectious period to derive estimatesfor R9"N#[ These\ each weighted by an estimated stan!dard error of R9"N#:zN "Becker 0884#\ can be used

1[1 FORCE OF INFECTION AND NEARESTin a linear regression of R9 against N to reject the

Þ 0887 British NEIGHBOUR MIXINGpseudo mass action hypothesis R9"N# � N but not toEcological Society

In this section we de_ne a function for the force of reject the mass action hypothesis R9"N# � const[ "BestJournal of AnimalEcology\ 56\ 43Ð57 infection and present evidence for the relative mag! weighted linear _t] log09R9 � 9=589Ð9=980 log09N\ SE

Days

Pat

ch

0 50 100 150 200 250 300

0

5

10

15

20

25

Infecteds

100

200

300

400

500

600

700

Fig[ 1[ Typical model simulation with parameter values as in Table 0 and m � 9=1 and r � 9=0\ with 49 999 animals distributed among 14 patches\showing number infected "Ei ¦ Ii# in each patch i after initial infection on day 9 in patch 01[ Vertical bars indicate new infections occurring in a patchwith no current infections[

of slope 9=9565 and the slope is not signi_cantly patch do not make contacts with another patch lesslikely\ might be of the form Sj"Ij:Nj#^ we _x here ondi}erent from zero\ P � 9=16#[

The force of infection for an individual in haulout the _rst for de_niteness[We expect that between!patch mixing is small com!i is made up as follows]

pared to within!patch mixing[ In the limit r � 9\ themodel is reduced to a single patch with a standardli � b$"0 − r#

Ii

Ni

¦ rSj�i−0\i\i¦0Ij

Sj�i−0\i\i¦0Nj% eqn 4SEIR model[ It is straightforward to observe thatsmall infections in the deterministic model will onlyso that r re~ects the relative frequency of between!increase if R9 × 0 where R9 � b:"g ¦ a ¦ m# ×haulout mixing while b is a contact rate[ Patches ats:"s ¦ m#[ Since nondisease mortality is negligiblethe edge of the array only mix with their immediateduring the brief infectious period "so m ð s# in thisinternal neighbours] the boundary conditions are zeropaper we de_nerather than periodic[ This is a representation of a

nearest!neighbour mixing process\ which describesR9 �

b

g ¦ a ¦ m[ eqn 50877 data well and allows some analytical progress

"Swinton 0887#[ While the discussion of within!patchmixing above gives us some con_dence in the _rst term By applying stochastic threshold theorems "Bailey

0864^ Svensson 0884#\ we expect there to be a thresh!in eqn 4\ the second is inevitably more speculative] itÞ 0887 British

is consistent with a picture in which all of the seals in old at R9 � 0 in the corresponding stochastic modelEcological Societythe neighbourhood are in a common {mixing pool|[ separating the cases where a major epidemic is likelyJournal of Animal

Ecology\ 56\ 43Ð57 An alternative assumption\ where contacts with one or unlikely[ It is possible to extend this analysis to the

59 multiple patch case in both deterministic "Dieckmann\ the generation time of infection "the sum of the infec!tious and latent periods#[ Figure 2 shows the numberPDV and seals Heesterbeck + Metz 0889# and stochastic settings

"Ball\ Mollison + Scalia!Tomba 0886# but the de_! of days between each new recorded outbreak from the0877 epidemic[ Most of the patches are separated bynition in eqn 5 remains the relevant one when the

initial infection is con_ned to a single one of a set di}erences of less than 09 days\ and inserting TT � 09days into this formula combined with the parameterof weakly coupled patches[ We use R9 merely as a

convenient dimensionless measure of transmissibility\ estimates below yields an estimate of r � 9=94[ This_gure should be viewed with caution\ because it isin that we use the estimates above as a means to

estimate b[ derived by assuming not only that N is large but alsothat infection spreads through the continuous timeCan estimates of b and R9 drawn from seal popu!

lations of 69Ð0499 be extrapolated to ever larger popu! mechanism described by the deterministic di}erentialequations\ which is likely to be a bad description oflations in our search for a population size that allows

persistence< The answer clearly depends on how the events within the _rst one or two generation periodsof infection when numbers of infectives are small[host population responds to increases in population

size] in particular how density changes as population Nevertheless\ it is of interest that it is of the sameorder of magnitude as the value r ¼ 9=0 which arisessize changes and the relationship between density and

transmission\ but also the likelihood of a population by attempting to mimic the observations "Fig[ 0# withthe numerical simulations "Fig[ 1# by eye[ Thus\ wealready spatially structured splitting o} new sub!

populations as it grows[ There is some theoretical use baseline interpatch mixing of the order of 09) ofthat within a patch[ Base parameters used are sum!justi_cation for maintaining a mass!action trans!

mission with constant b\ based on mechanistic marized for convenience in Table 0^ departures fromthis parameter set are described for individual runs[assumptions about the contact process "Heesterbeek

+ Metz 0882#[ de Jong et al[ "0884# examined theempirical evidence in the literature and found fourcomparisons between theory and data of which none 2[ Extinction times for models withoutfalsi_ed the mass!action assumption and three out replacementof four falsi_ed the pseudo!mass action[ Those four

The twin assumptions of a discrete annual birth ratecomparisons did not include the PDV epidemic data\and nearest!neighbour mixing allow us to make awhich\ as we have seen also provides evidence for masssubstantial amount of analytical progress with theaction and against pseudo mass action[ Theoreticalmodel system[ It is obvious that an infection that fadestechniques for distinguishing between these processesout before the end of one year cannot persist in thein data\ and for locating mixing processes as a pointpopulation no matter how high the birth rate[ Accord!on a continuum rather than as a choice between twoingly\ we begin by studying the properties of the timealternatives\ are in their infancy though\ and in par!to "inevitable# extinction in systems with no replace!ticular it is important to recognize that rejectingment of susceptibles[ We shall see subsequently thatpseudo mass action as a relevant scaling is not thethese properties will have a strong bearing on per!same as validating mass action[ Nevertheless\ sincesistence times for models with replacement[both the directly relevant empirical data and the theor!

The problem of determining the distribution ofetical arguments point in the same direction we chooseextinction times for the standard epidemic model in atransmission functions of the mass action form[ Asingle patch was _rst solved by Barbour "0864# fordirect consequence of this is that the basic repro!large N in a model with no latent class[ Conditionalductive rate R9 can be independent of the populationon a major epidemic occurring\ and under thesize N in the sense that we can plausibly take theapproximation f � 9 corresponding to almost all indi!transmission coe.cient b in eqn 5 as independentviduals eventually becoming infected\ his Theorem 1of N[implies that for a generation time of G\ the expectedextinction time TE is given by

1[2 ESTIMATION OF COUPLING STRENGTHTE

G:

R9

R9 − 0log N eqn 6As well as estimating within!haulout mixing\ the

observed 0877 data can be used to give us rough esti!mates of the between!haulout mixing by treating the at large N[ More recent results of Svensson "0884#

extend this analysis to include the e}ect of a latentlocations named in Fig[ 0 as individual patches[ Wecan estimate the between patch mixing r either by period[

How should this expression be modi_ed for mul!comparing the simulation output with the obser!vations in Fig[ 0 or by using the theoretical result tiple patches< Figure 3 shows the results of a simu!

lation for parameter values relevant to PDV "save"Swinton 0887# that TT\ the expected time for a transitÞ 0887 British

to occur between two patches is independent of N for that\ for clarity\ the interpatch coupling is 099!foldEcological SocietyN large and is approximately related to the between weaker than that estimated above#[ The single patchJournal of Animal

Ecology\ 56\ 43Ð57 patch mixing r by r � "0:1R9# e−"R9−0#TT:G where G is runs show the approximately linear growth of TE with

50

J Swinton et al[

Fig[ 2[ Di}erences between successive new outbreaks on the east "open squares^ mean 6=7 days# and west "closed squares^ mean7=3 days# wings of the 0877 epidemic "i[e[ sites above and below Anholt in Fig[ 0#[

Table 0[ Basic and derived parameters

Estimated parameter Symbol Typical value Discussion

Death rate m 9=0Ð9=1 year−0 Section 0[2Latent period L 9=91 years "¼6 days# Section 0[3Infectious period D 9=91 years "¼6 days# Section 0[3Case mortality m 9=0Ð9=5 per infected animal Section 0[3Basic reproductive ratio R9 1=7 Section 1[1Replacement birth rate r 9=1 year−0 Section 0[2Inter!patch mixing r 9=94Ð9=0 Section 1[2

Derived parameter Symbol Derived value Discussion

Loss of latency s s � 0:L Section 1[0Recovery rate g g � "0 − m#:D Section 1[0Mortality rate a a � m:D Section 1[0Generation time G G � D ¦ L Section 1[0Transmission rate b b � R9:D Section 1[1

log N as predicted[ For multipatch runs\ the situation epidemic is established\ and the second of size N1 inwhich it is not[ The force of infection for an individualis more complicated[ For small N\ it is unlikely that

infection from the initial patch spreads successfully in the second patch is assumed to scale like 0:N0\ sothe risk that any member of the second patch becomesinto the second patch\ and so the extinction time is

independent of the number of patches n[ For large N infected scales like N1:N0[ If the relative patch sizesremain the same as N � N0 ¦ N1 increases\ this risk"here\ bigger than around 094#\ the transit time to the

next patch is approximately constant\ and so the total remains independent of N[ This argument is describedin greater detail elsewhere "Swinton 0887#\ where antime to extinction is the time to spread through one

patch plus this transit time multiplied by the number explicit expression is derived for the transit time andÞ 0887 British

of transits[ That the transit time is independent of N the threshold patch size for transits to occur[ TheEcological Societycan be seen by considering the time taken to transit biological reason for the existence of a threshold patchJournal of Animal

Ecology\ 56\ 43Ð57 between two patches\ the _rst of size N0 in which the size lies in the mass action assumption that individuals

51

PDV and seals

Fig[ 3[ Lower panel] extinction times without births as a function of patch size for a model with 0\ 1\ 4 or 09 patches[ Medianof 099 simulations at each population size[ Parameters s � 14 year−0\ g � 49 year−0\ a � 49 year−0\ b � 179 year−0\ r � 9=990[Top panel] fraction\ f\ of the time that infection goes extinct in _rst patch before spreading to second patch in the two!patchmodel[

make only a _xed number of infectious contacts\ inde! annual persistence "AP# which means that infectionwill\ with probability greater than 49)\ persist in thependent of population size[ Thus\ individual risk does

not scale with population size\ but the risk of a one! population for longer than 0 year[ Since births onlyoccur annually and we assume that the introductiono} population event such as a transit does\ since it

compounds the individual risk for each member of of infection is immediately after the _rst birth season\AP is clearly independent of the demographic par!the population[

The nonmonotonicity of the extinction time curves ameters governing recruitment\ and critical par!ameters for AP can be calculated from the argumentsis a product of the phase transition near Nc between

the two regimes of large and small N[ As N decreases of the previous section or read o} a graph like Fig[ 3[AP is obviously a minimum requirement for LP andbut remains above the critical Nc\ transits take longer

and longer to occur "and are thus less likely to occur provides a guide to the necessary requirements for LP[Suppose that the critical patch size for AP in a singleat all#[ Thus\ the total time stretches by this transit

time "scaled by the number of patches# although its patch is Ny[ At the end of a year\ the number of newsusceptibles entering the population is rN[ A crudevariance increases[ As N decreases down through Nc\

the transit time can increase no more\ but the per! estimate for the critical community size for LP couldthen be N � Ny:r\ so that the infection takes at leastpatch extinction rate increases\ so that the median

time to extinction of the whole system decreases[ 0 year to get through the newborn susceptibles andthe cycle can repeat itself[ This can only be a crudeFinally\ for N ³ Nc\ transits do not occur at all[estimate because of two con~icting corrections cor!responding to the two components making up the

3[ Persistence thresholds for models withpersistence time in a multipatch model[ The _rst com!

replacementponent\ the time it takes to spread within a patch\is increased because the basic reproductive ratio isWe now come to the central question of the paper]

how do persistence thresholds\ in the presence of decreased by a factor of r] the diluting e}ect of thepresence of about "0−r#N immunes[ The second com!replacement\ depend on population size< We dis!

tinguish two forms of persistence] the most important ponent\ the time it takes to spread to all the patches\is decreased for a rather di}erent reason[ In general\is long!term persistence "LP# which we de_ne here to

mean that\ with probability greater than 49)\ infec! the birth season will happen when epidemics are stillÞ 0887 British

tion will persist in the population for at least 099 years[ raging\ at various levels\ in a number of patches\ soEcological SocietyIn order to study the properties of LP\ it is useful to that the assumption of a single introduction of infec!Journal of Animal

Ecology\ 56\ 43Ð57 introduce a much weaker de_nition of persistence] tion in one patch\ used to _nd Ny\ does not hold[ Thus\

52 the number of patches available to transit to and start that the infection came close to achieving persistencesince had it lasted for one or two generations more itJ Swinton et al[ new infections in is reduced[ Thus\ we would expectcould have spread to a whole new birth cohort[ Figureto _nd a larger gap between the critical patch size for4 suggests that this was not so because the size ofAP and LP in a multipatch than in a single patch model[the new birth cohort was too small] slightly largerNevertheless\ when about 19) of the population arepopulation sizes that tip the persistence time over 0replaced each year\ we do _nd that at the populationyear still do not persist[size when persistence occurs\ the corresponding model

A deterministic model of infection in a single patchwithout births has an extinction time in the 0Ð3 yearof the form I � bSI:N−gI suggests that infection canrange[only persist endemically if the fraction susceptible at

3[0 CRITICAL POPULATION SIZE FOR PDV the beginning of the year "just after the birth cohort#satis_es R9S:N × 0[ For R9 � 1=7 that requires a birth

Figure 4 shows how the time to extinction varies as acohort to make up a fraction greater than about

function of patch size for the base estimates we took0:1=7 � 24) of the population[ At a maximum

for PDV[ The key implication of this _gure is that thereplacement rate of just 19)\ this suggests that infec!

critical community size for this model population "attion can never persist in a single patch\ no matter how

least 097# is several orders of magnitude greater thanlarge the population size\ so that there is no critical

the known North Sea population "up to 094# or evencommunity size in this case[ When there are multiple

world population "up to 095# of harbour seals\ andpatches\ infection needs to be avoided in each patch

that persistence of infection for more than a year orfor several years to allow the susceptible fraction time

so would have been extremely unlikely[ These simu!to recover^ in the meantime it must be maintained

lations suggest that the persistence time of infectionelsewhere in the system[

with about 14 patches of 1999 animals each is aboutThus\ for small R9\ weak coupling\ or a small annual

a year "compare Fig[ 0#[ The possible determinants ofbirth rate\ the critical community size is determined

this critical community size can be organized underlargely by the spatial structure of the population] there

three headings] demographic parameters controllinghave to be enough patches to enable a long sequence

the birth of susceptibles^ epidemic parameters con!of successive colonizations of other patches between

trolling the spread within patches^ and spatial par!any two major epidemics in one patch[ For larger R9\ameters controlling the spatial structuring of the epi!strong coupling\ or a larger birth rate\ it is the timing

demic\ which we now consider in turn[of susceptible recruitment that is essential\ and this ismore sensitive to the demographic assumptions[

3[1 SENSITIVITY TO DEMOGRAPHYThe e}ect of birth rate can be seen in Fig[ 5\ which

In the 0877 epidemic\ infection was circulating for at compares the results of Fig[ 4 with simulations thatraise the birth rate parameter r to 9=2 and a yet moreleast 8 months and perhaps longer^ it might be argued

Þ 0887 BritishEcological SocietyJournal of Animal Fig[ 4[ Median extinction times of 099 simulations at each population size using base parameter values in Table 0\ for nearest

neighbour mixing with the initial case of infection introduced in the central patch[Ecology\ 56\ 43Ð57

53

PDV and seals

Fig[ 5[ Demography and persistence[ As Fig[ 4\ with 0\ 09\ 14\ 49\ 099 patches as before "now unlabelled# but also showing themedian extinction times for the higher birth rate cases r � 9=2 and r � 0=9[ Once persistence times are longer than a year\ thehigher birth rate allows infection to persist a little longer in each patch and reduces the patch size at which the transition topersistence occurs[

extreme set in which it is assumed that r � 0\ so that tip the extinction time over 0 year there is a morerapid transition to persistence[ R9 is only a summaryall animals which die in the course of the year are

replaced at the pupping season[ The primary e}ect of parameter of the epidemiological dynamics and wealso carried out a number of simulations using theincreasing r is to decrease the critical community size]

the mechanism by which this occurs can be seen only parameter values of "Heide!Jo�rgenson + Ha�rko�nen0881# with a similar R9 but a shorter latent period] theto be e}ective once the patch sizes are large enough

to ensure patch to patch transmission[ Even at the results did not di}er in substance from those presentedbelow[unrealistically high birth rate r � 0\ however\ the criti!

cal community size remains large[This discussion has so far ignored the impact of

3[3 SPATIAL STRUCTURING AND PERSISTENCEinfection!induced mortality[ Infections with a highdegree of lethality in a population maintained at a Figure 5 also reveals some of the e}ect of spatial sub!

structure[ As a single population is subdivided into 09given size will have a higher replacement rate of sus!ceptibles and thus might generally be expected to _nd subpatches\ the extinction time decreases at large N]

this is because the time taken to spread through eachit easier to persist] this e}ect can be shown to exist atlarge r but is not signi_cant at realistic r at which individual patch decreases[ As the degree of sub!

division increases\ this e}ect is outweighed by thethe density!dependent susceptible recruitment rate isalready saturated[ A reduction in population size will number of transits the infection must take between

each patch[ Figure 7 shows how this e}ect depends onalso promote transmission under the mass!actionassumption but the e}ect of this will be small at these the strength of interpatch coupling times[ For extinc!

tion times\ this is in agreement with the theory onparameter values "Diekmann\ de Koeijer + Metz0885b#[ patches without replacement "Swinton 0887# that

when the intensity of interpatch coupling r is reduced\the expected transit time from patch to patch at large

3[2 SENSITIVITY TO EPIDEMIOLOGYN is predicted to increase like log 0:r\ while the criticalpatch size at which transits occur at all is predictedThe previous section argued that for a higher R9\

critical sizes for annual persistence "AP# would be to scale like 0:r[ Although this produces very longtransient infections lasting up to a decade\ it has littlemore closely related to those for long!term persistence

Þ 0887 British"LP#[ This is supported by Fig[ 6] although the extinc! e}ect on persistence] as coupling becomes weaker andEcological Societytion times below 0 year are briefer than in Fig[ 4\ as weaker\ it is the ability of the infection to persist withinJournal of Animal

Ecology\ 56\ 43Ð57 predicted by eqn 6\ once there are enough patches to a single patch which determines global persistence and

54

J Swinton et al[

Fig[ 6[ Epidemiology and persistence] sensitivity to raising R9[ Parameter values as Fig[ 4 but with b � 499 year−0 so thatR9 � 09[ The higher R9 speeds the epidemic within individual patches\ so the slopes of the curves before persistence are ~atterthan in Fig[ 4\ but also makes it easier for the infection to persist in postepidemic patches\ so that the transition to persistenceoccurs at lower patch sizes once there are enough patches[

Fig[ 7[ Reduced interpatch coupling changes extinction times but not persistence sizes[ Critical population sizes for persistenceas in Fig[ 5 with either 19 or 099 patches\ and interpatch coupling r � 9=0 "strong#\ r � 9=90 "medium#\ r � 9=990 "weak#[

that\ as we have seen\ requires very large population Sea are whether such a threshold population size existsat all and\ if so\ whether it is larger or smaller thansizes[the current population size[ The results of the naturalexperiment of 0877 tell us that it\ if it exists\ is probably

Þ 0887 British 4[ Conclusionlarger[ In this paper we have gone further and foundEcological Society

The most important questions about the critical com! that at the best point estimates\ the critical communityJournal of AnimalEcology\ 56\ 43Ð57 munity size for PDV in harbour seals in the North size is so large "097 animals\ several orders of mag!

55 nitude greater than the known world population size# transmission provides a mechanism which wouldweakly couple a reservoir of infection in harp seals toPDV and seals that it can only be considered to exist in the unlikely

belief that observed seal population biology and cor! the harbour seal population[ The approach outlinedin this paper is particularly appropriate for evaluatingresponding model assumptions can be meaningfully

extrapolated to these huge extremes[ this mechanism[ A particularly interesting feature ofobserved PDV dynamics is the contrast between theMore meaningful than a particular number\

though\ is the con_rmation that the level of the thresh! rapid and intense North Sea epidemic and a less dra!matic\ {smouldering|\ infection in the western Atlantic]old is rather more sensitive to the demographic factors

controlling the supply of susceptibles than epidemic one possible if speculative explanation for this is thatthe western epidemic is taking place in a host meta!ones[ This arises because the epidemic is one which is

capable of reaching the majority of susceptibles in population equally strongly coupled but where eventsin neighbouring patches have become less closely cor!each birth cohort[ We have also demonstrated that

for this example the very existence of the critical com! related than in the eastern epidemic[ The {rescue e}ect|of metapopulation theory "Brown + Kodric!Brownmunity size at reasonable levels depends on the

assumption of spatial substructuring] without this\ 0866# suggests that even at high levels of coupling\it should be easier for an infection to persist in ainfection spreads too rapidly through the limited size

birth cohort[ decorrelated spatial structure than where there is aninitial\ highly localized invasion "Earn\ Rotani +A necessary caveat to this discussion is the recent

observation "Keeling + Grenfell 0886# that modifying Grenfell 0886#[ There is clearly a need for a bettertheoretical analysis of the implications of this cor!the assumption of exponentially distributed incu!

bation times can markedly alter predicted critical com! relation structure for the persistence of infection\ anda closer epidemiological examination of the nature ofmunity sizes for measles[ The speci_c setting of that

work includes seasonal forcing and age!structure in between species transmission of morbilliviruses[analysing endemic persistence at the bottom of sea!sonal troughs\ while we are concerned primarily with

Acknowledgementspersistence through the _rst trough after invasion ofa wholly susceptible population\ but further work is This research was supported by the Natural Environ!

ment Research Council^ B[T[G[ also acknowledgesclearly necessary along these lines[ A further issuelikely to be of importance in persistence is the nature support from the Wellcome Trust[ We thank the ref!

erees for their helpful comments[of between!patch coupling discussed brie~y in Section1[1] emerging methods for quantifying regional dis!persal\ at least at a reproductive level\ may be very

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weak\ the number of patches is of crucial importance Bailey\ N[T[J[ "0864# The Mathematical Theory of Infectiousfor prolonging the infection until the next birth period\ Disease and its Applications[ Gri.n\ London[

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Vaughan\ R[W[ "0864# Seals in Orkney[ The Natural Environ!Thompson\ P[M[ "0878b# The Common Seal[ Shire Naturalment of Orkney "ed[ R[ Goodier#\ pp[ 84Ð86[ Nature Con!History\ Aylesbury[servancy Council\ Edinburgh[Thompson\ P[M[\ Cornwell\ H[J[C[\ Ross\ H[M[\ Hall\ A[J[

Visser\ I[K[G[\ van de Bildt\ M[W[G[\ Brugge\ H[N[\ Reijn!+ Corpe\ H[M[ "0883a# Lack of evidence for the continuedders\ P[J[H[\ Vedder\ E[J[\ Kuiper\ J[\ de Vries\ P[\presence of phocine distemper virus in harbour seals fromWalvoort\ H[C[\ UytedeHaag\ F[G[C[M[ + Osterhaus\North!East Scotland[ The behaviour and ecology of commonA[D[M[E[ "0878# Vaccination of harbour seals "Phoca vit!and grey seals in the Moray Firth[ Final contract report toulina# against phocid distemper with two di}erent inac!the Scottish O.ce Agriculture and Fisheries Department\tivated canine distemper virus "CDV# vaccines[ Vaccine\University of Aberdeen[6\ 410Ð5[Thompson\ P[M[\ Cornwell\ H[J[C[\ Ross\ H[M[ + Miller\

Visser\ I[K[G[\ Vedder\ E[J[\ van de Bildt\ M[W[G[\ OÝrvell\D[ "0881# Serologic study of phocine distemper in a popu!C[\ Barrett\ T[ + Osterhaus\ A[D[M[E[ "0881# Canine dis!lation of harbor seals in Scotland[ Journal of Wildlife Dis!temper virus ISCOMS induce protection in harbour sealseases\ 17\ 10Ð6["Phoca vitulina# against phocid distemper but still allowThompson\ P[M[ + Harwood\ J[ "0889# Methods for esti!subsequent infection with phocid distemper virus!0[mating the population size of common seals\ Phoca vitu!Vaccine\ 09\ 324Ð7[lina[ Journal of Applied Ecology\ 16\ 813Ð27[

Visser\ I[K[G[\ Vedder\ E[J[\ Vos\ H[W[\ van de Bildt\Thompson\ P[M[\ Kovacs\ K[M[ + McConnell\ B[J[ "0883b#M[W[G[ + Osterhaus\ A[D[M[E[ "0882# Continued pres!Natal dispersal of harbor seals "Phoca vitulina# from breed!ence of phocine distemper virus in the Dutch Wadden Sea

ing sites in Orkney\ Scotland[ Journal of Zoology\ 123\population[ Veterinary Record\ 022\ 219Ð1[

557Ð62[Thompson\ P[M[ + Miller\ D[ "0889# Summer foraging Received 17 February 0886

Þ 0887 BritishEcological SocietyJournal of AnimalEcology\ 56\ 43Ð57