The persistence of a SIS disease in a metapopulation

JANET E. FOLEY, PATRICK FOLEY and NIELS C. PEDERSEN

School of Veterinary Medicine, Center for Companion Animal Health, University of California, Davis, CA

95616, USA and Department of Biological Sciences, California State University, Sacramento, CA 95670, USA

Summary

1. Deterministic models predict that susceptible-infective-susceptible (SIS) disease,

where there is no immunity to reinfection following recovery, will become in®-

nitely persistent in a host population. We explored the incorporation of stochasti-

city into SIS models; modelled interacting host-disease agents in metapopulations;

and examined model predictions in a real system involving viral infection in

domestic cats.

2. SIS models incorporating stochasticity predicted that disease persistence would

be ®nite and dependent on the host population size, provided the host population

was isolated. However, the disease may persist by dynamic spread among inter-

acting host metapopulations.

3. Feline enteric coronavirus (FECV) dynamics in domestic cats were well pre-

dicted by stochastic metapopulation models.

4. The models we present are mathematically tractable, generalizable, and

mechanistically realistic. The ®ndings from the cat-virus system are immediately

applicable to the management of cattery populations and could be adapted to

inform eradication programmes for other infectious diseases in animal and

human populations. The most practical methods to eradicate feline enteric cora-

novirus would be to remove small catteries (islands) from interactions with large

catteries (mainlands) and to convert mainlands to islands by depopulation.

Key-words: birth and death processes, feline enteric coronavirus, infectious disease,

spatial spread of disease.

Journal of Applied Ecology (1999) 36, 555±563

Introduction

Host individuals are `islands' of transient disease,

among which islands disease agents move to survive.

Disease outbreaks may terminate, in small patches

of hosts, when there are insu�cient susceptible hosts

to maintain an epidemic. Deterministic models of

epidemic disease describe this local extinction of dis-

ease using the SIRS (for Susceptible 4 Infective 4

Recovered 4 Susceptible) approach (Kermack &

McKendrick 1927; May & Anderson 1984).

Numerous authors have also incorporated stochasti-

city into epidemic models (Bartlett 1956; Bharucha-

Reid 1956; Bailey 1975), although most such models

are impossible or di�cult to solve. Incorporation of

stochasticity is critical for small host populations or

when dynamics are of the SIS (Susceptible 4

Infective 4 Susceptible) form, because deterministic

SIS models always provide for endemic disease

above some exact threshold population size

(Edelstein-Keshet 1988). In reality, thresholds for

endemicity of SIS disease may vary stochastically.

When stochastic models are employed to describe

SIS disease in small host populations, predicted epi-

demics are always ®nite in duration. Given stochas-

tic dynamics, persistence of disease agents depends

on either an inert reservoir in the environment or

dynamic spread of the agent among small local

populations, i.e. a metapopulation in the broad

sense. Gyllenberg, Hanski & Hastings (1997) appre-

ciated that the deterministic dynamics of SIRS and

metapopulation models are analogous, and

Ferguson, May & Anderson (1997) described a

simulation of measles persistence in a grid-like meta-

population. The three criteria for diseases well suited

to metapopulation analysis are: (i) existing in nature

in discrete host populations; (ii) spreading by coloni-

zation of these host populations; and (iii) sometimes

experiencing extinction within local populations.

Correspondence author: Dr Janet Foley, School of

Veterinary Medicine, Center for Companion, Animal

Health, University of California, Davis, CA 95616, USA

(fax: 530 7527701; e-mail jefoley@ucdavis.edu).

Journal of

Applied Ecology

1999, 36,

555±563

# 1999 British

Ecological Society

In this paper we model the persistence of an SIS,

a viral infectious disease of felids (feline enteric cor-

onavirus; FECV), that ful®ls the three criteria given.

The virus is endemic in catteries (which are small,

discrete local populations of hosts; Foley et al.

1997), is highly infectious, and can become locally

extinct if cat groups are small, given stochastic

dynamics. Local FECV dynamics are modelled

using a birth and death process model coupled with

a deterministic metapopulation model. The model is

®tted with parameters from empirical data. There

are two important applications of this work: the

appreciation that the expected extinction time of a

disease can be used directly in the global metapopu-

lation model, and as an example of how such mod-

els account for the persistence and spread of

infectious disease in nature. Lastly, we give recom-

mendations for the management of FECV based on

the model and show how this system is a template

for the development of eradication strategies for

similar diseases in discrete, interacting host popula-

tions.

Feline enteric coronavirus natural history andepidemiology

Domestic cats occur world-wide in feral cat colonies,

as indoor±outdoor pets and in indoor-only catteries,

either for humane purposes, pets or commercial

breeding. They are susceptible to a relatively benign

enteric coronavirus (FECV) that thrives in intestinal

epithelium and produces mild gastro-enteritis

(Pedersen et al. 1981). Feline coronaviruses are con-

tracted primarily during exposure to infectious cat

faeces in the environment, but also via ingestion or

inhalation during cat-to-cat contact. FECV is gener-

ally fragile once outside the cat's body, but may sur-

vive for as long as 7weeks if protected from heat,

light and desiccation. The transmission of the FECV

is frequent and di�cult to detect clinically.

Prevalence of FECV infections is related to cattery

size and density: in multiple-cat homes with ®ve or

more cats, approximately 100% of the cats have

been exposed (Foley et al. 1997). Once exposed

(often as kittens), cats may periodically shed FECV

in faeces for weeks to a few months and after recov-

ery are not immune. Thus they are likely to become

rapidly reinfected. Active FECV infection is diag-

nosed by detection of FECV RNA in faeces by

reverse-transcriptase polymerase chain reaction

(RT-PCR) (Poland et al. 1996; Foley et al. 1997).

Cats develop positive antibody titres in serum about

7 days after exposure to FECV, but the titres remain

elevated for months to years, even when the cat is

recovered and not shedding virus.

Materials and methods

METAPOPULATION MODEL

The strict-sense Levins metapopulation comprises a

set of identical, equally accessible interacting patches

(Levins 1969), with dynamics given by:

dp

dt� mp�1ÿ p� ÿ ep eqn 1

in which p(t) is the fraction of patches occupied at

time t by (in this case) FECV, m is the migration

rate among patches, and e is the per-patch extinc-

tion rate. Equivalently, p(t) is the probability that a

particular host patch is infected by the disease. If

the metapopulation has extremely variable patch

sizes, then mainland±island models are more appro-

priate, in that some patches become permanent

refuges (or foci) for disease (MacArthur & Wilson

1967; Hanski 1991). The dynamics are then:

dp

dt� m�1ÿ p� ÿ ep: eqn 2

For FECV, endemically infected catteries are

mainlands and transiently infected catteries are

islands. Intermediate patch size variability is

included in the model by using a structured metapo-

pulation in which p(t) depends on the patch size, N

(Gyllenberg, Hanski & Hastings 1997). The para-

meters that govern the dynamics of p(N,t) for all

models are the extinction rate, e(N), and the coloni-

zation rate, m(N), i.e. the rate of disease introduc-

tion into susceptible local populations. The

predicted Levins metapopulation equilibrium, p*, is:

p� � 1ÿ e=m eqn 3

while the island±mainland p* is:

p� � m

m� e: eqn 4

If e exceeds m, then a strict-sense Levins metapopu-

lation would go extinct across all patches, i.e. FECV

would disappear within that metapopulation of cat-

teries. In contrast, a mainland of endemic disease

would by de®nition remain infected and would

allow for positive occupancy probabilities on all

islands accessible to the mainland.

LOCAL POPULATION MODEL

Local population extinction rates of FECV are

obtained as the solution of a continuous time birth±

death process Markov chain (Feller 1971; Karlin &

Taylor 1975; Nisbet & Gurney 1982) with discrete

individual accounting, demographic stochasticity

(Mollison 1981) and continuous time. The number

of infectives, I, can be a value from 0 to N (total

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population size). Then B(I)dt is the probability of an

infection in some small time period dt. D(I)dt is the

probability of recovering from disease in dt. If there

are I(t) infectives at time t, then the probability that

I will increase by 1 in time dt is (dropping terms in

dt2):

P�I4I� 1� � bI�Nÿ I�dt � B�I�dt: eqn 5

Here b is the rate of transmission, i.e. the number ofnew infections produced per day after the introduc-

tion of one infective cat into a susceptible popula-

tion. The probability that I will decrease by 1 is:

P�I4Iÿ 1� � gIdt � D�I�dt eqn 6

where g is the rate of recovery, i.e. the number of

days after infection before a cat is no longer shed-

ding infectious virus. The probability that I will stay

the same is:

P�I4I� � 1ÿ �bI�Nÿ I� gI�dt: eqn 7

The expected value (or mean value) of I for this

time period is:

EI�t� dt� � I�t� � EdI

� I�t� � bI�Nÿ I�t��dtÿ gIdt: eqn 8

The time to extinction of FECV using this birth±

death process and starting at one infective individual

satis®es:

Te�I0 � 1� �XNi�1

qi eqn 9

where:

qi � 1

D�1� �1

g, if i � 1 eqn 10

qi � B�iÿ 1�:::B�1�D�i�:::D�1� if i>1: eqn 11

The derivation may be found in Nisbet & Gurney

(1982), Karlin & Taylor (1975) or Gardiner (1985).

For SIS disease dynamics, we insert values for births

and deaths obtained from equations 5 and 6 to

obtain the persistence time of FECV in a local host

population starting with I(0)=1,

Te�Io � 1� �XNi�1

1

g

�bg

�iÿ1 �Nÿ 1�!i�Nÿ i�! : eqn 12

The extinction rate of FECV is 1/Te. Parameters

needed to apply the metapopulation model are b, g,N (or e), p and m.

LOCAL POPULATION PARAMETER

ESTIMATION

Transmission rates (b) were estimated from expo-

sures to FECV in previously uninfected cats.

Groups of speci®c pathogen-free (SPF) domestic

short-haired cats and Persian cats were maintained

in viral containment facilities at the UC Davis

Veterinary Retrovirology Laboratory in ratios of 1

susceptible cat: 1 infective cat (Pedersen et al. 1981)

or four susceptible cats: 1 infective cat, and in

Havana Brown and domestic short-hair cats in pri-

vate homes in ratios of 4 : 1 and 2 : 1. After exposure

to FECV, cats were determined to be infective by

detection of FECV RNA within faeces by RT-PCR

(Foley et al. 1997), or anti-coronavirus IgG

(Pedersen 1995) in serum if the cat had been anti-

body-negative before the experimental manipula-

tion. Previous experiments indicated an approximate

4-day period of latency from initial exposure to

PCR-positive faeces (when cats are infective) and

cats do not seroconvert until 7 days after exposure

(Foley et al. 1997). Therefore the time from exposure

to infectivity was adjusted by subtracting 3 days if

the infection was detected by seroconversion.

Latency was not incorporated into the present mod-

els because an added state (exposed, not infective)

would increase model complexity signi®cantly and

the magnitude of the latent period of 4 days is small

compared with the typical duration of the suscepti-

ble state and infective states (both months).

Moreover, three cats were observed in the present

study to become faecal PCR-positive by day 2, sug-

gesting a very brief period of latency under some

conditions. The maximum likelihood estimator

(MLE) of bI is the reciprocal of the mean date till

®rst infection, assuming an exponential distribution

(Johnson, Kotz & Balakrishnan 1994). With this

MLE and I=1 (for these experiments), the most

robust estimate of latency is the minimum value of

the time to infection (2 days).

The recovery rate, g, was determined by maintain-

ing seropositive, faecal PCR-positive cats in isola-

tion until faecal shedding ceased for 3 consecutive

weeks. Then g=1/(last day faecal positive). The

95% con®dence intervals for b and g were obtainedas described elsewhere (Johnson, Kotz &

Balakrishnan 1994). The duration of immunity was

evaluated by reintroducing 13 recovered cats into an

endemically infected home in which 100% of the 40

resident cats were seropositive and 75±80% were

shedding FECV in faeces.

METAPOPULATION PARAMETER

ESTIMATION

An empirical estimate of p was obtained by deter-

mining how many catteries of size N=1,

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2, . . . 6+had at least one seropositive cat at time t.

The extinction rate was estimated as the reciprocal

of epidemic duration (Te), obtained from the birth±

death process local model. Alternatively, e(N) could

have been observed empirically, which would have

required a large sample size for reasonable con®-

dence.

The colonization rate (m) was estimated by subdi-

viding colonization into four independent events,

and attributing to each event a probability that this

event would result in the introduction of infection

into a naive population. These events were: adop-

tion of a new cat (ma), a cat visit (to another home,

veterinary o�ce or cat show) (mv), exposure to

fomites via the owner (typically because of occupa-

tional exposure to cats) (ms) and exposure to outside

cats while roaming (mo). The probability or risk

coe�cient (a) that a single performance of each

event would introduce FECV into a naive popula-

tion was estimated as the `best guess' by researchers,

veterinarians and cattery managers familiar with

FECV in the ®eld. For example, adopting an FECV

PCR-positive cat is a colonization, with a =1. The

approximate values of a used in the present model

are summarized in Table 1.

Components of colonization were assumed to be

additive, so that the overall colonization rate was:

m=aama+avmv+asms+aomo. eqn 13

Values for ma, mv, ms, and mo were estimated

from questionnaires in which cat owners were

asked: the annual frequency of travel to veterinar-

ians, cat shows, and other homes; the number and

source of any new cats in the home over the last

2 years; whether cats were strictly indoors or indoor/

outdoor; whether any cats had visited the home

over the last 2 years; and whether members of the

household worked with cats. The best estimate for

each m(N) was obtained from the linear model

regressing m (from questionnaires) on N.

Results

PARAMETER ESTIMATES

A decay function of the percentage of uninfected

cats (Fig. 1) indicated that the waiting time until

infection was exponentially distributed (d=0´1569,

P=0´383). Thus transmission could be modelled as

a Poisson process. The mean date of ®rst infection

was 23´04 days, resulting in an estimate of b of 0´043and 95% CI of 0´065±0´032.

Maintenance of infected cats in isolation revealed

that 8´2% of cats never ceased shedding coronavirus

Table 1. Risk coe�cients for various events leading to colonization of a population with FECV

Activity Risk coe�cient (a)

Adopting new cat from shelter 0´9

Adopting new cat from pedigreed breeder 0´9

Adopting new cat, other home p(N) of source home*

Adopting new cat, feral 0´1

Adopting new cat, stray 0´4 (highly variable)

Veterinary o�ces and cat shows visits 0´05

Visits to other households p(N) of the home visited

Owner's occupational exposure 0´05

*p(N) is the observed fraction of homes of size N with FECV infections, data shown in Table 2a, b.

Fig. 1. Exponential decay function of the proportion of cats experimentally exposed to FECV that remained uninfected

over time.

558Disease

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while under observation; these cats were classed as

chronic shedder cats (SI) cats and were excluded

from the calculation of recovery rate. For the 13

cats observed recovering, the mean period until

recovery was 37´2 days, with g=0´0269 and 95% CI

of 0´017±0´061.

PERSISTENCE OF FECV

The expected persistence time of FECV in catteries

increased rapidly with cattery size (Fig. 2). In fact,

there was a threshold population size of approxi-

mately ®ve, above which no reasonable values of band g allowed for extinction of FECV, i.e. no man-

agement strategies to reduce infections and promote

rapid recovery would allow for elimination of the

endemic disease in a year or less. Conversely, in a

small cattery with no chronic shedders, FECV epi-

demics would be unlikely to become persistent, even

in the face of delayed recovery and rapid virus

transmission. Very close to the threshold, changes of

b and g contribute to making Te manageable, but

the most important determinant of Te far from

threshold is N (Fig. 3). If any cat in a population is

a chronic shedder (SI), then Te=1, independent of

N.

Given the presence of persistently infected cat-

teries, an island±mainland metapopulation model

accounts best for global FECV persistence. The

empirical patch occupancy p(N) of indoor-only cat-

teries sharply increased from about 0´1 at N=1±

100% for Nr 5 (Table 2a), which agrees well with

the birth±death process model expectations, and

con®rms that catteries with r 5 cats act as main-

lands. Also given in Table 2 are the calculated values

of e(N) from the local birth±death process model

Table 2. (a) Metapopulation parameter and variable values for indoor-only catteries with birth and death process dynamics

and g=0´0269, b=0´043; (b) metapopulation values for indoor±outdoor catteries

(a) Indoor-only catteries

Cats per cattery 1 2 3 4 5

Number of catteries 12 2 6 6 4

e(N) (year±1) 9´812 5´456 2´281 0´671 0´140

m(N) (year±1) 0´606 1´212 1´818 2´424 3´03

Expected number of infected 0´69 0´36 2´66 4´69 3´82

Observed number of infected 1 0 3 4 4

Expected p*(N) 0´058 0´182 0´444 0´783 0´956

p(N) ± observation 0´08 0´0 0´5 0´67 1´0

(b) Indoor±outdoor catteries

Cats per cattery 1 2 3 4 5

Number of catteries 6 2 4 2 3

e(N) (year±1) 9´812 5´456 2´281 0´671 0´140

m(N) ± quest (year±1) 0 0 1´19 2´75 1´2

m(N) ±model (year±1) 0´019 0´167 1´03 0 3´584

Expected number of infected 0 0 1´37 1´61 2´69

Observed number of infected 1 1 3 0 3

Expected p*(N) 0 0 0´343 0´804 0´895

p(N) ± observation 0´16 0´5 0´75 0 1

Fig. 2. Expected time to extinction (Te) of an outbreak of FECV as a function of cattery size (N), assuming a birth and

death process SIS model, with b=0´043 and g=0´0269.

559J.E. Foley,

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0·01

Beta

Day

s to

ext

inct

ion

1 × 1014

1 × 1013

1 × 1012

1 × 1011

1 × 1010

1 × 109

1 × 108

1 × 107

1 × 106

1 × 105

1 × 104

1 × 103

1 × 102

1 × 101

1 × 100

1 × 1014

1 × 1013

1 × 1012

1 × 1011

1 × 1010

1 × 109

1 × 108

1 × 107

1 × 106

1 × 105

1 × 104

1 × 103

1 × 102

1 × 101

1 × 100

0·020·03

0·04

0·05 0·05

0·03

0·020·01

Gamm

a

Day

s to

ext

inct

ion

(c) N = 10

0·01

Beta

Day

s to

ext

inct

ion

1 000 000

100 000

10 000

1000

100

10

1

1 × 106

1 × 105

1 × 104

1 × 103

1 × 102

1 × 101

1 × 100

0·020·03

0·04

0·05 0·05

0·03

0·020·01

Gamm

a

Day

s to

ext

inct

ion

(b) N = 5

0·01

Beta

Day

s to

ext

inct

ion

1 × 104

1 × 103

1 × 102

1 × 101

1 × 100

1 × 104

1 × 103

1 × 102

1 × 101

1 × 100

0·020·03

0·04

0·05 0·05

0·030·02

0·01

Gamm

a

Day

s to

ext

inct

ion

(a) N = 3

0·04

Fig. 3. Predicted times to extinction (days) of a stochastic epidemic birth and death process in a cattery with three, ®ve and

10 cats. Beta is transmission rate and gamma is recovery rate. See text for model.

560Disease

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(with b=0´043, g=0´027) and calculated estimates

of m(N) from questionnaires for indoor-only cat-

teries. Reported values of m(N) were obtained from

the linear model regressing observed m on N. The

empirical estimate of p(N) was compared with the

predicted value of p*(N), assuming a Levins model

and the values of m(N) and e(N) given in Table 2a.

The data were also shown as numbers of observed

and expected infected catteries, to simplify compari-

sons. In all ®ve cases, the ®t was remarkably good

given the small sample size: the rounded-o� esti-

mates of expected infected catteries were equal to

the observed number of infected catteries.

Indoor±outdoor catteries had generally higher

values of p(N) than indoor-only catteries, re¯ecting

the increased exposure of indoor±outdoor cats to

infective cats and fomites (Table 2b). The observed

number of infected indoor±outdoor catteries was

not as close to expected as for indoor-only catteries,

probably because owners could not estimate accu-

rately on questionnaires how often cats came into

contact with infectious cats or fomites while out-

doors. We compared the observed values of m(N)

from questionnaires with those predicted from the

Levins model equilibrium with known e and p. The

results indicated that the estimate of m underesti-

mated the predicted m, i.e. there was signi®cant

colonization not accounted for in the questionnaire.

The critical value for FECV metapopulation persis-

tence, i.e. m(N)> e(N), occurs in both indoor-only

and indoor-outdoor catteries at N=4. Therefore,

the threshold cattery size to be a mainland (N=5)

was close to the critical N for persistence of a Levins

metapopulation, indicating that the FECV system

most closely resembles an island±mainland version

of a metapopulation.

Discussion

As we have demonstrated with the FECV model,

metapopulation dynamics contribute to the ongoing

incidence and persistence of patchily distributed

infectious diseases. In small populations, demo-

graphic stochasticity can produce local disease

extinction, so that global disease persistence depends

on the existence of mainland populations and fre-

quent migration among small populations. In an

unstructured metapopulation of small populations,

metapopulation persistence occurs only when the

colonization rate exceeds the local extinction rate.

Otherwise the disease agent goes extinct globally. In

an island±mainland system such as FECV in cat-

teries, extinction on small islands occurs only if all

migration ceases. In the FECV±cattery system,

homes with r 5 cats function as mainlands.

The output from the FECV metapopulation

model may be used to develop management strate-

gies to control and eradicate FECV in nature. From

the perspective of a single small cattery, the most

e�ective methods of controlling FECV would be to

remove the cattery from the metapopulation (by

eliminating all sources of immigration of FECV)

and to convert the cattery from a mainland to an

island, by reducing the population to four or fewer

cats. We did not incorporate latency into the para-

meter estimates, but doing so would not signi®cantly

change expected dynamics. Rather, transmission

would be slightly more e�cient, which would

slightly increase the expected persistence time.

However, expected persistence times are already

extremely high except below a stringent threshold in

population size; the threshold value is not sensitive

to the addition of minor latency and minor changes

of beta (b) and gamma (g). There is no su�cient

increase in gamma (g) that could allow for FECV

extinction in larger catteries. In contrast, major

reductions in transmission rate would be expected to

reduce FECV persistence in catteries signi®cantly

over the present threshold size; such a change in the

number of new transmissions [beta (b)] would prob-

ably be accomplished in the near future by a vac-

cine. As such technologies are developed, the

expected change in dynamics could be explored

using this model.

The important assumptions in the metapopulation

model were that all patches are equally accessible

and the assumptions implicit in the estimation of

migration rate. It is probably not true for real inter-

acting host populations that all patches are equally

accessible. However, simulations show that metapo-

pulation dynamics (at least near equilibrium) are

not very sensitive to deviations from the equal acces-

sibility assumption (Hanski, Foley & Hassell 1996).

Moreover, catteries are interconnected in a more

equally accessible way than many animal local

populations in which geographical proximity sets

limits to recolonization events.

The assumptions implicit in the calculation of m

included: (i) that all sources of colonization were

included in the subdivision of m; (ii) that the weights

attributed to each event were reasonable; and (iii)

that the component events were approximately inde-

pendent of each other and interaction terms could

be disregarded. There was evidence that owners of

indoor±outdoor cats signi®cantly underestimated m

in questionnaires. A simpler and more robust

method of estimating m would have been to observe

naive populations of various sizes, and quantify the

number of disease introductions in time. Such an

approach requires a very large spatial and temporal

scale of study because colonization events for any

system are rare or at least unevenly distributed in

space and time (Ims & Yoccoz 1997). This approach

has rarely been performed for any studies of meta-

populations in nature. A further di�culty in esti-

mating m comes from variability in rates of

561J.E. Foley,

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Ecology, 36,

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colonization across populations and time.

Conceptually, the e�ect of variable m is to reduce

the equilibrium level of disease (Levins 1969). It is

likely that variability in m is signi®cant in real cat

populations relative to the scale of movement of

FECV. Further research estimating m and its varia-

bility is warranted.

Metapopulation models may be extended to other

disease systems with high turnover, such as the com-

mon cold, gonorrhoea and phocine distemper.

Humans on the island of Tristan da Cunha (Shibli

et al. 1971) and in the small arctic community of

Spitsbergen (Paul & Freese 1933) were regularly and

discretely colonized by the common cold (an SIRS

disease) when visited by ships. A structured metapo-

pulation analysis is appropriate for the analysis of

the common cold in discrete human populations.

Globally endemic gonorrhoea may persist due to the

presence of at least one persistently infected sub-

group, i.e. a mainland (Lajmanovich & Yorke

1976). Hethcote (1976) generalized this result, show-

ing that global disease persistence could occur

among `subpopulations' each with R0<<1 (where

R0r 1 refers to the minimum initial population size

that will sustain an epidemic) provided there was

su�cient migration among groups. Phocine distem-

per, also an SIRS disease, occurred in discrete har-

bour seal populations in the North Sea that varied

in size from 30 to 3500 seals (Dietz, Heide-

Jorgensen & Harkonen 1989). Local epidemics of

phocine distemper varied in duration from 42 to

115 days over 17 populations with no signi®cant

relationship between population size and epidemic

duration (suggesting the possible contribution of

demographic stochasticity).

In contrast to disease metapopulations, strict-

sense Levins metapopulations may be uncommon in

plant and animal systems (Harrison 1991; Harrison

& Taylor 1997). Butter¯ies (Thomas & Hanski

1997) and pool frogs (Sjogren Gulve 1994) furnish

the few well-studied examples we have. Most discre-

tely distributed species have considerable variation

in local population size, and are thus structured

metapopulations (Harrison & Quinn 1989; Harrison

1991; Gyllenberg, Hanski & Hastings 1997). Such

structured metapopulations are harder to analyse

than the classical Levins metapopulations, but sim-

pli®cations are possible. As shown in this paper, it is

possible to identify local mainland populations that

remain persistent. The smaller populations wink on

and o� at a much more rapid rate. In fact, disease±

host systems make better examples of classical meta-

populations than some previously studied large

organisms.

Even in the case where host distributions are not

discrete, metapopulation analysis may be useful.

The alternative modelling approaches, including

non-spatial models such as the original Kermack±

McKendrick models, cannot capture the spatial

patchwork patterns of disease persistence.

Percolation models (essentially stochastic cellular

automata) do allow for space and chance (Durrett

& Levin 1994), but this ingenious and insightful the-

ory is hard to apply to real population dynamics.

Continuously distributed hosts may be modelled

with reaction±di�usion (using partial di�erential

equations) to predict the wave of disease advance

(Mollison & Kuulasmaa 1985; Murray 1989; van

den Bosch, Metz & Diekmann 1990; Mollison

1991). Reaction±di�usion models show disease ¯ow-

ing over a host landscape and then disappearing (in

theory). In practice such disease may persist due to

low densities of infectious subpopulations of hosts.

The purely continuous models do not account for

these persistent `slow fuses' in a host landscape

through which disease has already passed.

Many causes of patchy distributions of susceptible

hosts may occur in nature, such as variability in vac-

cine usage. Widespread vaccination for measles cre-

ated pronounced alterations in local and global

disease dynamics in cities in England (Grenfell,

Bolker & Kleckowski 1995). Metapopulation

approaches using transiently patchy susceptible and

infectious local populations o�er the best hope of

predicting disease persistence.

The metapopulation approach is an analytically

tractable method to study disease dynamics in dis-

cretely distributed hosts, and should provide insight

into continuously distributed hosts where suscepti-

ble, infectious and resistant patches arise during epi-

demics. Whenever local disease extinction occurs,

we must understand global infection dynamics to

account for patterns of disease persistence and emer-

gence. This understanding will be critical in any

attempts to manage or eradicate disease.

Acknowledgements

The authors wish to acknowledge the invaluable

technical assistance of Amy Poland, Cindi Ramirez,

Je� Carlson, Kim Floyd-Hawkins, Jennifer Norman

and Rosemary Panduro. Alan Hastings, Ed

Caswell-Chen, and Walter Boyce contributed to the

conceptual development. This research was sup-

ported by grants to Janet Foley from the Winn

Feline Foundation, Morris Animal Foundation,

Solvay Animal Health, the San Francisco

Foundation, and the Center for Companion Animal

Health, School of Veterinary Medicine, University

of California, Davis.

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Received 6 February 1998; revision received 30 April 1999

563J.E. Foley,

P. Foley &

N. C. Pedersen

# 1999 British

Ecological Society

Journal of Applied

Ecology, 36,

555±563