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Bubonic Plague: A Metapopulation Model of a ZoonosisAuthor(s): M. J. Keeling and C. A. GilliganSource: Proceedings: Biological Sciences, Vol. 267, No. 1458 (Nov. 7, 2000), pp. 2219-2230Published by: The Royal Society

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r4 THE ROYAL doi 0.1098/rspb.2000.1272

MIRJUOC I ETY

Bubonic plague: a metapopulation modelof a zoonosisM. J. Keeling'* and C. A.

Gilligan2'DepartmentfZoology,nd 2DepartmentfPlant Sciences, niversityfCambridge,owning treet, ambridge B23E?, UK

Bubonic plague (Yersiniaestis) s generally thought f as a historicaldisease; however, t is stillresponsiblefor around 1000-3000 deaths each year worldwide.This paper expands the analysis of a model forbubonic plague that encompasses the disease dynamics in rat, flea and human populations. Some keyvariables of the deterministicmodel, including theforceof nfection ohumans, are shown tobe robusttochanges in the basic parameters, although variation in the flea searchingefficiency,nd the movementrates of rats and fleas will be considered throughout he paper. The stochasticbehaviour of the corre-spondingmetapopulation model is discussed,withattention ocused on the dynamicsof rats and the forceof nfection t thelocal spatial scale. Short-lived ocal epidemics n ratsgoverntheinvasionof thediseaseand produce an irregular patternof human cases similarto thoseobserved.However, the endemic beha-viour in a fewrat subpopulations allows the disease to persist for many years. This spatial stochastic

model is also used to identify he criteriafor the spread to human populations in termsof the ratdensity.Finally, he full stochasticmodel is reduced to theform fa probabilistic ellular automaton,whichallowstheanalysis ofa large number of replicated epidemics n large populations. This simplifiedmodel enablesus to analyse the spatial properties of rat epidemics and the effects f movementrates,and also to testwhether the emergent metapopulation behaviour is a property of the local dynamics rather than theprecise details ofthe model.

Keywords: Black Death; epidemic; stochasticity; ersistence; patial model; animal diseases

1. INTRODUCTION

Bubonic plague (caused by the bacteriumYersinia estis),also known as theBlack Death, devastatedpopulations inEurope from the 14th-16th centuries, killing between

one-thirdand one-half of the entirepopulation (Langer1970). Not surprisingly,herehas been a vast amount ofresearch into the disease from a historical perspective,cataloguing the panic and social changes that it caused(Shrewsbury1970; Appleby 1980; McEvedy 1988; Risse1992; Scott et al. 1996). However, bubonic plague is farfrombeing confined o history; hepandemic in the early1900s killed many millions (Commission for the Investi-gation of Plague in India 1906; Hirst 1938; Sharif 1951;Curson & McCracken 1990; Risse 1992) and even todaytheWorld Health Organization reports1000-3000 casesof plague every year. Recent surveys indicate thatbubonic plague is widespread throughout he wild rodent

population in the United States, southernAsia, southernAfrica and South America; thus, even in areas whichhave sufferedfew human cases there may still be thepotentialfor large-scalehuman epidemic.

Two factors dd extra importance to the study of thisdisease: itsrecentre-emergence n India and Africa,andthe evolution of multi-drug-resistant trains of thebacterium.Over the last decade therehas been a signifi-cant increase in the numberofreportedcases of bubonicplague, withprolongedhuman epidemicsarisingin threemain areas: Surat, Mozambique and Madagascar(Kumar 1995; Barreto et al. 1994; Boisier etal. 1997). In1997 it was reported that a drug-resistant train of the

bacterium had evolved in Madagascar (Galimand etal.1997). These two aspects have led many to speculate

that this historical disease may re-emerge to becomean important public health problem (Pinner 1996;McCormick 1998; Gratz 1999).

In a previous paper (Keeling & Gilligan 2000), weintroduced a stochastic metapopulation model for

bubonic plague. Here we consider the full deterministicdynamics and stochastic behaviour in more detail. Inparticular, we concentrate on the rat dynamics at thelocal spatial scale, showing that the invasions arecontrolled by short-lived pidemics,whereas the persis-tence of the disease is governed by a few endemic sub-populations. The sensitivity f key model output variablesto changes in parameters is discussed, with attentionthroughout the paper focusing on the role of fleasearching efficiency nd movement of rats and fleas, andtheir ffects n the potential forhuman cases.

We first haracterizethe epidemic patternsobservedinthe historical data at three time-scalesrangingfrom one

to 60 years. The full deterministicmodel is formulated,linking rat, flea and human populations and describingthe spread of the disease fromrats to humans. The sensi-tivityof this model to parameter values is discussed insome detail. In ? 5 more biological complexity s intro-duced, making themodel both stochastic nd spatial, andobtaining rregularhuman epidemicscharacteristic fthehistorical data sets. Using this model we postulate thatbubonic plague persistseven in quite small rodentpopu-lations and therefore he observed historicaldynamicsarecaused by fluctuationsn the numberof rat cases ratherthan random importsof infection. n ? 6 we focus atten-tion on the potential riskto humans, and by examining

the occurrence of bubonic plague in otherwild rodentswe are able to estimate lower bounds for the fleasearching efficiencynd hence the threshold forhumancases. Finally, to test the robustness of our complexAuthorfor orrespondence([email protected]).

Proc. . Soc. ond. (2000) 267, 2219-2230 2219 ?) 2000 The RoyalSocietyReceived June 2000 Accepted4 August 2000







2220 M. J.Keelingand C. A. Gilligan Buboniclague: metapopulationodelfa zoonosis

x 1044

(a)

= 3.5cd

'U-o

0)1. 2.5

~0

date

9 (c)

a 1.

0

Jan Feb Mar Apr May Jun Jul Augtime

stochasticmodel, a cellular automaton s developed in ? 7.This displays ll characteristics f the full tochasticmodel,but nablesus to assess arger cale spatial patterns.

2. HISTORICAL DATA

Figure shows thedynamicsofbubonic plague in threeverydifferentocations (London in the middle ages, and

Bijapur, India, and Sydney, Australia, in the 20thcentury)and over three very different ime-scales. Allthreegraphs show similar properties.Human epidemicsoccur sporadically and for a limited period, with longdisease-freeeriods etween pidemicsfigurea,b). Also,the highlystochastic nature of this system s apparent(figure 1c),with variable duration and intensity nd noclear evidence of a regular deterministic pattern(figure a).

3. THE MODEL

1590yp 1600wor 610e 1620; 1630et 1640; 1650eal. 996) lasifyng hmandasetherucpil,ifc

tiuso (ec)vrd sn h oaino oe 17)h

140(b)

, 120 -

e 100 I

g 80 -

c 60 -

E 40 -0u

(4-4

2 20

,0

1935 1936 1937 1938 1939 1940 1941 1942 1943date

Figure1. Plaguecases from round theworld. a) Thenumber fdeaths n London attributed o theplaguefrom 590 to 1650 Graunt 1662).These data are annualfigures,nd hence how he eastdetail; however,hedata set covers he ongest ime-period.t should lso berealized hatduring his eriodoftimediagnosiswasuncertain; hereforehese esultshouldbe viewed squalitativenformationnly. b) The number fcommunitiesoutof432) intheBijapurdistrict f ndiathatwere nfectedytheplague (Sharif 951).The datawerecompiledmonthly,nd cover n eight-year-periodfrom 935 to 1943 ncluding nemajoroutbreak. c) Theonset fcases nSydneyn 1903 Curson& McCracken1990),whentherewere301 reportedases. Thesedailydata show arge tochastic luctuations,o a two-weekmoving veragewas calculated solid ine).

number ofsusceptible SH) and infectious IH) individualsis givenby

dSH _&H-= -IHSHIH,)dt (1)

dIH HSH IH - mHIH,dt

where the subscriptH shows that we are consideringthehuman population, 3H is the contact rate and mH is themortality rate. In this model the disease is consideredfatal to humans, so there s no recoveredclass. Althoughthe pneumonic form of the disease is highlycontagiousand is transmittedreadily between humans, historicalevidence shows that by far the greatest proportion ofcases have been caused by transmission from rats tohumans via fleas (Appleby 1980; Busvine 1993). Bubonicplague is therefore ermed a zoonosis, being a disease ofanimals that can be passed on to humans (zoonosesinclude other publicly feared diseases such as rabies,ebola, Lyme disease, and hantavirus).Full understanding

and prediction of the number of bubonic plague casesrequires a model of the entire transmission route, withequations for the rat, flea and human populations(Keeling & Gilligan 2000). This leads to an SIR-typemodel for the rat population in a similar manner to

Proc. . Soc.Lond. (2000)







Buboniclague: metapopulationodelfa zoonosis M. J.Keeling and C. A. Gilligan 2221

equations (1), describingthe number ofsusceptible, nfec-tious and resistant ats:

d rRSR I -R) + rRRR - P) -dRSR

SR K

-ARRF[l

-exp(- aTR)],TR (2)

dI= 3RS

F[l - exp(- aTR)] - (dR+ mR)IR,dt TR

d rRRR (- + mRgRIR -dRRR,

where TR = SR + JR RR (the total ratpopulation size).Both susceptible SR) and resistant RR) ratsbreed with anet reproductiverate of rR and a carryingcapacity KR;however, proportion of the offspringnheritresistanceto the disease fromtheir parents.All rats have a naturaldeath rate of dR. Transmissionof infection s via infectedfleas,of which there are F. The preciseformof the infec-tion term corresponds to infected fleas randomlysearching for a new rat host, for some given time-period(cf.Nicholson & Bailey 1935). If theyfind host and it issusceptible, hen with a givenprobability he ratbecomesinfected. herefore R is the transmission ate to rats anda is a measure of the searching efficiency f the fleas.Finally, rats leave the infectedclass at a rate mR and afractiongR of these survive to become resistant; theremainder die and release their infected fleas back intothe environment.

The dynamics of the flea population are modelled byN, the average number of fleas ivingon a rat (called theflea index), and F, the number of free nfectious leas thatare searchingfor host:

d= rFNIl -X +d

F[I -exp (- aTR)],dtKFI TR (3)dFd = (dR+ mR( -gR))IRN-dFF.dt

In the absence of bubonic plague, the flea index N obeysa logistic growthmodel, with carryingcapacity KF. Theotherterm n thedifferentialquation forN is due to freefleas finding new rat host and thereforencreasingtheaverage flea ndex. Free nfected leas are released into theenvironmentevery time an infected rat dies, and on

average this should releaseN fleas. Freefleasare assumedtodie from tarvation t a ratedF.

The final section ofthe model is the humanpopulation.Ignoring densitydependence in the birth and death rateswe obtain

dSH = rH SH +RH) - dHSH AHSHFexp(- aTR),dt

-= 13HSHFexp(- aTR) - (dH+ mH)IH, (4)dt

dt = MHgHIH- dHRH

Although this is again an SIR-type model, the infectionterm in equations (4) is proportional to the number offleas that do not find a new rat host. It is important torealize that the human dynamics do not affect he diseasebehaviour; the human cases are merely a by-productof

Table 1. Parameters sed n the epizooticmodel or bubonicplague

(Those parameters marked with an asterisk have beenestimated from either laboratory experiments or fieldobservations:1, Buckle & Smith (1994); 2, Hirst (1938);3, Wheeler & Douglas (1945); 4, Macchiavello (1954);

5, Hinnebusch t al. (1998); 6, Bacot (1915). The carryingcapacityfor rats and the fleassearching fficiencyt) arefundamentalarameters f themodel, nd sensitivityo theseis discussed n ? 6. The remainingparametershave beenchosen withinbiologicallyrealisticbounds,and the modeldynamicshave little ensitivityo the precisevalues used. Itshould be noted thatthe humanparameters re not used inthe calculation f thepotential orce f nfectionH. All ratesaremeasured er year.)

parameter value meaning

rR1 5 rat'sreproductiveatep*2 0.975 probabilityf nherited esistanceKRt 2500 rat's arryingapacityd/* 0.2 deathrateof atso*3-5 4.7 transmissionatemR 20 (infectiouseriod)-1

g*R2 0.02 probabilityfrecoveryPR 0.03 movement ate fratsat 4 x 10-3 flea earching fficiencyrF 20 flea's eproductiveatedF6 10 deathrateoffleasKF*2 3.29 -> 11.17, flea's arryingapacity errat

mean6.57IF 0.008 movement ateof leasrH 0.045 reproductiveateofhumansdH 0.04 deathrate fhumans/3H 0.01 transmissionate o humansmH 26 (infectiouseriod)-1gH 0.1 probabilityf ecovery

the progressionof the disease in the rodentcommunity.Therefore, or implicity, e can ust model the number offleas that fail to find a host, H (Keeling & Gilligan2000), where

IH= Fexp(-aTR). (5)

Hence, H is thenumber of nfected leaswhichcould feedon and infect human host;we call H thepotentialforce

of nfection ohumans.Note that thetrue force f nfectionforhumans should be proportionalto (but less than) H,

because not all the available fleaswill successfully ind ndinfect human. As the human demographic parametersare dependent upon the particular community beingmodelled, throughout he rest f thispaper we concentrateon the behaviour of H and do not explicitlymodel thehuman population, thusgivingour resultsgreater gener-ality.The parametervalues used in the model are derivedwhereverpossible fromexperimentsor fieldobservations(table 1). Other parameters are set within biologicallyrealistic bounds and are checked for sensitivitywithrespect o the disease dynamics.

4. DETERMINISTIC RESULTS

Figure 2a shows how H varies with the number of freefleas, F, and the total ratpopulation TR. It is clear that

Proc.R. Soc. Lond. B (2000)







2222 M.J.Keelingand C. A. Gilligan Bubonic lague: metapopulationodel fa oonosis

(a) (b)

103210

. 10~

100 a1

80 -C o

~~Plhn0 < 10 15 2 2 0 5 4

20~~~~~~~~~~~~~~~~~~~~~~~60-

toa4rt302-5 20 2 0 5 4

POPulatioN0 50 0 time years)

Figure 2. Results fromthe deterministicmodel forbubonic plague. (a) The potential forceof nfection o humans, XH, given

F free-living nfected fleas and a total rat population of TR. XH increases linearly with the number of fleas, but decreasesexponentially with thenumber of rats. (b) The number of nfectiousrats (solid line) and the potential force of nfection ohumans XH (dashed line) for the deterministicmodel, startingwith a totally susceptible rat population. Note that after thefirstepidemic outbreak, which lasts about one year, the value ofXH never exceeds unity, suggesting that human cases would be veryrare. Parameter values are given in table 1.

the real danger to humans occurs when there are manyinfected leas, such thatthere s a large reservoir f infec-tion, but few rat hosts,so that fleas are forced to feed onhumans.This is the situation aftera major epidemic (ormore precisely, n epizootic) in the rat population. Fromthis simpleobservation, ne would expecthuman cases tobe maximized by short-lived iolentepizooticswith high

virulence. This should produce many fleas but fewsurviving ats.The deterministic ehaviour of the disease in rats and

the potential force of infection,H, is given in figure 2b,using the parameter values in table 1. From a totallysusceptible population, the disease in rats undergoesdamped oscillations to a fixedpoint.The numberof casesin humans,which should be proportional to the potentialforce of infection H, lags behind the rat cases by aboutfour weeks (this correspondswell with historical data;Curson & McCracken 1990). It should also be noticedthat the size of each human epidemic is damped fasterthan the epizootic in rats. This means only large

outbreaks n the rodent population,far from quilibrium,can cause a substantialnumber of cases in humans.

(a) Sensitivity toparameter valuesAlmost all the model parametershave been takenfrom

the iterature, nd have been determinedbyeitherexperi-ment or observation table 1); however,understanding hesensitivity f the model to each parameter enables us tounderstandthe roles ofgeographical or temporalhetero-geneityin parameter values. Figure 3 shows the sensi-tivityof the deterministicmodel to 11 parameters thatcharacterize the behaviour of rats and fleas; the para-meters controlling he human population can be ignored

by simplyconsideringthe potential force ofinfection,H,rather than modelling the human dynamics explicitly.Three model outputs are examined: the equilibriumnumberof rat cases, theperiod of thefirst pizootic cycle,and thepotentialforceof nfection uringthis cycle.

For each output variable V of the model and any para-meter P we define the sensitivity conventionallyasfollows:

log VP

P-*Olog

Po DV (6)V(P0) DP'

where P0 is the default value of the parameter and thepartialderivatives evaluatedat thispoint.From hisdefini-tion we see that the sensitivitymeasuresthe proportionalchange in the outputvariable, V,for small proportionalchange in theparameterP. When V is proportionalto theparameterP, the ensitivity isequal tounity.

We can reverse the definition 6) and examine therelative change to any output variable as we alter theparameters:

V(P)~~~PP) 1_V PO)(p)

It is clear that large values of the sensitivity lead tofaster than linear changes in V as the parameter alters,and this s undesirable.

From multiple simulations,we note that even whenparametersare changed by a factor of two the essentialpatternof sensitivity emains,showingthat S is a robustmeasure of the effects f parameterchange. Only KF, thecarryingcapacity of fleas per rat, has any effect hat ismuch stronger han linear, although rR, 1 - p, KR and a

all have effects n the numberof rat or human cases thatare close to linear (figure3). Out of thesefivemost sensi-tiveparameters,onlya and KR have not been determinedby experimentor observation (see table 1).While the ratcarryingcapacity,KR, is clearly going to vary between

Proc.R. Soc.Lond.B (2000)







Bubonicplague: metapopulationodel fazoonosis M.J. Keelingand C. A. Gilligan 2223

1.5 I I 1 1 1 1

U,,,

0.5

rR I-p dR MR gR PR KR rF dp KF aparameter

Figure . Sensitivityfthedeterministic odel to theratandfleaparameters. hreepropertiesf themodel are measured: henumber frat cases at equilibrium,R/mR (closed bars), thepotential orce f nfectionohumans verthefirstpizooticwave(grey ars), and theperiod fthefirst pizooticwave (open bars).

populations, it is probable that the flea searching effi-ciency, , will be universal. Therefore, n ?? 5-8, we shallalways considerwhetherour results re robust to changes

in the parameter a.

5. GREATER BIOLOGICALREALISM

Although the deterministicmodel possesses many ofthe short-term ynamicscharacteristic f bubonic plague,with biologically reasonable parameters only fixed-pointbehaviour is observed; this is contrary to the historicaldata. Greater biological realism is included in the modelby introducing temporal forcing, individuality andspatially distinct subpopulations. These three forms ofheterogeneity ead to more realistic patternsof humancases, and long-termglobal persistenceof the disease in

the rodents.The flea index is known to vary seasonally throughout

the year (Hirst 1938); this is modelled by seasonallyforcing he flea carrying apacity:

Kr = ir 1 + E%rsin2wseason)

where 0 <season < 1 measures the time of year. Thisseasonalityinvariably eads to annual cycles n the deter-ministicmodel, although these cycles are not sufficient(nor even necessary) to produce the violent epidemicsobserved (figurela). The seasonality in temperatureorhumidity, nd hence flea activity, s considered to be

responsiblefor hedecrease in the numberof cases duringthe winter months n Britain (Raggett 1982; Scott et al.1996) and during the dry season in India (Hirst 1938).We have therefore ncluded seasonality for the sake ofcompleteness,althoughextensiveexploratory imulations

(not given here) show that its influenceon the dynamicsis only observed at the short time-scale.At time-scales ofone year or more, the effects f the seasonal fluctuations

are averaged out.The dynamics are now made stochastic, uch that eachevent happens at random but with rates determined bythe underlying differential equations (2)-(4). In thisformulationwe no longer see fixed-pointbehaviour, butstochastically driven large-scale epidemics arise (seeBartlett 1956; Renshaw 1991). Stochastic models allow usto examine the question of disease persistence (Bartlett1957; Grenfell 1992; Keeling 1997), but the unrealisticcomplete mixing of infected fleas and susceptible ratsleads to rapid extinctions. o prevent this, heterogeneitiesmust be allowed to develop; hence the entire system sspatially discretized into a set of locally coupled sub-

populations (see Grenfell et al. 1995; Keeling 1997;Grenfell & Harwood 1997). Coupling between adjacentsubpopulations is modelled by the slow random move-ment of rats and free-livingfleas at rates PR and PF,

respectively.Figure 4 shows two basic statistics at the stochastic

subpopulation level startingwith various proportionsofthe rat population being susceptible, for different aluesof the searching efficiency . Figure 4a summarizes thesurvival times of the disease, starting with a singleinfectious rat; figure4b shows the potential number ofhuman cases: the qualitative behaviour of both quanti-ties is fairly robust to quite large changes in the

searching efficiency . The greatest threat to humansoccurs when the rat population is highly susceptible,and in such populations the disease occurs as a short-lived (one- to two-year) epizootic. When the level ofsusceptibles is lower (25-50%) human cases are rare,








2224 M. J.Keeling and C. A. Gilligan Bubonicplague:metapopulationodel fa zoonosis

1 (a)

0.9

0.8

0.7

0.6

0.5

0.4

0.3-

0.2-

0.1 -

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

initialproportion f susceptiblerats

Figure 4. Results fromthe stochasticmodel, fora rodent subpopulation with various proportionsofsusceptible rats. (a) The

probability that when the disease is introduced it persistsfor more than two or tenyears, i.e. whether t is a short-livedepidemic(< 2 years) or a persistent ndemic (> 10 years) (b) The scaled potential force of nfection, RHover the entireduration of the

epidemic (note that they-axis is a logarithmicscaling). The parameter a is included, because the likelihood ofa flea findinghumanhost hould ncreasenproportiono ts earching fficiency.crosses, = 0.001; asterisks, = 0.002;diamonds,a = 0.004;opencircles, = 0.008;opensquares, = 0.016.)

but the disease frequentlyenters an endemic, highlypersistent phase, surviving for about ten years. Asexpected, if the level of susceptibles is very low thedisease fails to invade because the effective eproductive

ratio is less than unity.These three simple observations,which are robust to parameter changes, are the key tothe success of the stochastic spatial model and allow uslater to form a simple cellular automaton approximationfor the spatial dynamics.

These stochastic dynamics, when placed in a meta-population framework, emonstrate high level ofpersis-tence and realistic global behaviour (Keeling & Gilligan2000). A population of60 000 rats (typicalof a medium-

sized town) is divided into 5 x 5 subpopulations withnearest-neighbouroupling governedby the rats and fleamovement atesofAR = 0.03 per yearand AF = 0.008 peryear, respectively figure ). At a fewof the sites hediseasepersistsat low levels for ong periods of time,and from

Proc.R. Soc. Lond.B (2000)







Buboniclague: metapopulationodelfa oonosisM.J. Keeling nd C. A. Gilligan 2225

(a) 104

1002

0 10 20 30 40 50 60 70 80 90 100(b) 1 1 l 1 I , , 1

102

0 10 20 30 40 50 60 70 80 90 100time years)

(c)

0=01

Figure 5. Time-series resultsfrom the stochasticmetapopulation model, comprising25 subpopulations in a 5 x 5 grid;nearest-neighbourouplings controlled v therat and fleamovement ates ZRand HZF.Allgraphs re from single imulationf100 years; the results are typical and are not particularlysensitiveto the exact parameter values. (a) The number of nfectious

rats, R(,on a log-scale; the initial low number of cases is due to transientdynamics. (b) The value ofAZH the forceof nfection o

humans), again on a log-scale. It is clear that while infectiousrats are always present,the force of nfection o humans is muchmore erratic.The bottom set of25 graphs (c) shows the number of nfectious rats in each of the subpopulations over 100 years;many subpopulations show long periods ofendemicity,which is i'nstark contrastto the epidemic behaviour.

these ources t spreads o other ubpopulationsreatingwaves f hort-livedpidemics.n this opulation f60 000rats, ubonic lague asily ersistsn the odent opulationfor ver100yearswithout heneedfor xternalmports fnew cases. However, he potential orce f nfection,H,

suggests hat occasional large-scale human outbreaksshould be observed bout once every ten years. Suchdynamics resentnew nterpretationn historicalata.

By far hevastmajorityfhistoricalnformationn thespread fbubonic lague s concerned ith henumber fhuman ases, and these utbreaks endto be short ived,even n large ommunitiesSharif 951; hrewsbury970;

Twigg 993).Untilnow the tandardssumptionasbeenthat ach human utbreak astriggeredysome xternalsource, for example, infectedrats arriving by ship(Appleby1980; Slack 1980). While this s undoubtedlytruefor manysmallpopulations, he model developed

hereoffersn alternativexplanation.n large owns ndcities t is likely hatthe plaguewas endemic n somesectionsof the rat population and this could triggersporadic pidemicsnother reas; such pattern fbeha-viour was speculated orbubonicplague n India duringthe early 20th century Sharif1951). his persistencenthe rat populationmay explainwhyhumanepidemicswere till xperienced,ven ncities uch s Venice,whenstringent uarantinemeasureswere in effect Appleby1980). To date,muchof the historicalnterpretationasconcentratedn human ases, gnoringhe true pizooticin rodents,nd thereforeeglectinghefull ynamics.

6. POTENTIALFOR MODERNHUMAN PIDEMICS

Estimates fthecurrentmortalityrom ubonic lagueare in the range of 1000-3000 deaths per year,with








2226 M.J.Keeling nd C. A. Gilligan Bubonic lague:a metapopulationodel fa zoonosis

sustainedhumancases in many urban areas, e.g. Mozam-bique (Barreto et al. 1994), Madagascar (Boisier et al.1997; Chanteau 1998) and Surat (Kumar 1995; Saxena &Verghese 1996)). According to World Health Organiza-tion reports, although human cases are rare in manyareas, bubonic plague is actually widespread throughoutthe wild rodent community. Examples of this situationare southernAfrica, the Middle East and the UnitedStates (Kimsey et al. 1985; Rosser 1987; Craven et al.1993). However, due to the large number of urbanrodents, the potential exists for much larger outbreakswithintownsor cities; hence is it important to considerthe risk fspreadto the human population.

In ??4 and 5, which were concerned with historicalcases, we were interestedn the persistenceof the diseasein communities lready infectedwith theplague. Here, incontrast,we wish to model the invasion into a disease-free modern urban population. Large-scale infection ofhumans depends on two distinct factors: the disease

spreading from the wild rural rodent population torodents n the cities,and then the chance of an outbreakin urban rodents ausing cases in humans.The first tageof thisprocess s difficult o predictand will depend on avariety of external factors. In particular, the mixingbetween urban and rural rodents will be influencedbyweather, oodavailability nd the habitatsurrounding hetown or city.For the second stage, the stochasticmodelalready developed can be used to estimate the totalpoten-tial force of infection over the entire rodent epizootic(AH = f2Hdt) once the disease has entered the urban ratpopulation.

As with all metapopulation models, there s some arbi-

trariness n the area of each subpopulation. Clearly thecarrying capacity (KR) of each subpopulation shouldincrease with the area. However, the rate at which freefleas find a rodent host should be proportional to thedensity f rats and not the absolute number. Therefore,changingthe area ofa subpopulationshould increase thecarrying capacity and decrease the searching rate suchthat the product aKR remains constant.This behaviouris clear from imulations (figure 6); it can also be seenthat a large human outbreak is possible whenever0.5 < aKR < 20. These limitscorrespondto the situationwhereAH t KR.This formwas chosen so thatAH scaleslinearlywith the area of a subpopulation and each rat

producesone infected flea which fails to find a suitablerodenthost and thereforemay bite humans. Because thebehaviour s solely dependent upon the productaKR andnot their ndividual values, it is clear that the model wehave developed is not dependent upon the scale of thesubpopulations. he upper limit (aKR = 20) ca-nonly beachieved by very high densities of rodents,so althoughabove this limit AH is less than KR, the potential formanyhuman cases is still high; we can therefore oncen-trate xclusivelyn the owerbound.

We note that aKR can be related to the basic repro-ductiveratio of thedisease, RO:

Ro= d [1 -exp(-aKR)].

From standard theory, at epizootics can occur wheneverRo0> 1, which corresponds to aKR > 0.39. This means

iT

10-4023102 103New ork)104

carryingapacity R

Figure . Contourplotof thepotential orce f nfectionohumans elative o thenumber frats AH/KR) against herat carrying apacity KR) and theflea earching fficiency

(a). The regionwhereAH > KR is shaded grey, nd isboundedbythinblack contours. he grey ontours re whenAHKR = 0.01, 0.1, 0.2 and 0.5,and show the teepnessf hetransition. he dashed ines orrespondoaKR= 0.5 andaKR= 20,which orrespond loselywith heAH = KR

contour. he cross ndicates he etofparameters sed ntheprevious imulations. n thex- andy-axes re given he esti-matedrodent ensityn NewYork, nd theplausible owerbound on the earching fficiencyat a scale of1km2.

that there is only a very small parameter regime(0.39 < aKR < 0.5) in which the disease can invade therodentpopulation, but the chance ofgettinghuman cases

is rare. If bubonic plague entersan urban rat populationthere s a strongpotentialforhuman infection.The invasion threshold for the disease (Ro > 1

? aKR > 0.39) allows us to estimate the crucial para-meter a, by examining the incidence of sylvatic plague(bubonic plague in wild populations). Epidemics ofbubonic plague have been recorded in a varietyof wildrodent populations, including chipmunks, prairie dogs,ground squirrels and mice (Barnes 1982; Menkens &Anderson1991;Davis 1999). By definition, heseoutbreakscan onlyoccur when Ro > 1; therefore, y looking at thedensityof populations which sustained a large outbreak,we can estimate a lower bound on a, although the true

value is probablymuchhigher.Table 2 givesthe estimateddensities and the calculated minimum value of a forseveral rodent species in North America that havesuffered lague outbreaks.

Even with the lowest values of a calculated, there s stillthepotentialfor an outbreak n modern urban cities. It iswidely believed that in large cities, such as New York,there is at least one rat forevery human being; whichputs the rodentdensityat around 4500km-t. With suchhigh densities of rodents, the arrival of the disease islikelyto trigger large epizootic with obvious problemsforhuman health.

7. A CELLULAR AUTOMATON PPROXIMATION

Taofurther est the robustness f our results,we intro-duce a cellular automaton approximation (Durrett &Levin 1994; Keeling 1999), which is derived from the








Buboniclague: metapopulationodel fa zoonosis M. J. Keeling and C. A. Gilligan 2227

Table 2. Estimates fpopulation ensityfromWalker& NVowak999) forfour species f wild rodent hat xperienceutbreaksof bubonic lague

(By assuming that at these densitiesRo? 1 a lower bound on the searching efficiency, , can be calculated.)

species density km2) calculated minimuma

easternAmerican chipmunk 500-1000 3.91 x IO-4to 7.82 x IO-'westernAmerican chipmunk around 1500 2.61 x 10-4

black-tailed prairie dog 640-1560 2.51 x 10-4 to 6.11 x 10-4

ground squirrels 400-2000 1.96 x 10-4 to 9.78x l0-4

local dynamics of the fully stochasticspatial model. Inkeeping with the approach of Hassell et al. (1991), wesuggest that if this much-reduced model can producequalitatively imilar dynamics, then these dynamics mustbe attributable o the underlying ehaviour and not speci-ficsof the modelling approach. The greater computationalspeed of this cellular automaton model also allows us toaddress questions of spatial clustering nd the effects fmovement ates n more detail.

As previously stated, four factors contribute to thestochastic dynamics of the disease within the spatialrodentmetapopulation:

(i) in the absence of the disease, the proportion ofsusceptible ats ncreases;

(ii) epidemics and endemics can infect neighbouringsusceptiblepopulations;

(iii) highly susceptible subpopulations give rise to short-lived epidemics;

(iv) moderately usceptible ubpopulationscan often ead

to endemicpersistence.

We let each site in the two-dimensionalcellular auto-maton be in one of three basic states, hort-lived pidemic(E), persistent ndemic (P) and uninfected r susceptible(S). The susceptible tate sfurther ivided ntoN substates;in the absence of infection ach site moves sequentiallythroughthe susceptible ubstates, romSo to SN-1, whereSN-1 correspondsto a fully usceptiblepopulation. Othertransitionrates are given in table 3 and can all be esti-mated from he stochasticbehaviourof a subpopulation.

The greater computational speed of the cellular auto-maton model allows analysis of replicated epidemics over

long times. Like the full stochastic metapopulationmodel, the cellular automaton also shows regions ofendemic behaviour (black sites) which give rise tosporadic waves of epidemics (dark grey sites) (figure 7a).It is clear from this snapshot that both endemic andepidemic sites show a high degree of clustering.Thepresence of endemic regions greatly increases the prob-ability of global persistence of the disease; withoutendemic regions (Q, = 0) we observe a single wavespreading throughthe population which rapidly extin-guishes tself.

Measuring the mean-variance relationship for avarietyof sampling windows allows a rigorous studyof

thespatial data from he cellular automatonmodel.Whenthe observed variance against the mean on a log-logscale, a power-lawrelationship s expected (Taylor 1961;Keeling etal. 1997; Keeling & Grenfell1999). It is clearthat for all window sizes the power is greaterthan unity

Table 3. Transitionrobabilitiessed nthe ellularutomaton

(T is the probability f an epizootictriggeringases in theneighbouringubpopulations,nd Q is the probability hatthese riggeredasesgiveriseto a persistentndemic.Finally,due to the reduced ases n theendemic hase, theprobabilityofthis riggeringeighbouringases s reducedbya factor .)

transition probability

S, --->,+ I for ll i < NE -->SO IP --iS.l dS, > E TO- Q1) if in neighbourhoodS, - E qT( - Ql) if inneighbourhoodS, - P TQ if in neighbourhoodS, - P qTQ, if inneighbourhood

(figure 7b), suggesting that epidemics are clustered at avariety of scales, and outbreaks are correlated across

large distances (up to 1000 lattice sites). In fact, forwindows argerthan 300 x 300 thepower-lawasymptotesto two,whichwould imply synchrony fepidemicsat thisscale.

Finally, we can use this cellular automaton model toconsidertheeffects fcouplingbetweenneighbouring ellsTi. The transmission robabilitiesTi are increasingfunc-tionsof theparametersAR and AF which control hespatialspread in the full model. As shown in figure7c, there sclearlya threshold evel ofcoupling (at ca. TN1 = 0 75)below which the disease fails to spread. As the couplingincreaseswe find hat there s a monotonic ncrease in thenumber of endemic sites, although the number of

epidemic sites reaches a maximum when the coupling isaround TN--1 0.55. It would thereforeappear thatdrastically reducing the spatial spread of the rats andfleasmaybe an effectivemeans ofcontrolling he disease.

8. CONCLUSIONS

Besides being a disease of extreme historical impor-tance, bubonic plague is still prevalent today.Thereforean accurate predictivemodel has strong public healthapplicationsand also helpsto explain historicaldata.

Bubonic plague is amongst a handful of zoonoses(animal-based diseases) that have captured the public's

imagination.This may in part be due to the seeminglyspontaneousoccurrence of these diseases in human popu-lations, and their high mortalityrates. Here we haveshown that a mathematical understandingof the risksfromsuch diseases requires models for both the human

Proc.R. Soc.Lond.B (2000)







2228 M.J.Keeling nd C. A. Gilligan Buboniclague: metapopulationodelfa oonosis

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and animal populations.We note,however, hatbubonicplague s quite exceptional mongst oonoses ecause therisk o humans s not implymaximized y highnumberof animal cases, butby a combination fmany ases andhighmortality.herefore,lthough hemethodology aybe applicable o many ther iseases,we believe hat he

precise esults illbe highly pecific obubonic lague.A careful consideration f the biologyof bubonicplague has led us to reject he standard IR model, nddevelopa more complex et ofequationswhich ncludethe dynamics f the rat, flea and human populations.Although his etof equations ontains largenumber fparameters,many have already been estimatedfromlaboratory xperiments nd field observations Bacot1915;Hirst1938;Wheeler& Douglas 1945; Macchiavello1954; Hinnebusch t al. 1998). By considering H, thepotential orce f nfectionohumans, he three ifferen-tialequations nd five arameters hich elate o humanpopulationsan be subsumednto his ritical arameter,

which considerably implifies he model. Sensitivityanalysis hows hat hree undamental odeloutput ari-ables (the equilibrium umber f rat cases, the periodand the potential force of infection uring the firstepizootic cycle) are only weakly dependentupon themajority f the parameter alues. Numerous imulationsof the full stochasticmodel show that the qualitativebehaviours maintained venwhenparameter alues arechangedby a factor f two. The results f the cellularautomatonmodel add extra credence o this argument,showing hat t s thegeneral orm f the ocal dynamicsratherhantheprecise etails f the model hatproducesthe mergent etapopulationehaviour.

The set of fivedifferentialquations hatdescribe hedynamics f the rat and fleapopulations isplay ampedoscillations endingto a fixed point; this is clearlycontraryto observations. o capture the observeddynamicswe need to make the model stochastic ndspatial.By ntroducingtochasticity,hefixed ointbeha-viour s destroyednd twocontrastingets ofdynamicsare observedn the ratpopulation:persistentndemicsand short-livedpizootics. he spatialheterogeneitiesnthe ratmetapopulation eanthat oth ypes fdynamicscan be simultaneouslyresent; he short-livedpizooticsdrive human cases, whereas the endemicpopulationsallow the disease to persist.Our models suggest hat

relatively mall rodent populationscan sustain thedisease, and hence quarantinemeasureswill onlybeeffectiven the mallest ommunities.

By concentratingn the stochastic ynamics t thesubpopulationevel, he risk fa severehumanoutbreakcan be estimated nd related to Ro. By examiningoutbreaks n otherrodent pecies,we have been able tocalculate lowerbounds on the searching fficiency.Althoughuchestimates ill be imprecise,hedensity furban rats s so muchgreater han the density f wildrodents hatwe can be confidenthat fthe disease wereto reach an urbanpopulation n epizooticwouldoccur.Wenote hat hevalue of a usedthroughouthispaper s

for some arbitrary ubpopulation area; it is the combinedvalue of ilCR that is dynamically important, and this isindependentof the area.

To estimate the riskof a human epidemicwould requirean accurate parameterization of the chance that an

Proc. . Soc. ond. (2000)







Bubonic lague: metapopulationodelfa oonosis M. J. Keeling and C. A. Gilligan 2229

infected lea (in the absence of a suitable rodenthost) canfind nd infect human. Such detailed data will be highlydependentupon living conditions, anitation nd buildingdesign.We believe that although the riskof the diseasemoving fromrural to urban rodents s probably ow, oncethis step has been made the risk and consequences to

humans are sufficientlyarge to warrant oncern.Given that human cases of bubonic plague are adistinctpossibility,t is vital that we consider preventionmeasures. Surprisingly, implementing rodent controlmeasures soon afterhuman cases have been observed is avery dangerous policy. Human cases only occur whenthere is a large reservoirof infectious fleas; thereforereducing the numberof rats at this time would lead to alarger total force of infectionfor humans than if theepidemic in rodents were leftto run its natural course(Keeling & Gilligan 2000). The only potential methodsof preventinghuman outbreaks of the plague are toreduce the numberofratfleasusing insecticides Maupin

et al. 1991;Beard etal. 1992), vaccinate the rodentsor tokeep therat numbers ow enough (Buckle & Smith1994) sothat hepotentialforce f nfection ohumans snegligible.

This work was fundedbyThe Royal Society (MJ.K. andC.A.G.) and the LeverhulmeTrust (C.A.G.) and BBSRC(C.A.G.), and arose from he King'sCollege ResearchCentreproject on spatially extendeddynamics.We are grateful o manypeople fortheir help and insights, ncluding Steve Betts, BryanGrenfell,PeterJones, Kevin MacCracken, David MacDonald,Fred Nichols, Jim Oeppen, Richard Smith, Jonathan Swintonand the two anonymousreferees.

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