Analytic approximation of spatial epidemic models of foot and mouth disease

Theoretical Population Biology 73 (2008) 349–368www.elsevier.com/locate/tpb

Analytic approximation of spatial epidemic models offoot and mouth disease

Paul E. Parham∗, Brajendra K. Singh, Neil M. Ferguson

MRC Centre for Outbreak Analysis and Modelling, Department of Infectious Disease Epidemiology, Imperial College London, Norfolk Place,London W2 1PG, UK

Received 13 September 2007Available online 19 January 2008

Abstract

The effect of spatial heterogeneity in epidemic models has improved with computational advances, yet far less progress has been madein developing analytical tools for understanding such systems. Here, we develop two classes of second-order moment closure methods forapproximating the dynamics of a stochastic spatial model of the spread of foot and mouth disease. We consider the performance of such ‘pseudo-spatial’ models as a function of R0, the locality in disease transmission, farm distribution and geographically-targeted control when an arbitrarynumber of spatial kernels are incorporated. One advantage of mapping complex spatial models onto simpler deterministic approximations lies inthe ability to potentially obtain a better analytical understanding of disease dynamics and the effects of control. We exploit this tractability byderiving analytical results in the invasion stages of an FMD outbreak, highlighting key principles underlying epidemic spread on contact networksand the effect of spatial correlations.c© 2008 Elsevier Inc. All rights reserved.

Keywords: Epidemic modelling; Spatial models; Moment closure; Pair approximation; Foot and mouth; Disease control

1. Introduction

The 2001 UK epidemic of foot and mouth disease (FMD)represents an early example in the use of mathematicalmodelling to inform policy decisions in a livestock diseaseoutbreak. FMD is a highly-transmissible viral infection thatmay be transmitted via direct and indirect contact; examplesinclude nose-to-nose contact, infection via intermediate vectorssuch as farm vehicles, people or other animals and excretionor exhalation from infectious animals. Aerosol emissionmay contribute significantly to infection dynamics dependenton environmental conditions such as the weather, soilpH and relative humidity (Bachrach et al., 1957; Cottral,1969; Donaldson and Ferris, 1975; Oliveira et al., 1999;Stark, 1999; Turner et al., 2000). Frequently evident inmany FMD outbreaks are a range of characteristic spatialscales in disease transmission, potentially varying betweendirect contact (e.g. nose-to-nose) to long-distance aerosol

∗ Corresponding author.E-mail address: [email protected] (P.E. Parham).

0040-5809/$ - see front matter c© 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.tpb.2007.12.010

transmission (Gloster, 1982; Barnett and Cox, 1999; The RoyalSociety, 2002). Nonetheless, data on the 2001 UK outbreak andthe 1967/68 epidemic suggest that the majority of transmissionwas ‘local’, defined as an infection event within 3 km of itssource and where more than one possible transmission routemay be identified (around 80% and 90% respectively (Gibbenset al., 2001)). From a modelling perspective, it is evident thatmean-field approaches (based on a fixed force of infectionbetween all farm pairs) are an over-simplification; the stronglocality in transmission and the small, but non-zero probabilityof infection over significant distances should be accounted forin realistic models of spatiotemporal spread.

Extensive reviews of the 2001 UK epidemic have beenpublished elsewhere (Kao, 2002; Green and Medley, 2002;The Royal Society, 2002; Taylor, 2003; Keeling, 2005)and significant insight obtained from mathematical models,ranging from detailed individual-based stochastic spatialmicrosimulations (Morris et al., 2001) to more parsimoniousindividual-based models (Keeling et al., 2001, 2003) toconceptually more complex (but mathematically simpler)approaches (Ferguson et al., 2001a,b). An important attributeof mathematical models is the ability to simulate ‘what-if’

350 P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368

scenarios and the potential impact of intervention strategies tocontrol disease spread. It is vital that a model has the abilityto reliably evaluate the relative merits of potential controlpolicies, which in the context of FMD may involve cullingand/or vaccination strategies. In reality, control efforts may betargeted to geographic areas worse affected or at greatest risk ofsecondary infection and thus realistic and useful mathematicalmodels should be able to readily evaluate the effect of spatially-targeted strategies such as ring culling or ring vaccinationaround infection hotspots.

Complex, individual-based stochastic spatial microsimula-tions have provided much insight into many aspects of FMDdynamics, both qualitative and quantitative. Yet this also high-lights an important area where less progress has been madesince 2001, namely the need for ‘pseudo-spatial’ epidemicmodels, interpolating between tractable mean-field approachesand complex spatially-explicit stochastic models. A broad aimof this work is therefore to develop simpler models thatclosely resemble the dynamics of complex spatially-explicit ap-proaches, yet may also be amenable to obtaining an analyticalunderstanding into spatiotemporal dynamics.

Parham and Ferguson (2006) consider this problem bydeveloping two classes of moment closure methods toapproximate a fully spatial model, demonstrating a clearrelationship between the two for a simple SIR model. Inthe context of FMD, two distinct classes of moment closureformulations are possible, namely a spatially-explicit statisticalensemble approximation to the full model in continuous spaceand a network-based approach where the network structureis defined from the underlying spatial model and epidemicprocesses imposed. The latter class of spatial networks connectnodes with a probability according to their Euclidean separationand this approach has been considered elsewhere (Eames andKeeling, 2002; Read and Keeling, 2003; Keeling, 2005).

The ability to include neighbourhood-based control in mod-els of FMD has been considered in spatial microsimulations,but to date has not been considered in moment closure mod-els and this is crucial if such approximations are to be useful.Parham and Ferguson (2006) provide a useful means of relatingthe full model and both classes of closure approximations whenwe consider only one spatial process, but it is crucial to knowhow to generalise this for an arbitrary number of spatial pro-cesses, some or all of which may be spatially-localised. We aimhere to address this generalisation, but also make more explicitthe advantages of developing such an approach. To this end,three general objectives of this paper emerge: (1) to consider theextent to which continuous-space and pair approximation meth-ods on a network may approximate stochastic spatial epidemicmodels, (2) to consider the relative merits of spatial momentclosure methods versus pair-based approaches on contact net-works, and (3) to consider the extent to which one may obtainanalytical insight into the models and spatial epidemic modelsin general. The methods will be kept as generic as possible tohighlight general principles of interest, although particular rel-evance to FMD is considered throughout.

2. A between-farm model for FMD

2.1. The stochastic model

In line with most other mathematical models of FMD,we consider the farm to be the basic unit of interest andconsider between-farm dynamics rather than transmissionbetween individual animals. From a modelling perspective,this is justified from the resultant ability to consider a closedpopulation model (the addition and removal of farms almostalways takes place on far longer timescales than typical FMDoutbreaks) and the rapid within-farm spread caused by thehighly-transmissible nature of the virus. In the latter case,it is therefore generally more prudent to consider control atthe level of farms (i.e. culling or vaccinating farms) ratherthan individual animals to get ahead of the population-levelepidemic curve. Implicit in this assumption is the neglect offarm-level risk factors such as farm susceptibility and relativeinfectiousness (according to the animal species, mix and spatialextent of the farm for example). The inclusion of within-farmdynamics in a between-farm model represents an ongoing areaof research, both from a theoretical and applied perspective.

We develop an SEIR model here, where S, E, I and Rrepresent susceptible, exposed (but not infectious), infectiousand removed farms respectively. Disease transmission isparameterised by a contact rate between susceptible andinfectious farms β and a spatial infection kernel U (di j ) tocapture the decreasing probability of disease transmission withincreasing distance di j between farms i and j . In a continuous-space model, the total probability per unit time λInf(x, t) thata susceptible farm at location x becomes infected due to thepresence of all infectious farms in an area Ω is simply

λInf(x, t) = β

∫Ω

U (x − y)I (y) dy, (1)

where I (y) is the probability that a farm centred at y isinfectious at time t . We normalise U (di j ) such that for any farmin a population of size N∫

ΩU (r) dr = 1, (2)

where r = di j = |x − y| and dr = rdrdθ . Exposed farmsare assumed to become infectious with a fixed probability perunit time λExp(x, t) = ν and infectious farms removed fromthe population through natural loss of infectiousness at rateλNat(x, t) = σ . In reality, surveillance systems nearly alwaysresult in infectious premises (IPs) being removed by cullingbefore reaching the end of their natural infectious period andwe assume that this occurs at rate λIP(x, t) = αIP, a singlerate incorporating the probability of being culled per IP and theefficiency with which this process is carried out.

Two other control policies were implemented in 2001.Dangerous contact (DC) culling removes non-infectious farmssuspected of incubating FMD through direct or indirect contactwith one or more IP: examples include milk tankers, inspectorsor deliverymen travelling from farm to farm. A contiguouspremise (CP) cull removes farms neighbouring IPs based

P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368 351

on accurate adjacency data, geographic distance or someother geometrical consideration such as constructing Voronoidiagrams (Barber et al., 1996). Here, we model the process offarm removal due to geographic proximity to IPs by introducinga kernel M(di j ) to quantify the probability of removal for a non-infectious farm a distance di j from an IP, also normalised suchthat∫

ΩM(r) dr = 1 (3)

and we assume that this removal targets susceptible andexposed farms alike (infectious farms are already culled underIP culling). If this process occurs at rate αCP, the probabilitythat a susceptible or exposed farm is removed by this process issimply

λCP(x, t) = αCP

∫Ω

M(x − y)I (y) dy. (4)

Radial culling may be modelled in this approach by settingM(r) = M(r) as a step-function with critical distance Rc, theradius of the ring around each IP.

Whilst DC culling may be similarly modelled by introducinga function G(r) to quantify the probability that a susceptibleor exposed farm has already contacted an IP, this is merelya trivial extension of the CP approach above. In reality, DCsare identified by expert opinion on the ground and since thisinclusion only serves to complicate the methodology withoutoffering new insights, we focus hereafter on the model withIP and radial culling only (where we refer to radial or ringculling here, rather than true CP culling). Note, however, thatDC culling via the inclusion of an additional kernel or indeedthe inclusion of vaccination (spatially-targeted or otherwise)may be readily included in this framework at the cost onlyof introducing further complexity into the moment closureequations developed shortly. Future work may consider howa full range of policies may be realistically included in suchmodels, what insight may be obtained and how these insightscompare with full microsimulations.

In the context of approximating the ensemble behaviourof the full model, two classes of moment closure approachesmay be identified, namely a spatially-explicit continuous-space approach and a pair approximation method consideringepidemic processes imposed on a well-defined contact network.We develop each of these in turn.

2.2. Spatial moment closure

Spatial moment equations have been largely consideredin the plant disease, rather than livestock disease, literatureto date (Bolker and Pacala, 1997; Bolker, 1999; Bolker andPacala, 1999; Dieckmann and Law, 2000; Law and Dieckmann,2000; Bolker et al., 2000; Bolker, 2003; Brown and Bolker,2004). One derives a deterministic approximation to theensemble dynamics of the full model by tracking low-orderspatial statistics over time, thereby improving on mean-fieldapproaches by explicitly capturing farm correlations due tospatial processes in the underlying model. We make the usual

assumptions of spatial homogeneity and isotropy to simplifythe analysis and avoid the need to account for individual farmlocations.

Let us first distinguish between two types of average of afunction f = f (x, t). Let the ensemble average 〈 f (x, t)〉 bethe average of the function over all stochastic realisations of thefull model. For a model including a single spatial probabilitydistribution p(x), let us also define the spatial average f (x, t)as the average of f (x, t) weighted by p(x) summed over alllocations x in Ω , i.e.

f = f (x, t) = f (t) =

∫x∈Ω

p(x) f (x, t) dx, (5)

where the extension to more than one spatial argument of thefunction is trivial. For models incorporating more than onespatial kernel, one may generalise (5) such that the total spatialaverage with i kernels is given by

f = f (x, y, t)

=

∫x,y∈Ω

Ni∑

i=1Wi pi (x, y)

Ni∑i=1

Wi

f (x, y, t) dxdy, (6)

where pi (x, y) are the Ni spatial distributions with associatedweighting factors Wi assigning how much each kernelcontributes to the global average. We associate the weightingfactors Wi with the temporal rate or frequency of each event.

To develop the simplest moment closure model beyondthe mean-field approach, we adopt a second-order approachand track the mean densities X = X(t) (the fraction of thepopulation in state X averaged over all space) and covariancedensities where

cXY (x, y, t) = 〈(X (x)− 〈X (x)〉) (Y (y)− 〈Y (y)〉)〉

= 〈X (x)Y (y)〉 − 〈X (x)〉〈Y (y)〉. (7)

To derive the mean-density equations, we first write down theprobabilistic equations corresponding to the full model,

dS(x, t)

dt= −λInf(x, t)S(x, t)− λCP(x, t)S(x, t),

dE(x, t)

dt= λInf(x, t)S(x, t)− λExp(x, t)E(x, t)

− λCP(x, t)E(x, t),

dI (x, t)

dt= λExp(x, t)E(x, t)− λNat(x, t)I (x, t)

− λIP(x, t)I (x, t). (8)

One can then demonstrate (Appendix A.1) that substituting theevent probabilities from Section 2.1 into (8), realising from (6)that

X(t) = X (x, t) =

∫x∈Ω

X (x, t) dx (9)

since U (di j ) and M(di j ) are normalised from (2) and(3) respectively and taking the ensemble average over all

352 P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368

realisations gives the mean density equations

dS(t)

dt= −β

(S I +

∫U (r)cSI (r, t) dr

)−αCP

(S I +

∫M(r)cSI (r, t) dr

),

dE(t)

dt= β

(S I +

∫U (r)cSI (r, t) dr

)−αCP

(E I +

∫M(r)cE I (r, t) dr

)− νE,

dI (t)

dt= νE − (σ + αIP)I . (10)

Deriving the covariance equations requires a means ofapproximating triple expectations 〈X (x)Y (y)Z(z)〉 in terms ofsingle and joint expectations. Whilst one could write downan exact expression in terms of quadruple and higher-orderexpectations, employing spatial moment closure approachesbeyond second-order should be considered carefully and itis doubtful of their value when second-order approachesalready reside on the edge of tractability. The choiceof closure is often somewhat arbitrary and developing abetter theoretical and applied understanding of applyingsuch closures represents an important area for futureresearch.

Current moment closures vary considerably in theirsimplicity and robustness, with the latter often heavilydependent on the parameter regime; R0, the locality of theinfection process and initial disease prevalence may all affectthe choice of closure (Bolker, 1999; Filipe et al., 2004; Parham,2006). In contrast to other studies employing ‘power-1’ closure(formally obtained by setting all third-order cumulants tozero), we adopt ‘power-3’ closure. It is easily verified thatthe dynamics of spatial moment closure models under power-1, power-2 and power-3 closures differ by very little in manyparameter regimes (particularly close to mass-action), butpower-3 closure provides a superior robustness at low R0,more local transmission and for a smaller initial prevalence.Appendix A.1 derives the covariance equations under power-1 closure

∂cSS(r, t)

∂t= −2β

(I cSS + S(U ∗ cSI )

)− 2αCP

(I cSS + S(M ∗ cSI )

),

∂cSE (r, t)

∂t= −β

(S(U ∗ cE I )+ I cSE − S(U ∗ cSI )

− I cSS)− αCP

(S(M ∗ cE I )+ 2I cSE

+ E(M ∗ cSI ))− νcSE ,

∂cSI (r, t)

∂t= −β

(S(U ∗ cI I )+ S IU + I cSI

)−αCP

(S(M ∗ cI I )+ S I M + I cSI

)+ νcSE − (σ + αIP)cSI ,

∂cE E (r, t)

∂t= 2β

(I cSE + S(U ∗ cE I )

)− 2αCP

(I cE E + E(M ∗ cE I )

),

∂cE I (r, t)

∂t= β

(I cSI + S(U ∗ cI I )+ S IU

)−αCP

(I cE I + E(M ∗ cI I )+ E I M

)− ν(cE I − cE E )− (σ + αIP)cE I ,

∂cI I (r, t)

∂t= 2 (νcE I − (σ + αIP)cI I ) , (11)

whilst under power-3 closure, the covariance equationsadditionally incorporate higher-order terms too unwieldy tonote here (but considered in Appendix A.1). In regimes closeto mass-action, power-1 terms dominate the dynamics toa very good approximation, whilst more local transmissionresults in power-3 terms increasing in magnitude, leadingto both significant departure from power-1 dynamics (theassumption of vanishing third-order cumulants becomes poor)and the power-3 system remaining a robust approximationwhilst its power-1 counterpart breaks down. Power-3 closureis motivated here by the assumption of FMD surveillance suchthat modelling an outbreak will correspond to a very smallnumber of initially exposed and infectious farms.

2.3. Pair approximation on a network

Whilst one could develop a dynamic network modelanalogous to the full model in Section 2.1, we consider astatic contact network where links between farms are fixed intime and the average number of contacts per farm is decidedprior to the epidemic. We consider in Section 3.2 how thenetwork structure may be defined from the full spatial model.We consider a regular, fixed-edge network model where allfarms have the same average number of contacts n (so that thedegree distribution, the probability that a randomly chosen farmhas k links, is simply a delta function at k = n) and the infectionhazard between pairs of farms is identical for all pairs in thenetwork.

We consider the network parameterised such that thetransmission rate across a link is τ , exposed farms becomeinfectious at rate ν, infectious farms are removed through anatural loss of infectiousness at rate σ , IP culling occurs at rateµIP and susceptible and exposed farms are radially culled at rateµCP. The deterministic ensemble dynamics are derived fromthe master equation

dg(t)

dt=

∑ε

r(ε)1g(ε), (12)

where g(t) is the state variable of interest (usually an averagequantity or expectation), r(ε) is the rate of the event ε and1g(ε) the change this event causes in g(t) (Rand, 1999).Adopting the usual notation that [X ], [XY ] and [XY Z ] denotethe number of farms in state X , X–Y pairs and X–Y –Z triplesrespectively, it is readily shown that the network equations aregiven by

d[S]

dt= −(τ + µCP)[SI ],

d[E]

dt= τ [SI ] − ν[E] − µCP[E I ],

P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368 353

d[I ]

dt= ν[E] − (σ + µIP)[I ],

d[SS]

dt= −2(τ + µCP)[SSI ],

d[SE]

dt= τ ([SSI ] − [E SI ])− ν[SE]

−µCP ([SE I ] + [E SI ]) ,

d[SI ]

dt= −(τ + µCP) ([I S I ] + [SI ])+ ν[SE]

− (σ + µIP)[SI ],

d[E E]

dt= 2τ [E SI ] − 2µCP[E E I ] − 2ν[E E],

d[E I ]

dt= τ ([I S I ] + [SI ])− µCP ([I E I ] + [E I ])

+ ν ([E E] − [E I ])− (σ + µIP)[E I ],

d[I I ]

dt= 2ν[E I ] − 2(σ + µIP)[I I ], (13)

which is similar, but not identical, to the pair correlationmodel developed in 2001 (in the absence of vaccination andwhere all contacts are local) (Ferguson et al., 2001a,b). Thiswork and that of Parham and Ferguson (2006) considersthe correspondence of spatial and network approaches undermoment closure approximations to both, although a moreformal correspondence may be obtained by mapping (13) to theensemble behaviour of the spatial model prior to approximatingtriple expectations.

An important distinction between the spatial and networkapproaches lies in the means by which the correlations areincluded. In contrast to the additive approach in Section 2.2,correlations are defined multiplicatively on the network bydefining the correlation CXY between X–Y pairs as

CXY =N [XY ]

n[X ][Y ], (14)

where CXY = 1 corresponds to mass-action mixing, CXY > 1aggregation and CXY < 1 avoidance of farm types X and Y.In fact, the mapping in Section 3.3 illustrates how cXY (in thespatial model) and CXY from (14) are related and both areformally equivalent to Lloyd’s index of patchiness P (Lloyd,1967) quantifying the deviation from spatial randomnessfor a given spatial point process distribution. Multiplicativemoments often provide a more robust means of capturingcorrelation behaviour than additive moments, with the problemof negative populations avoided more often (among otherbenefits) (Keeling, 2000).

The pair correlation model is obtained by approximating thenumber of triples [XY Z ] in terms of only singles and pairs.As with spatial moment closure, a number of approximationsmay be developed, ranging in simplicity and formal motivation.Common closures include assuming a Poisson distribution forthe number of neighbours per farm, a multinomial distributionor a heuristic combination of the two (although other pairapproaches have also been developed, e.g. see Ellner et al.(1998) and Filipe et al. (2004) amongst others) and it is thelatter which is adopted here, consistent with epidemic modelselsewhere (Keeling et al., 1997; Rand, 1999). If we let the

transitivity φ represent the fraction of a farm’s contacts that arethemselves contacts of each other (the proportion of triples thatform triangles), we adopt the triple closure

[XY Z ] =

(n − 1

n

)[XY ][Y Z ]

[Y ]

((1 − φ)+ φ

N

n

[X Z ]

[X ][Z ]

)(15)

to close (13) at the level of pairs and where we assume that φencodes all information about higher-order structure.

3. Relationship between the models

3.1. Motivation

We have already seen how deterministic approximations tothe ensemble behaviour of stochastic spatial models may bederived in the form of spatial moment equations and we nowconsider the extent to which one may develop a spatially-implicit network approach that closely mimics the dynamics ofthe full model. There are a number of reasons for motivatingthis mapping:

• The dynamical equations are considerably simpler than boththe full model and spatial moment closure approaches, withstate variables in the latter being described by integro-differential equations compared to simpler ODEs describingthe evolution of the epidemic on a network.

• Pair approximations defined from the underlying model mayperform at least as well in capturing epidemic dynamicsas more complex spatial approaches, but retain the linkwith geographic space without the considerable complexityusually associated with its inclusion.

• Network moment closure approximations may be morerobust (in terms of breaking down when disease transmissionis very local, one has a small number of initially infectiousfarms or R0 is small) than their spatial counterparts (Parham,2006).

• The simpler mathematical form of the model is far moretractable, allowing a better analytical understanding ofspatiotemporal dynamics.

• Network moment closures may be easier to physicallymotivate from first principles than comparable spatialapproaches. Interpretation of even spatial power-1 closureis still poorly understood (and with power-2 and power-3 closures even less so), whilst some motivation exists inpair approximations by making an assumption about thedistribution of farms’ neighbours.

• The ODE description means that they are computationallyquicker, enabling quicker feedback to policy-makers in real-time scenarios.

Given that spatial moment closure approximations arederived directly from the full model, two important stepsremain to relate the three models, namely deriving quantitiesin the network model from the full model and considering howcomplex spatial moment closure models may be mapped ontothe pair approach. We consider each of these in turn.

354 P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368

3.2. The network model from first principles

Parham and Ferguson (2006) demonstrate that a normalisedspatial infection kernel U (r) allows a contact network offarms to be defined, where farms i and j are linked withprobability qi j = nU (x j − yi )/(N − 1). A key differencebetween a network approach and a fully spatial model lies inthe underlying contact process, namely that in the former case,there is a finite maximum on the number of farms that maybe contacted per IP in each time step; in the spatial approach,all (N − 1) farms may be contacted. This saturation propertyallows Parham and Ferguson (2006) to demonstrate that whenone has only one non-uniform spatial process and farms arelinked with probability nU (x j − yi ), the average number ofcontacts per farm n may be derived from U (r) as

1n

=1

Au=

∫Ω

U (r)2 dr, (16)

for unit uniform density ρ and where Au is the effective uniformneighbourhood size associated with the non-uniform kernelU (r) (also derived via a more heuristic approach in Bolker(1999); see also Wright (1946)). Generalising for ρ 6= 1 merelymultiplies the RHS of (16) by ρ.

The question then arises as to how this may be generalisedwhen the model contains an arbitrary number of spatialfunctions. If we let K (r) be the effective contact kernel in thiscase, it is clear that K (r) should be an intuitive extension fromthe one-kernel case, be normalised such that∫

ΩK (r) dr = 1 (17)

and account for the likelihood of each spatial event. To this endand incorporating the definition of spatial averages in systemswith more than one kernel (6), we define K (r) as

K (x, y) =

Ni∑

i=1Wi pi (x, y)

Ni∑i=1

Wi

, (18)

resulting in the effective kernel as a weighted average of theinfection and removal kernels here such that K (r) = (βU (r)+αCP M(r))/(β + αCP), readily shown to obey (17) via thenormalisation conditions (2) and (3). If we now link (i, j) pairswith probability nK (x j − yi ), we thus define by analogy theaverage number of neighbours per farm as

1n

=1

AK=

∫Ω

K (r)2 dr, (19)

(assuming unit density), whilst the transitivity of the network φis defined by analogy to Parham and Ferguson (2006) as

φ = 2πn∫∫∫

K (R)K (r ′)K (r) r ′dr ′rdrdθ, (20)

where R = |r − r′|.

Finally, by linking farms with probability qi j = nK (x j −

yi ), it is also clear by analogy that the equivalent network

infection process is defined by the constant hazard of infectionbetween connected farms τ = (β/n), whilst the removalprocess is similarly defined by µ = (αCP/n). Parham andFerguson (2006) suggest that this representation should providea good approximation when there is only limited infection ofsusceptible farms in the neighbourhood of IPs (R0 n), aswell as for increasingly large neighbourhood sizes (less localtransmission) when the dynamics of both models tend to mean-field behaviour.

3.3. Mapping under moment closure

The remaining issue is how to relate these spatial momentclosure models and pair correlation approaches. Let us followthe approach of Parham and Ferguson (2006) by firstly reducingthe spatial moment equations under power-1 closure (11) to aset of ODEs by averaging over all space. To aid this process, wedefine

c(p)XY (t) =

∫p(r)cXY (r, t) dr (21)

as the average of cXY (r, t) against a kernel p(r). Let usalso generalise the convolution approximation in Parham andFerguson (2006) such that

(p ∗ c)(r, t) ≈ kp(r, t)cXY (r, t), (22)

where we interpret kp(r, t) to be related to the conditionalexpectation of the event represented by p(r, t) occurringbetween a susceptible farm at position vector r and an X–Y pair(where farm X is located at the origin and Y at position vectorr′). One can calculate kp(r, t) from (22) by first assumingseparable covariances (so that cXY (r, t) = RXY (r)TXY (t),which removes the time-dependence of kp) and then averagingover all space (with the objective of calculating kp fromknowledge of p(r) alone; see Parham and Ferguson (2006)).We interpret kp as a clustering or connectivity parameterassociated with linking farms according to p(r), reflecting thespatial relationship between triples and pairs.

One can then reduce the integro-PDEs (11) to a set ofODEs if we also define by extension to (16) the overlapneighbourhood size Apq as

1Apq

=

∫p(r)q(r) dr (23)

for a system incorporating kernels p(r) and q(r). Taking ∂/∂tof (21) and using (16), (22) and (23) enables us to derive ODEscorresponding to the power-1 model (11) averaged over U (r)and M(r) separately (Appendix A.2).

Let us now define the total average covariance cXY (t) asthe weighted average of cXY (r, t) against both U (r) and M(r),which from (6) is simply

cXY (t) =

(βc(u)XY + αCPc(M)XY

β + αCP

), (24)

equivalently obtained by substituting the effective kernel K (r)from Section 3.2 into (21). Taking ∂/∂t of (24) and substituting

P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368 355

for the individual averages from Appendix A.2 allows us toderive ODEs for the total average covariances, which for thecovariance between susceptible and infectious farms gives

dcSI

dt= −(β + αCP)

((βkU + αCPkM

β + αCP

)ScI I + I cSI

+

β(β

AU+

αCPAU M

)+ αCP

(β

AU M+

αCPAM

)(β + αCP)2

S I

+ νcSE − (σ + αIP)cSI , (25)

where the new parameters k and A may both be calculated fromfirst principles.

Let us now consider how to map between the continuous-space model (11) and the pair approximation (13). Parham andFerguson (2006) suggest from dimensional considerations thata mapping may be obtained a priori in the absence of radialculling by [X ] ≡ Au X where Au is the effective neighbourhoodsize created by linking farms with a probability proportional toU (r). Let us therefore postulate the analogous mapping here

[X ] ≡ AX , (26)

where A is now the effective neighbourhood size (yetto be defined) when one includes more than one spatialkernel. Appendix A.3 demonstrates that this enables one todemonstrate that (26), together with the analogous secondmoment counterpart

[XY ] ≡ A2 XY ≡ A2(X Y + cXY ), (27)

leads to the simple conclusion that τ = (β/A) and µCP =

(αCP/A) (as well as trivially mapping the exposed andinfectious periods) as expected from our earlier first principlesarguments (with n = A).

One can now use (26) and (27), together with the parametermappings, to consider the similarity in the dynamics of (11) and(13). Appendix A.3 demonstrates that the pair Eq. (13) implythat

dcSI

dt≈ −(β + αCP)

(φScI I + I cSI +

S I

A

)+ νcSE − (σ + µIP)cSI (28)

as we get closer to mass-action, leading to the identificationsthat

φ = φMC =

(βkU + αCPkM

β + αCP

)(29)

and

1n

=1A

=

β(β

AU+

αCPAU M

)+ αCP

(β

AU M+

αCPAM

)(β + αCP)2

. (30)

Whilst the mapping for φ arises from the process of mappingunder moment closure and the approximations therein (madeexplicit here by labelling it φMC ), this definition of A may alsobe derived from more rigorous approaches: both generalisingthe heuristic arguments of (Bolker, 1999) (see Appendix A.4)

and substituting the effective kernel K (r) from Section 3.2 into(19) also yield (30).

4. Invasion dynamics

Before considering how well our approximations capturethe ensemble behaviour of the full model, we derive someanalytical results governing the spread of FMD. A fundamentalissue in infectious disease epidemiology is whether a givenoutbreak will cause an epidemic and thus investigating theinvasion of pathogens in a host population represents animportant consideration. In reality, it should be recognisedthat infection and the probability of an epidemic ensuing is ahighly-stochastic phenomenon. Nonetheless, we demonstratein Section 5 that the pair model may provide an excellentapproximation to the full model provided the spatial scale ofdisease transmission is not too local and the host populationnot too strongly clustered. Carrying out an invasion analysis of(13) therefore provides a good approximation to carrying out anequivalent analysis on the less tractable full model.

Invasion analysis of the mean-field version of (13) (in theabsence of radial culling) allows one to demonstrate that thenumber of exposed and infectious farms grow as [E](t) ≈

[E](0)er1t and [I ](t) ≈ [I ](0)er2t where r1 = r2 =12 (ν +

µIP)(√

1 −4νµIP(1−R0)

(ν+µIP)2− 1

). One can then demonstrate that

since R0 = (β/αIP),

R0 =

(1 +

r1

ξ

)(1 +

r2

αIP

)(31)

and it is trivial to show that for r1 and r2 > 0, we requireR0 > 1, the standard mean-field criterion for the outbreak tobecome an epidemic.

Let us now consider a similar analysis of the pair model (13).Rewriting in terms of the number of farms and the correlationsbetween them, together with the fact that τ = (β/n), µCP =

(αCP/n) and [S]/N ≈ 1 early on, we can approximate theinvasion dynamics by

d[E]

dt≈ β[I ]CSI − ν[E] − αCP

[E][I ]CE I

N,

d[I ]

dt≈ ν[E] − αIP[I ]. (32)

Consider now the early behaviour when the number of exposedand infectious farms grows exponentially according to [E](t) ≈

[E](0)er1t and [I ](t) ≈ [I ](0)er2t (where r1 6= r2 now). Weevaluate the correlations at their quasi-equilibria C∗

SI and C∗

E I ,which represent stabilised local correlations between pairs overa short timescale, essentially arising from the local saturation inthe number of contacts that can be made per IP per generation(and is thus independent of R0). Substituting into (32) and usingthe fact that R0 = (β/αIP) enables one to obtain the analogousresult to (31) as

R0 =(r2 + αIP)

ναIPC∗

SI

((r1 + ν)+ αCP

([I ]CE I

N

)∗), (33)

356 P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368

from which one can demonstrate that for an epidemic to ensue(r1 and r2 > 0), we have now the threshold condition

R0C∗

SI −αCP

ν

([I ]CE I

N

)∗

> 1 (34)

(which, of course, reduces to the mean-field result in theabsence of radial culling and spatial structure). The secondterm on the LHS accounts for the impact on the effectivereproduction number R of locally removing exposed farmsfrom the population through radial culling, whilst the first termquantifies the multiplicative effect on R0 of radially cullingsusceptible farms and the formation of spatial structure due tothe infection process. From the definition of CSI and (13), wehave

dCSI

dt= −(τ + µCP)CSI

(φ(n − 1)

[I ]CSI

N(C I I − 1)

−[I ]CSI

N+ 1

)+ ν

[E]

[I ](CSE − CSI ), (35)

together with

ddt

([I ]CE I

N

)= τCSI

([S][I ]

N [E]

(1 −

n

N[I ]CE I

)+ (n − 1)

[S][I ]2

N 2[E]CSI

)− µCP

[I ]CE I

N

(1 −

[I ]CE I

N

)+ ν

([E]CE E

N−

[I ]CE I

N

)− µIP

([I ]CE I

N

). (36)

A number of special cases may be identified.

4.1. IP culling only

In the case where we cull only IPs, the threshold condition(34) simply reduces to R0C∗

SI > 1, so we simply need todetermine C∗

SI . (35) may be simplified considerably early on.Firstly, C I I 1 due to the rapid build-up of correlationsbetween IPs as the epidemic grows exponentially and moreover,φ(n − 1)C I I 1 for the same reason, even when n and φ aresmall. Secondly, we can use the arguments of Keeling et al.(1997) to approximate CSE as

CSE ≈

(n − 1

n

)CSS ((1 − φ)+ φCSI ) , (37)

and this may be simplified further (in a randomly-distributedhost population) since CSS ≈ 1. This approximation for CSE isfound to perform extremely well for all φ > 0, both when wecull IPs only and when radial culling is additionally employed.One may then approximate (35) as

dCSI

dt≈ −τCSI

(1 + φ(n − 1)

([I ]C I I

N

)CSI

)+ ν

[E]

[I ]

((n − 1

n

)((1 − φ)+ φCSI )− CSI

), (38)

so that only [I ]C I I /N (non-trivial due to the rapid increase inthe number of IPs early on) and [E]/[I ] (not a fixed ratio sincer1 6= r2) need to be additionally evaluated. Two subcases maybe identified.

4.1.1. No spatial structure (φ = 0)Whilst this special case appears of little interest, it is

worth noting that not only will this permit insight into thedependence of the dynamics on n, analysis of contact tracingdata from the 2001 UK epidemic has suggested that φ was verysmall (Ferguson et al., 2001a) and thus φ = 0 may providean approximation to the full dependence of the dynamics onnetwork structure; we return to this more general case inSection 4.1.2.

When φ = 0, such that no contacts of a given farm arecontacts of each other, it is clear from (38) that evaluating[E]/[I ] will allow us to determine C∗

SI . Computing [E]/[I ]exactly, however, leads to an intractable result and we thusdevelop a heuristic approximation instead. When φ = 0 and allcorrelations are at their quasi-equilibria, one can argue that theprocess of a farm moving from the exposed to the infectiousclass implies that the farm simply now inherits a fraction(n − 1)/n of the neighbourhood it had previously and thus inthis parameter regime, we approximate(

[E]

[I ]

)∗

≈

(n − 1

n

). (39)

We can thus approximate (38) when φ = 0 as

dCSI

dt≈ −τCSI + ν

(n − 1

n

)((n − 1

n

)− CSI

), (40)

resulting in the quasi-equilibrium

C∗

SI =(n − 1)2

n((n − 1)+

(βν

)) (41)

since τ = (β/n). In agreement with analytical results for theequivalent SIR pair model where C∗

SI = (n − 2)/n (Keeling,1999), the potential for FMD to successfully invade is reducednonlinearly as the average number of neighbours per farmdecreases or, in the spatial equivalent, as the infection kernelbecomes more localised; the effect is more pronounced withincreasing contact rate β and/or a longer-latency period (wherean increasingly short latency period leads to (41) tending to(n − 1)/n). The effect of contact structure in the SEIR modelis stronger than in the SIR model whenever n > nc where nc =

(2(βν) − 1)/((β

ν) − 1). The effect of farms becoming exposed

but not infectious slows disease spread and thus diseases with afinite latency need to be more transmissible to be as effective.

4.1.2. Spatial structure (φ > 0)Let us now consider the general case where φ > 0 and the

invasion dynamics of CSI are governed by (38) such that wenow need to consider [E]/[I ] and [I ]C I I /N early on.

In the special case where φ = 0, we argued that([E]/[I ])∗ ≈ (n − 1)/n. Thus, let us approximate this ratio inthe general case as follows: for the fraction (1−φ) of triples thatdo not form triangles, ([E]/[I ])∗ ≈ (n −1)/n as before. For anexposed farm forming part of a closed triple and subsequentlybecoming infectious, let us approximate its new neighbourhood

P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368 357

as a fraction (n − φn)/n of its neighbourhood in the exposedstate (φn possible neighbours are removed) and thus

[E] ≈

((1 − φ)

(n − 1

n

)+ φ

(n − φn

n

))[I ],

so that([E]

[I ]

)∗

≈ (1 − φ)

(n − 1

n+ φ

). (42)

This approximation performs extremely well for φ . 0.5; onecan argue that in networks where more than half of all triplesform triangles, it is a significant over-simplification to assumethat φn neighbours are lost when farms move from the exposedto infectious state and (42) leads to the subsequent analyticalsolution over-estimating the effect of spatial correlations whenφ & 0.5.

To evaluate ([I ]C I I /N )∗, we have that

ddt

([I ]C I I

N

)= ν

(2[E]CE I

N−

[E]C I I

N

)− µIP

([I ]C I I

N

), (43)

which means, in turn, that we need to evaluate [E]CE I /Nand [E]C I I /N . In the former case, after imposing the invasionapproximations [S]/N ≈ 1, C I I 1 and φC I I 1, we have

ddt

([E]CE I

N

)≈ τCSI

(φ(n − 1)

([I ]C I I

N

)CSI + 1

)+ ν

([E]

[I ]

([E]CE E

N

)−

([E]CE I

N

)−

[E]

[I ]

([E]CE I

N

)). (44)

When all correlations are at their quasi-equilibria, one can arguethat [E]CE I /N , proportional to the expectation of finding anexposed farm given that there is an infectious farm within itsneighbourhood, will always be much greater than [E]CE E/N ,proportional to the expectation of finding an exposed farmgiven that an exposed farm resides nearby (there is a higherprobability of becoming infected when nearest neighboursare infectious) and one would expect this approximation toperform particularly well when φ is small. Using [E]CE I /N

[E]CE E/N , together with (42), reduce (44) further to

ddt

([E]CE I

N

)≈ τCSI

(φ(n − 1)

([I ]C I I

N

)CSI + 1

)− ν

([E]CE I

N

)(1 + (1 − φ)

(n − 1

n+ φ

)), (45)

so that at its quasi-equilibrium

([E]CE I

N

)∗

=

τCSI

(1 + φ(n − 1)

([I ]C I I

N

)CSI

)ν(

1 + (1 − φ)(

n−1n + φ

)) . (46)

Whilst we still need to calculate [E]C I I /N to evaluate[I ]C I I /N from (43), one can further argue that

C I I ∝ CE I (47)

when both correlations are at their quasi-equilibria. Theconstant of proportionality should be dependent on the averageduration of latency (relative to the total time spent infectedand infectious) and the approximation that exposed farmshave a scaled neighbourhood size (n − 1)/n upon becominginfectious (which reduces the E-I correlation and increases theI-I correlation) implies a positive constant of proportionality.Thus, since the E to I transition depends only on time andnot the network structure around a farm (so that we would notexpect the constant of proportionality to depend on φ (verifiedin numerical analysis)), we further approximate

C I I ≈

(n − 1

n

)(ν + µIP

ν

)CE I (48)

and thus([E]C I I

N

)≈

(n − 1

n

)(1 +

µIP

ν

)([E]CE I

N

). (49)

Substituting (49) and (46) into (43) allows us to calculate([I ]C I I /N )∗, whereupon substituting into (38), together with(42), finally gives C∗

SI as the solution of the cubic equation

C∗3SI +

(λ4

λ5

)C∗2

SI +

(λ1(λ3λ5 − τ)

λ2λ3λ5

)C∗

SI

+

(λ1λ4

λ2λ5

)= 0, (50)

where λ1 = µIP

(1 + (1 − φ)

(n−1

n + φ))

, λ2 = −τφ(n −

1)(

2 −

(n−1

n

) (1 +

µIPν

)), λ3 = ν(1 − φ)

(n−1

n + φ)

, λ4 =

(1 − φ)(

n−1n

)and λ5 = φ

(n−1

n

)− 1. Since we require

solutions such that C∗

SI > 0 for all n and φ, it is readily shownthat for all realistic R0, latent and infectious periods

C∗

SI =13

(2√

A2 − 3B sin(13

arcsin(

2A3− 9AB + 27C

(A2 − 3B)3/2

))− A

), (51)

where A, B and C are defined from (50) by analogy to thecharacteristic equation

C∗3SI + AC∗2

SI + BC∗

SI + C = 0, (52)

valid for φ 6= 0. Fig. 1 plots C∗

SI from (51) as a function ofn and φ for φ . 0.5, the approximate regime in which the[E]/[I ] approximation, and hence C∗

SI , is valid. It is clear thatthe effect of network structure is to monotonically decrease R0with both a decreasing neighbourhood size n and increasingφ, with the effect of n found to play a far more significantdynamical role. Even with a very poorly connected network(the case in 2001), rapid IP culling and the imposition ofmovement restrictions and biosecurity measures to reduce the

358 P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368

Fig. 1. C∗SI = C∗

SI (n, φ) from (51) for the case of IP culling only (with a 7day average infectious period) where β = 4/7 (corresponding to R0 = 4 in themean-field model).

average number of infectious contacts per susceptible farm cansignificantly reduce R0. Effective localisation of infection notonly directly affects R0, but also enables radial culls to bemore effective with sufficient culling radius. This is consideredfurther in Section 4.2. These results are also in qualitativeagreement with those for the simple SIR model in Keeling(1999). We note that the effects of network topology in the SIRmodel are generally found to play a more significant dampingrole than for the SEIR model, particularly with increasing φ.

4.2. IP plus radial culling

Let us now consider additional control through radialculling. It is clear that the threshold condition for the epidemicto take-off is now given by (34) and thus both C∗

SI and([I ]CE I /N )∗ must be evaluated. Whilst one can repeat theanalysis of Section 4.1 for φ > 0, it is easily verifiednumerically that both C∗

SI and ([I ]CE I /N )∗ both weaklydecrease monotonically with increasing φ as in the case ofIP culling only (see Fig. 1). Thus, the considerable additionalanalytical complexity (not shown) yields no new insight, but issimply a trivial extension of the results from Section 4.1 andthe case of zero transitivity below; hence, there is no need toexplicitly consider this case any further.

When φ = 0, we can follow an identical analysis toSection 4.1.1. Noting that both the [E]/[I ] approximation (39)and the Keeling et al. (1997) approximation for CSE (37)remain accurate for all reasonable αCP, substituting into (35),together with the fact that [I ]/N ≈ 0 early on, gives a trivialextension to the result for IP culling only, namely

C∗

SI =(n − 1)2

n((n − 1)+

(β+αCPν

)) , (53)

so that C∗

SI simply scales inversely with faster radial culling.To calculate ([I ]CE I /N )∗, we substitute the standard

invasion approximation [S]/N ≈ 1 to give from (36) that

ddt

([I ]CE I

N

)≈ τCSI

([I ]

[E]

((n − 1)

([I ]CSI

N

)+ 1

Fig. 2. The epidemic threshold (34) as a function of neighbourhood size nand the rate of radial culling αCP for a 7 day infectious period and β = 4/7(corresponding to R0 = 4 in the mean-field model).

− n

([I ]CE I

N

)))− µCP

([I ]CE I

N

)(1 −

([I ]CE I

N

))+ ν

(([E]CE E

N

)−

([I ]CE I

N

))− µIP

([I ]CE I

N

). (54)

Since [I ]/N ≈ 0 and [I ]CE I /N 1 around the quasi-equilibria, together with the fact that [E]CE E/N [I ]CE I /N(since the probability of becoming exposed is much less thanthe probability of becoming infectious with an exposed farmnearby), using (39) and rearranging gives the required quasi-equilibrium as(

[I ]CE I

N

)∗

=βC∗

SI

βnC∗

SI + αCP

(n−1

n

)+ (ν + µIP)(n − 1)

, (55)

where C∗

SI is given by (53). Substituting (53) and (55) intothe threshold condition (34) enables the effect of additionallyradially culling to be evaluated and Fig. 2 illustrates thedependence of the epidemic threshold on n and the rate ofculling αCP.

Additionally culling contiguous premises early on furtherreduces the likelihood of an epidemic taking off withdecreasing n, particularly with increasingly rapid culling andthis relationship becomes stronger with decreasing R0 sincethe likelihood of control is greater. Given limited resourcesper day to control disease spread, questions of optimumallocation arise and here issues such as whether it is moreefficient to cull fewer farms more rapidly or more farmsover a longer duration become important. It is clear that theoptimum strategy will depend crucially on R0 and n, withradial culling per se becoming more efficient with increasinglylocal disease transmission. Here, we find that culling a smallerneighbourhood more rapidly will present the most efficient useof resources, particularly at low R0. However, such a strategybecomes less effective as disease transmission becomes lesslocal (farms have a larger neighbourhood size) and indeed

P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368 359

localised culling is then a poor use of resources. Indeed,adopting this policy but failing to get ahead of the epidemiccurve may result in fewer farms left unaffected at the end ofthe epidemic than would otherwise have been the case withIP culling only (thousands of farms are culled for an increasein S(∞) of only several hundred for example) and in thiscase, other non-IP control should be considered. Thus, ringculling is a prudent strategy only when transmission is verylocal, either as a result of the outbreak itself or as a resultof effective movement restrictions, and this is consistent withanalysis elsewhere (Keeling et al., 2003).

5. Results

We now examine the parameter regimes in which bothclasses of moment closure models developed in Section 3accurately capture the ensemble behaviour of the full model.We consider the spatial moment closure model under power-3 closure and the pair correlation model under the mappingsdeveloped in Section 3.3. We set the initial conditions to modela hypothetical FMD outbreak in the UK with a total populationof 130,000 farms, of which 50 farms are initially infectious, 10exposed (but noninfectious) and the rest susceptible. The smallinitial prevalence necessitates imposition of power-3 spatialmoment closure, since power-1 closure has been shown tobreak down in such regimes (Bolker, 1999; Filipe et al., 2004;Parham, 2006). The shape of the infection kernel is chosen as anormalised offset power law of the form

U (r) =(1 − a)(2 − a)

2πψ2u

(1 +

rψu

)a (56)

(where a > 2) following analysis elsewhere suggesting that thisform best fits the 2001 UK FMD epidemic (see Chis-Ster andFerguson (2007)), together with the fact that the two-parameterform also allows greater flexibility to investigate the robustnessof our conclusions to both the scale and shape parameter ofthe infection process. Note that the offset form removes thenormalisation problem about r = 0. (56) corresponds to aneffective neighbourhood size

Au =4πψ2

u (1 − 2a)

(1 − a)(2 − a)2(57)

and we initially consider the case where a = 3 (but consider thesensitivity to this in Section 5.3). The full model is simulatedwith a discrete time-step of 0.1 days and we average over 50realisations. The average latent period is fixed as 3 days (Sellersand Forman, 1973).

We consider a number of factors in assessing the accuracywith which both moment closure models capture the full model,as well as comparing the dynamics between each other. Weexamine similarity in the dynamics as a function of R0 andthe spatial scale of transmission, as well as the dependenceon a in Section 5.3. We also consider the effect of varyingthe mean latent and infectious periods (the rate at which weIP cull in the latter case), as well as the dependence on theirunderlying delay distributions — we assume that both are

exponentially distributed in general, but comment also on theeffect of relaxing this assumption. Where we investigate theimpact of radial culling, we consider the effect of varying therate of removal and the choice of culling kernel.

The distribution of farms will also affect the spatialand temporal disease dynamics. We therefore consider thesecomparisons in both a Poisson-distributed farm population and‘patchy’ farm distribution with a higher density of farms incertain regions than others; this more closely resembles thedistribution of UK farms. Finally, we also consider the impactof boundary conditions in the full model simulation by runningthe model over both a finite area and an infinite space byemploying periodic boundary conditions.

5.1. Poisson-distributed farms

5.1.1. IP culling onlyWe consider first the case where IPs only are culled. Fig. 3

compares the average of 50 realisations of the full modelwith the mean-field approximation (obtained by setting allcovariances to zero in (11)), the spatial moment closure model(under power-3 closure) and the pair model. The absence of aradial cull means that it is the infection kernel U (r) that createsthe spatial network upon which we impose epidemic processes.The pair model is run for two values of the connectivity φ,namely the value obtained by fitting the model to the simulationaverage (φB F ) and that calculated from first principles from(20) (φ1st ).

Fig. 3(a)–(c) compare the results for increasingly localdisease transmission for the case where R0 = 4. Valuesin parentheses represent the epidemic final size captured byS(∞), the total number of susceptible farms remaining at theend of the outbreak. Whilst moment closure models improveconsiderably at capturing the full model with increasing n,the dynamics tend to the mean-field model and the superiorsimplicity and tractability of the latter means that the additionalspatial complexity in the former is unjustified. Since low-order moment methods become less reliable at short spatialscales, it is therefore for ‘intermediate’ spatial scales that suchapproximations are most useful. Error bars on the simulationaverages represent ±2 standard deviations from the mean.

At the intermediate scale of transmission in Fig. 3(a), boththe spatial moment equations and pair model capture thesimulation well throughout most of the epidemic, with theformer performing marginally better at capturing the invasiondynamics, the latter better toward the tail. Both perform lesswell at capturing the epidemic peak. One can heuristicallyargue that geographic regions of susceptible farms at longdistances from IPs may not become infected in the full modeldue to the stochastic nature of the infection process, whilstthe deterministic approximations fail to account for this andthus overestimate the peak. In this regime, we also findthat estimating the connectedness from first principles agreesalmost exactly with the best-fit value. However, Fig. 3(b)and (c) demonstrate that while φB F depends on R0 andψu (decreasing monotonically with both), φ1st (as calculatedfrom (20)) depends on neither; this issue is considered

360 P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368

Fig. 3. Comparing the ensemble dynamics of the full model with the spatialmoment equations and pair model (under the mappings in Section 3.3) run fortwo values of φ, namely φB F obtained by fitting the model to the simulationaverage and φ1st calculated from first principles from (20). We consider thecase of IP culling only (7 days on average after becoming infectious) when

R0 = 4 and where the infection kernel U (r) =

[πψ2

u

(1 +

rψu

)3]−1

. (a),

(b) and (c) compare the models at Au = 24, 12 and 6 respectively. Values inparentheses represent S(∞), the total number of farms remaining unaffected atthe end of the epidemic.

further in Section 5.3. Thus, (20) poorly estimates φB F whentransmission is strongly localised and R0 low. Note alsothat when n = 24, the pair models perform impressivelyat estimating S(∞), significantly improving on the spatialmoment equations. Nonetheless, the discrepancy betweenS(∞) in the full model and the pair model, even when fitted,

Fig. 4. Comparison as per Fig. 3(a) (n = 24) when R0 = 2.

indicates a limitation of such models in this respect; it is alimitation of the model itself here rather than knowledge of thenetwork parameters.

We find that φB F decreases as transmission becomes morelocal, but since φ1st is independent of ψu , (20) significantlyover-estimates φB F in such regimes and the resultant pairmodel poorly approximates the full model. Thus, whilst suchpair-based approximations may have the potential to accuratelyapproximate full stochastic spatial models, further analyticalprogress is required to better estimate φB F as a function ofneighbourhood size; this is considered further in Section 5.3.Such approaches will then perform better than complexintegro-differential equations, with the added benefit of beingconsiderably simpler and more tractable. The limitation of thepair model in accurately estimating the final size becomesincreasingly evident with decreasing kernel width. Hence,whilst improved analytical estimates of φB F will better capturethe dynamics, S(∞) is largely insensitive to φ, representing alimitation of the pair approach in this respect.

Fig. 3(a) should be contrasted with Fig. 4 making the samecomparison when R0 = 2. We find little difference in thequalitative conclusions drawn, with both the spatial momentequations and pair model with φB F performing similarly.Parham and Ferguson (2006) argue that the pair approximationwith n = Au performs well whenever R0 n (since thefinite maximum on the number of possible contacts plays aless significant dynamical role) and indeed comparison betweenFigs. 3(a) and 4 supports this argument, although this remainsdependent on accurately estimating φB F .

Identical analysis with variable rates of IP culling yieldidentical qualitative results. Faster IP culling merely scalesthe timescale of the epidemic and results in fewer infectedfarms in total. S(∞) is independent of αIP when there isno spatial structure (φ = 0) (as in the SIR model), butleads to larger S(∞) for φ > 0 at shorter infectiousperiods since an increasing density of local connections(farms are better connected to their neighbours’ neighbours)inhibits global disease spread by saturating the number oflocal contacts, leading to fewer infected farms overall. Thisbecomes increasingly evident with decreasing n. Our resultsalso demonstrate no significant variation with the boundaryconditions imposed on the simulation (true for all R0 and ψu)

P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368 361

Table 1Normalised M(r) and the associated neighbourhood size AM

Policy Normalised M(r) Neighbourhood size AM

Uniform (πR2c )

−1 πR2c

Inside-out (2π(Rc − ln(1 + Rc))(1 + r))−1 2π(1+Rc)(Rc−ln(1+Rc))2

(1+Rc) ln(1+Rc)−Rc

Outside-in 3r2πR3

c

8πR2c

9

and we thus consider only the case of non-periodic boundaryconditions hereafter.

5.1.2. IP plus radial cullingWe now additionally include a radial cull. The choice

of M(r) encodes further information about the probabilityof farms being culled based on their geographic distancefrom IPs and we consider three illustrative forms: (1) allsusceptible and exposed farms within the ring have an equalprobability of being culled (independent of their distancefrom the IP), (2) farms closest to IPs are culled with ahigher probability (an ‘inside-out’ approach), and (3) farmsare culled with a greater probability at the edge of the ringthan those nearby (an ‘outside-in’ policy). Examining the extentto which moment closure methods are able to approximatespatiotemporal dynamics when disease transmission is highly-localised and/or control geographically-targeted is vital and onemay argue that an ‘inside-out’ policy encodes more ‘spatialinformation’ than allocating all farms within the ring to beculled with equal probability; how well the conclusions inSection 5.1.1 apply as a function of additional spatial scalesin the model is an important question. Mathematically, wemodel these policies by choosing M(r) = constant, M(r) ∝

(1 + r)−1 and M(r) ∝ r (defined for r ≤ Rc and whereM(r) = 0 for r > Rc) for the uniform, inside-out andoutside-in cases respectively; the normalised forms, togetherwith their associated neighbourhood sizes, are given in Table 1.The overlap neighbourhoods are calculated from AU M =

1/∫Ω U (r)M(r) dr.

We compare a number of approximations to the full spatialmodel in Fig. 5. The spatial moment equations are comparedwith the pair model with n = A for φB F , φ1st and φMC(obtained from (29)). The pair model is also considered underthe alternative hypothesis of connecting farms in the networkwith a probability proportional to the infection kernel only (n =

Au) and we consider this model with φ1st and the associatedbest-fit value φB F .

Fig. 5(a) compares these models when Au = 24 when weadopt a uniform ring cull. The pair models with n = A and n =

Au (and their respective φB F values) perform well at capturingthe full model (even at the epidemic peak and particularlytowards the tail), with all other approximations performingwell only in certain regimes. At this scale of transmission,these pair models are relatively indistinguishable and furtheranalysis is required to conclude which choice of neighbourhoodsize performs better as infection becomes more local. Thepair model with φMC performs poorly throughout since (29)assumes a force of infection close to mass-action and a regime

Fig. 5. Comparison of the full model, spatial moment equations and pairmodels when we cull IPs plus non-IPs in a uniform ring (on average one dayafter IP culling and where Rc = 3 km) when R0 = 4 at Au = 24, 12 and6 in (a), (b) and (c) respectively. We compare the pair model under n = Awith φ = φB F , φ1st and φMC where possible, as well as considering thealternative hypothesis of connecting farms with probability U (r) only. Runsin (a) not shown in (b) and/or (c) indicate where the pair model breaks downwith decreasing n and increasing φ.

where third-order cumulants are zero, neither of which are wellsatisfied here.

Reducing ψu leads to similar results, with these two pairmodels and the spatial moment equations performing well,but the models with φ1st performing poorly by failing tocapture the decrease in φB F with more local transmission. Theapproximation with φMC breaks down during the invasion stage

362 P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368

of the outbreak (at around 10 days), highlighting a limitation ofthe pair approach when both n is small and φ large. Whilstthe n = A and n = Au models (with their respectiveφB F ) still perform near identically, a clear benefit of adoptingn = A emerges (together with the more formal motivationin Section 3). It is clear that increasingly local infection isassociated with decreasing φB F . It is also clear that decreasingAu will quickly lead to φB F = 0 with n = Au , while φB F > 0with n = A for a wider range of neighbourhood sizes and downto smaller n, leading to a better fit with the full model in regimeswhere we are often most interested. This is highlighted by theextreme case in Fig. 5(c) when Au = 6 and both neighbourhoodsize models have φB F = 0 (a fundamental limitation of the pairapproach here at such localised transmission), but where themodel with n = A performs significantly better, even capturingthe epidemic tail. All other pair models break down early on inthis regime due to large associated values of φ.

Identical qualitative results are found from adopting eitheran inside-out or outside-in approach and these results arealso independent of R0 and the disease latency and infectiousperiods; the simulation again demonstrates no significantdifference between periodic and non-periodic boundaryconditions.

5.2. Clustered farm distributions

Whilst assuming a Poisson-distributed host population mayprovide an accurate approximation in many systems, censusdata on the UK farm population indicates that this distributiondeviates significantly from spatial randomness. An importantquestion is therefore how well the results in Section 5.1 holdas the host distribution becomes less random and we observe a‘patchy’ or clustered distribution.

The Poisson cluster process provides a means of generatingsuch distributions in simulation models (Diggle, 2003). In apopulation of size N , n p = N/(1 + nc) ‘parents’ are chosenat random and we associate with each parent nc ‘children’scattered about their parent farm with a probability determinedby a ‘cluster kernel’ κ(r) (with associated scale parameterψκ ). We arbitrarily choose κ(r) to be the same functionalform as the infection kernel. We can conveniently controlhost clustering in the spatial moment closure model by settingthe initial covariance between susceptible farms such thatcSS(r, 0) = nc(nc −1)(κ ∗κ)(r) (Bolker, 1999). In the networkmodel, we can make use of the mappings in Section 3.3:substituting (26) and (27) into (14) in the limit n → N givesCSS(0) = 1 + cSS(0)/S(0)2. Note that the general relationshipbetween these statistics, CX X (t) = 1 + cX X (t)/X(t)2, isidentical to calculating Lloyd’s index of patchiness P in bothmodels (Lloyd, 1967), indicating that cXY and CXY provideconsistent measures of capturing correlations in their respectivemodels.

We carry out an identical analysis to Section 5.1 andcompare the dynamics of these models for the cases of ‘weakly’and ‘strongly’ clustered populations. To draw a parallel with thework of Bolker (1999), we define weak clustering to be wherenc = 4 farms per cluster (with scale parameter ψκ =

√2/5)

Fig. 6. Comparison between the models when farms obey a strongly clustereddistribution and R0 = 4. (a) and (b) consider culling IPs only at Au = 24 and12 respectively (cf. Fig. 3(a) and (c)), whilst (c) considers culling IPs plus anoutside-in radial cull at Au = 24. The value of φ1st in (c) lies in the high-connectivity regime where the pair model breaks down and is therefore notshown.

and nc = 9 (ψκ =

√2/5(1.5)2) for the case of strong clustering

(where ψκ is calculated using n = Au). We find that the resultsfor weak clustering lie between those for randomly-distributedfarms and the case of strong clustering and we thus present onlythe analysis for the strongly-clustered distribution here.

For the case of IP culling only (Fig. 6(a)), we find that whilstthe spatial moment equations still perform well at capturingthe invasion dynamics, in general they poorly capture the fullmodel, even at relatively large neighbourhood sizes and thussuch approaches are limited when the local density of contacts

P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368 363

becomes large. For the case of randomly-distributed farmswhen Au = 24, the pair model with φ1st performed extremelywell, whilst φ1st now significantly underestimates φB F by notaccounting for the non-uniform density profile (since φB F nowincreases with decreasing n) and thus the resultant pair modelpoorly captures the simulation. Consideration of how (20) maybe extended for non-uniform ρ(r) may be considered in futureresearch.

The value in developing analytical estimation of φ isdemonstrated by the pair model with φB F , which is found for awide range of Au to perform almost as well as for the case ofrandomly-distributed farms. Thus, the pair model is still able toperform significantly better than the spatial moment equationswhen the host population distribution demonstrates significantclustering, with the problem again reduced to developing betterestimates of φB F . The pair model performs slightly less wellat capturing early behaviour as the farm distribution becomesmore clustered and the spatial scale of disease transmissionbecomes very local (see Fig. 6(b) when Au = 12). Note inagreement with the analysis of Bolker (1999) that the effectof a highly-clustered host population is to initially acceleratedisease spread (relative to a random distribution), but later slowthe epidemic due to local contacts being saturated more quickly,resulting in a reduced epidemic peak, a longer overall outbreaktimescale and a significantly larger S(∞). These results allbecome more pronounced with decreasing n.

In the case of IP plus radial culling, however, both classesof moment closure models perform poorly at capturing the fullmodel, even at large neighbourhood sizes. Fig. 6(c) considers anIP plus outside-in ring cull when Au = 24 (comparable resultsare obtained with the uniform and inside-out approaches). Wefind that the pair model breaks down around φc ∼ 0.25 atAu = 24 (where φc decreases with decreasing Au) and thussince φB F > φc, the pair approximation performs extremelypoorly at capturing the full model. Note also that φ1st andφMC > φc, both when n = A and n = Au . Both classesof approximations are therefore found to perform poorly atcapturing the effect of spatially-targeted control in a highly-clustered host population, even at large R0 and when diseasetransmission is not particularly local.

A combination of our analysis here and analytical resultsfrom Section 4.2 demonstrate that in situations where acase for ring culling can be made, an inside-out approachgenerally represents the least efficient policy in a randomly-distributed farm population, failing to get ahead of the ‘within-ring’ epidemic unless culling is carried out extremely rapidly.In general, an outside-in approach generally represents themost efficient policy by sacrificing farms closest to IPs andminimising the cost to the population as a whole, althoughthere is often little difference between this and a uniformculling approach. However, an important exception ariseswhen the host population obeys a more clustered distribution.Analysis of the strongly clustered case demonstrates thatwhen culling contiguous premises is justified, an inside-out approach always provides a significantly more efficientstrategy, essentially because a greater fraction of potentialcontacts can be culled before becoming infected and thus we

can get ahead of the within-ring epidemic curve without over-culling an unnecessarily high number of farms (as in an outside-in approach) to achieve control. Thus, whilst it remains beyondthe scope of this paper to formulate optimal control strategiesmore generally, it is clear that the choice of strategy will dependon R0, n and the distribution of farms.

Finally, we also investigate the effect of relaxing theassumption of exponentially-distributed latent and infectiousperiods. We modify the deterministic moment closure modelsby dividing the exposed and infectious classes into r subclasseschained together, implicitly assuming a gamma distribution forthe waiting time in both compartments — r is then the shapeparameter of the gamma distribution (where r = 1 correspondsto an exponential distribution) and the mean duration of timespent in each infection stage (1/λ) remains unchanged (whereλ is the scale parameter of the gamma distribution). In brief, wefind that the results above display some dependence on r , withthe moment closure approximations potentially less accurate asr increases in regimes where previously they had performedwell. With IP culling only (in a Poisson-distributed farmpopulation), there is again little qualitative difference betweenthe models when Au = 24 (where the gamma-distributedmodels demonstrate a faster timescale of epidemic processesrelative to the exponential distribution, since farms spend moretime in the infectious class as r increases). However, we notethe important caveat that this arises in the pair model here withφB F = 0, thus highlighting a further drawback to the crudeparameterisation of the pair approximation (15). As infectionbecomes more local, both the spatial moment closure modeland pair approach with zero transitivity perform increasinglypoorly and lie outside the error bars on the simulation for largerneighbourhood sizes than previously found. These conclusionscontinue to hold when we additionally introduce radial cullingand a strongly clustered farm population (although φB F is non-zero in the latter case), but otherwise the general findings abovefor exponentially-distributed waiting times remain true.

5.3. Semi-empirical estimation of φB F

Sections 5.1 and 5.2 highlight parameter regimes where bothclasses of moment closure models may accurately capture thefull model when the kernel shape parameter a = 3. Two furtherissues arise from this analysis: firstly, since the first principlescalculation of φ from (20) is independent of model parameters(other than an implicit dependence on a), can we develop an‘ad hoc’ means of better estimating φB F and secondly, to whatextent do our results in Sections 5.1 and 5.2 change whena 6= 3?

We seek here a ‘semi-empirical’ means of estimating φB F toovercome the aforementioned difficulties with calculation from(20). Consider first the values of φB F when a = 3 in Fig. 7 asa function of n and R0. As n → ∞, we obtain the mass-actionlimit under which all classes of model converge and thus φ → 1in this limit. We find from Fig. 7 that when n > n0, where n0is the neighbourhood size below which the best-fit value of φ iszero (representing a limitation of the pair model in this respect),

364 P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368

Fig. 7. φB F = φB F (n) (open circles) for the case of IP culling only and wherewe consider R0 = 2, 4 and 6 for a = 3 and R0 = 4 when a = 4. We comparethese values with the semi-empirical formula (58) with (ξ, ω) values (2.4, 0.39)and (1.9, 0.49) for a = 3 and a = 4 respectively. The observed variation ofφB F with n is not captured by the first principles calculation from (20) (whichgives φB F = 0.33 for a = 3).

φB F can be well approximated by

φB F = 1 − ξn−ω, (58)

where ξ and ω > 0. Analysis of φB F = φB F (n) demonstratesonly a very weak dependence on R0 (see Fig. 7) and indeedfitting (58) for R0 = 2, 4 and 6 (when a = 3) yields ω ≈ 0.39for all three values of R0. Variation of ξ with R0 is also minimaland Fig. 7 plots (58) with the average value ξ ≈ 2.4, whichperforms well in all three cases. Thus, the pair model mostclosely matches the full model when the fraction of triples thatdo not form triangles (1 − φ) follows a power-law. The criticalneighbourhood size n0 below which the best-fit value of φ iszero is readily estimated as n0 = ξ1/ω.

We find that the power-law relationship (58) holds for all a(with appropriate values of ξ and ω). Fitting when a = 4 (forR0 = 4), for example, gives ω = 0.49 and ξ = 1.9 (see Fig. 7)and we find that φB F increases linearly with a to a very goodapproximation. However, whilst this approach enables us tobetter estimate φB F as a function of model parameters, we findthat the fit of the pair model to simulation becomes poorer as aincreases. Increasing a corresponds to a less dispersed infectionprocess, with the resultant prevalence curves displaying areduced maximum and a longer overall epidemic timescale(although the effect on final size is minimal). (Note thata & 20 provides an approximation for the behaviour with anexponential infection kernel). Numerical analysis demonstratesthat the pair model may provide an accurate approximationwhen a . 3.5, even when ψu is large. Whilst this rangeof a may appear limited, it is not surprising that the pairmodel performs increasingly poorly as the kernel becomes lessdispersed and developing approximations that perform well inthis regime remains an important issue. Chis-Ster and Ferguson(2007) report that an infection kernel of the form (56) providesthe best fit to the 2001 UK FMD epidemic and moreover thata ∼ 2.68 (after the imposition of movement restrictions on 23rdFebruary 2001). The 2001 outbreak therefore resides within

the range of a values in which the pair model may provide anaccurate approximation to the full simulation.

We note again that whilst (58) provides an accurate meansof estimating φB F independent of a, values beyond a ∼ 3.5 liein a parameter regime where both the pair and spatial momentclosure models perform very poorly at capturing the simulation.

6. Conclusions

Whilst considerable advances in computational power haveallowed substantial progress in understanding the role of spatialheterogeneity in epidemic models, far less progress has beenmade in developing analytical tools for understanding sucheffects. The need to develop simpler ‘pseudo-spatial’ modelsthat accurately capture locality in an infection process andthe impact of population-level and geographically-targetedcontrol, whilst permitting an analytical understanding ofdisease dynamics, remains an important problem. In this paper,we develop two classes of moment closure methods that providesecond-order deterministic approximations to the ensemblebehaviour of a fully spatial stochastic model of FMD. Wedemonstrate how these formally distinct methods may berelated, providing a framework within which all three modelsmay be rigorously mapped, together with the ability to definean approximating network topology in terms of the underlyingmodel itself.

We find that in a randomly-distributed host population,both classes of moment closure models are able to providean excellent approximation at intermediate spatial scalesof disease transmission provided control is not spatially-targeted, accurately capturing all aspects of the epidemic otherthan around the peak. In the case of geographically-targetedintervention and/or increasingly local disease transmission,the simple parametric pair approximation on the network isable to perform far better than spatial moment methods forappropriately chosen network connectedness φ. We develop auseful semi-empirical means of estimating the value of φ thatresults in maximum correspondence between the pair modeland full simulation when the infection kernel is not too local,demonstrating that this occurs when the fraction of triples thatdo not form triangles follows a power law. The pair modelwith correctly identified φ is able to substantially improve onspatial moment methods in regimes where we are often mostinterested, namely at low R0, when transmission is local and/orcontrol strategies are spatially-targeted.

This work also demonstrates, however, that low-ordermoment methods are fundamentally limited as spatial processesbecome more local due to the implicit neglect of higher-ordercorrelations. Whilst φ may account for higher-order structure incertain parameter regimes, in many cases where the use of suchmodels is justified and the effect of network structure important,the clustering coefficient provides a very crude parameter toreliably capture such topology.

Other important implications emerge from our results.Whilst invasion dynamics are generally well captured inrandomly-distributed populations, the final size is generally lesswell estimated, particularly as spatial scales become shorter. It

P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368 365

is also clear that in spatial systems incorporating more thanone non-uniform spatial kernel, all spatial functions shouldbe included in defining the effective neighbourhood size ofthe approximating network. Another important conclusion isthat the spatial moment closure approximation performs morepoorly as the population becomes increasingly clustered, whilstthe pair approximations may perform almost as well as inthe randomly-distributed case if φ is appropriately chosen andcontrol not spatially-targeted. If the latter condition is notsatisfied, however, the pair models also poorly capture thefull model, even at large neighbourhood sizes. We thereforeconclude that developing moment closure methods which bettercapture epidemic behaviour when host populations becomeincreasingly patchy also represents an important area forfuture research and this is consistent with work elsewhere(e.g. Thomson and Ellner (2003)). Host clustering may play avital role in governing spatial and temporal disease spread, aswell as influencing the choice of control policy; radial cullingwith an inside-out approach, for instance, is found to be a veryefficient strategy when farms are strongly clustered, with thebenefits scaling significantly with both the locality of diseasetransmission and R0.

An important advantage in developing more tractableapproximations to fully spatial models lies in the abilityto obtain analytical insight into model dynamics. Here, wedemonstrate through an invasion analysis how the standardmean-field criterion for an epidemic to take-off (R0 > 1)is modified when we incorporate spatial structure and radialculling, as well as deriving useful results for how diseasespread on the network depends on topological properties.By approximating spatial correlations around their quasi-equilibria, we demonstrate that the ability of FMD tosuccessfully invade is reduced nonlinearly with a decreasingnumber of neighbours per farm. We also demonstrate thatlarge φ inhibits disease spread, in essence since the greaterprobability of disease transmission from nearest neighboursreduces the ability of the disease to spread globally.Possible effects of radial culling are also discussed and itsappropriateness in a variety of scenarios considered. Futurework may derive other analytical results for such systems,directly relevant to understanding the dynamics and control ofFMD and other livestock diseases.

Acknowledgments

The authors would like to thank an anonymous referee forhis/her very helpful comments which improved this paper, aswell as BBSRC, DEFRA and MRC for financial support.

Appendix

A.1. Deriving the spatial moment equations

The dynamical equations corresponding to the full stochasticmodel are given by (8), whereupon substituting the ratesλInf(x, t), λCP(x, t), λExp(x, t), λNat(x, t) and λIP(x, t) gives

for the number of exposed farms

dE(x, t)

dt= βS(x, t)

∫y∈Ω

U (x − y)I (y) dy − νE(x, t)

−αCP E(x, t)∫

y∈ΩM(x − y)I (y) dy. (59)

Given that from (9)

dE(t)

dt=

∫x∈Ω

dE(x, t)

dtdx,

substituting (59), switching to plane polars and takingexpectations across the statistical ensemble gives

dE(t)

dt= β

∫U (r)〈S(x)I (y)〉 dr − ν

∫〈E(x)〉 dx

−αCP

∫M(r)〈E(x)I (y)〉 dr. (60)

Using (9), the definition of the covariance (7) and thenormalisation conditions (2) and (3) then reduce (60) to

dE(t)

dt= β

(S I +

∫U (r)cSI (r, t) dr

)−αCP

(E I +

∫M(r)cE I (r, t) dr

)− νE (61)

in a spatially homogeneous and isotropic landscape where r =

|r| and the covariances depend only on the distance betweenfarms, not their precise locations. Identical derivations maybe followed for the change in the number of susceptible andinfectious farms.

To derive the covariance equations, let us illustrate thederivation with the S-I covariance. From (7) and considerationof a short time interval 1t , we have

1cSI

1t=1

1t〈S(x)I (y)〉 −

1

1t

(S I)

(62)

(where we use the ergodic property of the system 〈X (x)〉 ≡

X (x)). Consider the first term on the RHS. Since

1S(x)I (y) = I (y)1S(x)+ S(x)1I (y),

taking 1/1t of both sides, substituting from (8) and takingexpectations over all realisations give

1〈S(x)I (y)〉1t

= −β

∫U (x − z)〈S(x)I (y)I (z)〉 dz

−αCP

∫M(x − z)〈S(x)I (y)I (z)〉 dz

+ ν〈S(x)E(y)〉 − (σ + αIP)〈S(x)I (y)〉. (63)

Note that (63) is still exact and it is this point at which mappingthe ensemble behaviour of the spatial and network models(prior to the imposition of moment closure approximations)should be exact, although this is not pursued further here.

Arguably the simplest closure one may adopt to approximatetriple expectations of the form 〈X (x)Y (y)Z(z)〉 is to define

M3 = 〈(X (x)− 〈X (x)〉)(Y (y)− 〈Y (y)〉)(Z(z)− 〈Z(z)〉)〉

(64)

366 P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368

and then set all third-order cumulants M3 to zero (power-1closure). Expanding the RHS of (64) gives the approximation

〈X (x)Y (y)Z(z)〉 = X〈Y (y)Z(z)〉 + Y 〈X (y)Z(z)〉

+ Z〈X (x)Y (y)〉 − 2X Y Z . (65)

If we now substitute (65) into (63) and use the definition ofthe covariance (7) and normalisation conditions (2) and (3),substituting this and the mean-density equations into (62) in thelimit 1t → 0 eventually leads to

∂cSI (r, t)

∂t= −β

(S(U ∗ cI I )+ S IU + I cSI

)−αCP

(S(M ∗ cI I )+ S I M + I cSI

)+ νcSE − (σ + αIP)cSI . (66)

The other covariance equations may be identically derived. Weadditionally note that for the autocovariances

1X (x)X (y) = X (y)1X (x)+ X (x)1X (y)

= 2X (x)1X (y) (67)

due to the assumption of translation invariance.The moment equations can also be derived under other

closures and we consider here the assumption of power-3closure

〈X (x)Y (y)Z(z)〉 =〈X (x)Y (y)〉〈X (x)Z(z)〉〈Y (y)Z(z)〉

〈X (x)〉〈Y (y)〉〈Z(z)〉. (68)

It is readily verified that if we now substitute (68) into (63), weobtain the power-1 equations plus higher-order terms, whichincrease in magnitude as we move away from mass-action. Forthe cSI covariance for instance, we obtain

∂cSI (r, t)

∂t= −β

(S(U ∗ cI I )+ S IU + I cSI︸ ︷︷ ︸

+cSI c(U )SI

S+ 2cSI U +

(UcSI ∗ cI I )

I

+cSI (U ∗ cI I )

I+

Uc2SI

S I+

cSI (UcSI ∗ cI I )

S I2

)

−αCP

(S(M ∗ cI I )+ S I M + I cSI︸ ︷︷ ︸+

cSI c(M)SI

S+ 2cSI M

+(McSI ∗ cI I )

I+

cSI (M ∗ cI I )

I+

Mc2SI

S I

+cSI (McSI ∗ cI I )

S I2

)+ νcSE − (σ + αIP)cSI , (69)

where the terms in underbraces arise in the power-1 model;we obtain similarly complex equations for the other covarianceequations (not shown). Thus, whilst we use the power-1 modelto give a flavour of the moment equations in the text, and indeedto demonstrate the methodology and mapping developed, weconsider the power-3 system throughout for numerical results,in essence due to the small initial prevalence assumed.

A.2. Averaging the spatial moment equations over separatespatial kernels

To average the moment equations over U (r), we first take∂/∂t of (21) to give

dc(u)XY

dt=

∫U (r)

(∂cXY (r, t)

∂t

)dr, (70)

so that substituting the covariance Eq. (11), using theconvolution approximation (22) and the neighbourhooddefinitions (16) and (23) gives

dc(u)SS (t)

dt= −2β

(I c(u)SS + ku Sc(u)SI

)− 2αCP

(I c(u)SS + kM Sc(u)SI

),

dc(u)SE (t)

dt= −β

(ku Sc(u)E I + I c(u)SE − ku Sc(u)SI − I c(u)SS

)−αCP

(kM Sc(u)E I + 2I c(u)SE + kM Ec(u)SI

)− νc(u)SE ,

dc(u)SI (t)

dt= −β

(ku Sc(u)I I +

S I

Au+ I c(u)SI

)

−αCP

(kM Sc(u)I I +

S I

AuM+ I c(u)SI

)+ νc(u)SE − (σ + αIP)c

(u)SI ,

dc(u)E E (t)

dt= 2β

(I c(u)SE + ku Sc(u)E I

)− 2αCP

(I c(u)E E + kM Ec(u)E I

),

dc(u)E I (t)

dt= β

(I c(u)SI + ku Sc(u)I I +

S I

Au

)

−αCP

(I c(u)E I + kM Ec(u)I I +

E I

AuM

)− ν

(c(u)E I − c(u)E E

)− (σ + αIP)c

(u)E I ,

dc(u)E E (t)

dt= 2

(νc(u)E I − (σ + αIP)c

(u)I I

)(71)

and we can follow an identical approach to average thecovariance equations across M(r) only. The total averagecovariance equations may then be derived using the fact that

dcXY (t)

dt=

1(β + αCP)

(β

dc(u)XY

dt+ αCP

dc(M)XY

dt

). (72)

A.3. Mapping the SEIR models under moment closure

If we assume a priori the first moment mapping (26) for thecase of two or more spatial kernels, we have

d[S](t)

dt≡ A

dS(t)

dt, (73)

so that substituting from (11) and (13) gives

P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368 367

(τ + µIP)[SI ] ≡ A[β( S I + c(u)SI )+ αCP( S I + c(M)SI )

]≡ A

[β

([S][I ]

A2 + c(u)SI

)+ αCP

([S][I ]

A2 + c(M)SI

)]. (74)

If we now equate for the transmission and culling terms as sep-arate identities, it is trivial to show that for dimensional consis-tency, τ ≡ βA/A2

≡ (β/A) and similarly µCP ≡ (αCP/A).Postulating the analogous second moment mapping a priori[XY ] ≡ A2 XY and substituting for τ and µCP in (74) gives(β

A+αCP

A

)A2( S I + cSI ) ≡ A

[β( S I + c(u)SI )

+ αCP( S I + c(M)SI )], (75)

which is immediately consistent for the terms in S I and whereequating for the covariances gives

cSI (t) =

(βc(u)SI + αCPc(M)SI

β + αCP

), (76)

the definition of the total average cSI (t) as the weighted sumof the separate averages (or equivalently cXY (r, t) averagedagainst K (r)).

Consider now the spatial system when disease transmissionis not too local and we are not too far from mass-action. In suchregimes, it is readily verified that it is the closure independentterms in the power-3 model that dominate the dynamics to avery good approximation and thus the power-1 system (11)provides a good approximation in this case. If we therefore taked/dt of (76), substituting the ODEs for the separate covariances(see (71)) and rearranging gives

dcSI (t)

dt= −(β + αCP)

(βkU + αCPkM

β + αCP

)ScI I + I cSI

+

β(β

AU+

αCPAU M

)+ αCP

(β

AU M+

αCPAM

)(β + αCP)2

S I

+ νcSE − (σ + αIP)cSI . (77)

Now consider the model imposed on the network. We havefrom the second moment mapping that [SI ] ≡ A2SI ≡

A2( S I +cSI ), so that taking d/dt of both sides and rearranginggives

dcSI (t)

dt≡

1

A2

(d[SI ]

dt−

d[S][I ]

dt

).

If we now substitute the network equations (13) under theclosure (15), using the moment mappings and rearranging givesfrom the pair model

dcSI (t)

dt= −(β + αCP)

(I cSI

1 +

cSI

S I

× φScI I

1 +

2cSI

S I+

c2SI

S2

I2

+

S I

A

1 +

cSI

S I

)νcSE − (σ + µIP)cSI . (78)

As transmission becomes less local and closer to the mean-fieldassumption, it is the first term in each of the curly brackets in(78) that dominate the dynamics to a very good approximation(cSI /S I 1) and (78) is approximately

dcSI

dt≈ −(β + αCP)

(φScI I + I cSI +

S I

A

)+ νcSE − (σ + µIP)cSI . (79)

Thus, comparison of (79) with the equivalent Eq. (77) deriveddirectly from the spatial model itself leads to the identifications

φ = φMC =

(βkU + αCPkM

β + αCP

)(80)

and

1n

=1A

=

β(β

AU+

αCPAU M

)+ αCP

(β

AU M+

αCPAM

)(β + αCP)2

(81)

where φ is labelled φMC to indicate that this mapping isapproximate and derived under moment closure. This approachalso suggests that this mapping for n is also approximate,although arguments in Section 3 and Appendix A.4 suggest thatthis definition can be derived from more rigorous arguments.

A.4. Heuristic derivation of A from first principles

We extend here the arguments of (Bolker, 1999) to derivethe expression for n = A from a more heuristic approach. Letus firstly connect farms with probability K (r) = (βU (r) +

αCP M(r))/(β + αCP) (from which it is easy to verify byintegrating over all space that

∫K (r) dr = 1 from the

normalisation conditions (2) and (3)). For the trivial case whereU (r) = U and M(r) = M (both are uniform kernels within thehabitat area Ω ), we have from the normalisation condition that∫ (

βU + αCP M

β + αCP

)dr = 1

⇒1A

=βU + αCP M

β + αCP. (82)

If we now consider the general case where U = U (r) andM = M(r), we have by analogy

1

A2 =

(βU (r)+ αCP M(r)

β + αCP

)2

,

whereupon integrating over all Ω gives∫ (βU (r)+ αCP M(r)

β + αCP

)2

dr =

∫1

A2 dr =1A, (83)

or using our definition of the neighbourhood sizes Au , AM andthe overlap neighbourhood AU M gives

1n

=1A

=

β(β

AU+

αCPAU M

)+ αCP

(β

AU M+

αCPAM

)(β + αCP)2

, (84)

identical to the mapping obtained under moment closure inAppendix A.3 and by defining A = 1/

∫K (r)2 dr.

368 P.E. Parham et al. / Theoretical Population Biology 73 (2008) 349–368

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