multiscale modelling of scrapie epidemiology

Ecological Modelling 179 (2004) 515–531

Multiscale modelling of scrapie epidemiologyII. Geographical level: hierarchical transfer ofthe herd model to the regional disease spread

B. Duranda,∗, M.A. Duboisb, P. Sabatierc, D. Calavasd,C. Ducrote, A. Van de Wiellef

a Unité Epidémiologie, AFSSA Alfort, 23 Av. du Gén. de Gaulle, BP 67, F94703 Maisons-Alfort Cedex, Franceb SPEC, CEN Saclay, Orme des Merisiers, F91191 Gif sur Yvette Cedex, France

c Unité BioMathématiques, INRA-ENV Lyon, 1 avenue Bourgelat, BP 83, F69280 Marcy l’Etoile, Franced Unité Epidémiologie Bovine, AFSSA, 31 Avenue Tony Garnier, BP 7033, F69342 Lyon Cedex 07, France

e Unité d’Epidémiologie Animale, INRA Theix, F63122 St. Genès Champanelle, Francef ADMA, 64000 Pau, France

Received 26 January 2004; accepted 18 May 2004

Abstract

Geographical diffusion of scrapie disease is modelled as the outcome of two processes: intra-herd dynamic and regionalspreading of genetic and infectious material between herds. The intra-herd dynamic of the disease is represented in this paper byan analytical iterative model with three state variables: the percentage of infected animals, the percentage of resistance alleles,and the percentage of hypersusceptibility alleles. Parameters of this analytical representation are estimated and validated usingan existing intra-herd model.

The main between-herd contamination path is known to be the trading of animals. Herds contacts on grazing grounds arealso suspected to allow disease transmission. Exposure of herds to these contamination paths varies depending on local pastoralhabits.

We model these contamination paths as diffusion processes in three 1-D spaces: the first represents the situation in winter,where herds are stationed in farms, but when commercial exchanges are also active; the second represents the summer conditionfor sedentary herds, which are grazing in the vicinity of their farms, but are in contact with other neighbouring herds, and thethird represents herds migrating in summer which are strongly exposed to other herds in mountain grazing grounds.

We thus obtain a model for the disease diffusion at a regional scale. Using this model it is possible to analyse the influence ofdifferent pastoral habits and control measures on the geographic spread of the disease. Furthermore, using both the herd-levelmodel and the regional-level model, we can test the consequences of individual-level hypotheses on regional-level dynamics;this is possible because of the hierarchical transfer of the herd model to the regional disease spread, and because both modelsare mechanistic.© 2004 Elsevier B.V. All rights reserved.

Keywords:Scrapie; Modelling; Between-herd transmission; Difference equations; Hierarchical transfer

∗ Corresponding author. Tel.:+33-1-49-77-13-34; fax:+33-1-43-68-97-62.E-mail address:[email protected] (B. Durand).

0304-3800/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2004.05.015

516 B. Durand et al. / Ecological Modelling 179 (2004) 515–531

1. Introduction

The objective of the second part of this two-partarticle is to use the intra-herd model described in thefirst part to propose a model of the scrapie dynamics ata geographical level, regional disease dynamics beingthe outcome of coupling within-herd disease dynamics(genetic and infectious) and between-herd interactions(genetic and infectious fluxes). Thus, the modelled unitis not the individual anymore but the herd.

Such a model is composed of two sub-models: awithin-herd model and a between-herd model.

The simplest way to get a within-herd model was touse directly the model described in the previous paper.This is not the solution which we used, partly for prac-tical reasons but mainly for methodological reasons.The intra-herd model described in the previous paperis based on a three-dimensional individual dynamicmodel: age, genotype and health state. It is thereforea rich and complex model, and its use with thousandsinstances being run simultaneously would have posedserious implementation and execution problems.

Moreover, the level of detail used in this model isnot relevant for disease modelling at a geographicallevel. For instance, among the three individual dimen-sions enumerated above, age does not seem to be rel-evant at the geographical level.

Therefore, instead of using directly the intra-herdmodel, we chose to derive from it an “aggregated”analytical model, using only relevant state variablesfor the regional-level model. From here, to avoid con-fusion, we will use the words “intra-herd model” or“general model” to refer to the model described in theprevious paper, and the words “aggregated model”to refer to the simplified model derived from theformer.

Because the aggregated model is a simplifiedversion of the general model, some of the controlvariables of the general model are hidden in the aggre-gated model. The aggregated model is built followingthree steps. We first choose which control variableswe will keep in the aggregated model: we are guidedby the nature of the processes which we have to modelat the regional level. Secondly, we choose which statevariables we need in the aggregated model and wedefine the corresponding equations, using a set of newcontrol parameters. The last step is then to estimatethese parameters values using data obtained from

the general model (with its complete set of controlvariables). Then, and only then the aggregated modelmay be used in the regional model.

More precisely, this hierarchical transfer from theherd level to the regional level is done following theconstraints below:

• dynamics produced by the aggregated model mustbe as close as possible to the dynamics produced bythe intra-herd model;

• all the relevant variables for between-herd interac-tions modelling (at the regional level) must be in-corporated into the aggregated model; and

• the aggregated model must be as simple as possible.

The aggregated model is first described below, andvalidation results show that this model respects thefirst (and to a large extent the third) of the three con-straints enumerated above. Then, we describe how thisaggregated model is used in the regional-level model,to respect the second of these constraints. Finally, pre-liminary results are shown, which reproduce the dis-ease history in two French Pyrenean sub-regions.

This work is the first published model of between-herd scrapie dynamics.

2. Within-herd disease dynamic

While building up the aggregated model, the mainconstraint we have to satisfy is that all the rele-vant state variables and control parameters for thegeographical-level model be incorporated into the ag-gregated model. Thus, the first problem to solve is thedefinition of this set of state variables and control pa-rameters. Then, the second problem is to write downequations describing the dynamics of these variables,satisfying two contradictory constraints: these equa-tions must be as simple as possible, but the dynamicsmust be as close as possible to the intra-herd modeldynamics.

The geographical-level dynamics of the disease isthe outcome of: (i) within-herd disease dynamics; (ii)interactions between herds; and (iii) control measures.Therefore, in order to solve the first problem we haveto enumerate between-herd interactions at the regionallevel, and also the control measures we want to model.

Most of animal disease control plans are based onthe identification of the infected herds, where specific

B. Durand et al. / Ecological Modelling 179 (2004) 515–531 517

control measures are applied. There is currently no val-idated laboratory test to detect the disease on a livinganimal: all diagnostic tests are based on the examina-tion of a specific part of the brain of suspect animals(Schreuder et al., 1997). Several new approaches havebeen proposed (Race et al., 1998; Schreuder et al.,1998) but they are not yet applicable for routine di-agnosis. Thus, realistic control measures (henceforthcontrol measures currently used in France) are basedon the observed mortality level. The aggregated modelmust incorporate the relevant variable(s) in order tocalculate these mortality levels.

We chose to model two kinds of between-herd in-teractions: genetic interactions and infectious interac-tions. The former are the male genetic flows betweenherds located in the modelled area. The latter are thecontamination of healthy herds by infectious herds fol-lowing disease contamination paths. At the geograph-ical level, we assume that the modelled area is closedfor both kinds of interactions.

Male genetic outflow from a given herd representsthe sale of rams, either born in this herd or born else-where but purchased earlier from another herd. In bothcases, half of the genetics of males results from the fe-male genetics of some herd. Thus, the female geneticstructure has to be incorporated into the aggregatedmodel. On the other hand, herds’ male genetic inflows– which are control parameters in the intra-herd model– must also be control parameters of the aggregatedmodel, the dynamics of which will be defined in thebetween-herd interactions model.

Infectious flows between herds also originate fromherds (where infectious animals are). Thus, some statevariable indicating the herd infection level has to beincorporated into the aggregated model.

To define the state variables and the control param-eters of the aggregated model, we chose to simplifythe intra-herd model neglecting one of the three in-dividual status dimensions it defines (the age of ani-mals) and simplifying the two others (their genotypesand health states).

Equations describing the dynamics of these statevariables were directly derived from the intra-herdmodel, neglecting highest degree terms.

The aggregated model is first formally describedbelow. Secondly, we explain how its parameters areestimated. Finally, validation results are shown, for agiven set of estimated parameters.

2.1. Model description

2.1.1. State variablesA herd is described using three state variables defin-

ing the genetic and health status of females (ewes andlambs) at the beginning of yeart.

The health status is described using a unique vari-able:

It : proportion of infected females

In the first part of this two-part article, the geneticstatus of animals was described using a three classesallelic model (resistance alleles denoted byα, hy-persusceptibility alleles denoted byβ, and intermedi-ate alleles denoted byω), which defines six classesof genotypes (fromα–α to β–β). In the aggregatedmodel, the description of this genetic structure is sim-plified and described using only two variables:

αt : proportion of resistance alleles among femalesβt : proportion of hypersusceptibility alleles among

females

These two variables describe completely the geneticstructure of a herd, at the allelic level, the proportionof intermediate alleles among females at the beginningof yeart being 1− αt − βt .

Two other variables, the dynamics of which is de-fined at the regional level, define the male genetic flowentering inside the herd during yeart (through ramsand artificial insemination):

αt : proportion of resistance alleles in the male ge-netic inflow

βt : proportion of hypersusceptibility in the male ge-netic inflow

Note that these two variables are control variablesfor the aggregated model, but are state variables at thegeographical level. The role of rams, other than thegenetic inflow, is neglected.

2.1.2. Dynamics of the proportion of infected animalsWe assume that the herd size, at the beginning of

each year, is maintained constant and equal to a theo-retical herd size denoted byN.

The dynamics of the proportion of infected femalesis described by the following relation:

https://www.researchgate.net/publication/13902180_Control_of_scrapie_eventually_possible?el=1_x_8&enrichId=rgreq-31fc28b4-d861-4181-b62f-ed885674870f&enrichSource=Y292ZXJQYWdlOzI1ODEwNTIzNTtBUzo5OTczMDk0NDE2Nzk0NUAxNDAwNzg5MTE4NjE5


It+1 = 1

N(xt − yt + zt) (1)

wherext is the number of infected females at the endof year t (before lambing and culling),yt the numberof infected animals culled at the end of yeart, andztthe number of infected ewe–lambs (born at the end ofyeart) introduced in the herd.

The proportion of infected animals at the end ofyear t (before lambing and culling) is:It + AtIt(1 −It) − MtIt , whereIt is the proportion of infected ani-mals and 1 –It is the proportion of susceptible animalsat the beginning of yeart. The expressionAtIt(1− It)

represents the proportion of horizontally infected an-imals during yeart (At being the horizontal transmis-sion parameter), andMtIt is the proportion of animalsdead of scrapie during yeart (Mt being the mortalityrate of infected animals due to scrapie).At andMt de-pend both upon the proportion of hypersusceptibilityand resistance alleles at the beginning of yeart:

At = a1 + a2αt + a3βt

Mt = m1 + m2αt + m3βt

Therefore, the number of infected females at theend of yeart, before lambing and culling (i.e.xt),is the product of the number of remaining animalsN(1 − MtIt) and the proportion of infected animalsdefined above:

xt = N(1 − MtIt)(It + AtIt(1 − It) − MtIt) (2)

The number of infected animals culled at the endof year t (i.e. yt) is the product of the culling rate ofinfected animals (denoted byQt) and the number ofthese animals at the end of yeart, before lambing andculling (xt):

yt = Qtxt (3)

Qt depends upon the yearly mortality rate of in-fected animals due to scrapie:

Qt = q1 + q2Mt

The proportion of infected ewe–lambs born at theend of yeart is the product of the proportion of in-fected animals at the end of yeart (before lambingand culling) defined above by the vertical transmissionrate (denoted byBt): (It + AtIt(1 − It) − MtIt) Bt .

Bt depends upon the proportion of hypersuscepti-bility and resistance alleles at the beginning of yeart:

Bt = b1 + b2αt + b3βt.

The number of renewal ewe–lambs (infected or not)introduced into the herd at the end of yeart is N(1 −(1 − MtIt)(1 − Rt)), where the expression(1 − (1 −MtIt)(1− Rt)) corresponds to the proportion of deador culled animals (which is equal to the proportion ofrenewal ewe–lambs to maintain a constant herd size),Rt being the culling rate for yeart, which depends uponthe mortality rate of infected animals due to scrapieduring the same year:

Rt = r1 + r2Mt

Therefore, the number of infected ewe–lambs (bornat the end of yeart) introduced in the herd (i.e.zt) isthe product of the proportion of infected ewe–lambsborn at the end of yeart by the number of renewalewe–lambs introduced into the herd at the end of yeart:

zt = (It + AtIt(1 − It) − MtIt)

×BtN(1 − (1 − MtIt)(1 − Rt)) (4)

Finally, substitutingxt , yt and zt in Eq. (1), usingEqs. (2)–(4)allows us to define the dynamics of theproportion of infected females:

It+1 = (It + AtIt(1 − It) − MtIt)((1 − Qt)

× (1 − MtIt) + Bt(1 − (1 − Rt)(1 − MtIt)))

(5)

2.1.3. Dynamics of the proportion of resistance andhypersusceptibility alleles

The dynamics of the ewes genetic structure is de-fined usingEqs. (6) and (7). The last terms of eachEqs. (6) and (7)represent the convergence of the ewesgenetic structure towards the male genetic structure,g being a control parameter for this convergence. Thenegative terms of each equations denote the influenceof the disease on the ewes genetic structure, due tothe higher (resp. lower) mortality rate of animals har-bouring hypersusceptibility (resp. resistance) alleles.Ut andVt are the parameters which control this influ-ence. Both depend upon the proportion of hypersus-


ceptibility and resistance alleles at the beginning ofyeart:

αt+1 = αt − UtαtIt + g(αt − αt) (6)

βt+1 = βt − VtβtIt + g(βt − βt) (7)

with

Ut = u1 + u2αt + u3βt

Vt = v1 + v2αt + v3βt

whereUt is the mortality differential for resistance al-leles,Vt the mortality differential for hypersusceptibil-ity alleles,g, u1, u2, u3, v1, v2, v3 are control variables.

2.1.4. Output variablesFinally, the three state variablesIt , αt , andβt allow

us to calculate, for each time step, the values of threeoutput variables. These variables are state variables inthe between-herd disease diffusion model:

• νt is the proportion of infected ewe–lambs born atthe end of yeart (which may, if they are sold, con-taminate the purchaser herd).

• χt is the proportion of excreting infected animalsat the beginning of the second semester (whichmay contaminate animals of other herds on grazinggrounds during summer).

• σt is the mean susceptibility of animals (the propor-tion of exposed animals getting infected).

The values of these variables are calculated usingEqs. (8)–(10):

νt = (It + AtIt(1 − It) − MtIt)Bt (8)

χt = It(x1 + x2αt + x3βt) (9)

σt = s1 + s2αt + s3βt (10)

wherex1, x2, x3, s1, s2 ands3 are control parameters.

2.2. Parameter estimation

The values ofs1, s2 and s3 are directly obtainedfrom the control parameters of the intra-herd model,using:s1 = τω−ω

1 , s2 = τα−α2 − τω−ω

1 , s3 = τβ−β

1 −τω−ω

1 , whereτω−ω1 is the transition rate for intermedi-

ate homozygotes, for the transition from state S (non-infected) to state E (non-excreting infected),τα−α

1 is

the value of the same rate for resistant homozygotes,andτ

β−β

1 its value for hypersusceptible homozygotes.Other parameters (a1, a2, a3, m1, m2, m3, b1, b2, b3,

q1, q2, r1, r2, g, u1, u2, u3, v1, v2, v3, x1, x2, x3) areestimated using linear regressions from data obtainedwith the intra-herd model.

The structure of the male genetic inflow is one ofthe most important parameters of the intra-herd model.In order to sample correctly the domain of this malegenetic inflow, the data we used for the estimation re-sults of a unique long duration simulation (500 years)of a single flock (without between-herd transmission),during which the male genetic inflow varied.

The value of the parameterg depends only on theculling structure (the variation of culling rate withage). The values of the other parameters depend alsoon the values of the other parameters of the intra-herdmodel.

The parameters estimation and the numerical simu-lations described below were conducted using a Scilabcomputer program (Scilab Group, 2000).

2.3. Model validation

The epidemiological status of sheep harbouring re-sistant genotypes in infected flocks is not known: ei-ther these animals are really uninfected (they do notcarry the scrapie agent), else they are preclinically in-fected animals, with a long incubation period (possi-bly longer than the life span of these animals). In thelatter case, two hypotheses arise again: either theseanimals are infected (they carry the scrapie agent) butnot infectious, else they are infected and may also beinfectious (they may excrete the scrapie agent and con-taminate healthy animals). This question is of greatimportance, because of its practical consequences onthe design of scrapie eradication plans.

The existence of preclinically infected animals isa logical consequence of the long incubation periodof scrapie and has been suspected for a long time(Sigurdsson, 1954). Today, the purchase of such ani-mals is considered to be the most common contamina-tion path of healthy flocks (Hoinville, 1996). However,the infectious status of preclinically infected animalshas never been demonstrated, even if experimentalwork on mice models (Dickinson et al., 1975; Bruce,1985) and epidemiological data (Ducrot and Calavas,1998) suggest that this infectious status exists.


Table 1Transition rates between individual health states

Genotype τ1a τ2

b τ3c

α–α (resistant+ resistant) 1 .01 0α–ω (resistant+ intermediate) 1 .01 .0495α–β (resistant+ hypersusceptible) 1 .9505 .0495ω–ω (intermediate+ intermediate) 1 .01 .99ω–β (intermediate+ hypersusceptible) 1 .9505 .99β–β (hypersusceptible

+ hypersusceptible)1 1 .99

a Proportion of exposed animals getting infected.b Transition rate from theE state (non-excreting infected) to the

I state(excreting infected), per semester (six months).c Transition rate from theI state to theR (dead animals) state,

per semester (death of infected animals).

We chose to model this hypothesis (the infectiousstatus of preclinically infected animals), using themain control parameters of the intra-herd model: thetransition rates between health states denoted byτ

x−y

1(the susceptibility level for the genotypic groupx–y,i.e. the transition rate from state S—non-infected—tostate E—non-excreting infected),τ

x−y

2 (the transitionrate from state E to state I—excreting infected—forthe genotypic groupx–y), andτ

x−y

3 (the mortality ratefor the genotypic groupx–y).

The values of these transition rates are given inTable 1. They model an hypothesis following which:(i) exposed animals always get infected (τ

x−y

1 is al-ways 1); (ii) homozygous intermediate individualsrarely become excreting infected (τω−ω

2 is low) and,if it is the case, die quickly (τω−ω

3 is high); (iii)once they are excreting infected, sheep having a re-sistance allele become sick much slower than othersdo (τα−y

3 � τ{ω,β}−{ω,β}3 ); and (iv) once they are

Table 2Definition and parameterisation of the aggregated model parameters

Notation Description Estimated values

At = a1 + a2αt + a3βt Horizontal transmission rate a1 .20 a2 .41 a3 −4.49Mt = m1 + m2αt + m3βt Mortality rate of infected animals m1 .02 m2 −.06 m3 .73Qt = q1 + q2Mt Culling rate of infected animals q1 .33 q2 −1.52Bt = b1 + b2αt + b3βt Vertical transmission rate b1 −.10 b2 .36 b3 1.12Rt = r1 + r2Mt Culling rate for the whole herd r1 .22 r2 −.28Ut = u1 + u2αt + u3βt Mortality differential for resistance alleles u1 −.02 u2 .09 u3 −.58Vt = v1 + v2αt + v3βt Mortality differential for hypersusceptibility alleles v1 .27 v2 −.04 v3 −.17g Convergence of ewes genetic structure

toward male genetic structure.13

χt = It(x1 + x2αt + x3βt) Proportion of excreting animals x1 −.05 x2 .22 x3 1.39σt = s1 + s2αt + s3βt Mean susceptibility s1 1.0 s2 .0 s3 .0

non-excreting infected, individuals having an hyper-susceptibility allele become excreting infected fasterthan others do (τβ−y

2 � τ{α,ω}−{α,ω}2 ).

We assume that five animals are exposed to infec-tion by each excreting infected animal during lamb-ing (K1 = 5) and that outside of the lambing period,one animal is exposed to infection by each excretinginfected animal (K2 = 1).

The estimated values of the parameters are pre-sented inTable 2.

The adequacy of the fit is evaluated comparingthe dynamics obtained from the intra-herd modeland from the aggregated model respectively, for longduration simulations (500 years), in a given flock(without between-herd transmission) during whichthe male genetic inflow varies: from an initial sit-uation where this male genetic inflow is composedof 25% of resistance alleles, 25% of hypersuscepti-bility alleles (and of 50% of intermediate alleles), asequence of genetic compositions is generated usingrandomly chosen yearly variations.

The values ofI, M, α andβ were measured at thebeginning of each simulated year using the intra-herdmodel on the one hand and the aggregated model onthe other hand. The comparisons between these dy-namics are shown inFigs. 1 and 2.

Results show that the aggregated model dynamicsare close to the dynamics obtained from the intra-herdmodel. Depending on the randomly generated malegenetic inflow, simulations may show a persistentdisease (Fig. 1), with varying mortality levels. Aninteresting point is that the disease may even becomeinapparent for long time periods (seeFig. 1: 21 years


Fig. 1. Comparison of the dynamics obtained from the aggregated model with the data obtained from the intra-herd model: persistence ofthe disease over a long time period. Crosses: data obtained from the intra-herd model, solid lines: dynamics calculated using the aggregatedmodel.

Fig. 2. Comparison of the dynamics obtained from the aggregated model with the data obtained from the intra-herd model: extinction ofthe disease. Crosses: data obtained from the intra-herd model, solid lines: dynamics calculated using the aggregated model.


in a row without any clinical case, from year 192 toyear 213). The disease may also completely disappearduring a simulation (Fig. 2). Similar results have beenfound in another within-herd stochastic simulationmodel (Webb and Hoinville, 2000).

3. Modelling inter-herd scrapie transmission

We model the regional dynamics of the disease asthe outcome of genetic fluxes between herds and ofinfectious fluxes originating from infected herds.

The former are modelled as the outcome of theactivity of some zootechnical selection structure ofbreeders, controlling male genetic fluxes betweenherds. The latter are between-herd contacts allowingdisease transmission.

As suggested by early reports (Stockman, 1913;Mc Fadyean, 1918), the most common contaminationmode of healthy herds is the purchase of preclinicallyinfected sheep (Sigurdarson, 1991; Hoinville et al.,2000). Experimental evidence shows also that the dis-ease can be transmitted by contact between infectedand healthy animals (Dickinson et al., 1964, 1974;Brotherson et al., 1968). Such results, together withepidemiological data (Chatelain et al., 1983; Sigurdar-son, 1991; Ducrot and Calavas, 1998) suggest that thedisease could be transmitted from infected to healthyflocks by contact between animals on grazing grounds.

The scrapie agent (and the other TSEs agents) isknown to be very resistant to chemical and physi-cal inactivation agents, both under natural conditions(Brown and Gajdusek, 1991), and in decontamina-tion procedures (Taylor, 1989). Epidemiological datahave shown that the agent could remain infectious forup to 3 years, without the presence of any sheep, insheepfolds previously used by heavily infected flocks(Sigurdsson, 1954). Such a contaminated environmentcould thus be a third contamination mode of healthyflocks. It provides also a possible mechanism of dis-ease transmission by contacts between herds.

Finally, the role of rats, dogs and birds have beenevoked to explain some scrapie outbreaks (Hoinvilleet al., 2000).

We chose to model two contamination paths: (i)the purchase of preclinically infected sheep and (ii)the contact between healthy and infected animals ongrazing grounds.

Thus, we neglect the possible reservoir role of theenvironment: unlike the inside-sheepfolds location(Sigurdsson, 1954; Sigurdarson, 1991), there is nostrong evidence of a reservoir role of the environmentfor outdoors locations (pastures). Similarly, we ne-glect the possible role of rats, dogs and birds becauseof the lack of strong evidence.

If every breeder may buy renewal females (toreplace dead animals or because of zootechnicalreasons), the herd exposition to infectious herds ongrazing grounds varies following pastoral habits. Wedistinguish two herd types:

• sedentary herdsuse, during summertime, pasturesin the vicinity of sheepfolds where they are in con-tact with neighbouring herds during the day, butcome back to the sheepfold every night,

• transhumant herdsare moved during summertime toa mountain grazing range where they are in contactwith other herds during 4–5 months.

Thus, during a year, each herd will have two differ-ent localizations: a winter localization (in sheepfolds)where trading takes place (purchase and sell of breed-ing animals) and a summer localization that varies fol-lowing herd type and where contacts between herdstake place.

Between-herd contacts follow obviously a proxim-ity principle: sedentary herds are in contact with neigh-bouring sedentary herds and transhumant herds are incontact with the herds moved to the same summerpasture. It seems reasonable to assume that tradingfollows also a proximity principle: a given herd willrather purchase animals from neighbouring herds, thananimals from more distant herds.

To model between-herd interactions, it is thereforenecessary to represent geographical locations.

We describe below the choices we have made aboutthis spatialization. The equations describing the aggre-gated model are also modified to model between-herdcontagion and genetic flows.

3.1. Spatialization

We consider a population ofM herds labelled from1 toM, and spread inN regions of a given natural geo-graphical area. Three 1-D discrete spaces are defined,corresponding respectively to winter herd localization,


Fig. 3. Schematic representation of the geographical locations of herds.M = 20 (number of herds),N = 2 (number of regions),NE = 4(number of mountain grazing ranges),Pbuf = 2 (size of the buffer areas between regions for vicinity grazing ranges),Pvois = 2 (size ofthe neighbourhood on vicinity grazing ranges).

summer localization of sedentary herds, and summerlocalization of transhumant herds (Fig. 3).

The notations used in the following are the same asthose used inSection 2, with an upper indice charac-terising the herd considered:Ix

t represents variableItfor herd numberx.

3.1.1. Selection structure of breedersThe selection structure of breeders is important for

the regional disease dynamics because it determineshow fast will spread both infection (though the tradeof ewes) and modifications of the genetic structure(through the trade of rams).

We assume that the selection structure is hierarchi-cal with three different levels:

• herds involved in a selection scheme denoted byS1herds,

• herds using the selection results obtained by theformer, denoted byS2 herds,

• other herds, denoted byS3 herds.

The number of herds in each category is supposedto be roughly constant over time.

We suppose that commercial relationships betweenbreeders follow these rules:

• S1 herds buy animals only from otherS1 herds,• S2 herds buy animals only fromS1 herds,• S3 herds buy animals only fromS2 herds.

To each herd we associate a state variableLxt repre-

senting the status of herdx for yeart. Because of thetime scale of the model (several centuries), the cate-gory of a given breeder cannot be considered as beingconstant. Therefore, we also define four control pa-rameters allowing us to compute the yearly transitionfrom one status to another (the allowed transitions –apart of transitions from one state to the same state –being:S1 ↔ S2 ↔ S3):

• Φ1: mean percentage ofS1 herds,• Φ2: mean percentage ofS2 herds,• ϕ1: yearlyS1 → S1 transition rate,• ϕ2: yearlyS2 → S2 transition rate.

3.1.2. Winter localizationPositH the geographic 1-D finite space of sizeM

which represents winter localization of herds. To eachherd x we associate a coordinateHx on this space,with: 1 ≤ Hx ≤ M. This space results in fact fromthe compaction into one ofN sub-spacesH1 to HN,corresponding to theN studied regions.


Inter-herd commercial relations such as purchaseof rams and ewes are done in this space: we assumethat the breeding herd is chosen on average in thevicinity of the buying herd. We noteCT

x the winterlocalization of the selling herd for buying herdx. CT

x

is therefore the nearest herd from herdx (in the H-space) respecting conditions defined inSection 3.1.1for the status of the herdx (S1, S2 or S3). Becauseherds status may change at every time step,CT

x mayconsequently be also modified.

3.1.3. Summer localization on vicinity grazing rangesPosit P the geographic 1-D finite discrete space

which represents summer localization of herds graz-ing in vicinity grazing ranges (VGRs). This space in-cludesN sub-spacesP1 to PN, corresponding to theNstudied regions. We suppose inter-herd contacts to bemore frequent within each region—to model this hy-pothesis, we defineN − 1 empty (herdless) buffer re-gions each of lengthPbuf at the interfaces between theN regions. To each herdx, we associate a coordinatePx on this space, with: 1≤ Px ≤ M + (N − 1)Pbuf.

We define a mapping functionFP from winter lo-calizations to summer localizations on VGRs, with,for each sedentary herdx: FP (Hx) = Px.

On a grazing range, each herd is in contact with itsneighbours inP. If Pvois is the size of this neighbour-hood, we can define the setCP

x of sedentary herdswhich herdx will be in contact with on its grazingrange:CP

x = {y, |Px − Py| ≤ Pvois andy �= x}.

3.1.4. Summer localization on mountain grazingranges

Posit NE the number of mountain grazing ranges,andE the geographic 1-D finite discrete space of sizeNE which represents the localization of mountain graz-ing ranges (MGRs). To each herdx, we associate acoordinateEx on this space, with: 1≤ Ex ≤ NE.

We define a mapping functionFE from winter lo-calizations to summer localizations on MGRs, with,for each transhumant herdx: FE (Hx) = Ex.

We suppose that MGRs are closed: on a given MGR,each herd will be able to encounter only those herdswhich are located on the same MGR. We can thereforedefine a setCE

x of transhumant herds which herdxwill be in contact with on its mountain grazing range:CE

x = {y, Ey = Ex andy �= x}.

3.2. Between-herd contagion

We consider two modes of herd contamination: thepurchase or contaminated ewes and the contact with aninfected herd either on vicinity pastures or on moun-tain grazing rangesEq. (11)is completed using termswhich represent these two contamination paths:

Ixt+1 = (Ix

t + Axt I

xt (1 − Ix

t ) − Mxt I

xt )

×((1 − Qxt )(1 − Mx

t Ixt )

+Bxt (1 − (1 − Rx

t )(1 − Mxt I

xt )))

+1xt pν(ν

yt − νx

t ) + κσxt (1 − Ix

t )∑i∈Cx

χit

#Ci

(11)

where:

1xt is the indicating variable thus defined:{

1xt = 1 if and only if herd x buys ewes during yeart

1xt = 0 otherwise

herdx buys ewes from herdy: y = CTx .

where:

pν is the number of ewes bought over the total num-ber of ewes;

κ is the contagion parameter on vicinity and moun-tain grazing ranges;

Cx is the set of herds which herdx encounters onvicinity and mountain grazing ranges:

for sedentary herds,Cx = CPx ; and

for transhumant herds,Cx = CEx

#Cx is the number of these herds for herdx;χit is the proportion of excreting infected animals atthe beginning of the second semester, for herdi(Eq. (9)).

3.3. Between-herd genetic flows

Eqs. (12) and (13)are modified to include the con-sequences of buying ewes on the genetic herd compo-sition:

αxt+1 = αx

t − Uxt α

xt I

xt + g(αx

t − αxt ) + 1x

t pν(αyt − αx

t )

(12)

βxt+1 = βx

t − Vxt αx

t Ixt + g(βx

t − βxt ) + 1x

t pν(βyt − βx

t )

(13)


For each herd, we define two additional state vari-ables:

• αxt : proportion of resistance alleles in rams of herd

x, used as reproductives during yeart; and• βx

t : proportion of hypersusceptibility alleles in ramsof herdx, used as reproductives during yeart.

We also define two variables describing the geneticstatus of rams from the artificial insemination center(AIC):

αIAt : proportion of resistance alleles among AICrams in yeart; and

βIAt : proportion of hypersusceptibility alleles amongAIC rams in yeart

The male genetic inflow in herdx during yeart isthen defined as follows:

αxt = αx

t (1 − λxt ) + λx

t αIAt (14)

βxt = βx

t (1 − λxt ) + λx

t βIAt (15)

whereλxt is the proportion of new-born ewes kept in

herdx borne from ewes inseminated from AIC duringyeart.

We suppose that the AIC uses exclusively ramscoming fromS1 herds (herds involved in a selectionscheme). The dynamics of resistance and hypersus-ceptibility among AIC rams is then given by:

αIAt+1 = αIA

t (1 − ρIA ) + ρIA

∑i∈S1t

αit

#S1t

(16)

βIAt+1 = βIA

t (1 − ρIA ) + ρIA

∑i∈S1t

βit

#S1t

(17)

whereρIA is the yearly renewal ratio of AIC rams,S1t

is the set ofS1 herds on yeart and #S1t is the numberof these herds.

We suppose the yearly culling rate of rams (death orslaughter of old rams) and the yearly replacement rateof rams (global outgoing) to be constant and identicalfor all herds. On a given year, a breeder can eitherbuy rams from outside, or (exclusively) add youngrams born in his herd. Rams which are bought froman external herd can be either juvenile rams born inthis external herd or having been themselves boughtpreviously from another herd.

The genetic evolution of rams in a given herd isthen computed according to:

αxt+1 = αx

t (1 − ρT ) + 2xt ρT ((1 − ρP)α

yt

+ ρPαyt − αx

t ) + (1 − 2xt )ρT (αx

t − αxt ) (18)

βxt+1 = βx

t (1 − ρT ) + 2xt ρT ((1 − ρP)β

yt

+ ρPβyt − βx

t ) + (1 − 2xt )ρT (βx

t − βxt ) (19)

where:

2xt is the indicator variable defined as:

2xt = 1 if and only if, during yeart, herdx

buys external rams2xt = 0 otherwise

ρT is the yearly replacement rate of rams (the meanproportion of rams a breeder sells each year:culling of old rams or sale of young rams to an-other breeder);

ρP is the yearly culling rate of rams (the mean pro-portion of rams culled each year, usually becausethey are too old). Note that the culling rate is in-cluded in the replacement rate. Thus, we mustalways have:ρP < ρT .

Herdx buys replacement rams from herdy: y = CTx .

4. Model exploitation

Scrapie has been known in France for a very longtime: the first description of this disease was pub-lished in 1732 (Pattison, 1988, citing a monograph ofM’Gowan, 1914); we can also quote Girard (1830).Since 1996, scrapie has been a notifiable disease inFrance. A surveillance network was created to detectaffected herds (Anonymous, 1997). This network hasshown that the most affected area was located in south-western France, in the department named Pyrénées-Atlantiques (PA) (Calavas et al., 1998).

PA is composed of two neighbouring regions sepa-rated by natural barriers (mountains): the Béarn (east-ern) and the Pays-Basque (western). Scrapie historyin PA is singular because the disease has successivelyaffected these two regions. In the 1950s, it seems thatthe disease was endemic in the Béarn but was notpresent in the Pays-Basque. Later, the disease progres-sively disappeared in the Béarn and moved westwardsto the Pays-Basque. Today, the majority of the scrapie


affected herds are located in the Pays-Basque region,where the disease is enzootic.

We apply the model to a situation similar to thesituation of the PA with two separated regions. Resultsshow that the model is able to reproduce the history ofscrapie in the PA, with two epidemic waves, affectingsuccessively the two regions.

4.1. Control parameters

We simulate a geographical area with 500 herds, di-vided in two regions, each containing half of the pop-ulation. We assume that in both regions, 50% of herdsare sedentary herds and 50% of herds are transhumantherds. For each herdx, initial valuesαx

0, βx0, αx

0 and

βx0 are randomly generated assuming a regional pro-

portion of hypersusceptibility and resistance alleles of20%. At t = 50, the disease is introduced in a uniqueS1 herd (a herd involved in a selection scheme) of thefirst region. The duration of a simulation is 400 timesteps (400 years).

The mapping functionFP from winter localizationsto summer localizations on VGRs ensures that herdsappear inP following the same order as in H: the co-ordinates of herds inP1 are identical to their coordi-nates inH1; for herds inPi (i > 1), coordinates areshifted to take buffer zones into account.

We assume that herds located in a given regionare moved to MGRs located somewhere in the sameregion. The mapping functionFE from winter local-izations to summer localizations on MGRs performsrandom permutations inside eachHi sub-spaces.TheseN random permutations are then compactedinto one space. Dividing this space intoNE parts ofequal size allows us to calculate coordinates onE.

We use the estimated values of the aggregated pa-rameters presented inTable 2. The values of the othercontrol parameters are listed inTable 3. They werederived using field data (Arranz, 1997) and experts’opinions.

The variable 1xt indicating if herdx buys ewes attime stept is randomly calculated at each time step,ensuring that the mean time lag between two time stepswith ewes purchase is of 5 years. The correspondingvariable for rams purchase 2x

t is set to one for all herdsand for the whole duration of the simulation (everyherd buys rams every year).

Table 3Values of the regional model control parameters

Notation Value Description

M 500 Number of herdsN 2 Number of regionsΦ1 .1 Mean percentage ofS1 herdsΦ2 .45 Idem forS2 herdsϕ1 .9 YearlyS1–S1 transition rateϕ2 .5 Idem forS2 herdsPbuf 5 Size of the between-regions empty

buffer zonesPvois 10 Neighbourhood size for contacts on

vicinity grazing rangesNE 25 Number of mountain grazing rangespv .04 Number of ewes bought over the

total number of ewesκ .005 Contagion parameter on vicinity

and mountain grazing rangesρT .25 Yearly replacement rate of rams

(sale and culling)ρP .125 Yearly culling rate of rams

The simulation is run without AIC which became acommon practice only recently.

4.2. Global dynamics

The time evolution of the yearly prevalence ratio(the percentage of infected herds) shows a bimodalshape, indicating two successive epidemic waves(Fig. 4a). The cumulative of the incidence ratio (thenumber of newly affected herds per year over thenumber of herds) also shows such a succession, withtwo distinct inflexion points (Fig. 4b).

Moreover, the cumulative of the incidence ratioshows that, at the end of the simulation, almost allof the herds have been infected, assuming a herd isconsidered as infected if one animal (or more) died ofscrapie during the simulation. If we consider a herdto be infected if at least five (or 10) animals died ofscrapie in a given year, the prevalence and incidenceratios are much lower (Fig. 4).

The time evolution of the hypersusceptibility andresistance alleles percentages in the simulated pop-ulation shows (Fig. 5) a progressive decrease of thehypersusceptibility allele and a slight increase of theresistance allele: scrapie induces a counter-selectionpressure over hypersusceptibility alleles and these al-leles are progressively replaced by intermediate andresistance alleles. Due to the fact that, at the beginning


Fig. 4. Time evolution of the yearly prevalence ratio and of the cumulative incidence ratio. (a) Yearly prevalence ratio, and (b) cumulativeincidence ratio.s =1: a herd is considered as infected if at least one animal dies of scrapie in a year,s = 5: a herd is considered asinfected if at least five animal die in a year,s = 10: a herd is considered as infected if at least ten animal die in a year.

Fig. 5. Time evolution of the alleles percentage in the simulatedgeographical area.α: resistance allele,β: hypersusceptibility allele,1 − α − β: intermediate allele.

at the simulation, the intermediate allele is the mostprevalent allele (60%) this replacement benefits moreto these alleles than to the resistance alleles.

It is interesting to note that, despite the fact that mostof the simulated herds are affected, hypersusceptibility

Fig. 6. Dynamics of the number of infected (a) and dead (b) animals in space (x-axis: longitudinal positions at wintertime) and time(y-axis: years), in two simulated regions, after the introduction of the disease atx = 0. Vertical dashed line: separation between the tworegions, horizontal dashed line: year of introduction of the disease at the locationx = 0 (arrow).

alleles do not disappear completely, but remain presentat a low rate.

4.3. Spatial dynamics

The spatiotemporal dynamics of the number of ani-mals dead of scrapie (Fig. 6a) shows that the epidemicsfirst spreads in the left region (where the disease hasbeen introduced at the 50th time step) and progres-sively disappears there before to touch the right regionwhere it spreads again and finally disappears. Thus,the bimodal shape of the prevalence ratio (Fig. 4) isexplained by two spatially distinct epidemic waves.This result is in agreement with scrapie history in thePA.

From the introduction of the disease in the left re-gion to the last death in the right region, the epidemiclasts approximately 270 years. Within each of the tworegions, epidemics lasts approximately 150 years.


Fig. 7. Dynamics of the percentage of resistance (a) and hypersusceptibility (b) alleles in space (x-axis: longitudinal positions at wintertime)and time (y-axis: years), in two simulated regions, after the introduction of the disease atx = 0. Vertical dashed line: separation betweenthe two regions, horizontal dashed line: year of introduction of the disease at the locationx = 0 (arrow).

If we consider that herds are known to be infectedonly if more than five animals die in a year (takinginto account only the darkest areas inFig. 6b), thisduration is approximately shorter by a factor of 2.

It is also interesting to note that infection takes amuch longer time than mortality before disappearing(Fig. 6a). There is a silent period for each herd, afterthe last death, during which the herd remains infected.This silent period also exists at the regional level: foreach region, there are still infected herds decades afterthe disease has disappeared.

Spatiotemporal dynamics of hypersusceptibility al-leles and of resistance alleles (Fig. 7) show both theeffect of the selection structure and the effect of thegenetic component of the epidemic waves.

At the very beginning of the simulation, the geneticstructure of herds (randomly chosen) is very hetero-geneous. Before the introduction of the disease andthe appearance of a selective pressure,Fig. 5 showsfor both alleles a roughly constant global percentage.However,Fig. 7 shows the appearance of spatiotem-poral clusters with more or less resistance and/or hy-persusceptibility alleles. These clusters are due to theworking of the selection structure and to the fact thatrams are purchased from herds located in the vicinityof wintertime localizations.

After the 50th year, the selective pressure inducedby scrapie applies successively in the two regions andinduces an increase of resistance alleles and a de-crease of hypersusceptibility alleles. Within each re-gion, these modifications are parallel to the apparitionof the epidemic wave and they explain the decrease of

the epidemic wave: the counter-selection of hypersus-ceptibility alleles and the selection of resistance andintermediate alleles induces an increasing resistanceof herds and, finally, the extinction of the disease.

5. Discussion

We have presented a regional model of scrapiespread. The intra-herd component of this model isobtained from the model presented in (Sabatier etal., 2002) using hierarchical transfer: an aggregatedintra-herd model is derived from the complete intra-herd model, reducing carefully the number of controland state variables. The parameters of the aggregatedmodel are then estimated using simulations resultsproduced by the complete model. Comparisons ofthe dynamics produced by both models show thatthe aggregated model fits quite closely the completemodel. This aggregated model may then be used in aregional-level model of scrapie spread.

This regional-level model has been designed fol-lowing a mechanistic approach, with a relatively de-tailed analysis of the processes underlying scrapie re-gional diffusion: between-herd genetic fluxes (whichdepend on the selection structure) and between-herdinfectious fluxes (sale of animals and contacts on pas-tures). Control parameters will allow us to simulate ina sensitivity analysis study the effect of various controlmeasures: genetic selection, slaughter, animal move-ments restrictions etc. (work in progress).


5.1. Discussion of the aggregation method

The problem of scale change in physics is wellknown: switching from the dynamics of a particle tothe behaviour of a large collection of particles gavebirth to statistical physics which is one of the mostactive branches of present day physics. It is oftentempting to use these methods when modelling com-plex systems in life sciences, and many interestingresults have been obtained using such an approach.However, it is not always possible because: (i) ob-jects in life sciences tend to be more complex than amolecule or a material point, and (ii) the collectivebehaviour of interest does no involve a very largenumber of individual objects, thus rendering meanfield approaches irrelevant. A possible method con-sists in transferring only the relevant state variablesfrom the individual to the collective level. The so-called “aggregation methods” have been developedto deal with this class of problems. They have beenused for a long time, and many interesting recentdevelopments could be quoted—see, e.g.Iwasa et al.(1987), Iwasa et al. (1989), Auger and Roussarie(1994). We used a pragmatic approach, firstly iden-tifying the variables which were likely to have animpact at the regional level, then deducing how thesevariables were related to the state variables at the herdlevel. For instance, instead of the distribution func-tions of the genotypes, the percentage of resistanceand hypersusceptibility alleles seemed to containall the relevant information on the flocks genetics.The time evolution of the aggregated variables wasderived from intuitively obvious simple relations,and the ensuing analytical aggregated model wastested versus the complete herd model. The aggre-gation error (in the meaning ofO’Neill and Rust,1979) was then verified to be negligible.

5.2. Discussion of the results

Between-herd infection paths taken into accountare the trade of breeding animals and the contact be-tween herds on pastures (vicinity or mountain grazinggrounds).

Preliminary exploitation results show how themodel is used, starting from individual-level hypothe-ses to study their consequences on the regional dis-ease dynamics. The main individual-level hypothesis

we have made is related to the epidemiological statusof infected animals harbouring resistant genotypes:today this status is not known, and we have assumedthat some of these infected animals could also be in-fectious during long periods before the apparition ofthe disease (or the culling of these animals).

The time scale used in the regional model is ofthe order of several centuries. The scrapie surveil-lance network has been created in France in 1996,only 4 years ago. Quantitative epidemiological dataare thus very sparse and cover only a very shorttime series compared to those simulated using theregional model. However, empirical data exist aboutthe history of scrapie in some French regions. Pre-liminary exploitation results show that the regionaldynamics obtained from the model (and based on thepreceding individual-level hypothesis) are in agree-ment with these data: simulations reproduces twoepidemic waves which were observed in two Frenchsouth-eastern regions, Béarn and Pays Basque.

On the one hand, as it has been recently observedin the latter region, disease is predicted to spread rel-atively fast within a region, because infectious fluxesare both local (vicinity grazing ranges) and distant(trade of ewes, mountain grazing ranges). On the otherhand, disease is predicted to spread slowly betweenregions, because contacts between herds are less fre-quent.

As it has been observed in the Béarn region, themodel predicts that within a region, the disease dis-appears after the epidemic wave: the mortality differ-entials for resistance (Ut) and hypersusceptible alleles(Vt) model a selective pressure induced by the dis-ease on the former allele, and a counter selection ofthe latter allele. Because the horizontal and verticaltransmission parameters (At andBt) depend upon therespective proportion of the susceptibility alleles, dis-ease transmission decreases when the hypersuscepti-bility alleles proportion decreases. Moreover, becausethe mortality rate (Mt) also depends upon the respec-tive proportion of the susceptibility alleles, the clin-ical disease is predicted to disappear faster than in-fection, which may persist several years after the lastcases.

It is interesting to note that neither hypersusceptibil-ity alleles, nor intermediate alleles disappear, despitethe fact that resistance alleles have a selective advan-tage over them. Furthermore, after the disease has


disappeared, the predicted proportions of the differ-ent alleles are relatively close to the values observedtoday in the Béarn region on AIC rams (Aguerre,1998, unpublished document), with a persistence ofhypersusceptibility alleles at a low level, and sim-ilar proportions for the intermediate and resistancealleles.

The predicted cumulative incidence ratio obtainedusing the model is very high at the end of the simu-lations, with almost all herds having been infected.Such a high value may not seem plausible: at thenational level, postal surveys conducted in GreatBritain (Hoinville et al., 2000) and in the Netherlands(Schreuder et al., 1993) showed that 15% (resp. 6%)of farmers thought that at least one case of scrapiehad occurred in their flocks.

However, scrapie symptoms are not always typi-cal: Clark and Moar (1992), in an epidemiologicalstudy conducted in the Shetlands isles showed that,in infected flocks, 26% of found dead animals werescrapie-positive, and that 16% of affected animalsdid not exhibit clinical signs. Therefore it seemsreasonable to think that, if there are only few casesin a herd, the disease may not be suspected. If weconsider a herd to be infected if at least five (orten) animals died of scrapie in a given year, theprevalence and incidence ratios are much lowerand are more representative of epidemiological data(Fig. 4).

6. Conclusions

Scrapie has been known for a long time, but manyquestions about this disease remain unanswered. Wementioned above the problem of the epidemiologicalstatus of infected animals harbouring resistant geno-types. Another important question is related to thebetween-individuals contamination paths. The diseasetransmission by contact between animals has beenexperimentally demonstrated. Some biological mech-anisms underlying this transmission have also beendemonstrated (e.g. the ingestion of placentasPattisonet al., 1972). Other mechanisms are under study inresearch labs (e.g. nematodes, hay mitesWisniewskiand Sigurdarson, 1996; Carp et al., 2000). How-ever, neither the relative importance of these mecha-nisms in disease transmission under field conditions,

nor the dynamics of individual infectiousness areknown.

The preceding problems are related to the individ-ual level. Other questions are related to the regionallevel and arise with the design of eradication plans.Some countries have tried to eradicate scrapie withlimited success, e.g. USA (Detwiler, 1992) and Ice-land (Sigurdarson, 1991). Only scrapie-free countries(Australia, New Zealand) could eradicate outbreakscaused by imported sheep (Detwiler, 1992). Today,new approaches of scrapie control based on the ge-netic selection of resistant rams are under study inThe Netherlands (Schreuder et al., 1997) and have be-gun in Great Britain (Anonymous, 2000). However,all that can be achieved using such control measuresis the extinction of the clinical disease: in order toobtain the true eradication of the disease (includingthe agent), much more would be required, particularlymore knowledge about the epidemiological status ofinfected individuals harbouring resistant genotypes.

Starting from regional-level problems, we are backto individual-level questions. Problems arising at bothlevels are thus clearly linked together.

The objective of the models we have presented inthis two-part article is precisely to produce a tool al-lowing the handling of such problems. Using this tool,individual-level questions may be modelled as hy-potheses in the intra-herd model, and herd-level ques-tions may be modelled using control parameters of theregional model. These hypotheses may then be testedboth at the herd level and at the regional level. Further-more, because both models are mechanistic models,hierarchical transfer allows us to evaluate the conse-quences at the regional level of hypotheses relating tothe individual level.

Following this direction, it should be possible, usingthese models:

• to exclude some individual-level hypotheses andthus to suggest experimental research subjects;

• to search for the existence of control measures thatare efficient whatever hypothesis is made, and thusto suggest prophylactic orientations both to farmersand to public animal-health authorities.

Such a systematic exploitation remains to be done,using current quantitative data to try deriving prospec-tive results.


Acknowledgements

This work is supported by the Comité françaisd’experts “Prions & ESST”, ACC5, Projet “Etude dela transmission de la tremblante”.

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multiscale modelling of scrapie epidemiology

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