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A modelling framework to describe the spread of scrapie between sheep flocks in Great Britain
Preventive Veterinary Medicine 67 (2005) 143–156 www.elsevier.com/locate/prevetmed
A modelling framework to describe the spread of scrapie between sheep flocks in Great Britain Simon Gubbins* Centre for Epidemiology and Risk Analysis, Veterinary Laboratories Agency, New Haw, Addlestone, Surrey KT15 3NB, UK Received 11 August 2003; received in revised form 29 July 2004; accepted 25 August 2004
Abstract My aim was to develop a stochastic, spatial model describing the spread of scrapie between sheep flocks in Great Britain; I wanted a model, which could subsequently be used to assess the efficacy of different control strategies. The structure of the model reflects the demography of the British sheep flock, including a description of the contact structure between flocks. The dynamics of scrapie were incorporated through two probabilities associated with each flock: of acquiring infection and of experiencing a within-flock outbreak following exposure. The acquisition of infection depends on whether or not a flock buys-in sheep and, if it does, whether or not it trades with an affected flock. Once a flock is exposed, the probability of a within-flock outbreak occurring and its duration depend on the basic reproductive number, the prion-protein (PrP) genotype profile and the flock size. The model was validated using regional data from two postal surveys conducted in 1998 and 2002, which demonstrated that the model captures the spatial dynamics of scrapie (at least at a regional level). Moreover, the predicted distribution for the duration of a within-flock outbreak reflects the duration of outbreaks reported in the literature. Using the model to predict long-term trends in the proportion of affected flocks suggested that, even without control measures beyond the removal of animals with clinical signs, scrapie ultimately will disappear from the national flock, though it is likely to be decades before the disease is eliminated. However, there were scenarios consistent with the available data which suggested that scrapie could remain endemic within the British sheep flock. Consequently, it is essential to take this uncertainty in the long-term dynamics of scrapie into account when considering the efficacy of control strategies. Although control strategies were not explicitly
* Present address: Institute for Animal Health, Compton, Newbury, Berkshire RG20 7NN, UK. Tel.: +44 1635 578411; fax: +44 1635 577237. E-mail address: [email protected]. 0167-5877/$ – see front matter. Crown Copyright # 2004 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.prevetmed.2004.08.007
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examined, the model suggests two aspects important for control: larger flocks remain affected for longer and provide infection for other, smaller flocks and animal movements must be traceable. Crown Copyright # 2004 Published by Elsevier B.V. All rights reserved. Keywords: Transmissible spongiform encephalopathy; Scrapie; Sheep; Stochastic model; Demography; Disease control
1. Introduction Recently, much attention has been given to the epidemiology of scrapie within sheep flocks (Elsen et al., 1999; Baylis et al., 2000, 2002; Redman et al., 2002). A number of mathematical models have been developed to describe the within-flock dynamics and control of scrapie (Stringer et al., 1998; Hagenaars et al., 2000). These models have been used in several ways: to analyse outbreaks (Woolhouse et al., 1998, 1999; Matthews et al., 2001; Hagenaars et al., 2003); to calculate the basic reproductive number for scrapie (Matthews et al., 1999); and to investigate persistence patterns within flocks (Hagenaars et al., 2001). In contrast, relatively little attention has been given to the spread of scrapie between flocks. Simple flock-level models (Gravenor et al., 2001; Kao et al., 2001) have been developed using information collected from an anonymous postal survey conducted in 1998 (McLean et al., 1999; Hoinville et al., 2000). More recently, flock-level models have been developed to investigate the potential spread of BSE in British sheep (Ferguson et al., 2002; Kao et al., 2002). These models describe only changes in the number or proportion of affected flocks over time and do not attempt to describe the spatial distribution of affected flocks or the impact of this distribution on the subsequent spread. However, these factors are important because, first, affected flocks are not uniformly distributed (Hoinville et al., 2000; Sivam et al., 2003) and, second, acquisition of infection is primarily through the movement of infected animals between flocks (Ducrot and Calavas, 1998; McLean et al., 1999; Hoinville et al., 2000; Hopp et al., 2001) which will depend on the flocks’ locations. I developed a stochastic, spatially explicit model to describe the spread of scrapie between sheep flocks in Great Britain (GB) that, subsequently, can be used to assess the impact of different control strategies. The structure of the model broadly reflects the conclusions of a study into risk factors associated with scrapie at a flock level in Great Britain (McLean et al., 1999). Those analyses suggested that a minimal model for the spread of scrapie between flocks should include flock size and the proportion of sheep homebred and examine the interplay between the transmission of scrapie within and between flocks. The risk-factor study identified only the broad structure for a model. Consequently, my aim was to present the modelling approach used in detail. In particular, I identified the demographic and epidemiological data required to describe the sheep population and the key parameters needed to characterise the transmission of scrapie within and between flocks. Of particular importance is a description of the mixing patterns between flocks (that is, who trades with whom). Two approaches previously have been used to address this question in the context of scrapie: one assumes all flocks are equally likely to acquire
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infection (Kao et al., 2001) and the other involves a detailed description of potential pathways for transmission (Webb and Sauter-Louis, 2002). For the model described here, a method of intermediate complexity was developed to describe mixing patterns using data on sheep movements and a simplified description of the British sheep industry.
2. Methods 2.1. Model structure The unit of population in the model is the flock. Flocks are divided into two classes: unaffected (i.e. no animals are infected with scrapie) and affected (i.e. at least one animal is infected with scrapie). Each flock has two probabilities associated with it: (i) of acquiring infection; and (ii) of experiencing a within-flock outbreak following exposure. The scrapie status of each flock is updated on a yearly basis. 2.2. Acquisition of infection The principal mechanism by which infection is acquired is the movement of infected animals between flocks (Ducrot and Calavas, 1998; McLean et al., 1999; Hoinville et al., 2000; Hopp et al., 2001). Consequently, the probability that an unaffected flock j acquires infection in year t is equivalent to the probability that the flock buys-in sheep and trades with at least one affected flock. This is given by: 0 1 Y lj ðtÞ ¼ bj @1 ð1 fWjk ÞA; (1) k 2 IðtÞ
where bj is the probability that flock j buys in sheep, I(t) is a list of affected flocks in year t, f is the probability of transmission between flocks given contact and Wjk is an element of the matrix representing who trades with whom (see Section 2.3). The probability that a flock buys-in sheep (bj) was estimated was estimated by fitting a binomial logistic-regression model to data collected as part of a postal survey conducted in 2002 (Sivam et al., 2003). Independent variables included in the model were region, farm type (hill/upland/lowland), flock type (purebred/commercial) and flock size. Farm type describes the type of land on which sheep are grazed (ranging from the harsh hill environment, through the less-harsh upland areas to the relatively benign lowland areas). Flock type reflects the primary business of the flock: production of ewes or rams for breeding (purebred) or production of lambs for human consumption (commercial). 2.3. Who trades with whom? The British sheep industry is unique and complex; it involves a stratified system which has evolved over hundreds of years to make best use of the land available for sheep farming (Pollott, 1998). Here, a simplified description was used in which the major sectors were the three farm types (Pollott, 1998; Arnold et al., 2002). Sheep were assumed to move within
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each sector (i.e. from hill to hill; upland to upland; and lowland to lowland) but movement between sectors was restricted so that sheep only are moved from hill to upland flocks or from upland to lowland flocks. The principal source of data on sheep movements is statutory movement records kept by all flocks. These record the number of sheep moved and the postal address to or from which the sheep were moved. Few farmers were willing to provide copies of these records. Moreover, photocopying the flock-movement books, computing grid references from postal addresses and entering the records on a database for analysis was time consuming. Consequently, it was possible to collect records from only seven farms (located in different regions of England and Wales) and for a single year (1999, the year in which the study began). Most movements were over relatively short distances (<20 km), but some were over several hundred kilometres (Fig. 1c). The number of different locations to or from which sheep were moved for each farm varied from 6 to 51. The contact structure for flocks is described by the matrix representing who trades with whom (W). An element of this matrix (Wjk) is equal to the probability of moving an animal from flock k to j (cjk) if flock j potentially trades with flock k and zero if it does not. The matrix, W, was constructed as follows. (i) Each farm was assigned a set of movement records drawn uniformly from the seven sets collected. (ii) The number of contacts for each farm was drawn from a Poisson distribution used to summarise the data, the mean of which was equal to the mean number of contacts for the seven sets of movement records. (iii) For farm j, a farm k was selected uniformly and, if farms j and k were of appropriate types to trade, the contact from k to j was accepted (i.e. Wjk = cjk) with probability cjk. (iv) If the contact was accepted and farm k was of the same type as farm j, the contact also goes from j to k (i.e. Wkj = ckj) with probability ckj. (v) Steps (iii) and (iv) were repeated until all contacts between farms had been assigned. The probability of moving an animal from flock k to j (cjk) was assumed to be equal to the probability that a sheep is moved at least a distance x to the farm. Consequently, cjk = 1 Fj(xjk) where Fj is the empirical cumulative distribution function for the movement records used for farm j (Fig. 1c) and xjk is the distance between flocks j and k.
2.4. Within-flock outbreaks When a flock acquires infection, the question arises of whether or not scrapie subsequently will spread within the flock. A range of factors influence the maintenance of scrapie, including farm- and flock-management practices (McLean et al., 1999) and the prion-protein (PrP) genotype profile of the flock (Baylis et al., 2000). Rather than attempt to include all these factors explicitly in the model, a simpler approach was adopted in which within-flock outbreaks are characterised by two parameters: the within-flock basic reproductive number, R0 (see, e.g., Anderson and May, 1986), and the outbreak duration (TE). Differences amongst flocks were incorporated as differences in these parameters. In
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Fig. 1. (a and b) Agricultural census data on sheep flocks in GB: (a) density of sheep flocks. Regions marked are those used in the 2002 postal survey; and (b) distribution of flock sizes (number of breeding ewes and rams over 1 year old). (c) Sheep movements in GB: empirical cumulative distribution functions for movement data from seven farms in GB.
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addition, I assumed that the only action taken to control scrapie is the removal of animals which show clinical signs. Following the introduction of an infected animal to the flock, the probability that a within-flock outbreak occurs is given by: ( 1 R0 > 1 1 Pr ðoutbreakÞ ¼ ; (2) R0 0 R0 1 where R0 is defined above (Diekmann and Heesterbeek, 2000). If an outbreak occurs, its duration (TE) is given by:
TE ¼
R0 G lnðsNÞ; R0 1
(3)
where R0 is defined above, G is the incubation period and N is the flock size (Barbour, 1975). The factor s is the flock-level susceptibility which corrects the flock size to allow for differences in the risk of scrapie amongst PrP genotypes. This was defined to be: X s¼ ri ni ; (4) i
where ni is the proportion of animals of PrP genotype i in the flock and ri is the risk of scrapie in genotype i measured relative to that with the highest risk. When a flock acquired infection, it was assigned values for each parameter (R0, G and s) drawn uniformly from plausible ranges. A range for R0 of 2.5–14.0 was used, which is consistent with an observed within-flock outbreak (Hagenaars et al., 2003) and includes the point estimate of 3.9 computed for a different outbreak (Matthews et al., 1999). Although the age at onset of clinical signs is observed readily, the age at infection is not; consequently, the mean incubation period (G) was difficult to estimate. A range for G of 1.6–5.9 years was used, where the lower limit is an estimate of the incubation period based on a modelling study of a within-flock outbreak (Matthews et al., 2001) and the upper limit is the upper 95% confidence limit for the mean age at onset for 1740 notified scrapie cases. Using data on PrP genotype frequencies for 15 scrapie-affected flocks (Tongue et al., 2004a) and estimates for the risk of scrapie in different PrP genotypes (Tongue et al., 2004b), the flock-level susceptibility (s), was assumed to range from 0.002 to 0.14.
2.5. Sheep flocks in GB The location and size of flocks were obtained from agricultural-census data. Sheep flocks are found throughout GB with the highest densities in the north west and south west of England and in Wales (Fig. 1a). Flock sizes vary from only a few sheep to thousands of sheep with a median flock size of 285 breeding animals (Fig. 1b). Farm and flock type for each holding, which are not collected as part of the census, were drawn from a multinomial distribution in which the probability that a flock is a given type was estimated from the proportion of flocks of each type in a region (derived from the 2002 postal survey).
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2.6. Distribution of affected flocks Scrapie has been a notifiable disease in the UK since 1993. Although statutory notifications provide information on the number and location of flocks which report scrapie cases, there is substantial under-reporting (Hoinville et al., 2000; Sivam et al., 2003). Postal surveys conducted in 1998 and 2002 provide the number of respondents reporting at least one clinical case in their flock during the 12 months preceding the survey stratified by region (Hoinville et al., 2000; Sivam et al., 2003; see Fig. 1a for regions). The results of the 1998 survey were used to set the initial distribution of affected flocks, while those for the 2002 survey were used to validate the model. 2.7. Parameter estimation No independent estimates were available for the probability of transmission between flocks given contact (f). To estimate this parameter, a likelihood-based approach was adopted in which the probability of observing the results of the 2002 postal survey for a given parameter value was estimated. The 2002 postal survey provided the number of affected flocks (Ij) and the total number of flocks (Nj) in region j which responded to the survey. The probability (P(f,vk)) of observing these results for model realization vk and parameter value f is given by: Y Nj Ij Nj Ij Pðf; vk Þ ¼ pj ðvk ; fÞ ð1 pj ðvk ; fÞÞ ; (5) Ij j
where pj(vk,f) is the proportion of affected flocks in region j for that realization. The likelihood (L(f)) is calculated by averaging the probability (5), over all realizations of the model (Gelfland and Smith, 1990). Hence, L(f) is estimated by: LðfÞ
K 1X Pðf; vk Þ; K k¼1
(6)
where K is the number of realizations simulated for each parameter value. 2.8. Implementing the model Exploratory analyses were undertaken to determine the number of realizations needed to provide robust results without being prohibitively expensive (in terms of computation time). For parameter estimation, the likelihood for each value of f was calculated by simulating 5000 realizations of the model (i.e. K = 5000 in Eq. (6)) in batches of 100 with a different demographic and contact structure assigned for each batch (see Sections 2.3 and 2.5). Time courses for the model were computed by simulating 100 realizations of the model, with the new demographic and contact structures assigned every 10 realizations. For all realizations, the initial distribution of affected flocks was set according to the results of the 1998 postal survey. The impact of the 2001 foot-and-mouth disease epidemic in GB was included by assuming that a flock was culled as a result of FMD controls with probability estimated from the responses to the 2002 postal survey for each region.
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3. Results 3.1. Parameter estimation and model validation Computing the likelihood for the probability of transmission between flocks (f) (Fig. 2a), yielded a maximum-likelihood estimate (MLE) for f of 0.0002 (95% confidence interval (CI): 0.0–0.014). Comparison of the observed and expected values for the proportion of affected flocks in each region showed that the expected values lie in the confidence interval for the estimates based on the results of the 2002 postal survey (Fig. 2b). However, the results suggest that the model tends to underestimate the proportion of affected flocks in south-west England (SW) and the West Midlands (WM) and overestimate the proportion of affected flocks in North Wales (NWa) (Fig. 2b). A formal x2 test to assess goodness-of-fit indicated an acceptable fit of the model to the data (x2 = 12.9, d.f. = 11, P = 0.30). The model for within-flock outbreaks predicted a median outbreak duration of 7.5 years, though outbreaks can last much longer (>40 years) (Fig. 2c). The median duration is similar to an estimate for the mean duration of 7.3 years obtained using a different model fitted to the results of the 1998 postal survey (Gravenor et al., 2001; Ferguson et al., 2002). Outbreak durations have been reported for high-incidence flocks that range from at least 4– 26 years (Woolhouse et al., 2001; Redman et al., 2002). Consequently, the distribution for the duration of within-flock outbreaks predicted by the model reflects that observed in the field. 3.2. Sensitivity analysis Uncertainty in the parameters describing within-flock outbreaks was dealt with by sampling from a range of plausible values rather than assuming point estimates. The probability of transmission between flocks given contact was estimated so that the model is consistent with the observed proportions of affected flocks in each region of GB. However, the sensitivity of the model to changes in the description of contacts between flocks requires further consideration. In particular, the contact structure is based on movement records for only seven flocks. To assess the sensitivity of the model to the movement records used, time courses were simulated using the MLE and upper 95% confidence limit (CL) for f in which one set of movement records was omitted in turn. The results for each set of time courses were compared using Kruskal-Wallis tests to detect significant (P = 0.05) differences and, where significant differences were found, using multiple contrasts to identify them (Zar, 1999). The results for the simulations using the MLE (f = 0.0002) showed no significant differences. This is not surprising given that the probability of transmission is very small and, hence, there is likely to be little spread. Differences were identified, however, for the simulations using the upper 95% CL (f = 0.014). This allowed the movement records to be grouped according to their impact on the predictions for the proportion of affected flocks when compared with those for the model where no records were omitted. Omitting the records for farms 1, 5 or 6 resulted in a higher proportion of affected flocks; there were no differences when the records for farms 3 or 7 were omitted; and omitting the records for
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Fig. 2. Parameter estimation and model validation. (a) Loglikelihood for the probability of transmission between flocks given contact (f), normalized so the maximum is zero. (b) Comparison of the observed and expected percentage of affected flocks stratified by region. Observed values are shown as estimates (white squares) and 95% confidence intervals (error bars). Expected values (grey bars) are the mean of 5000 realizations of the model using the maximum likelihood estimate. Regions are shown in Fig. 1a. (c) Predicted distribution for the duration of a within-flock outbreak (in years) derived using the model described in Section 2.6.
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farms 2 or 4 resulted in a lower proportion of affected farms. Farms 1, 5 and 6 have relatively few contacts and omitting their movement records results in an increase in the mean number of contacts in the model (see Section 2.3) and, hence, the potential for transmission. Farm 2 has fewer movements over shorter distances (<50 km; Fig. 1c) than the other six farms and, hence, omitting its movement records reduced the median distance over which sheep are moved (thus, reducing the potential for spread). Farm 4 had many contacts and omitting its movement records reduced the mean number of contacts in the model (and, thus, reduced the potential for spread). 3.3. Scrapie in GB: endemic or epidemic? It has been argued recently that scrapie in GB is essentially a single 200-year-long epidemic and, consequently, the disease eventually should disappear from the national flock even without any intervention (Bradley, 2001; Woolhouse et al., 2001). To test this hypothesis, the model was simulated for 100 years using three values for the probability of transmission given contact (f): the MLE; the upper 95% CL; and the largest value consistent with the 2002 postal survey results at the 95% level as judged by a x2 goodnessof-fit test. The simulations showed that, for values of f in the 95% CI, the model predicted that scrapie eventually would die out from the national flock—though it will take decades to do so (Fig. 3a and b). However, there were values of f consistent with the available data for which the model predicted that scrapie will remain endemic in the British sheep population (Fig. 3c).
4. Discussion Validating a spatial model is difficult even where good quality data are available. It is even-more problematic for scrapie in GB where under-reporting means that detailed spatial data are lacking. The approach I adopted was to use data from two postal surveys conducted in 1998 and 2002 to validate the model using data collected at a regional level (Fig. 2b). This showed that the model captures the spatial dynamics of scrapie, at least at a regional level. Comparison of the predicted distribution for the duration of a within-flock outbreak (Fig. 2c) with available data also indicated that the model reflects what is observed in the field. The key parameter in the model for describing the spread between flocks was the probability of transmission given contact (f). Realistic values for f are likely to be close to the prevalence of infection within an affected flock (that is, the probability that an animal selected from the flock (e.g. for purchase or sale) is infected). A plausible range for the prevalence of infection is 0.8–1.2%, based on a median within-flock incidence in GB of 0.4 cases per hundred ewes per year (Hoinville et al., 2000) and an estimate of 2–3 infected ewes per case (Woolhouse et al., 1998; Matthews et al., 2001). This range is contained within the 95% CI for f, which suggests that the estimates for f obtained from the model are realistic. The data used to fit and validate the model cover only a short time-scale (4 years) relative to that of scrapie. This could bias estimates for f towards lower values and, hence,
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Fig. 3. Predicted long-term trends for the percentage of affected flocks in Great Britain and their dependence on the probability of transmission given contact (f): (a) f = 0.0002 (MLE); (b) f = 0.014 (upper 95% CL); and (c) f = 0.03 (largest value for which the model is consistent with the 2002 postal survey at the 95% level). Each figure shows the median (solid line) and 2.5th and 97.5th percentiles (dashed lines) for the percentage of affected flocks.
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account for why the MLE is close to zero (Fig. 2a). Furthermore, using the model to predict long-term trends in the proportion of affected flocks suggested that it was likely that scrapie ultimately would be eliminated (Fig. 3a and b)—but that there were also scenarios in which scrapie remained endemic and which were consistent with the available data (Fig. 3c). Spatial data on affected flocks over a longer time-scale should provide more-robust parameter estimates and, hence, allow more-robust predictions to be made for longer-term trends in the proportion of affected flocks. More importantly, it will be essential to take this uncertainty in the long-term dynamics of scrapie into account when considering the efficacy of control strategies. In the model, the movement of infected animals is assumed to be the principal mechanism by which scrapie is transmitted between flocks. Consequently, data on sheep movements are an essential input. Because of difficulties associated with the collection of the data, however, movement records were collected for only seven flocks and for a single year. More recent data on sheep movements with a wider coverage were collected as part of the 2002 postal survey in which farmers were asked how frequently they purchased sheep from farms or markets in their own county or in another county. Responses to this question suggest that farmers are more likely to purchase sheep locally (i.e. within their county)— but some are willing to buy animals from sources further away (i.e. in another county). Indeed, several farmers even reported purchasing sheep from overseas. These results provide some qualitative support for the description of the distances over which sheep are moved (Fig. 1c). The survey results also suggested there is regional variation in the pattern of sheep movements, where, for example, farmers in south-west England (SW) and the West Midlands (WM) were more likely to purchase sheep from sources in other counties. This variation may account, in part, for under-estimation of the proportion of affected flocks in both these regions (Fig. 2b). Moreover, the sensitivity analysis highlighted the impact on the model of changes in the mean number of contacts and the distances over which sheep are moved. Consequently, should more and better quality data on sheep movements become available, it will be important to revisit the description of the contact structure between flocks in the model. In the model, the only action taken by farmers to control scrapie was assumed to be the removal of animals with clinical signs. Although the control of scrapie was not explicitly considered, the results of the model suggest two aspects of the spread of scrapie that have implications for control. First, larger flocks typically are affected for longer (see Eq. (3)) and, hence, large scrapie-affected flocks should be targeted in any control strategy. Second, the importance of the movement of animals between flocks for the spread of scrapie means it is essential that sheep movements can be traced. Moreover, the impact of control strategies readily can be incorporated in the model through the parameters associated with within-flock outbreaks. In particular, changes in the PrP genotype profile of a flock resulting from selective breeding for TSE resistance can be incorporated as changes in the flock-level susceptibility (s). This will be investigated elsewhere (Gubbins and Webb, 2004). Dangerous to keep in unless we have already accepted it. The model was developed to describe the spread of scrapie between sheep flocks in Great Britain and, consequently, a number of features of the model are likely to be specific to GB. However, the underlying framework is generic and identifies the questions that need to be addressed if the model is to be applied to other countries.
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Acknowledgements This work was funded by the Department for Environment, Food and Rural Affairs (Defra). I am grateful to all the farmers who supplied the movement records used in this paper. The collection of the census and movement data was instigated by Cerian Webb (now at the University of Cambridge) at an earlier stage of the work. Rachel Eglin, Mike Boulton and Mohammad Ali (all VLA) collated the movement records.
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Kao, R.R., Gravenor, M.B., Baylis, M., Bostock, C.J., Chihota, C.M., Evans, J.C., Goldmann, W., Smith, A.J.A., McLean, A.R., 2002. The potential size and duration of an epidemic of bovine spongiform encephalopathy in British sheep. Science 295, 332–335. Matthews, L., Woolhouse, M.E.J., Hunter, N., 1999. The basic reproduction number for scrapie. Proc. R. Soc. Lond. B 266, 1085–1090. Matthews, L., Coen, P.G., Foster, J.D., Hunter, N., Woolhouse, M.E.J., 2001. Population dynamics of a scrapie outbreak. Arch. Virol. 146, 1173–1186. McLean, A.R., Hoek, A., Hoinville, L.J., Gravenor, M.B., 1999. Scrapie transmission in Britain: a recipe for a mathematical model. Proc. R. Soc. Lond. B 266, 2531–2538. Pollott, G.E., 1998. Sheep breeds and breeding in Britain 1996-7. Sheep Yearbook 1998. Meat and Livestock Commission, Milton Keynes, pp. 37–53. Redman, C.A., Coen, P.G., Matthews, L., Lewis, R.M., Dingwall, W.S., Foster, J.D., Chase-Topping, M.E., Hunter, N., Woolhouse, M.E.J., 2002. Comparative epidemiology of scrapie outbreaks in individual sheep flocks. Epidemiol. Infect. 128, 513–521. Sivam, K., Baylis, M., Gravenor, M.B., Gubbins, S., Wilesmith, J.W., 2003. Results of a postal survey in 2002 into the occurrence of scrapie in Great Britain. Vet. Rec. 153, 782–783. Stringer, S.M., Hunter, N., Woolhouse, M.E.J., 1998. A mathematical model of the dynamics of scrapie in a sheep flock. Math. Biosci. 153, 79–98. Tongue, S., Wilesmith, J.W., Cook, C.J., 2004a. Frequencies of prion protein (PrP) genotypes and distribution of ages in 15 scrapie-affected flocks in Great Britain. Vet. Rec. 154, 9–16. Tongue, S.C., Pfeiffer, D.U., Warner, R., Elliott, H., del Rio Vilas, V., 2004b. Estimation of the relative risk for developing clinical scrapie: the role of prion protein (PrP) genotype and selection bias. Vet. Record (in press). Webb, C.R., Sauter-Louis, C., 2002. Investigations into the contact structure of the British sheep population. In: Menzies, F.D., Reid, S.W.J. (Eds.), Vicenary Proceedings, Cambridge, 3–5 April 2002. Society for Veterinary Epidemiology and Preventive Medicine, Edinburgh, pp. 10–20. Woolhouse, M.E.J., Stringer, S.M., Matthews, L., Hunter, N., Anderson, R.M., 1998. Epidemiology and control of scrapie within a sheep flock. Proc. R. Soc. Lond. B 265, 1205–1210. Woolhouse, M.E.J., Matthews, L., Coen, P., Stringer, S.M., Foster, J.D., Hunter, N., 1999. Population dynamics of scrapie in a sheep flock. Phil. Trans. R. Soc. Lond. B 354, 751–756. Woolhouse, M.E.J., Coen, P., Matthews, L., Foster, J.D., Elsen, J.-M., Lewis, R.M., Haydon, D.T., Hunter, N., 2001. A centuries-long epidemic of scrapie in British sheep? Trends Microbiol. 9, 67–70. Zar, J.H., 1999. Biostatistical Analysis. Prentice Hall, NJ.
A modelling framework to describe the spread of scrapie between sheep flocks in Great Britain Simon Gubbins* Centre for Epidemiology and Risk Analysis, Veterinary Laboratories Agency, New Haw, Addlestone, Surrey KT15 3NB, UK Received 11 August 2003; received in revised form 29 July 2004; accepted 25 August 2004
Abstract My aim was to develop a stochastic, spatial model describing the spread of scrapie between sheep flocks in Great Britain; I wanted a model, which could subsequently be used to assess the efficacy of different control strategies. The structure of the model reflects the demography of the British sheep flock, including a description of the contact structure between flocks. The dynamics of scrapie were incorporated through two probabilities associated with each flock: of acquiring infection and of experiencing a within-flock outbreak following exposure. The acquisition of infection depends on whether or not a flock buys-in sheep and, if it does, whether or not it trades with an affected flock. Once a flock is exposed, the probability of a within-flock outbreak occurring and its duration depend on the basic reproductive number, the prion-protein (PrP) genotype profile and the flock size. The model was validated using regional data from two postal surveys conducted in 1998 and 2002, which demonstrated that the model captures the spatial dynamics of scrapie (at least at a regional level). Moreover, the predicted distribution for the duration of a within-flock outbreak reflects the duration of outbreaks reported in the literature. Using the model to predict long-term trends in the proportion of affected flocks suggested that, even without control measures beyond the removal of animals with clinical signs, scrapie ultimately will disappear from the national flock, though it is likely to be decades before the disease is eliminated. However, there were scenarios consistent with the available data which suggested that scrapie could remain endemic within the British sheep flock. Consequently, it is essential to take this uncertainty in the long-term dynamics of scrapie into account when considering the efficacy of control strategies. Although control strategies were not explicitly
* Present address: Institute for Animal Health, Compton, Newbury, Berkshire RG20 7NN, UK. Tel.: +44 1635 578411; fax: +44 1635 577237. E-mail address: [email protected]. 0167-5877/$ – see front matter. Crown Copyright # 2004 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.prevetmed.2004.08.007
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examined, the model suggests two aspects important for control: larger flocks remain affected for longer and provide infection for other, smaller flocks and animal movements must be traceable. Crown Copyright # 2004 Published by Elsevier B.V. All rights reserved. Keywords: Transmissible spongiform encephalopathy; Scrapie; Sheep; Stochastic model; Demography; Disease control
1. Introduction Recently, much attention has been given to the epidemiology of scrapie within sheep flocks (Elsen et al., 1999; Baylis et al., 2000, 2002; Redman et al., 2002). A number of mathematical models have been developed to describe the within-flock dynamics and control of scrapie (Stringer et al., 1998; Hagenaars et al., 2000). These models have been used in several ways: to analyse outbreaks (Woolhouse et al., 1998, 1999; Matthews et al., 2001; Hagenaars et al., 2003); to calculate the basic reproductive number for scrapie (Matthews et al., 1999); and to investigate persistence patterns within flocks (Hagenaars et al., 2001). In contrast, relatively little attention has been given to the spread of scrapie between flocks. Simple flock-level models (Gravenor et al., 2001; Kao et al., 2001) have been developed using information collected from an anonymous postal survey conducted in 1998 (McLean et al., 1999; Hoinville et al., 2000). More recently, flock-level models have been developed to investigate the potential spread of BSE in British sheep (Ferguson et al., 2002; Kao et al., 2002). These models describe only changes in the number or proportion of affected flocks over time and do not attempt to describe the spatial distribution of affected flocks or the impact of this distribution on the subsequent spread. However, these factors are important because, first, affected flocks are not uniformly distributed (Hoinville et al., 2000; Sivam et al., 2003) and, second, acquisition of infection is primarily through the movement of infected animals between flocks (Ducrot and Calavas, 1998; McLean et al., 1999; Hoinville et al., 2000; Hopp et al., 2001) which will depend on the flocks’ locations. I developed a stochastic, spatially explicit model to describe the spread of scrapie between sheep flocks in Great Britain (GB) that, subsequently, can be used to assess the impact of different control strategies. The structure of the model broadly reflects the conclusions of a study into risk factors associated with scrapie at a flock level in Great Britain (McLean et al., 1999). Those analyses suggested that a minimal model for the spread of scrapie between flocks should include flock size and the proportion of sheep homebred and examine the interplay between the transmission of scrapie within and between flocks. The risk-factor study identified only the broad structure for a model. Consequently, my aim was to present the modelling approach used in detail. In particular, I identified the demographic and epidemiological data required to describe the sheep population and the key parameters needed to characterise the transmission of scrapie within and between flocks. Of particular importance is a description of the mixing patterns between flocks (that is, who trades with whom). Two approaches previously have been used to address this question in the context of scrapie: one assumes all flocks are equally likely to acquire
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infection (Kao et al., 2001) and the other involves a detailed description of potential pathways for transmission (Webb and Sauter-Louis, 2002). For the model described here, a method of intermediate complexity was developed to describe mixing patterns using data on sheep movements and a simplified description of the British sheep industry.
2. Methods 2.1. Model structure The unit of population in the model is the flock. Flocks are divided into two classes: unaffected (i.e. no animals are infected with scrapie) and affected (i.e. at least one animal is infected with scrapie). Each flock has two probabilities associated with it: (i) of acquiring infection; and (ii) of experiencing a within-flock outbreak following exposure. The scrapie status of each flock is updated on a yearly basis. 2.2. Acquisition of infection The principal mechanism by which infection is acquired is the movement of infected animals between flocks (Ducrot and Calavas, 1998; McLean et al., 1999; Hoinville et al., 2000; Hopp et al., 2001). Consequently, the probability that an unaffected flock j acquires infection in year t is equivalent to the probability that the flock buys-in sheep and trades with at least one affected flock. This is given by: 0 1 Y lj ðtÞ ¼ bj @1 ð1 fWjk ÞA; (1) k 2 IðtÞ
where bj is the probability that flock j buys in sheep, I(t) is a list of affected flocks in year t, f is the probability of transmission between flocks given contact and Wjk is an element of the matrix representing who trades with whom (see Section 2.3). The probability that a flock buys-in sheep (bj) was estimated was estimated by fitting a binomial logistic-regression model to data collected as part of a postal survey conducted in 2002 (Sivam et al., 2003). Independent variables included in the model were region, farm type (hill/upland/lowland), flock type (purebred/commercial) and flock size. Farm type describes the type of land on which sheep are grazed (ranging from the harsh hill environment, through the less-harsh upland areas to the relatively benign lowland areas). Flock type reflects the primary business of the flock: production of ewes or rams for breeding (purebred) or production of lambs for human consumption (commercial). 2.3. Who trades with whom? The British sheep industry is unique and complex; it involves a stratified system which has evolved over hundreds of years to make best use of the land available for sheep farming (Pollott, 1998). Here, a simplified description was used in which the major sectors were the three farm types (Pollott, 1998; Arnold et al., 2002). Sheep were assumed to move within
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each sector (i.e. from hill to hill; upland to upland; and lowland to lowland) but movement between sectors was restricted so that sheep only are moved from hill to upland flocks or from upland to lowland flocks. The principal source of data on sheep movements is statutory movement records kept by all flocks. These record the number of sheep moved and the postal address to or from which the sheep were moved. Few farmers were willing to provide copies of these records. Moreover, photocopying the flock-movement books, computing grid references from postal addresses and entering the records on a database for analysis was time consuming. Consequently, it was possible to collect records from only seven farms (located in different regions of England and Wales) and for a single year (1999, the year in which the study began). Most movements were over relatively short distances (<20 km), but some were over several hundred kilometres (Fig. 1c). The number of different locations to or from which sheep were moved for each farm varied from 6 to 51. The contact structure for flocks is described by the matrix representing who trades with whom (W). An element of this matrix (Wjk) is equal to the probability of moving an animal from flock k to j (cjk) if flock j potentially trades with flock k and zero if it does not. The matrix, W, was constructed as follows. (i) Each farm was assigned a set of movement records drawn uniformly from the seven sets collected. (ii) The number of contacts for each farm was drawn from a Poisson distribution used to summarise the data, the mean of which was equal to the mean number of contacts for the seven sets of movement records. (iii) For farm j, a farm k was selected uniformly and, if farms j and k were of appropriate types to trade, the contact from k to j was accepted (i.e. Wjk = cjk) with probability cjk. (iv) If the contact was accepted and farm k was of the same type as farm j, the contact also goes from j to k (i.e. Wkj = ckj) with probability ckj. (v) Steps (iii) and (iv) were repeated until all contacts between farms had been assigned. The probability of moving an animal from flock k to j (cjk) was assumed to be equal to the probability that a sheep is moved at least a distance x to the farm. Consequently, cjk = 1 Fj(xjk) where Fj is the empirical cumulative distribution function for the movement records used for farm j (Fig. 1c) and xjk is the distance between flocks j and k.
2.4. Within-flock outbreaks When a flock acquires infection, the question arises of whether or not scrapie subsequently will spread within the flock. A range of factors influence the maintenance of scrapie, including farm- and flock-management practices (McLean et al., 1999) and the prion-protein (PrP) genotype profile of the flock (Baylis et al., 2000). Rather than attempt to include all these factors explicitly in the model, a simpler approach was adopted in which within-flock outbreaks are characterised by two parameters: the within-flock basic reproductive number, R0 (see, e.g., Anderson and May, 1986), and the outbreak duration (TE). Differences amongst flocks were incorporated as differences in these parameters. In
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Fig. 1. (a and b) Agricultural census data on sheep flocks in GB: (a) density of sheep flocks. Regions marked are those used in the 2002 postal survey; and (b) distribution of flock sizes (number of breeding ewes and rams over 1 year old). (c) Sheep movements in GB: empirical cumulative distribution functions for movement data from seven farms in GB.
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addition, I assumed that the only action taken to control scrapie is the removal of animals which show clinical signs. Following the introduction of an infected animal to the flock, the probability that a within-flock outbreak occurs is given by: ( 1 R0 > 1 1 Pr ðoutbreakÞ ¼ ; (2) R0 0 R0 1 where R0 is defined above (Diekmann and Heesterbeek, 2000). If an outbreak occurs, its duration (TE) is given by:
TE ¼
R0 G lnðsNÞ; R0 1
(3)
where R0 is defined above, G is the incubation period and N is the flock size (Barbour, 1975). The factor s is the flock-level susceptibility which corrects the flock size to allow for differences in the risk of scrapie amongst PrP genotypes. This was defined to be: X s¼ ri ni ; (4) i
where ni is the proportion of animals of PrP genotype i in the flock and ri is the risk of scrapie in genotype i measured relative to that with the highest risk. When a flock acquired infection, it was assigned values for each parameter (R0, G and s) drawn uniformly from plausible ranges. A range for R0 of 2.5–14.0 was used, which is consistent with an observed within-flock outbreak (Hagenaars et al., 2003) and includes the point estimate of 3.9 computed for a different outbreak (Matthews et al., 1999). Although the age at onset of clinical signs is observed readily, the age at infection is not; consequently, the mean incubation period (G) was difficult to estimate. A range for G of 1.6–5.9 years was used, where the lower limit is an estimate of the incubation period based on a modelling study of a within-flock outbreak (Matthews et al., 2001) and the upper limit is the upper 95% confidence limit for the mean age at onset for 1740 notified scrapie cases. Using data on PrP genotype frequencies for 15 scrapie-affected flocks (Tongue et al., 2004a) and estimates for the risk of scrapie in different PrP genotypes (Tongue et al., 2004b), the flock-level susceptibility (s), was assumed to range from 0.002 to 0.14.
2.5. Sheep flocks in GB The location and size of flocks were obtained from agricultural-census data. Sheep flocks are found throughout GB with the highest densities in the north west and south west of England and in Wales (Fig. 1a). Flock sizes vary from only a few sheep to thousands of sheep with a median flock size of 285 breeding animals (Fig. 1b). Farm and flock type for each holding, which are not collected as part of the census, were drawn from a multinomial distribution in which the probability that a flock is a given type was estimated from the proportion of flocks of each type in a region (derived from the 2002 postal survey).
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2.6. Distribution of affected flocks Scrapie has been a notifiable disease in the UK since 1993. Although statutory notifications provide information on the number and location of flocks which report scrapie cases, there is substantial under-reporting (Hoinville et al., 2000; Sivam et al., 2003). Postal surveys conducted in 1998 and 2002 provide the number of respondents reporting at least one clinical case in their flock during the 12 months preceding the survey stratified by region (Hoinville et al., 2000; Sivam et al., 2003; see Fig. 1a for regions). The results of the 1998 survey were used to set the initial distribution of affected flocks, while those for the 2002 survey were used to validate the model. 2.7. Parameter estimation No independent estimates were available for the probability of transmission between flocks given contact (f). To estimate this parameter, a likelihood-based approach was adopted in which the probability of observing the results of the 2002 postal survey for a given parameter value was estimated. The 2002 postal survey provided the number of affected flocks (Ij) and the total number of flocks (Nj) in region j which responded to the survey. The probability (P(f,vk)) of observing these results for model realization vk and parameter value f is given by: Y Nj Ij Nj Ij Pðf; vk Þ ¼ pj ðvk ; fÞ ð1 pj ðvk ; fÞÞ ; (5) Ij j
where pj(vk,f) is the proportion of affected flocks in region j for that realization. The likelihood (L(f)) is calculated by averaging the probability (5), over all realizations of the model (Gelfland and Smith, 1990). Hence, L(f) is estimated by: LðfÞ
K 1X Pðf; vk Þ; K k¼1
(6)
where K is the number of realizations simulated for each parameter value. 2.8. Implementing the model Exploratory analyses were undertaken to determine the number of realizations needed to provide robust results without being prohibitively expensive (in terms of computation time). For parameter estimation, the likelihood for each value of f was calculated by simulating 5000 realizations of the model (i.e. K = 5000 in Eq. (6)) in batches of 100 with a different demographic and contact structure assigned for each batch (see Sections 2.3 and 2.5). Time courses for the model were computed by simulating 100 realizations of the model, with the new demographic and contact structures assigned every 10 realizations. For all realizations, the initial distribution of affected flocks was set according to the results of the 1998 postal survey. The impact of the 2001 foot-and-mouth disease epidemic in GB was included by assuming that a flock was culled as a result of FMD controls with probability estimated from the responses to the 2002 postal survey for each region.
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3. Results 3.1. Parameter estimation and model validation Computing the likelihood for the probability of transmission between flocks (f) (Fig. 2a), yielded a maximum-likelihood estimate (MLE) for f of 0.0002 (95% confidence interval (CI): 0.0–0.014). Comparison of the observed and expected values for the proportion of affected flocks in each region showed that the expected values lie in the confidence interval for the estimates based on the results of the 2002 postal survey (Fig. 2b). However, the results suggest that the model tends to underestimate the proportion of affected flocks in south-west England (SW) and the West Midlands (WM) and overestimate the proportion of affected flocks in North Wales (NWa) (Fig. 2b). A formal x2 test to assess goodness-of-fit indicated an acceptable fit of the model to the data (x2 = 12.9, d.f. = 11, P = 0.30). The model for within-flock outbreaks predicted a median outbreak duration of 7.5 years, though outbreaks can last much longer (>40 years) (Fig. 2c). The median duration is similar to an estimate for the mean duration of 7.3 years obtained using a different model fitted to the results of the 1998 postal survey (Gravenor et al., 2001; Ferguson et al., 2002). Outbreak durations have been reported for high-incidence flocks that range from at least 4– 26 years (Woolhouse et al., 2001; Redman et al., 2002). Consequently, the distribution for the duration of within-flock outbreaks predicted by the model reflects that observed in the field. 3.2. Sensitivity analysis Uncertainty in the parameters describing within-flock outbreaks was dealt with by sampling from a range of plausible values rather than assuming point estimates. The probability of transmission between flocks given contact was estimated so that the model is consistent with the observed proportions of affected flocks in each region of GB. However, the sensitivity of the model to changes in the description of contacts between flocks requires further consideration. In particular, the contact structure is based on movement records for only seven flocks. To assess the sensitivity of the model to the movement records used, time courses were simulated using the MLE and upper 95% confidence limit (CL) for f in which one set of movement records was omitted in turn. The results for each set of time courses were compared using Kruskal-Wallis tests to detect significant (P = 0.05) differences and, where significant differences were found, using multiple contrasts to identify them (Zar, 1999). The results for the simulations using the MLE (f = 0.0002) showed no significant differences. This is not surprising given that the probability of transmission is very small and, hence, there is likely to be little spread. Differences were identified, however, for the simulations using the upper 95% CL (f = 0.014). This allowed the movement records to be grouped according to their impact on the predictions for the proportion of affected flocks when compared with those for the model where no records were omitted. Omitting the records for farms 1, 5 or 6 resulted in a higher proportion of affected flocks; there were no differences when the records for farms 3 or 7 were omitted; and omitting the records for
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Fig. 2. Parameter estimation and model validation. (a) Loglikelihood for the probability of transmission between flocks given contact (f), normalized so the maximum is zero. (b) Comparison of the observed and expected percentage of affected flocks stratified by region. Observed values are shown as estimates (white squares) and 95% confidence intervals (error bars). Expected values (grey bars) are the mean of 5000 realizations of the model using the maximum likelihood estimate. Regions are shown in Fig. 1a. (c) Predicted distribution for the duration of a within-flock outbreak (in years) derived using the model described in Section 2.6.
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farms 2 or 4 resulted in a lower proportion of affected farms. Farms 1, 5 and 6 have relatively few contacts and omitting their movement records results in an increase in the mean number of contacts in the model (see Section 2.3) and, hence, the potential for transmission. Farm 2 has fewer movements over shorter distances (<50 km; Fig. 1c) than the other six farms and, hence, omitting its movement records reduced the median distance over which sheep are moved (thus, reducing the potential for spread). Farm 4 had many contacts and omitting its movement records reduced the mean number of contacts in the model (and, thus, reduced the potential for spread). 3.3. Scrapie in GB: endemic or epidemic? It has been argued recently that scrapie in GB is essentially a single 200-year-long epidemic and, consequently, the disease eventually should disappear from the national flock even without any intervention (Bradley, 2001; Woolhouse et al., 2001). To test this hypothesis, the model was simulated for 100 years using three values for the probability of transmission given contact (f): the MLE; the upper 95% CL; and the largest value consistent with the 2002 postal survey results at the 95% level as judged by a x2 goodnessof-fit test. The simulations showed that, for values of f in the 95% CI, the model predicted that scrapie eventually would die out from the national flock—though it will take decades to do so (Fig. 3a and b). However, there were values of f consistent with the available data for which the model predicted that scrapie will remain endemic in the British sheep population (Fig. 3c).
4. Discussion Validating a spatial model is difficult even where good quality data are available. It is even-more problematic for scrapie in GB where under-reporting means that detailed spatial data are lacking. The approach I adopted was to use data from two postal surveys conducted in 1998 and 2002 to validate the model using data collected at a regional level (Fig. 2b). This showed that the model captures the spatial dynamics of scrapie, at least at a regional level. Comparison of the predicted distribution for the duration of a within-flock outbreak (Fig. 2c) with available data also indicated that the model reflects what is observed in the field. The key parameter in the model for describing the spread between flocks was the probability of transmission given contact (f). Realistic values for f are likely to be close to the prevalence of infection within an affected flock (that is, the probability that an animal selected from the flock (e.g. for purchase or sale) is infected). A plausible range for the prevalence of infection is 0.8–1.2%, based on a median within-flock incidence in GB of 0.4 cases per hundred ewes per year (Hoinville et al., 2000) and an estimate of 2–3 infected ewes per case (Woolhouse et al., 1998; Matthews et al., 2001). This range is contained within the 95% CI for f, which suggests that the estimates for f obtained from the model are realistic. The data used to fit and validate the model cover only a short time-scale (4 years) relative to that of scrapie. This could bias estimates for f towards lower values and, hence,
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Fig. 3. Predicted long-term trends for the percentage of affected flocks in Great Britain and their dependence on the probability of transmission given contact (f): (a) f = 0.0002 (MLE); (b) f = 0.014 (upper 95% CL); and (c) f = 0.03 (largest value for which the model is consistent with the 2002 postal survey at the 95% level). Each figure shows the median (solid line) and 2.5th and 97.5th percentiles (dashed lines) for the percentage of affected flocks.
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account for why the MLE is close to zero (Fig. 2a). Furthermore, using the model to predict long-term trends in the proportion of affected flocks suggested that it was likely that scrapie ultimately would be eliminated (Fig. 3a and b)—but that there were also scenarios in which scrapie remained endemic and which were consistent with the available data (Fig. 3c). Spatial data on affected flocks over a longer time-scale should provide more-robust parameter estimates and, hence, allow more-robust predictions to be made for longer-term trends in the proportion of affected flocks. More importantly, it will be essential to take this uncertainty in the long-term dynamics of scrapie into account when considering the efficacy of control strategies. In the model, the movement of infected animals is assumed to be the principal mechanism by which scrapie is transmitted between flocks. Consequently, data on sheep movements are an essential input. Because of difficulties associated with the collection of the data, however, movement records were collected for only seven flocks and for a single year. More recent data on sheep movements with a wider coverage were collected as part of the 2002 postal survey in which farmers were asked how frequently they purchased sheep from farms or markets in their own county or in another county. Responses to this question suggest that farmers are more likely to purchase sheep locally (i.e. within their county)— but some are willing to buy animals from sources further away (i.e. in another county). Indeed, several farmers even reported purchasing sheep from overseas. These results provide some qualitative support for the description of the distances over which sheep are moved (Fig. 1c). The survey results also suggested there is regional variation in the pattern of sheep movements, where, for example, farmers in south-west England (SW) and the West Midlands (WM) were more likely to purchase sheep from sources in other counties. This variation may account, in part, for under-estimation of the proportion of affected flocks in both these regions (Fig. 2b). Moreover, the sensitivity analysis highlighted the impact on the model of changes in the mean number of contacts and the distances over which sheep are moved. Consequently, should more and better quality data on sheep movements become available, it will be important to revisit the description of the contact structure between flocks in the model. In the model, the only action taken by farmers to control scrapie was assumed to be the removal of animals with clinical signs. Although the control of scrapie was not explicitly considered, the results of the model suggest two aspects of the spread of scrapie that have implications for control. First, larger flocks typically are affected for longer (see Eq. (3)) and, hence, large scrapie-affected flocks should be targeted in any control strategy. Second, the importance of the movement of animals between flocks for the spread of scrapie means it is essential that sheep movements can be traced. Moreover, the impact of control strategies readily can be incorporated in the model through the parameters associated with within-flock outbreaks. In particular, changes in the PrP genotype profile of a flock resulting from selective breeding for TSE resistance can be incorporated as changes in the flock-level susceptibility (s). This will be investigated elsewhere (Gubbins and Webb, 2004). Dangerous to keep in unless we have already accepted it. The model was developed to describe the spread of scrapie between sheep flocks in Great Britain and, consequently, a number of features of the model are likely to be specific to GB. However, the underlying framework is generic and identifies the questions that need to be addressed if the model is to be applied to other countries.
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Acknowledgements This work was funded by the Department for Environment, Food and Rural Affairs (Defra). I am grateful to all the farmers who supplied the movement records used in this paper. The collection of the census and movement data was instigated by Cerian Webb (now at the University of Cambridge) at an earlier stage of the work. Rachel Eglin, Mike Boulton and Mohammad Ali (all VLA) collated the movement records.
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