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Modelling the national scrapie eradication programme in the UK
Mathematical Biosciences 174 (2001) 61±76 www.elsevier.com/locate/mbs
Modelling the national scrapie eradication programme in the UK Rowland R. Kao
a,*
, Michael B. Gravenor a, Angela R. McLean
b
a
b
Institute for Animal Health, Compton, nr. Newbury Berks RG20 7NN, UK Department of Zoology, University of Oxford, South Parks Road, Oxford, Oxon OX1 3PS, UK
Received 8 February 2001; received in revised form 5 September 2001; accepted 14 September 2001
Abstract In accordance with a policy to eliminate all transmissible spongiform encephalopathies from the food chain, a national untargeted ram breeding programme to eliminate scrapie in the UK is in the ®nal stages of planning. Here we formulate a model of ¯ock-to-¯ock scrapie transmission, in order to consider the eect of a targeted breeding programme which is in the early stages of consideration. We estimate the size of the susceptible ¯ock population, and discuss implications for potential control programmes. Targeting all rams and ewes in highly susceptible ¯ocks rather than rams in all ¯ocks will eradicate scrapie more quickly, and so is likely to be bene®cial as long as suitable penalties or incentives are available to facilitate their identi®cation. A more restricted programme aimed only at highly aected ¯ocks would be much easier to implement and crucially will eradicate scrapie just as quickly. This will leave behind a residue population of susceptible sheep, which could then be gradually removed by a more general breeding programme. Ó 2001 Published by Elsevier Science Inc. Keywords: Metapopulation; Prion; Scrapie; Sheep; Trading; TSE
1. Introduction Scrapie is an invariably fatal neurological disease of sheep that is the ®rst recognized transmissible spongiform encephalopathy (TSE). It has been endemic in the UK national sheep ¯ock for several centuries. While of some economic concern, it has never been associated with any human diseases. However, sheep have been shown to be susceptible to infection with bovine *
Corresponding author. Fax: +44-1635 577 237. E-mail address: [email protected] (R.R. Kao).
0025-5564/01/$ - see front matter Ó 2001 Published by Elsevier Science Inc. PII: S 0 0 2 5 - 5 5 6 4 ( 0 1 ) 0 0 0 8 2 - 7
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spongiform encephalopathy (BSE) by the oral route [1], and the likely contamination of meat and bone meal feed with BSE-infected material during the late 1980s makes the natural infection of sheep with BSE a very real possibility. As sheep that have been experimentally infected with BSE show clinical signs that are nearly indistinguishable from scrapie, should a natural BSE infection be discovered in the sheep population, a scrapie eradication programme on a national scale would be required. This concern has lead to a policy recommendation that all TSEs should be removed from the food chain. Control of scrapie is hampered by a lack of data on transmission, a long incubation period [2] and the absence of an eective pre-clinical diagnostic test. However, scrapie is strongly associated with genetic susceptibility to the disease [3]. Thus selective breeding for resistance to scrapie could form the basis of an eradication programme, and this has recently been proposed by the UK Department for the Environment, Food and Rural Aairs (DEFRA, formally the Ministry of Agriculture, Fisheries and Food). This breeding programme will encourage all farmers to breed from sheep of low susceptibility, eventually creating a scrapie-resistant national ¯ock. Whole-¯ock genotyping is not feasible for the entire population, so the current ®rst phase of the MAFF proposal involves genotyping only rams in the breeding pool [4]. Later phases of the eradication scheme await further re®nement and consultation. Here we develop a mathematical model of ¯ock-to-¯ock scrapie transmission to explore various control scenarios. While there have been mathematical studies of scrapie at the ¯ock level [5±7], and of BSE across cattle herds [8,9], the ¯ock-to-¯ock spread of scrapie has not yet been addressed. Unfortunately, the mathematical description of a spatially extended community such as the UK national sheep ¯ock is typically complicated. By considering the community to consist of a `metapopulation' of weakly interacting sub-populations or `patches' (i.e. the ¯ocks), these complexities can often be reduced while preserving essential properties of the system. In the classic de®nition of metapopulations all patches are the same and interact equally [10]. Flocks of domesticated animals are well suited to this type of analysis, as trading is only loosely dependent on proximity, unlike the migratory behaviour which often characterizes the interactions between wildlife populations. We construct a model of ¯ock-to-¯ock scrapie transmission around two assumptions. The ®rst assumption is that trading is the most signi®cant mode of transmission. Infection through cograzing is proven, but typically requires extensive perinatal contact [11]. Thus it is unlikely to be the most important route of ¯ock-to-¯ock transmission unless ¯ocks often combine at lambing. On the other hand, even though scrapie is a noti®able disease in the UK, the long incubation period, coupled with under-reporting [12] and misdiagnosis [13] are strong indicators that the trading of infected sheep is mainly responsible for the maintenance and spread of scrapie in the national ¯ock. The second assumption is that a subpopulation of `high risk' ¯ocks is mainly responsible for ¯ock-to-¯ock spread. Baylis et al. [14] describe a case-control study of scrapieaected and unaected farms (`the farm study'). They found a statistically signi®cant greater proportion of susceptibles, but only in the younger age cohorts (0±2 years of age) of the scrapieaected ¯ocks. As young sheep are unlikely to have experienced scrapie-associated losses, the likely implication is that their genotype pro®le predisposed them to a scrapie epidemic. While McLean et al. [13] identi®ed other risk factors for predisposition, they are likely to be secondary to genetics. If the second assumption is correct, a breeding programme targeted at all sheep in a suciently small high risk ¯ock population could be implemented for a similar eort to a programme
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targeted at breeding rams in all ¯ocks (the ratio of breeding rams to breeding ewes is about 1±40 [13]). If these ¯ocks or a signi®cant proportion of them could be readily found, targeting the high risk ¯ocks would more quickly reduce the susceptibility of ¯ocks most responsible for maintaining scrapie. Identifying all high risk ¯ocks may prove dicult. We therefore use our model to explore the implications of targeting either all high risk ¯ocks whether they already harbour scrapie or not, or the ¯ocks which harbour a serious scrapie problem, as these are likely to form an identi®able subset of the high risk population. 2. Model structure 2.1. The basic model The model considers several important features discussed above (Fig. 1(a)). The total number of ¯ocks (N) is ®xed. Most ¯ocks have no scrapie and are considered susceptible (S). By buying-in scrapie-infected sheep, ¯ocks become exposed (E). These ¯ocks are characterized by relatively low prevalence of scrapie. Buying and selling occurs through either the trading of sheep from one ¯ock to another (directly or through a sheep auction), or the dispersal of a ¯ock resulting in the sale of all of its breeding stock (Fig. 1(b)). If a scrapie-infected sheep transmits the disease to another in the same ¯ock, then a within-¯ock outbreak is assumed to occur, and the ¯ock is considered to have a high disease incidence and be aicted (A). These ¯ocks have relatively high prevalence of scrapie, and are most responsible for spread of the disease. Both exposed and aicted ¯ocks can recover and revert to the susceptible class. All ¯ocks can disperse and be immediately replaced, with replacement ¯ocks either susceptible or exposed. Trading is unbiased with respect to the scrapie status of ¯ocks, and to the sheep within ¯ocks. This model structure is fundamentally of the form of the SEI epidemic model (for example, [15]), with disease transmission rates determined by the proportion of scrapie-infected sheep available for purchase. Though the de®nitions of `exposed' and `aicted' are intended to mirror the de®nitions `scrapie-challenged' (scrapie only in bought-in sheep) and `scrapie-born' (scrapie in home-born sheep) as de®ned in [13], there are distinctions between the two sets of de®nitions. Most importantly, scrapie-challenged ¯ocks may contain large numbers of bought-in sheep that are infected with scrapie either before or after entering the ¯ock, and so these ¯ocks may contribute signi®cantly to ¯ock-to-¯ock scrapie transmission. To examine the high risk population hypothesis, there are two categories of ¯ocks: category one does not become aicted (the low risk ¯ocks), while in category two, aiction following exposure to scrapie is common (the high risk ¯ocks). As suggested earlier, we assume that the principle dierence between the two ¯ock types is their composition with respect to frequency of polymorphisms in the PrP gene. The generation of new high risk ¯ocks will be a combination of buying in and breeding from sheep with susceptible genotypes, and possibly the adoption of management practices such as lambing in pens which promote the transmission of scrapie [13]. Recall that MacDonald [16] de®ned the basic reproduction ratio, R0 , to be the expected number of secondary cases caused by a typical infected individual in an initially disease-free equilibrium population. A value of R0 < 1 implies that the disease cannot persist (e.g. [15]), and so the difrisk < 1 and ferences between the two ¯ock types imply that for within-¯ock transmission, Rlow 0
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Fig. 1. (a) General ¯ock structure. Flocks are distinguished by infection status (susceptible/exposed/aicted), and ability to maintain within-¯ock transmission (resistant sub-population 1, susceptible sub-population 2). Flocks may recover from scrapie if all scrapie-infected sheep are sold or die o. Flocks of type 1 are converted to type 2 and vice versa. (b) Breeding stock may be bought or sold from any one ¯ock to any other ¯ock. Stock sold from exposed or aicted ¯ocks may be infected with scrapie. Breeding stock from a dispersing ¯ock may be sold to many ¯ocks, while a new ¯ock may be formed from many ¯ocks. Both activities may result in exposure of ¯ocks to scrapie. (c) Modelling a genotyping programme. A targeted scheme could target only aicted ¯ocks as they are relatively easy to identify, or could target all high risk ¯ocks if they can be identi®ed. If replacement stock is bred or bought only from resistant lambs, an aicted ¯ock can be converted to low risk in 2 years. In an untargeted scheme, rams of all ¯ocks are genotyped. All high risk ¯ocks are aected, but conversion to low risk ¯ocks may take up to 8 years. In both cases, it is assumed that all new ¯ocks are formed from resistant stock. risk Rhigh > 1. For the purposes of this model we assume this implies that A1 0, even though this 0 is not strictly correct. The total population N N1 N2 is ®xed, with Ni Si Ei Ai . If a given ¯ock is as likely to buy from one ¯ock type as another, then the probability of becoming infected when buying a single lot is
pE1 ;E2 ;A2
pA A2 pE E1 E2 ; N1 N2
1
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where pE=A is the probability of acquiring at least one scrapie-infected sheep when buying from an exposed/aicted ¯ock. When a new ¯ock is formed, we must consider that multiple lots of sheep will be bought. If we consider only ewes (the ratio of rams to ewes is about 1±40 [13]), random purchase of lots of sheep from the entire ¯ock population gives the probability that a new ¯ock will include at least one infected sheep to be dE1 ;E2 ;A2 1
1
n
pE1 ;E2 ;A2 f ;
where pE1 ;E2 ;A2 is de®ned in Eq. (1), and nf is the number of lots bought to create a new ¯ock. While we have some knowledge of the risk factors associated with scrapie aiction [13] and indications of the dierences between a few scrapie-aected and unaected ¯ocks [14], we do not know the relationship between genotype pro®le and risk level, and in particular do not know the relationship between R0 and genotype pro®le. The situation is further complicated by ignorance of the ¯ock-level genotype pro®le of the national ¯ock. Therefore we postulate a conversion rate hL N1 ; N2 from low risk to high risk. We assume that increasing the number of low risk ¯ocks will decrease the conversion rate to high risk and increase the conversion rate to low risk ¯ocks, due to a decreased pool of susceptible sheep, i.e. ohL 6 0; oN1
ohL P 0; oN2
2
where the second statement follows immediately from the ®rst due to conservation of the total number of ¯ocks. Similarly, for the equivalent quantity hH , ohH P 0; oN1
ohH 6 0: oN2
3
The types of newly formed ¯ocks will also depend on the number of existing ¯ocks of each type, with a proportion ,L N1 ; N2 starting as low risk ¯ocks, and the remainder as high risk. As for hL , o,L 6 0; oN1
o,L P 0: oN2
Then a set of equations describing this model is dS1 dt
s DS1 pE1 ;E2 ;A2 wE1 D 1
DS1
hL S1 hH S2
u1 A2 ;
dE1 s DS1 pE1 ;E2 ;A2 dt dS2 dt
dE1 ;E2 ;A2 ,L N1 N2
4 w DE1 DdE1 ;E2 ;A2 ,L N1 N2
s DS2 pE1 ;E2 ;A2 wE2 D 1
dE1 ;E2 ;A2 1
hL E1 hH E2 A2 ;
,L N1 N2
DS2 hL S1
u2 A2 ;
dE2 s DS2 pE1 ;E2 ;A2 w DE2 dt dA2 Cf E2 u DA2 hH A2 : dt
5 hH S2 6
Cf E2 DdE1 ;E2 ;A2 1
,L N1 N2 ;
7
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We de®ne the remaining variables: s, the rating of buying and selling of lots, through trading between active ¯ocks; D, the rate at which ¯ocks are dispersed; w, the rate at which exposed ¯ocks recover from scrapie; ui , the rate at which aicted ¯ocks recover from scrapie to become a susceptible ¯ock of type i (with u1 u2 u), and · Cf , the rate of progression of exposed high risk ¯ocks to aiction. For individuals (here, individual ¯ocks) with discreet infectious states in a heterogeneous population, Diekmann et al. [17] gave a more mathematical de®nition of the basic reproduction ratio (R0 ) to be the spectral radius of the next generation matrix M mij , where element ij is the typical number of cases of infection class j created by a single ¯ock in infection class i over its lifetime, given an otherwise susceptible population. In cases of interest to us, this is the magnitude of the dominant eigenvalue of M. The next generation matrix in this case is given by 2 sDN p Dd , N N 3 sDN2 p1;0;0 Dd1;0;0 1 ,L N1 N2 hL 1 1;0;0 1;0;0 L 1 2 0 wDhL wDhL 6 sDN1 p0;1;0 Dd 7 sDN2 p0;1;0 Dd0;1;0 1 ,L N1 N2 Cf 0;1;0 ,L N1 N2 hH 7: 8 M6 4 wDCf hH wDCf hH 5 wDCf hH sDN1 p0;0;1 Dd0;0;1 ,L N1 N2 hH sDN2 p0;0;1 Dd0;0;1 1 ,L N1 N2 0 uDhH uDhH
· · · ·
Eq. (8) is used to calculate the eigenvalues (and thus R0 ) used later in this paper. 2.2. Parameters in the basic model All parameters are discussed below. Data from a recently comprehensive anonymous postal survey of scrapie (`the postal survey') across the UK sheep industry [12,13,18±20] provide the best available information about the movement of sheep between ¯ocks and is used to parametrize our model. We assume that the average ¯ock lifetime is similar to the length of time a farmer remains in the sheep industry. A simple model of the farmer population is dNfarmer bfarmer dt
dfarmer Nfarmer ;
where Nfarmer is the current number of sheep farmers, bfarmer is the incidence of new farmers, and dfarmer is the farmer retirement rate. Assuming that Nfarmer changes very slowly, we estimate from the postal survey that 1=dfarmer 40 yr, giving a ¯ock dispersal rate of D 1=40 yr 1 . If the age structure is in equilibrium, then on average the ®rst age cohort of a ¯ock is replaced every year. From [13], the ®rst age cohort represents 18.7% of the total ¯ock. On average, 59.8% of a ¯ock is home born; then the proportion of the ¯ock that is purchased per year is s D 0:075 yr 1 , where s is the rate of trading between ¯ocks. Therefore s 0:050 yr 1 . Breeding sheep are bought either as individuals (usually rams) or in lots containing several sheep typically of similar quality (ewes). Here we assume that a typical ¯ock contains 400 ewes and 10 rams, produces 800 lambs per year, and buys and sells breeding stock in lots of 25. The probability of acquiring an infected sheep is based on combinatorics arguments. Using the above ¯ock and lot sizes, we calculate the probability that there is at least one infected
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sheep found in a lot from an exposed or aicted ¯ock with the hypergeometric distribution, using pE=A 1
p 0; n; i; f;
where, if p m; n; i; f is the probability that m infected animals will appear in a lot of size f given iE=A infected sheep for sale per exposed/aicted ¯ock selling a total of n sheep, then the probability that a lot will contain no infected animals is p 0; n; i; f
n i! n f! : n! n i f!
The mean number of infected animals per ¯ock will typically be non-integer. When a lot is picked from many ¯ocks, we approximate using the continuous expression p 0; n; i; f
C n i 1C n C n 1C n i
f 1 ; f 1
where C n is the Gamma function, with nC n C n 1 and C 1 1. From the postal survey, there are between 1 and 2 clinically infected sheep found in an average exposed ¯ock in 1 year, and between 5 and 6 clinically infected sheep found in an average high incidence ¯ock. We use the former ®gure as a lower estimate for the number of potentially infectious sheep in an exposed ¯ock. However, the excess unexplained mortality of sheep on scrapie aected farms [13], and the paucity of older susceptible sheep on some farms with a high incidence of scrapie indicates that aicted ¯ocks may harbour more scrapie than indicated by the number of clinical cases. A rough estimate showing that the number of sheep which may have died of scrapie-related causes may be up to 30% (based on scrapie-related mortality estimates from [14]). Of course, there is no indication if these sheep are also infectious, and so we choose a conservative estimate that for every clinically infected sheep found, there are 10 potentially infectious sheep in the ¯ock (about 15%). The probability of acquiring scrapie when buying a single lot of ewes from an exposed ¯ock is pE 0:09, while for an aicted ¯ock, pA 0:98. The ¯ock recovery rates from scrapie are estimated from the postal survey. The duration of outbreaks were found to be 2 years for exposed ¯ocks and 15 years for aicted ¯ocks, leading to estimated maximum recovery rates of w 1=2 yr 1 , and u1 u2 hH 1=15 yr 1 [19]. From the available data, it is unknown what proportion of recovered ¯ocks become low risk. Susceptible genotypes appear to oer no competitive selection advantage over resistant [21], and so we would expect that in the absence of a breeding programme, the proportion of high risk ¯ocks would not increase, as would be the case if a competitive advantage existed in a scrapie-free environment. In the presence of scrapie, preliminary simulations indicate a high net conversion rate from high risk to low risk would have eliminated scrapie on a time scale inconsistent with the known course of the epidemic, where the prevalence of scrapie is slowly rising [18]. Therefore the net conversion rate must be slow. Therefore we make the conservative assumption that natural conversion rates are negligible, i.e. hH u1 1, and therefore u2 1=15 yr 1 . We note that from conditions 2 and 3, assuming otherwise will only improve the performance of a breeding for resistance programme as N2 is reduced. The proportion of new high risk ¯ocks is also an important consideration since encouraging the removal of whole high incidence ¯ocks is a likely consequence of any incentive scheme. In the
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absence of information on the relationship between genotype pro®le and within-¯ock R0 , we assume that the proportion of newly formed low risk ¯ocks is directly related to the proportion of low risk ¯ocks in the pool, i.e. N1 : ,L N1 N2 For a given set of parameters and assumed starting value of N2 =N1 the model output simulates the changes in the numbers of ¯ocks in each subpopulation over time. The proportion of all high risk ¯ocks is unknown. However, it can be estimated as the value of N2 =N1 that generates known prevalences of scrapie, in combination with all known parameter values. Gravenor et al. [19] calculate the equilibrium proportions of scrapie-aected ¯ocks to be 2.6%, and of ¯ocks with home-born scrapie to be 0.9%. Therefore our crucial output from the model is the value of N2 =N1 that is consistent with these prevalences and the parameter values discussed above. However, in part due to sparseness of appropriate farmer responses, the rate of progression from exposed to aicted in high risk ¯ocks (Cf ) is dicult to estimate. This parameter is related to the incubation period of the disease and to management practices, and is related to what Hagenaars et al. [7] refer to as the within-¯ock `mean generation time', which they derive an expression for, but do not estimate. A value greater than 1 per year implies a bought-in infected sheep is soon infectious and transmits infection during the ®rst lambing after purchase. If Cf is very small (for example, less than 0.1 per year) this implies that on average a single infected sheep is highly unlikely to transmit infection, and multiple exposures may be required. Because of the pivotal nature of this parameter, we explore it at length in the results below.
3. Control programme A breeding programme aimed at reducing the number of susceptible sheep per ¯ock will increase the conversion of high risk ¯ocks to low risk (Fig. 1(c)). High risk ¯ocks are converted to low risk ¯ocks at an additional rate m: In a targeted programme, whole ¯ock replacement would require at least 2 years (m 0:5 yr 1 ) for replacement lambs to reach maturity, assuming sucient replacements are available. An untargeted programme based on ram genotypes only is likely to create resistance more slowly. In the sheep genome, valine at codon 136 is strongly associated with scrapie susceptibility. Baylis et al. [14] found 32% of sheep in two scrapie-aected ¯ocks were valine carriers. In the ®rst two age cohorts, before scrapie-associated mortality is likely to be signi®cant, this ratio was about 50%. By comparison, the ratio of about 13.5% in two scrapie-free ¯ocks was not signi®cantly dierent for the ®rst two age cohorts compared to the rest, as would be expected. If 50% is representative of the genotype ratio for an unaicted high risk ¯ock, then breeding only from scrapie-resistant rams will result in the genotype proportion falling from 50% to less than 13.5% in about 8 years. Some scrapie-free ¯ocks may have a greater prevalence of susceptible genotypes, increasing the conversion time. In aicted ¯ocks, losses due to scrapie may further reduce the proportion of susceptibles. This would allow more rapid conversion to low risk status. We thus consider the impact of a variety of conversion rates for an untargeted programme, ranging from 2 to 8 years.
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4. Results 4.1. R0 All results were calculated using Mathematica version 4.0, as described in [22]. We note that for non-zero conversion rate u2 , the only equilibrium this system has occurs when the number of ¯ocks with scrapie is zero, however from [18,21] we can assume that we are well past the rapidly changing early stages of the epidemic and for the slow conversion rates assumed, that the proportion of aicted ¯ocks is in a quasi-steady-state, i.e. relatively stable over the time frame of interest. Fig. 2 shows the size of the high risk population for varying values of the progression rate Cf that are consistent with known estimates of scrapie infection prevalence. The range of Cf values considered (one per month to one per 10 years) corresponds to a high risk population ranging from about 3% to 20% of the national ¯ock, illustrating the importance of determining Cf . To examine the impact of this population size on control, we calculate variations in the basic reproduction ratio (R0 ) using Eq. (8). The value of R0 is a rough indicator of the eort that would be required to control the disease. A range of R0 (from about 1.1 to 1.4) is consistent with known prevalences and likely parameter values. Fig. 3(a) shows that a smaller high risk population is consistent with rapid within-¯ock transmission, and therefore a higher R0 . Not only does the smaller high risk population reduce the eort in a targeted control programme, the larger R0 value implies that any control programme (including those not based on genotyping) would have to be more eective to eradicate scrapie. For all reasonable values for Cf , the rate of progression to aiction, prevalence of all aected ¯ocks is above 2.6% as expected.
Fig. 2. Estimates of the current size of the high risk population are shown for dierent values of Cf , given known prevalence of aicted ¯ocks. The numbers of exposed, aicted and high risk susceptible ¯ocks is shown as stacked curves. The mean time for a bought-in infected sheep to transmit to other members of the ¯ock (Cf 1 ) ranges from 1 month to 10 years. The size of the national ¯ock is assumed to be 50 000. The coloured area shows all ¯ocks containing scrapie-infected sheep. The remaining ¯ocks are low risk susceptible (not shown).
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Fig. 3. Dependence on R0 . (a) Variation of R0 with the proportion of the population that is high risk. Smaller N2 implies larger R0 . In this case, targeting is more feasible, and because R0 is larger, scrapie may be more dicult to control by other means. (b) R0 as a function of the probability of buying at least one scrapie sheep when buying a lot of 25 sheep from an aicted ¯ock. (c) R0 as a function of the trading rate s (proportion of ¯ock traded per year), with a ®xed total dispersal related trading of D 0:025 yr 1 . Three curves in both (a) and (b) are for dierent values of the progression time to aiction.
The impact of parameter changes on R0 is also a good indicator of the parameter sensitivity of the model. The sensitivity of these conclusions to parameters for which only rough estimates are available (pE ; pA ; D; s) is minimal. However, while the trading rate is well established (see Section 2.2), the model at present does not consider the marked heterogeneities in trading structure. McLean et al. [13] report that most farmers either buy-in the majority of their stock or mainly breed their own stock. The eect of trading on R0 is signi®cant ± little eort would be required to either decrease the probability (p) of acquiring infection through trading (Fig. 3(b)) or restrict trading to the point at which scrapie could not persist (Fig. 3(c)), however such restrictions on trade may not be feasible. The addition of heterogeneity may further change R0 , indicating that further analysis of ¯ock structure is required. 4.2. Genotyping programmes: non-targeted versus targeting high risk or targeting aicted ¯ocks Fig. 4 shows the time course of an epidemic governed by Eq. (5), following the introduction of a single exposed high risk ¯ock. Initially, the disease progression rate can be approximated by R0 u1 u2 hH ; where the generation time is T 1= u1 u2 hH ). The slow rate of
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Fig. 4. Example epidemic over 50 years. The epidemic starts with a single aicted ¯ock in a national ¯ock of 50 000. Times of progression to aicted status is 1 and 10 years in (a) and (b), respectively.
progression is therefore consistent with the low value of R0 and long generation times in the parameter estimates. Because susceptible ¯ocks only slowly become aected, disease control eort can at least initially ignore susceptible ¯ocks, and concentrate on aicted ones. In Fig. 5, we compare the time course of an epidemic under a programme which targets all sheep in all high risk ¯ocks, a programme which targets all sheep in aicted ¯ocks, and a non-targeted genotyping programme. Either targeting programme quickly converts aicted high risk ¯ocks into low risk ¯ocks, so it is initially more eective in reducing scrapie prevalence. Within 2 years, targeting reduces the proportion of aicted ¯ocks to less than half that under an untargeted programme. However, in the simple targeting programme aimed only at aicted ¯ocks, once the proportion of susceptible ¯ocks falls below that necessary for the disease to persist (i.e. R0 < 1) the aicted
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Fig. 5. Reduction in high risk ¯ocks over 10 years for two types of targeting and a non-targeted breeding programme (total of 50 000 ¯ocks). Targeting aicted ¯ocks rapidly reduces ¯ock scrapie prevalence, but it does not ®nd scrapiefree high risk ¯ocks (N2 never reaches zero). In contrast, a non-targeted scheme will eventually create a scrapie resistant national ¯ock. Targeting all high risk ¯ocks does not eradicate scrapie any more quickly than targeting aicted ¯ocks, but does rapidly eliminate the high risk population. In (a), the time of progression to aiction is 1 year, in (b) it is 10 years.
¯ocks will disappear, but some high risk (susceptible) ¯ocks will remain. In contrast both the more comprehensive targeted programme aimed at all high risk ¯ocks and the non-targeted programme will eventually reduce the high risk ¯ocks to zero. Surprisingly, we note that changing the progression rate to aiction (Cf ), does little to change the rate at which genotyping removes aicted ¯ocks. This is because the conversion rate m is rapid compared to the rate at which susceptible ¯ocks are exposed and eventually become aicted. If this is the case, then if a high proportion of aicted ¯ocks can be found, targeting
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will be eective no matter what Cf is, as long as the conversion rate to low risk ¯ocks is high enough. Targeting all high risk ¯ocks also does little to change the rate of aicted ¯ock removal, but of course does remove all susceptible high risk ¯ocks as well. 5. Discussion The ®rst phase of Britain's scrapie control programme is currently planned as an untargeted genotyping of rams in all ¯ocks. Here, we have considered the implications of genotyping targeted at all sheep but in aicted ¯ocks only. Not only will the eort expended diminish with time as the proportion of aicted ¯ocks approaches zero, but because aicted ¯ocks are rapidly removed, scrapie will disappear more quickly. Targeting is most attractive when the high risk population is small, not only because it reduces the cost and eort required, but because it implies a larger value of R0 which also makes disease control more dicult under any programme. Further, genotyping all sheep in target ¯ocks quickly creates a valuable source of resistant sheep, as long as they can be accredited as scrapie free. Identi®cation of all high risk ¯ocks (rather than just those that are aicted) is likely to require extensive genotyping, since even low risk ¯ocks may harbour some susceptible genotypes [14], and except at low values of the rate of progression to aicted Cf the proportion of all sheep in the high risk ¯ocks is much greater than the proportion of rams in all ¯ocks (estimated 3±20% high risk ¯ocks compared to about 2.5% rams in all ¯ocks). On the other hand, targeting ¯ocks owned by farmers who report multiple cases of scrapie to MAFF would approximate the targeting of aicted ¯ocks discussed here, and would only require genotyping a relatively small number of sheep (estimated less than 1% aicted ¯ocks ± Section 2.2). Scrapie is noti®able to MAFF, but under-reporting may make targeting dicult, especially for farmers with multiple cases of scrapie (farmers with scrapie were eight times less likely to report to MAFF than anonymously [12]). Accurate reporting is an essential component of a targeting programme and would have to be addressed. This may be aided by better knowledge of the characteristics of high risk ¯ocks [13] and especially of the genotype pro®le of a typical high risk ¯ock [14]. Despite the attendant diculties, an incentive programme aimed at encouraging the reporting of scrapie cases and breeding for resistance would be much easier to implement than the alternative of attempting to locate all high risk ¯ocks, would result in the eradication of scrapie nearly as quickly as a scheme aimed at all high risk ¯ocks, and much more quickly than an untargeted scheme. If such a scheme were to be implemented, an accompanying research programme to validate the central hypothesis of an identi®able high risk population would be essential. What is surprising is that generation of new high risk ¯ocks and the existence of many exposed ¯ocks makes little dierence to the overall ecacy of eradication. This is a result of the slow time scale of the epidemic, much slower than the projected time for the breeding programme to be eective. An argument for a general breeding programme is that it will eventually eliminate the entire high risk ¯ock population, whereas a targeted programme aimed at aicted ¯ocks will only reduce it to a level for which R0 will become one. As the eort involved in a targeted programme diminishes quickly over time, one suggestion would be to expend eort on both programmes, with the intention of phasing out the targeting scheme after the ®rst few years.
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This analysis uses the best data available to estimate parameters important for ¯ock-to-¯ock transmission of scrapie through trading. The model suppositions could be validated in at least two ways. First, the existence of signi®cant dierences between scrapie-aected and scrapie-free ¯ocks should be determined from further analysis of the farm study. Second, the size of the high risk population would be illustrated once a substantial proportion of the national ¯ock is genotyped, though risk factors other than genotype pro®le (such as outlined in [13]) will need to be accounted for. However two checks can be made if the data were available. First, using the proportion of aicted ¯ocks from the postal survey, the farm study genotype pro®les can be checked for consistency with the pro®le of all sheep which have been genotyped by MAFF. Second, the within-¯ock R0 can be used to evaluate the progression rate Cf and establish an independent value of the size of the high risk population N2 . Matthews et al. [23] quote R0 to be 3.9 in an experimental scrapie ¯ock. However this ¯ock has an unusual genotype pro®le and an R0 for natural scrapie in ordinary ¯ocks must be determined. Our model provides a broad illustration of the national sheep ¯ock in the UK, simplifying the continuum of ¯ock genotype pro®les by classifying them as either high risk or low risk. Various heterogeneities, such as sex and trading structure can be incorporated into the model by adding more ¯ock classes such as open/closed, and including separate classes for rams and ewes. Preferential trading of at risk sheep will increase probability of infection transmission, but not the fundamental model dynamics. While any control programme will aect many parameters in the model and not just one, these simple illustrations of control programmes do give broad hints as to the likely ecacy of a programme targeting particular parameters. Co-grazing and environmental transmission may aect any control recommendations. However, if trading is the predominant factor, then trading patterns and the distribution of infection across ¯ocks must be better understood. Whether or not other factors are important for scrapie eradication, trading is certainly critical for disease spread on the national level. Under the model assumption that exposure to a single infected sheep is the same as being exposed to many, it is in the buyers' best interests to buy all their sheep from a single ¯ock, as within-¯ock prevalence is high, but the prevalence of aected ¯ocks is low. For example, if one ¯ock of 10 is completely infected, then if 10 sheep are bought, there is a 10% probability of buying exactly 10 infected sheep if all the sheep are bought from a single source, whereas there is a 100% probability of buying exactly one, if a sheep is bought from every ¯ock. Using the data from the postal survey, if one were buying 16 lots in an unbiased manner from the national ¯ock (as one might if starting up a ¯ock from scratch), the probability a ¯ock will start out as an exposed one is 15%. The assumption of unbiased trading is likely to require further scrutiny, and thus we must investigate trading structure in the UK sheep ¯ock. Speci®cally, what is a typical ¯ock `biography'? To ascertain this, we must ask a variety of questions: Who trades with whom? How many trading partners does a ¯ock have? How does the trading age structure change ¯ock age structure? How long does a ¯ock exist for? Knowing trading patterns will also tell us how trading aects the ¯ock genotype pro®le. Whether or not targeting is worthwhile depends on the ultimate aim of a control programme. If the most important consideration is reducing the number of susceptible sheep to zero, then a nontargeted programme is preferred. As it stands, targeting all high risk ¯ocks may be dicult, however if cost and speed of result are important, simply targeting ¯ocks with multiple cases of scrapie is likely to be worthwhile, especially in the early years. Here we provide a framework for
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including trading patterns in a model of ¯ock-to-¯ock disease spread. By concentrating on the characteristics of trading between ¯ocks, the need to better understand patterns of trading is highlighted. Acknowledgements We wish to thank Dr M. Baylis for help with the genotype data and both he and Dr E.F. Houston for useful discussions and commentary on the manuscript. The anonymous referees have also provided many useful comments which have been incorporated in the manuscript. References [1] J.D. Foster, J. Hope, H. Fraser, Transmission of bovine spongiform encephalopathy to sheep and goats, Vet. Rec. 133 (1993) 339. [2] L.A. Detwiler, Scrapie, Rev. Sci. Tech. l'Oce Int. Epizooties 11 (1992) 491. [3] P.B.G.M. Belt, I.H. Muileman, B.E.C. Schreuder, J.B. -de Ruijter, A.L.J. Gielkens, M.A. Smits, Identi®cation of ®ve allelic variants of the sheep PrP gene and their association with natural scrapie, J. Gen. Virol. 76 (1995) 509. [4] Anonymous, National Scrapie Plan for Great Britain: Consultation on Proposals for Phase 1: Ram Genotyping Scheme, Ministry of Agriculture, Fisheries and Food, London, UK, 2000 (Publication No. PB5145). [5] S.M. Stringer, N. Hunter, M.E.J. Woolhouse, A mathematical model of the dynamics of scrapie in a sheep ¯ock, Math. Biosci. 153 (1998) 79. [6] M.E.J. Woolhouse, L. Matthews, P. Coen, S.M. Stringer, J.D. Foster, N. Hunter, Population dynamics of scrapie in a sheep ¯ock, Philos. Trans. Roy. Soc. London Series B 354 (1999) 751. [7] T.J. Hagenaars, C.A. Donnelly, N.M. Ferguson, R.M. Anderson, The transmission dynamics of the aetiological agent of scrapie in a sheep ¯ock, Math. Biosci. 168 (2000) 117. [8] C.A. Donnelly, N.M. Ferguson, A.C. Ghani, M.E.J. Woolhouse, C.J. Watt, R.M. Anderson, The epidemiology of BSE in cattle herds in Great Britain. I. Epidemiological processes, demography of cattle and approaches to control by culling, Philos. Trans. Roy. Soc. London Series B 352 (1997) 781. [9] N.M. Ferguson, C.A. Donnelly, M.E.J. Woolhouse, R.M. Anderson, The epidemiology of BSE in cattle herds in Great Britain. II. Model construction and analysis of transmission dynamics, Philos. Trans. Roy. Soc. London Series B 352 (1997) 803. [10] R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bull. Entomol. Soc. Am. 15 (1969) 237. [11] J.L. Hourrigan, A.L. Klingsporn, Scrapie: studies on vertical and horizontal transmission, in: C.J. Gibbs Jr. (Ed.), Bovine Spongiform Encephalopathy: The BSE Dilemma, Springer, Berlin, 1996. [12] L.J. Hoinville, A.R. McLean, A. Hoek, M.B. Gravenor, J. Wilesmith, Scrapie occurrence in Great Britain, Vet. Rec. 145 (1999) 405. [13] A.R. McLean, A. Hoek, L.J. Hoinville, M.B. Gravenor, Scrapie transmission in Britain: a recipe for a mathematical model, Proc. Roy. Soc. London B 266 (1999) 2531. [14] M. Baylis, F. Houston, W. Goldmann, N. Hunter, A.R. McLean, The signature of scrapie: dierences in the PrP genotype pro®le of scrapie-aected and scrapie-free UK sheep ¯ocks, Proc. Roy. Soc. London B 267 (2000) 2029. [15] R.M. Anderson, R.M. May, in: Infectious Diseases of Humans, Oxford University, Oxford, 1991, p. 228. [16] G. MacDonald, The analysis of equilibrium in malaria, Trop. Dis. Bull. 49 (1952) 813. [17] O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz, On the de®nition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990) 365. [18] M.B. Gravenor, D. Cox, L.J. Hoinville, A. Hoek, A.R. McLean, Temporal trends in scrapie transmission rate between farms in Britain: no evidence for a correlation with the BSE epidemic, Nature 406 (2000) 584.
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[19] M.B. Gravenor, D. Cox, L.J. Hoinville, A. Hoek, A.R. McLean, The ¯ock-to-¯ock force of infection for scrapie in Britain, Proc. Roy. Soc. London B 268 (2001) 587. [20] L.J. Hoinville, A. Hoek, M.B. Gravenor, A.R. McLean, Descriptive epidemiology of scrapie in Great Britain: results of a postal survey, Vet. Rec. 146 (2000) 455. [21] M.E.J. Woolhouse, P. Coen, L. Matthews, J.D. Foster, J.M. Elsen, R.M. Lewis, D.T. Haydon, N. Hunter, A centuries long epidemic of scrapie in British sheep, Trends Microbiol. 9 (2001) 67. [22] S. Wolfram, The Mathematica Book, 4th Ed., Wolfram Media/Cambridge University, USA, 1999. [23] L. Matthews, M.E.J. Woolhouse, N. Hunter, The basic reproduction number of scrapie, Proc. Roy. Soc. London B 266 (1999) 1085.
Modelling the national scrapie eradication programme in the UK Rowland R. Kao
a,*
, Michael B. Gravenor a, Angela R. McLean
b
a
b
Institute for Animal Health, Compton, nr. Newbury Berks RG20 7NN, UK Department of Zoology, University of Oxford, South Parks Road, Oxford, Oxon OX1 3PS, UK
Received 8 February 2001; received in revised form 5 September 2001; accepted 14 September 2001
Abstract In accordance with a policy to eliminate all transmissible spongiform encephalopathies from the food chain, a national untargeted ram breeding programme to eliminate scrapie in the UK is in the ®nal stages of planning. Here we formulate a model of ¯ock-to-¯ock scrapie transmission, in order to consider the eect of a targeted breeding programme which is in the early stages of consideration. We estimate the size of the susceptible ¯ock population, and discuss implications for potential control programmes. Targeting all rams and ewes in highly susceptible ¯ocks rather than rams in all ¯ocks will eradicate scrapie more quickly, and so is likely to be bene®cial as long as suitable penalties or incentives are available to facilitate their identi®cation. A more restricted programme aimed only at highly aected ¯ocks would be much easier to implement and crucially will eradicate scrapie just as quickly. This will leave behind a residue population of susceptible sheep, which could then be gradually removed by a more general breeding programme. Ó 2001 Published by Elsevier Science Inc. Keywords: Metapopulation; Prion; Scrapie; Sheep; Trading; TSE
1. Introduction Scrapie is an invariably fatal neurological disease of sheep that is the ®rst recognized transmissible spongiform encephalopathy (TSE). It has been endemic in the UK national sheep ¯ock for several centuries. While of some economic concern, it has never been associated with any human diseases. However, sheep have been shown to be susceptible to infection with bovine *
Corresponding author. Fax: +44-1635 577 237. E-mail address: [email protected] (R.R. Kao).
0025-5564/01/$ - see front matter Ó 2001 Published by Elsevier Science Inc. PII: S 0 0 2 5 - 5 5 6 4 ( 0 1 ) 0 0 0 8 2 - 7
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spongiform encephalopathy (BSE) by the oral route [1], and the likely contamination of meat and bone meal feed with BSE-infected material during the late 1980s makes the natural infection of sheep with BSE a very real possibility. As sheep that have been experimentally infected with BSE show clinical signs that are nearly indistinguishable from scrapie, should a natural BSE infection be discovered in the sheep population, a scrapie eradication programme on a national scale would be required. This concern has lead to a policy recommendation that all TSEs should be removed from the food chain. Control of scrapie is hampered by a lack of data on transmission, a long incubation period [2] and the absence of an eective pre-clinical diagnostic test. However, scrapie is strongly associated with genetic susceptibility to the disease [3]. Thus selective breeding for resistance to scrapie could form the basis of an eradication programme, and this has recently been proposed by the UK Department for the Environment, Food and Rural Aairs (DEFRA, formally the Ministry of Agriculture, Fisheries and Food). This breeding programme will encourage all farmers to breed from sheep of low susceptibility, eventually creating a scrapie-resistant national ¯ock. Whole-¯ock genotyping is not feasible for the entire population, so the current ®rst phase of the MAFF proposal involves genotyping only rams in the breeding pool [4]. Later phases of the eradication scheme await further re®nement and consultation. Here we develop a mathematical model of ¯ock-to-¯ock scrapie transmission to explore various control scenarios. While there have been mathematical studies of scrapie at the ¯ock level [5±7], and of BSE across cattle herds [8,9], the ¯ock-to-¯ock spread of scrapie has not yet been addressed. Unfortunately, the mathematical description of a spatially extended community such as the UK national sheep ¯ock is typically complicated. By considering the community to consist of a `metapopulation' of weakly interacting sub-populations or `patches' (i.e. the ¯ocks), these complexities can often be reduced while preserving essential properties of the system. In the classic de®nition of metapopulations all patches are the same and interact equally [10]. Flocks of domesticated animals are well suited to this type of analysis, as trading is only loosely dependent on proximity, unlike the migratory behaviour which often characterizes the interactions between wildlife populations. We construct a model of ¯ock-to-¯ock scrapie transmission around two assumptions. The ®rst assumption is that trading is the most signi®cant mode of transmission. Infection through cograzing is proven, but typically requires extensive perinatal contact [11]. Thus it is unlikely to be the most important route of ¯ock-to-¯ock transmission unless ¯ocks often combine at lambing. On the other hand, even though scrapie is a noti®able disease in the UK, the long incubation period, coupled with under-reporting [12] and misdiagnosis [13] are strong indicators that the trading of infected sheep is mainly responsible for the maintenance and spread of scrapie in the national ¯ock. The second assumption is that a subpopulation of `high risk' ¯ocks is mainly responsible for ¯ock-to-¯ock spread. Baylis et al. [14] describe a case-control study of scrapieaected and unaected farms (`the farm study'). They found a statistically signi®cant greater proportion of susceptibles, but only in the younger age cohorts (0±2 years of age) of the scrapieaected ¯ocks. As young sheep are unlikely to have experienced scrapie-associated losses, the likely implication is that their genotype pro®le predisposed them to a scrapie epidemic. While McLean et al. [13] identi®ed other risk factors for predisposition, they are likely to be secondary to genetics. If the second assumption is correct, a breeding programme targeted at all sheep in a suciently small high risk ¯ock population could be implemented for a similar eort to a programme
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63
targeted at breeding rams in all ¯ocks (the ratio of breeding rams to breeding ewes is about 1±40 [13]). If these ¯ocks or a signi®cant proportion of them could be readily found, targeting the high risk ¯ocks would more quickly reduce the susceptibility of ¯ocks most responsible for maintaining scrapie. Identifying all high risk ¯ocks may prove dicult. We therefore use our model to explore the implications of targeting either all high risk ¯ocks whether they already harbour scrapie or not, or the ¯ocks which harbour a serious scrapie problem, as these are likely to form an identi®able subset of the high risk population. 2. Model structure 2.1. The basic model The model considers several important features discussed above (Fig. 1(a)). The total number of ¯ocks (N) is ®xed. Most ¯ocks have no scrapie and are considered susceptible (S). By buying-in scrapie-infected sheep, ¯ocks become exposed (E). These ¯ocks are characterized by relatively low prevalence of scrapie. Buying and selling occurs through either the trading of sheep from one ¯ock to another (directly or through a sheep auction), or the dispersal of a ¯ock resulting in the sale of all of its breeding stock (Fig. 1(b)). If a scrapie-infected sheep transmits the disease to another in the same ¯ock, then a within-¯ock outbreak is assumed to occur, and the ¯ock is considered to have a high disease incidence and be aicted (A). These ¯ocks have relatively high prevalence of scrapie, and are most responsible for spread of the disease. Both exposed and aicted ¯ocks can recover and revert to the susceptible class. All ¯ocks can disperse and be immediately replaced, with replacement ¯ocks either susceptible or exposed. Trading is unbiased with respect to the scrapie status of ¯ocks, and to the sheep within ¯ocks. This model structure is fundamentally of the form of the SEI epidemic model (for example, [15]), with disease transmission rates determined by the proportion of scrapie-infected sheep available for purchase. Though the de®nitions of `exposed' and `aicted' are intended to mirror the de®nitions `scrapie-challenged' (scrapie only in bought-in sheep) and `scrapie-born' (scrapie in home-born sheep) as de®ned in [13], there are distinctions between the two sets of de®nitions. Most importantly, scrapie-challenged ¯ocks may contain large numbers of bought-in sheep that are infected with scrapie either before or after entering the ¯ock, and so these ¯ocks may contribute signi®cantly to ¯ock-to-¯ock scrapie transmission. To examine the high risk population hypothesis, there are two categories of ¯ocks: category one does not become aicted (the low risk ¯ocks), while in category two, aiction following exposure to scrapie is common (the high risk ¯ocks). As suggested earlier, we assume that the principle dierence between the two ¯ock types is their composition with respect to frequency of polymorphisms in the PrP gene. The generation of new high risk ¯ocks will be a combination of buying in and breeding from sheep with susceptible genotypes, and possibly the adoption of management practices such as lambing in pens which promote the transmission of scrapie [13]. Recall that MacDonald [16] de®ned the basic reproduction ratio, R0 , to be the expected number of secondary cases caused by a typical infected individual in an initially disease-free equilibrium population. A value of R0 < 1 implies that the disease cannot persist (e.g. [15]), and so the difrisk < 1 and ferences between the two ¯ock types imply that for within-¯ock transmission, Rlow 0
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Fig. 1. (a) General ¯ock structure. Flocks are distinguished by infection status (susceptible/exposed/aicted), and ability to maintain within-¯ock transmission (resistant sub-population 1, susceptible sub-population 2). Flocks may recover from scrapie if all scrapie-infected sheep are sold or die o. Flocks of type 1 are converted to type 2 and vice versa. (b) Breeding stock may be bought or sold from any one ¯ock to any other ¯ock. Stock sold from exposed or aicted ¯ocks may be infected with scrapie. Breeding stock from a dispersing ¯ock may be sold to many ¯ocks, while a new ¯ock may be formed from many ¯ocks. Both activities may result in exposure of ¯ocks to scrapie. (c) Modelling a genotyping programme. A targeted scheme could target only aicted ¯ocks as they are relatively easy to identify, or could target all high risk ¯ocks if they can be identi®ed. If replacement stock is bred or bought only from resistant lambs, an aicted ¯ock can be converted to low risk in 2 years. In an untargeted scheme, rams of all ¯ocks are genotyped. All high risk ¯ocks are aected, but conversion to low risk ¯ocks may take up to 8 years. In both cases, it is assumed that all new ¯ocks are formed from resistant stock. risk Rhigh > 1. For the purposes of this model we assume this implies that A1 0, even though this 0 is not strictly correct. The total population N N1 N2 is ®xed, with Ni Si Ei Ai . If a given ¯ock is as likely to buy from one ¯ock type as another, then the probability of becoming infected when buying a single lot is
pE1 ;E2 ;A2
pA A2 pE E1 E2 ; N1 N2
1
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65
where pE=A is the probability of acquiring at least one scrapie-infected sheep when buying from an exposed/aicted ¯ock. When a new ¯ock is formed, we must consider that multiple lots of sheep will be bought. If we consider only ewes (the ratio of rams to ewes is about 1±40 [13]), random purchase of lots of sheep from the entire ¯ock population gives the probability that a new ¯ock will include at least one infected sheep to be dE1 ;E2 ;A2 1
1
n
pE1 ;E2 ;A2 f ;
where pE1 ;E2 ;A2 is de®ned in Eq. (1), and nf is the number of lots bought to create a new ¯ock. While we have some knowledge of the risk factors associated with scrapie aiction [13] and indications of the dierences between a few scrapie-aected and unaected ¯ocks [14], we do not know the relationship between genotype pro®le and risk level, and in particular do not know the relationship between R0 and genotype pro®le. The situation is further complicated by ignorance of the ¯ock-level genotype pro®le of the national ¯ock. Therefore we postulate a conversion rate hL N1 ; N2 from low risk to high risk. We assume that increasing the number of low risk ¯ocks will decrease the conversion rate to high risk and increase the conversion rate to low risk ¯ocks, due to a decreased pool of susceptible sheep, i.e. ohL 6 0; oN1
ohL P 0; oN2
2
where the second statement follows immediately from the ®rst due to conservation of the total number of ¯ocks. Similarly, for the equivalent quantity hH , ohH P 0; oN1
ohH 6 0: oN2
3
The types of newly formed ¯ocks will also depend on the number of existing ¯ocks of each type, with a proportion ,L N1 ; N2 starting as low risk ¯ocks, and the remainder as high risk. As for hL , o,L 6 0; oN1
o,L P 0: oN2
Then a set of equations describing this model is dS1 dt
s DS1 pE1 ;E2 ;A2 wE1 D 1
DS1
hL S1 hH S2
u1 A2 ;
dE1 s DS1 pE1 ;E2 ;A2 dt dS2 dt
dE1 ;E2 ;A2 ,L N1 N2
4 w DE1 DdE1 ;E2 ;A2 ,L N1 N2
s DS2 pE1 ;E2 ;A2 wE2 D 1
dE1 ;E2 ;A2 1
hL E1 hH E2 A2 ;
,L N1 N2
DS2 hL S1
u2 A2 ;
dE2 s DS2 pE1 ;E2 ;A2 w DE2 dt dA2 Cf E2 u DA2 hH A2 : dt
5 hH S2 6
Cf E2 DdE1 ;E2 ;A2 1
,L N1 N2 ;
7
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We de®ne the remaining variables: s, the rating of buying and selling of lots, through trading between active ¯ocks; D, the rate at which ¯ocks are dispersed; w, the rate at which exposed ¯ocks recover from scrapie; ui , the rate at which aicted ¯ocks recover from scrapie to become a susceptible ¯ock of type i (with u1 u2 u), and · Cf , the rate of progression of exposed high risk ¯ocks to aiction. For individuals (here, individual ¯ocks) with discreet infectious states in a heterogeneous population, Diekmann et al. [17] gave a more mathematical de®nition of the basic reproduction ratio (R0 ) to be the spectral radius of the next generation matrix M mij , where element ij is the typical number of cases of infection class j created by a single ¯ock in infection class i over its lifetime, given an otherwise susceptible population. In cases of interest to us, this is the magnitude of the dominant eigenvalue of M. The next generation matrix in this case is given by 2 sDN p Dd , N N 3 sDN2 p1;0;0 Dd1;0;0 1 ,L N1 N2 hL 1 1;0;0 1;0;0 L 1 2 0 wDhL wDhL 6 sDN1 p0;1;0 Dd 7 sDN2 p0;1;0 Dd0;1;0 1 ,L N1 N2 Cf 0;1;0 ,L N1 N2 hH 7: 8 M6 4 wDCf hH wDCf hH 5 wDCf hH sDN1 p0;0;1 Dd0;0;1 ,L N1 N2 hH sDN2 p0;0;1 Dd0;0;1 1 ,L N1 N2 0 uDhH uDhH
· · · ·
Eq. (8) is used to calculate the eigenvalues (and thus R0 ) used later in this paper. 2.2. Parameters in the basic model All parameters are discussed below. Data from a recently comprehensive anonymous postal survey of scrapie (`the postal survey') across the UK sheep industry [12,13,18±20] provide the best available information about the movement of sheep between ¯ocks and is used to parametrize our model. We assume that the average ¯ock lifetime is similar to the length of time a farmer remains in the sheep industry. A simple model of the farmer population is dNfarmer bfarmer dt
dfarmer Nfarmer ;
where Nfarmer is the current number of sheep farmers, bfarmer is the incidence of new farmers, and dfarmer is the farmer retirement rate. Assuming that Nfarmer changes very slowly, we estimate from the postal survey that 1=dfarmer 40 yr, giving a ¯ock dispersal rate of D 1=40 yr 1 . If the age structure is in equilibrium, then on average the ®rst age cohort of a ¯ock is replaced every year. From [13], the ®rst age cohort represents 18.7% of the total ¯ock. On average, 59.8% of a ¯ock is home born; then the proportion of the ¯ock that is purchased per year is s D 0:075 yr 1 , where s is the rate of trading between ¯ocks. Therefore s 0:050 yr 1 . Breeding sheep are bought either as individuals (usually rams) or in lots containing several sheep typically of similar quality (ewes). Here we assume that a typical ¯ock contains 400 ewes and 10 rams, produces 800 lambs per year, and buys and sells breeding stock in lots of 25. The probability of acquiring an infected sheep is based on combinatorics arguments. Using the above ¯ock and lot sizes, we calculate the probability that there is at least one infected
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sheep found in a lot from an exposed or aicted ¯ock with the hypergeometric distribution, using pE=A 1
p 0; n; i; f;
where, if p m; n; i; f is the probability that m infected animals will appear in a lot of size f given iE=A infected sheep for sale per exposed/aicted ¯ock selling a total of n sheep, then the probability that a lot will contain no infected animals is p 0; n; i; f
n i! n f! : n! n i f!
The mean number of infected animals per ¯ock will typically be non-integer. When a lot is picked from many ¯ocks, we approximate using the continuous expression p 0; n; i; f
C n i 1C n C n 1C n i
f 1 ; f 1
where C n is the Gamma function, with nC n C n 1 and C 1 1. From the postal survey, there are between 1 and 2 clinically infected sheep found in an average exposed ¯ock in 1 year, and between 5 and 6 clinically infected sheep found in an average high incidence ¯ock. We use the former ®gure as a lower estimate for the number of potentially infectious sheep in an exposed ¯ock. However, the excess unexplained mortality of sheep on scrapie aected farms [13], and the paucity of older susceptible sheep on some farms with a high incidence of scrapie indicates that aicted ¯ocks may harbour more scrapie than indicated by the number of clinical cases. A rough estimate showing that the number of sheep which may have died of scrapie-related causes may be up to 30% (based on scrapie-related mortality estimates from [14]). Of course, there is no indication if these sheep are also infectious, and so we choose a conservative estimate that for every clinically infected sheep found, there are 10 potentially infectious sheep in the ¯ock (about 15%). The probability of acquiring scrapie when buying a single lot of ewes from an exposed ¯ock is pE 0:09, while for an aicted ¯ock, pA 0:98. The ¯ock recovery rates from scrapie are estimated from the postal survey. The duration of outbreaks were found to be 2 years for exposed ¯ocks and 15 years for aicted ¯ocks, leading to estimated maximum recovery rates of w 1=2 yr 1 , and u1 u2 hH 1=15 yr 1 [19]. From the available data, it is unknown what proportion of recovered ¯ocks become low risk. Susceptible genotypes appear to oer no competitive selection advantage over resistant [21], and so we would expect that in the absence of a breeding programme, the proportion of high risk ¯ocks would not increase, as would be the case if a competitive advantage existed in a scrapie-free environment. In the presence of scrapie, preliminary simulations indicate a high net conversion rate from high risk to low risk would have eliminated scrapie on a time scale inconsistent with the known course of the epidemic, where the prevalence of scrapie is slowly rising [18]. Therefore the net conversion rate must be slow. Therefore we make the conservative assumption that natural conversion rates are negligible, i.e. hH u1 1, and therefore u2 1=15 yr 1 . We note that from conditions 2 and 3, assuming otherwise will only improve the performance of a breeding for resistance programme as N2 is reduced. The proportion of new high risk ¯ocks is also an important consideration since encouraging the removal of whole high incidence ¯ocks is a likely consequence of any incentive scheme. In the
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absence of information on the relationship between genotype pro®le and within-¯ock R0 , we assume that the proportion of newly formed low risk ¯ocks is directly related to the proportion of low risk ¯ocks in the pool, i.e. N1 : ,L N1 N2 For a given set of parameters and assumed starting value of N2 =N1 the model output simulates the changes in the numbers of ¯ocks in each subpopulation over time. The proportion of all high risk ¯ocks is unknown. However, it can be estimated as the value of N2 =N1 that generates known prevalences of scrapie, in combination with all known parameter values. Gravenor et al. [19] calculate the equilibrium proportions of scrapie-aected ¯ocks to be 2.6%, and of ¯ocks with home-born scrapie to be 0.9%. Therefore our crucial output from the model is the value of N2 =N1 that is consistent with these prevalences and the parameter values discussed above. However, in part due to sparseness of appropriate farmer responses, the rate of progression from exposed to aicted in high risk ¯ocks (Cf ) is dicult to estimate. This parameter is related to the incubation period of the disease and to management practices, and is related to what Hagenaars et al. [7] refer to as the within-¯ock `mean generation time', which they derive an expression for, but do not estimate. A value greater than 1 per year implies a bought-in infected sheep is soon infectious and transmits infection during the ®rst lambing after purchase. If Cf is very small (for example, less than 0.1 per year) this implies that on average a single infected sheep is highly unlikely to transmit infection, and multiple exposures may be required. Because of the pivotal nature of this parameter, we explore it at length in the results below.
3. Control programme A breeding programme aimed at reducing the number of susceptible sheep per ¯ock will increase the conversion of high risk ¯ocks to low risk (Fig. 1(c)). High risk ¯ocks are converted to low risk ¯ocks at an additional rate m: In a targeted programme, whole ¯ock replacement would require at least 2 years (m 0:5 yr 1 ) for replacement lambs to reach maturity, assuming sucient replacements are available. An untargeted programme based on ram genotypes only is likely to create resistance more slowly. In the sheep genome, valine at codon 136 is strongly associated with scrapie susceptibility. Baylis et al. [14] found 32% of sheep in two scrapie-aected ¯ocks were valine carriers. In the ®rst two age cohorts, before scrapie-associated mortality is likely to be signi®cant, this ratio was about 50%. By comparison, the ratio of about 13.5% in two scrapie-free ¯ocks was not signi®cantly dierent for the ®rst two age cohorts compared to the rest, as would be expected. If 50% is representative of the genotype ratio for an unaicted high risk ¯ock, then breeding only from scrapie-resistant rams will result in the genotype proportion falling from 50% to less than 13.5% in about 8 years. Some scrapie-free ¯ocks may have a greater prevalence of susceptible genotypes, increasing the conversion time. In aicted ¯ocks, losses due to scrapie may further reduce the proportion of susceptibles. This would allow more rapid conversion to low risk status. We thus consider the impact of a variety of conversion rates for an untargeted programme, ranging from 2 to 8 years.
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4. Results 4.1. R0 All results were calculated using Mathematica version 4.0, as described in [22]. We note that for non-zero conversion rate u2 , the only equilibrium this system has occurs when the number of ¯ocks with scrapie is zero, however from [18,21] we can assume that we are well past the rapidly changing early stages of the epidemic and for the slow conversion rates assumed, that the proportion of aicted ¯ocks is in a quasi-steady-state, i.e. relatively stable over the time frame of interest. Fig. 2 shows the size of the high risk population for varying values of the progression rate Cf that are consistent with known estimates of scrapie infection prevalence. The range of Cf values considered (one per month to one per 10 years) corresponds to a high risk population ranging from about 3% to 20% of the national ¯ock, illustrating the importance of determining Cf . To examine the impact of this population size on control, we calculate variations in the basic reproduction ratio (R0 ) using Eq. (8). The value of R0 is a rough indicator of the eort that would be required to control the disease. A range of R0 (from about 1.1 to 1.4) is consistent with known prevalences and likely parameter values. Fig. 3(a) shows that a smaller high risk population is consistent with rapid within-¯ock transmission, and therefore a higher R0 . Not only does the smaller high risk population reduce the eort in a targeted control programme, the larger R0 value implies that any control programme (including those not based on genotyping) would have to be more eective to eradicate scrapie. For all reasonable values for Cf , the rate of progression to aiction, prevalence of all aected ¯ocks is above 2.6% as expected.
Fig. 2. Estimates of the current size of the high risk population are shown for dierent values of Cf , given known prevalence of aicted ¯ocks. The numbers of exposed, aicted and high risk susceptible ¯ocks is shown as stacked curves. The mean time for a bought-in infected sheep to transmit to other members of the ¯ock (Cf 1 ) ranges from 1 month to 10 years. The size of the national ¯ock is assumed to be 50 000. The coloured area shows all ¯ocks containing scrapie-infected sheep. The remaining ¯ocks are low risk susceptible (not shown).
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Fig. 3. Dependence on R0 . (a) Variation of R0 with the proportion of the population that is high risk. Smaller N2 implies larger R0 . In this case, targeting is more feasible, and because R0 is larger, scrapie may be more dicult to control by other means. (b) R0 as a function of the probability of buying at least one scrapie sheep when buying a lot of 25 sheep from an aicted ¯ock. (c) R0 as a function of the trading rate s (proportion of ¯ock traded per year), with a ®xed total dispersal related trading of D 0:025 yr 1 . Three curves in both (a) and (b) are for dierent values of the progression time to aiction.
The impact of parameter changes on R0 is also a good indicator of the parameter sensitivity of the model. The sensitivity of these conclusions to parameters for which only rough estimates are available (pE ; pA ; D; s) is minimal. However, while the trading rate is well established (see Section 2.2), the model at present does not consider the marked heterogeneities in trading structure. McLean et al. [13] report that most farmers either buy-in the majority of their stock or mainly breed their own stock. The eect of trading on R0 is signi®cant ± little eort would be required to either decrease the probability (p) of acquiring infection through trading (Fig. 3(b)) or restrict trading to the point at which scrapie could not persist (Fig. 3(c)), however such restrictions on trade may not be feasible. The addition of heterogeneity may further change R0 , indicating that further analysis of ¯ock structure is required. 4.2. Genotyping programmes: non-targeted versus targeting high risk or targeting aicted ¯ocks Fig. 4 shows the time course of an epidemic governed by Eq. (5), following the introduction of a single exposed high risk ¯ock. Initially, the disease progression rate can be approximated by R0 u1 u2 hH ; where the generation time is T 1= u1 u2 hH ). The slow rate of
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Fig. 4. Example epidemic over 50 years. The epidemic starts with a single aicted ¯ock in a national ¯ock of 50 000. Times of progression to aicted status is 1 and 10 years in (a) and (b), respectively.
progression is therefore consistent with the low value of R0 and long generation times in the parameter estimates. Because susceptible ¯ocks only slowly become aected, disease control eort can at least initially ignore susceptible ¯ocks, and concentrate on aicted ones. In Fig. 5, we compare the time course of an epidemic under a programme which targets all sheep in all high risk ¯ocks, a programme which targets all sheep in aicted ¯ocks, and a non-targeted genotyping programme. Either targeting programme quickly converts aicted high risk ¯ocks into low risk ¯ocks, so it is initially more eective in reducing scrapie prevalence. Within 2 years, targeting reduces the proportion of aicted ¯ocks to less than half that under an untargeted programme. However, in the simple targeting programme aimed only at aicted ¯ocks, once the proportion of susceptible ¯ocks falls below that necessary for the disease to persist (i.e. R0 < 1) the aicted
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Fig. 5. Reduction in high risk ¯ocks over 10 years for two types of targeting and a non-targeted breeding programme (total of 50 000 ¯ocks). Targeting aicted ¯ocks rapidly reduces ¯ock scrapie prevalence, but it does not ®nd scrapiefree high risk ¯ocks (N2 never reaches zero). In contrast, a non-targeted scheme will eventually create a scrapie resistant national ¯ock. Targeting all high risk ¯ocks does not eradicate scrapie any more quickly than targeting aicted ¯ocks, but does rapidly eliminate the high risk population. In (a), the time of progression to aiction is 1 year, in (b) it is 10 years.
¯ocks will disappear, but some high risk (susceptible) ¯ocks will remain. In contrast both the more comprehensive targeted programme aimed at all high risk ¯ocks and the non-targeted programme will eventually reduce the high risk ¯ocks to zero. Surprisingly, we note that changing the progression rate to aiction (Cf ), does little to change the rate at which genotyping removes aicted ¯ocks. This is because the conversion rate m is rapid compared to the rate at which susceptible ¯ocks are exposed and eventually become aicted. If this is the case, then if a high proportion of aicted ¯ocks can be found, targeting
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will be eective no matter what Cf is, as long as the conversion rate to low risk ¯ocks is high enough. Targeting all high risk ¯ocks also does little to change the rate of aicted ¯ock removal, but of course does remove all susceptible high risk ¯ocks as well. 5. Discussion The ®rst phase of Britain's scrapie control programme is currently planned as an untargeted genotyping of rams in all ¯ocks. Here, we have considered the implications of genotyping targeted at all sheep but in aicted ¯ocks only. Not only will the eort expended diminish with time as the proportion of aicted ¯ocks approaches zero, but because aicted ¯ocks are rapidly removed, scrapie will disappear more quickly. Targeting is most attractive when the high risk population is small, not only because it reduces the cost and eort required, but because it implies a larger value of R0 which also makes disease control more dicult under any programme. Further, genotyping all sheep in target ¯ocks quickly creates a valuable source of resistant sheep, as long as they can be accredited as scrapie free. Identi®cation of all high risk ¯ocks (rather than just those that are aicted) is likely to require extensive genotyping, since even low risk ¯ocks may harbour some susceptible genotypes [14], and except at low values of the rate of progression to aicted Cf the proportion of all sheep in the high risk ¯ocks is much greater than the proportion of rams in all ¯ocks (estimated 3±20% high risk ¯ocks compared to about 2.5% rams in all ¯ocks). On the other hand, targeting ¯ocks owned by farmers who report multiple cases of scrapie to MAFF would approximate the targeting of aicted ¯ocks discussed here, and would only require genotyping a relatively small number of sheep (estimated less than 1% aicted ¯ocks ± Section 2.2). Scrapie is noti®able to MAFF, but under-reporting may make targeting dicult, especially for farmers with multiple cases of scrapie (farmers with scrapie were eight times less likely to report to MAFF than anonymously [12]). Accurate reporting is an essential component of a targeting programme and would have to be addressed. This may be aided by better knowledge of the characteristics of high risk ¯ocks [13] and especially of the genotype pro®le of a typical high risk ¯ock [14]. Despite the attendant diculties, an incentive programme aimed at encouraging the reporting of scrapie cases and breeding for resistance would be much easier to implement than the alternative of attempting to locate all high risk ¯ocks, would result in the eradication of scrapie nearly as quickly as a scheme aimed at all high risk ¯ocks, and much more quickly than an untargeted scheme. If such a scheme were to be implemented, an accompanying research programme to validate the central hypothesis of an identi®able high risk population would be essential. What is surprising is that generation of new high risk ¯ocks and the existence of many exposed ¯ocks makes little dierence to the overall ecacy of eradication. This is a result of the slow time scale of the epidemic, much slower than the projected time for the breeding programme to be eective. An argument for a general breeding programme is that it will eventually eliminate the entire high risk ¯ock population, whereas a targeted programme aimed at aicted ¯ocks will only reduce it to a level for which R0 will become one. As the eort involved in a targeted programme diminishes quickly over time, one suggestion would be to expend eort on both programmes, with the intention of phasing out the targeting scheme after the ®rst few years.
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This analysis uses the best data available to estimate parameters important for ¯ock-to-¯ock transmission of scrapie through trading. The model suppositions could be validated in at least two ways. First, the existence of signi®cant dierences between scrapie-aected and scrapie-free ¯ocks should be determined from further analysis of the farm study. Second, the size of the high risk population would be illustrated once a substantial proportion of the national ¯ock is genotyped, though risk factors other than genotype pro®le (such as outlined in [13]) will need to be accounted for. However two checks can be made if the data were available. First, using the proportion of aicted ¯ocks from the postal survey, the farm study genotype pro®les can be checked for consistency with the pro®le of all sheep which have been genotyped by MAFF. Second, the within-¯ock R0 can be used to evaluate the progression rate Cf and establish an independent value of the size of the high risk population N2 . Matthews et al. [23] quote R0 to be 3.9 in an experimental scrapie ¯ock. However this ¯ock has an unusual genotype pro®le and an R0 for natural scrapie in ordinary ¯ocks must be determined. Our model provides a broad illustration of the national sheep ¯ock in the UK, simplifying the continuum of ¯ock genotype pro®les by classifying them as either high risk or low risk. Various heterogeneities, such as sex and trading structure can be incorporated into the model by adding more ¯ock classes such as open/closed, and including separate classes for rams and ewes. Preferential trading of at risk sheep will increase probability of infection transmission, but not the fundamental model dynamics. While any control programme will aect many parameters in the model and not just one, these simple illustrations of control programmes do give broad hints as to the likely ecacy of a programme targeting particular parameters. Co-grazing and environmental transmission may aect any control recommendations. However, if trading is the predominant factor, then trading patterns and the distribution of infection across ¯ocks must be better understood. Whether or not other factors are important for scrapie eradication, trading is certainly critical for disease spread on the national level. Under the model assumption that exposure to a single infected sheep is the same as being exposed to many, it is in the buyers' best interests to buy all their sheep from a single ¯ock, as within-¯ock prevalence is high, but the prevalence of aected ¯ocks is low. For example, if one ¯ock of 10 is completely infected, then if 10 sheep are bought, there is a 10% probability of buying exactly 10 infected sheep if all the sheep are bought from a single source, whereas there is a 100% probability of buying exactly one, if a sheep is bought from every ¯ock. Using the data from the postal survey, if one were buying 16 lots in an unbiased manner from the national ¯ock (as one might if starting up a ¯ock from scratch), the probability a ¯ock will start out as an exposed one is 15%. The assumption of unbiased trading is likely to require further scrutiny, and thus we must investigate trading structure in the UK sheep ¯ock. Speci®cally, what is a typical ¯ock `biography'? To ascertain this, we must ask a variety of questions: Who trades with whom? How many trading partners does a ¯ock have? How does the trading age structure change ¯ock age structure? How long does a ¯ock exist for? Knowing trading patterns will also tell us how trading aects the ¯ock genotype pro®le. Whether or not targeting is worthwhile depends on the ultimate aim of a control programme. If the most important consideration is reducing the number of susceptible sheep to zero, then a nontargeted programme is preferred. As it stands, targeting all high risk ¯ocks may be dicult, however if cost and speed of result are important, simply targeting ¯ocks with multiple cases of scrapie is likely to be worthwhile, especially in the early years. Here we provide a framework for
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including trading patterns in a model of ¯ock-to-¯ock disease spread. By concentrating on the characteristics of trading between ¯ocks, the need to better understand patterns of trading is highlighted. Acknowledgements We wish to thank Dr M. Baylis for help with the genotype data and both he and Dr E.F. Houston for useful discussions and commentary on the manuscript. The anonymous referees have also provided many useful comments which have been incorporated in the manuscript. References [1] J.D. Foster, J. Hope, H. Fraser, Transmission of bovine spongiform encephalopathy to sheep and goats, Vet. Rec. 133 (1993) 339. [2] L.A. Detwiler, Scrapie, Rev. Sci. Tech. l'Oce Int. Epizooties 11 (1992) 491. [3] P.B.G.M. Belt, I.H. Muileman, B.E.C. Schreuder, J.B. -de Ruijter, A.L.J. Gielkens, M.A. Smits, Identi®cation of ®ve allelic variants of the sheep PrP gene and their association with natural scrapie, J. Gen. Virol. 76 (1995) 509. [4] Anonymous, National Scrapie Plan for Great Britain: Consultation on Proposals for Phase 1: Ram Genotyping Scheme, Ministry of Agriculture, Fisheries and Food, London, UK, 2000 (Publication No. PB5145). [5] S.M. Stringer, N. Hunter, M.E.J. Woolhouse, A mathematical model of the dynamics of scrapie in a sheep ¯ock, Math. Biosci. 153 (1998) 79. [6] M.E.J. Woolhouse, L. Matthews, P. Coen, S.M. Stringer, J.D. Foster, N. Hunter, Population dynamics of scrapie in a sheep ¯ock, Philos. Trans. Roy. Soc. London Series B 354 (1999) 751. [7] T.J. Hagenaars, C.A. Donnelly, N.M. Ferguson, R.M. Anderson, The transmission dynamics of the aetiological agent of scrapie in a sheep ¯ock, Math. Biosci. 168 (2000) 117. [8] C.A. Donnelly, N.M. Ferguson, A.C. Ghani, M.E.J. Woolhouse, C.J. Watt, R.M. Anderson, The epidemiology of BSE in cattle herds in Great Britain. I. Epidemiological processes, demography of cattle and approaches to control by culling, Philos. Trans. Roy. Soc. London Series B 352 (1997) 781. [9] N.M. Ferguson, C.A. Donnelly, M.E.J. Woolhouse, R.M. Anderson, The epidemiology of BSE in cattle herds in Great Britain. II. Model construction and analysis of transmission dynamics, Philos. Trans. Roy. Soc. London Series B 352 (1997) 803. [10] R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bull. Entomol. Soc. Am. 15 (1969) 237. [11] J.L. Hourrigan, A.L. Klingsporn, Scrapie: studies on vertical and horizontal transmission, in: C.J. Gibbs Jr. (Ed.), Bovine Spongiform Encephalopathy: The BSE Dilemma, Springer, Berlin, 1996. [12] L.J. Hoinville, A.R. McLean, A. Hoek, M.B. Gravenor, J. Wilesmith, Scrapie occurrence in Great Britain, Vet. Rec. 145 (1999) 405. [13] A.R. McLean, A. Hoek, L.J. Hoinville, M.B. Gravenor, Scrapie transmission in Britain: a recipe for a mathematical model, Proc. Roy. Soc. London B 266 (1999) 2531. [14] M. Baylis, F. Houston, W. Goldmann, N. Hunter, A.R. McLean, The signature of scrapie: dierences in the PrP genotype pro®le of scrapie-aected and scrapie-free UK sheep ¯ocks, Proc. Roy. Soc. London B 267 (2000) 2029. [15] R.M. Anderson, R.M. May, in: Infectious Diseases of Humans, Oxford University, Oxford, 1991, p. 228. [16] G. MacDonald, The analysis of equilibrium in malaria, Trop. Dis. Bull. 49 (1952) 813. [17] O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz, On the de®nition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990) 365. [18] M.B. Gravenor, D. Cox, L.J. Hoinville, A. Hoek, A.R. McLean, Temporal trends in scrapie transmission rate between farms in Britain: no evidence for a correlation with the BSE epidemic, Nature 406 (2000) 584.
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[19] M.B. Gravenor, D. Cox, L.J. Hoinville, A. Hoek, A.R. McLean, The ¯ock-to-¯ock force of infection for scrapie in Britain, Proc. Roy. Soc. London B 268 (2001) 587. [20] L.J. Hoinville, A. Hoek, M.B. Gravenor, A.R. McLean, Descriptive epidemiology of scrapie in Great Britain: results of a postal survey, Vet. Rec. 146 (2000) 455. [21] M.E.J. Woolhouse, P. Coen, L. Matthews, J.D. Foster, J.M. Elsen, R.M. Lewis, D.T. Haydon, N. Hunter, A centuries long epidemic of scrapie in British sheep, Trends Microbiol. 9 (2001) 67. [22] S. Wolfram, The Mathematica Book, 4th Ed., Wolfram Media/Cambridge University, USA, 1999. [23] L. Matthews, M.E.J. Woolhouse, N. Hunter, The basic reproduction number of scrapie, Proc. Roy. Soc. London B 266 (1999) 1085.