Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol168.Issue2.Dec2000:

The transmission dynamics of the aetiological agent of scrapie

in a sheep ¯ock

T.J. Hagenaars

*,1

_{, C.A. Donnelly}

1

_{, N.M. Ferguson}

1

_{, R.M. Anderson}

1

Wellcome Trust Centre for the Epidemiology of Infectious Disease (WTCEID), Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3FY, UK

Received 2 December 1999; received in revised form 22 June 2000; accepted 1 September 2000

Abstract

We formulate and investigate the properties of a model framework to mimic the transmission dynamics of the aetiological agent of scrapie in a sheep ¯ock. We derive expressions for summary parameters that characterize transmission scenarios, notably the basic reproduction numberR0 and the mean generation timeTg. The timescale of epidemic outbreaks is expressed in terms ofR0and cumulants of the generation time distribution. We discuss the relative contributions to the overall rate of transmission of horizontal and vertical routes during invasion and in endemicity. Simpli®ed models are used to obtain analytical insight into the characteristics of the endemic state. Ó _{2000 Elsevier Science Inc. All rights reserved.}

Keywords:Scrapie; Transmission dynamics; Basic reproduction number; Epidemic model; Vertical transmission

1. Introduction

Scrapie is a transmissible spongiform encephalopathy (TSE) in sheep (for a recent review, see Ref. [1]). Like other TSEs (see, e.g. [2,3]), scrapie a€ects the central nervous system and is fatal. It has a variable incubation period with a mean of several years and is believed to be naturally transmitted by horizontal and vertical routes. However, the precise transmission mechanisms of the aetiological agent of scrapie are poorly understood at present, and so is its transmission dynamics. The factors relevant to scrapie epidemiology are numerous: detailed evidence exists for genetically determined susceptibility di€erences [4,5], for both vertical and horizontal

transmis-www.elsevier.com/locate/mbs

*

Corresponding author. Tel.: +44-1865 271 264; fax: +44-1865 281 241. E-mail address:thomas.hagenaars@zoo.ox.ac.uk (T.J. Hagenaars).

1

Present address: Department of Infectious Disease Epidemiology, Imperial College School of Medicine, St Mary's Campus, Norfolk Place, London W2 1PG, UK.

0025-5564/00/$ - see front matter Ó _{2000 Elsevier Science Inc. All rights reserved.}

sion of scrapie, and experimental results in mice suggest the possibility of a carrier genotype [6±8]. Furthermore, evidence exists that the environment can act as a reservoir of infectious particles [9], and see Ref. [1].

As a result of the BSE epidemic and its link with new variant CJD (vCJD) in humans, control of scrapie and other possible TSEs in sheep has become a priority in the UK and other European countries. Diagnostic tests for preclinical scrapie infection are under development [10,11]. The potential hazard to human health from diseased sheep arises from the fact that many sheep in Britain have been fed contaminated meat and bone meal (MBM) prior to the ruminant feed ban in 1988 and the ®nal ban on the use of MBM in all animal feeds in 1996. As a result, given the successful experimental transmission of BSE to sheep by oral challenge with infected bovine brain tissue [12], there is the hypothetical but very real possibility that a BSE-like spongiform en-cephalopathy has established itself in the sheep population, presenting itself as scrapie. If this TSE were horizontally transmissible, as appears to be the case for scrapie, it would be much more dicult to control than BSE in cattle, where the elimination of the feed-borne infection route was sucient to interrupt and control the epidemic in Britain [13±16].

Recently enhanced e€orts [17] to collect epidemiological data for scrapie are likely to facilitate more detailed analyses of the transmission dynamics of this disease using mathematical models. Potentially useful mathematical models tend to be fairly complex, for example due to the inclusion of age structure, age-dependent susceptibilities, transmission via both horizontal and vertical routes as well as the population genetics of di€erent susceptibility classes. As a result, calculating the basic reproduction number (R0) and other quantities that characterize the model's dynamical

regimes requires some care. In this paper we propose a theoretical framework in which the ¯ock-level dynamics of scrapie can be studied for a wide range of underlying between-animal trans-mission scenarios. We characterise di€erent transtrans-mission scenarios using the basic reproduction number and the generation time distribution, and investigate how these measures relate to the duration of epidemics. We discuss the interplay between horizontal and vertical transmission routes. Simpli®ed models are used to gain insights into the characteristics of endemic states, in-cluding the e€ect of disease on the genotype distribution and the occurrence of recurrent incidence peaks.

2. Model framework

In formulating our theoretical framework, we take into account the following processes: direct and indirect (via environment) horizontal transmission; vertical transmission; genotypes that di€er in susceptibility to scrapie infection as well as in the distribution of the incubation period; age-dependence in susceptibility and rates of slaughter; dependence of horizontal and vertical infectiousness on time since infection or time to onset; seasonality in lambing.

o_Xc

o_t …t;a† ‡ o_Xc

o_a …t;a† ˆd…a†s…t†B

c_{t† ÿi}c_{t;a† ÿl…a†X}c_{t;a†; …1†}

o_Nc

o_t …t;a† ‡ o_Nc

o_a …t;a† ˆd…a†s…t†B c_{t† ÿ}

Z a

0

fc…s†ic_{tÿs;aÿs†dsÿl…a†N}c_{t;a†; …2†}

dI

dt…t† ˆkdh…t† ÿgI…t†; …3†

ic…t;a† ˆd…a†BcY…t† ‡k

c

…a;t†Xc…t;a†; …4†

kc…t;a† ˆgc_h…a†…vI…t† ‡kdh…t††; …5†

kdh…t† ˆ

X

c0

Z Z a0

0

Ac_dh0 …t;s;a0†ic0

…tÿs;a0†dsda0; …6†

Bc…t† ˆ B0b c_t†

P

c0bc 0

…t†; …7†

bc…t† ˆX c0

Z

a…a0†Gcc0_Nc0_t;a0_†da0_{; …8†}

Bc_Y…t† ˆgc_vX c0

Z Z a0

0

Acc_v0…t;s;a0†ic0_tÿs;a0_†dsda0_{; …9†}

Here Eqs. (1) and (2) describe the change in Xc_{t;a†, the number of susceptible (or resistant)} animals with genotypecof agea, andNc_{t;a†, the total number of animals with genotypecof age}

a. Eq. (3) represents the change of the infectivity loal I of the environment. Bc _{denotes the total} birth rate after factoring out a periodic normalized seasonality pro®le s…t†, of genotypec lambs (corresponding to an increase in the number of animals of age aˆ0, as denoted by the `delta function' d…a†), B0 a constant overall birth rate, BcY…t† the birth rate of infected lambs with

ge-notype c, ic_{t;a†the incidence of new infections in genotype c animals of agea, l…a† the rate of} routine culling of animals of agea, resulting in an equilibrium survival probabilityS…a†in absence of disease, andfc_{s†the incubation period distribution for genotypecanimals.k}c

…t;a†denotes the force of infection for the horizontal transmission route (including both direct transmission, cor-responding to the infection riskkdh, and transmission via the environment, proportional tovI…t†),

v a parameter that moderates the transmissibility of scrapie via the environment (relative to the direct horizontal transmissibility), and 1=gthe characteristic time of decay for the infectivity of the environmental reservoir. Furthermore, gc_h…a† is the relative susceptibility/exposure to horizontal infection of an animal of ageaand genotypec,gc_vthe relative susceptibility to vertical infection of an animal of genotypec,Ac_dh…t;s;a†the expected force of direct horizontal infection experienced at timetby an animal with unit susceptibility caused by an animal with genotypecthat was infected

sunits of time ago at agea, andAcc0

v …t;s;a†the expected birth rate (after taking out the seasonality

s…t†) of infected genotypeco€spring, assuming unit susceptibility, born at timetto an animal with genotypec0 _{that was infectedsunits of time ago at age a. In Eq. (8), a…a† is a lambing rate, and}

Gcc0 _{the proportion of genotype c lambs born to genotype c}0 _{ewes. For a single locus-two allele}

model we havec2 fRR;RS;SSg(whereRis the resistance andSthe susceptibility allele), and for random mating, the proportions Gcc0 _{are elements of the matrix (see [18])}

Gˆ …Gcc0

wherep is the frequency of the Rallele in the rams.

We can write Ac_dh…t;s;a† as the product of the probability S…a‡s†=S…a† that the primary in-fective survives from routine culling at least until agea‡sconditional on its survival to agea, an expected net infectiousnessbc…s†(`net' because this infectiousness is weighed by the probability of not yet having died from the disease a timesafter infection), and, assuming mass-action scaling of the contact rate with population size, a factor 1=N…t†(withN…t†the total number of animals in the ¯ock at timet)

Ac_dh…t;s;a† ˆS…a‡s†

S…a†N…t†b

c

…s†: …11†

In stochastic implementations of the model framework,N…t† is replaced byN…t† ÿ1 in the above expression (see [19]). Similarly,Acc0

v …t;s;a† can be written as the product of the probabilityS…a‡

s†=S…a†that the primary infective survives at least until agea‡sconditional on its survival to age

a, its expected net vertical infectiousness c_{s†, the age-dependent lambing rate B}

0a…a‡s†=

…P

c0bc 0

Acc_v0…t;s;a† ˆS…a‡s†

A number of simplifying assumptions has been made in the above equations. In particular, in Eq. (5) it is assumed that the age and genotype distribution of secondary infections via the horizontal transmission routes (direct and indirect) do not depend on the age or genotype of the primary infection from which they originate, i.e. horizontal mixing is separable with respect to age and genotype. In Eq. (6) it is also assumed that the infectiousness only depends on genotype and incubation stage, and not on age. Furthermore, we have assumed that the incubation period distribution is independent of the age at infection. In this paper we will concentrate on the case of an open ¯ock, here de®ned (as in Ref. [20,21]) by the assumption that the ewes in the ¯ock are mated with rams from outside which have constant allele frequencyp, i.e. independent of the allele frequency in the ¯ock.

2.1. A PDE model

By modeling the force of infection at time t in terms of the number of infected individuals (possibly strati®ed by `infection load' or `incubation stage') at that same time, one can devise SI-type models (see [22]), that consist of ordinary di€erential equations or of partial di€erential equations (PDEs). Here, we discuss a PDE model where we treat the incubation process as a movement through a discrete number of incubation stages. Similar models for the transmission dynamics of scrapie have been developed by Woolhouse et al. [21,23], Stringer et al. [20], and Matthews et al. [24].

Assuming that the progression of disease in time (i.e. incubation and pathogenesis) does not depend on the route of acquisition, and taking the same number of incubation stages for each (susceptible) genotype, our PDE model is obtained by replacing Eqs. (2), (6), and (9) by

incubation stage k. Animals leaving the ®nal incubation stage kf reach onset of scrapie. If we

choose the transition rates to be independent of the stagek, we obtain a gamma distribution for the incubation period, given by the probability density function

…mc_†n

…nÿ1†!t

…nÿ1†_exp…ÿ…mc_†t†:

In the models of Woolhouse and co-workers [20±24], the infectiousness of an infected animal is taken to be proportional to an infection load, exponentially increasing with time; variation in the initial infection load gives rise to a distributed incubation period.

3. The basic reproduction number and generation time distribution

The basic reproduction number R0 is de®ned here as the expected number of secondary

in-fections arising from a single primary infection in a naive population. It measures the capability of the infectious agent to establish itself and spread in the host population [22,25]. In particular, in deterministic models, infection will establish within the host population when R0P1. In more

realistic stochastic models (in which demographic and infection events are modeled as random processes that occur with a certain rate), ifR0<1 no major disease outbreaks can occur. Control

strategies aim to reduce the basic reproduction number of a disease, preferably to below 1. For a structured population, the calculation for R0 has been outlined in [26,27,25]. In [24], the

calcu-lation ofR0 was presented before for the partial-di€erential equation model for scrapie employed

in [21,23,20,24]. For a derivation ofR0 for structured populations in situations where birth rate,

survivorship and transmission coecient vary over time, see [28].

3.1. R0 in absence of vertical transmission

For the special case of horizontal transmission only (both direct and via the environment), assuming separable mixing, the expression forR0 reduces to [27, Eq. (20)], and translates to

R0 ˆ

in our notation. We recall thatBc…0† is the number of new susceptible individuals of genotypec

born per time unit,S…a† is the survival probability to age a, and gc_{a† is the age-dependent} sus-ceptibility; the product S…a†Bc_0†gc_{a† represents the distribution of primary infections over age} and genotype. We have introduced Ac_h…t;s;a† as the rate of horizontal infection, including both direct transmission and transmission via the environment, of a susceptible individual with unit susceptibility by an individual of genotypecthat was infectedsunits of time ago while its age was

a. For our model of the environmental infectivity reservoir, integrating out the passage of the agent through the reservoir, we obtain

Ac_h…t;s;a† ˆ 1

3.2. Factors determining the intensity of horizontal transmission

Eqs. (18) and (11) form the mathematical representation of how the horizontal transmission potential of scrapie is determined by four key factors:

1. `Mean' susceptibility of animals in a ¯ock (as represented by the factorBc_0†gc_{a†). The larger} the fraction of animals that are (partly) resistant, the smaller the (e€ective) contact rate of a primary infective with susceptible animals. If susceptibility is age-dependent, R0 will be largest

when peak susceptibility occurs at young age, since this maximizes the average susceptibility of (secondary) infected animals during disease invasion.

2. Flock demography(`S…a‡s†'). In a typical sheep ¯ock, a majority of animals does not survive more than four years, whereas some animals might survive up to 10 yr of age or older. Since the mean incubation period of scrapie is a few years, this large turnover in animals will limit disease transmission, especially under scenarios where late-stage infected animals are most infectious and hence responsible for most of the transmission.

3. Infectiousness as a function of time since infection(`bc…s†'). For a given maximum level of infec-tiousness, the more con®ned infectiousness is to animals in late stages of incubation, the smaller the transmission potential.

4. Nature of the environmental reservoir (`v=g'). Transmission can be promoted either by a long half-life (small g) or by a high transmissibility of the agent via the environment (large v).

3.3. Including vertical transmission

When allowing for non-zero vertical transmission, the calculation of R0 becomes more

com-plicated for two reasons. The ®rst is that the age distribution of vertically infected animals is di€erent from the age distribution of horizontally infected animals, so that the average over age in the calculation of the reproduction number has to be done separately (see also [29]). The second reason is that for vertical transmission, the rates depend on the combination of genotypes of primary infection (ewe) and secondary infection (lamb), i.e. mixing is not separable. Consecutive generations of infected animals are now related via a generation matrixM_{g, andR0is given by the}

largest eigenvalue of this matrix. For a single-locus two-allele system, allowing for a possible carrier infection state for the homozygous resistant animals, this is a 66 matrix, since six dif-ferent types of infected animals need to be considered (three genotypes times two transmission routes). Ordering these types as…y_hc;y_vc†, where the subscripts h and v denote `horizontally infected' and `vertically infected', respectively, this matrix has the form

M_gˆ Mhh Mhv

Mvh Mvv

; …20†

where the elements of the four 33 submatrices read:

M_hhcc0 ˆBc0…0†

Z Z

S…a†gc_h0…a†Ac_h0…0;s;a†dadsM_hhc0; …21†

M_hvcc0 ˆBc0…0†

Z

S…a†g_hc…a†da

Z

M_vhcc0 ˆgc_v

Generally, one needs to resort to numerical root-®nding methods for obtaining the largest ei-genvalue of the matrix M_{g (see [24]). If we assume that the genotype dependency of the}

suscep-tibility is the same for vertical and horizontal transmission,Mhhis proportional toMhv andMvh is

proportional to Mvv. In this case the characteristic equation forMg for a single-locus two-allele

system is e€ectively of fourth order (two zero eigenvalues). If the RRgenotype is truly resistant against infection, the problem simpli®es further into a cubic eigenvalue equation, so thatR0can be

derived analytically.

4. Horizontal and vertical transmission and their interplay

In this section we discuss how horizontal and vertical transmission can contribute to infection propagation, both through their separate as well as combined action. This is best elucidated by considering the analytically most tractable case where only one genotype is susceptible to

infec-tion. In this case the generation matrix is (essentially) a 22 matrix with elements

Mhh;Mhv;Mvh;Mvv, and

We note that this applies to any disease that can be transmitted via horizontal as well as vertical routes; the same is true for all results discussed in this section. From this expression forR0, we see

that disease propagation takes place via three `pathways': one has `pure' pathways hh and vv as well as an alternating pathway hvh (represented by the 4MhvMvh term). We note that in a

non-expanding sheep ¯ock, vertically infected animals will on average infect at most 1_±p animals vertically each, i.e.Mvv61ÿp. For a given transmission probability from a maximally infectious

ewe, vertical transmission is most e€ective when the incubation period is long compared to the mean life span and animals are strongly infectious throughout incubation (allowing ewes to pass on infection year after year). The absolute contribution toR0of transmission pathways not purely

horizontal, R0ÿMhh, is largest in the limit, where MhhMvv (e.g. when the horizontal

trans-mission coecient is large), where it is entirely due to the alternating transtrans-mission pathway and given byR0ÿMhhˆMhvMvh=Mhh. In fact, this limit is relevant under many scenarios, since it will

be approached already for `modest' values ofR0 even whenMvvis close to unity. As is illustrated

in Fig. 2, for the examples of Mvvˆ0:2 and 0.4, the limit is reached or almost reached already

onceR0 >1.

During the initial phase of disease invasion, the infectives are distributed over transmission routes as yh=yv ˆ …R0ÿMvv†=Mvh ˆMhv=…R0ÿMhh†, where yh…yv† is the fraction of infected

ani-mals that acquired infection via the horizontal (vertical) route. In endemic equilibrium, the disease propagation is described by ane€ectivegeneration matrixM

g(taking into account, amongst other

largest eigenvalue equals unity. Thus the endemic distribution of infective individuals is then given by

y_h=y_v ˆ …1ÿM_vv†=M_vh Pp=M_vh : …26†

Fig. 2. (a) Contribution toR0 of the vertical transmission route (R0ÿMhh), shown in units ofjMhvMvh=Mhh) as a

function of the horizontal transmissionMhh, forMvvˆ0:2 and di€erent values ofj. The arrows indicate whereR0ˆ1.

We note that for vertical transmission there is no infection saturation as occurs for horizontal transmission, so that ratios of the `vertically transmitting' matrix elements M

vv=MvvandMvh =Mvh

are generally bigger than their horizontally transmitting counterparts. Whethery_h=y_v <yh=yv or

not will be determined by the combined e€ect of three di€erences between invasion phase and endemicity: the increased (in endemicity as compared to the invasion phase) infection prevalence amongst ewes will enhance the fraction of vertically acquired infections; if heterozygotes are susceptible as well, a reduction in the fraction of homozygous susceptible animals (due to disease-induced mortality) will reduce the fraction of vertically acquired infections (M_vv <Mvv); a lowering

of the average age at infection via the horizontal route might either increase or decrease the fraction of vertically acquired infections, depending on its e€ect on the number of births while infectious for horizontally infected animals (M

vh $Mvh). We note that for pˆ0, one hasMvv ˆMvv. If in this

case vertical transmission is perfect (Mvvˆ1), Eq. (26) gives yh ˆ0. In this case, all animals are

infected; see also [30]. If M

vvˆMvv and the ratio Mvh =Mvh is increasing with overall horizontal

transmission coecient (for example when the lowering of the average age at infection of hori-zontally infected animals has the e€ect of enhancing the number of births while infectious), it follows from Eq. (26) thaty

h=yv decreases with increasing horizontal transmission eciency. This

explains the ®nding in [30] that the highest equilibrium vertical transmission rates are seen when horizontal transmission is very ecient (In [30], the increase with R0 of Mvh =Mvh in fact arises

because, due to the carrying-capacity form assumed for population growth, the birth rate (in particular to infected ewes) is enhanced in response to an increased (disease-induced) mortality).

5. Generation time and epidemic timescales

Besides the basic reproduction number, another useful quantity in characterizing transmission scenarios is the mean generation time Tg, the expected time between a primary and a secondary

and vmax is an eigenvector corresponding to the maximum eigenvalue of M. In the absence of

We note that as the generation time for direct horizontal transmission is limited by the mean lifetime of animals, the generation time for transmission via the environment is not. In Eq. (32) this is re¯ected in the fact that the second term is independent ofS…a†.

The typical timescale for disease invasion in a ¯ock is determined by R0 together with the

generation time distribution. In the extreme case that there is no variation in the generation time (i.e. the time between a primary and secondary infection is always the same), the real-time growth raterduring the initial phase of invasion is given byrˆln…R0†=Tg. In the more interesting case of

a distributed generation time, corrections to this expression forr can be obtained using a cum-ulant expansion of the generation time distribution, as shown in Appendix A. To second order in ln…R0†, we obtain

g† is the variance of the generation time distribution.

6. Characteristics of endemic states

We now turn to the exploration of endemic states, in particular to the endemic disease prev-alence and genotype frequencies. Since, in the presence of disease, susceptible animals have a lower life expectancy and on average fewer o€spring than animals resistant to infection, the susceptible allele frequency will be reduced during an outbreak of scrapie. This e€ect has been described and studied using a PDE model in [20]. Here we will gain analytical insight into this matter, and into the characteristics of the endemic state in general, using simpli®ed di€erential equation models.

6.1. A simpli®ed model

dX1

where the subscripts 1,2,3 denote the genotypesRR;RS;SS, respectively, lis the birth and death rate in absence of scrapie,the vertical transmission rate, andmis the rate of extra mortality for animals infected with scrapie. Disease-induced mortality is assumed to be compensated for by recruitment of animals from outside the ¯ock, whose genotype frequencies correspond to the constant external allele frequency p, as represented by the gain terms proportional tomY in Eqs. (35)±(37). Assuming the ¯ock is in Hardy±Weinberg equilibrium initially, the basic reproduction

number for the above model is given by R0 ˆRh0 ‡R

0 ˆl…1ÿp†=…l‡m†are the contributions of horizontal transmission and vertical transmission,

respectively.

6.2. Infection prevalence

Due to the simpli®ed form of the above model, it is possible to solve for the endemic equi-librium analytically. In absence of vertical transmission (ˆ0), the equilibrium fraction of sus-ceptible animals is a factor 1=R0 smaller than the initial fraction…1ÿp†

2

. Maternal transmission further reducesX

3, however in a linear manner

X₃=N ˆl‡mÿl…1ÿp†

The equilibrium prevalence Y_{=N is proportional to the di€erence between the initial and the}

equilibrium fraction of susceptible animals, with a proportionality factor that depends on p

Y=N ˆ l disease-induced mortality has no e€ect on endemic prevalence. Forp6ˆ0, the endemic prevalence is reduced when increasing the degree of disease-induced mortality (measured by here bym=l).

6.3. Reduction in the frequency of the susceptibility allele

For the disease-induced increase in the frequency of the resistant allele, we ®nd

The increase depends linearly on the vertical transmission coecient . We illustrate how it de-pends onpandum=lin Fig. 4. We note that the increase in the resistant allele frequency will be larger in the case where heterozygotes are susceptible as well, since in that case the life expectancy and reproductivity of all genotypes carrying the susceptibility allele is reduced.

Fig. 4. Relationship between the `disease-induced' increase in the resistant allele frequency and the initial frequency, for di€erent values ofum=l.

Fig. 3. Relationship between endemic prevalenceY_{=Nand initial frequencypof the resistant allele, for di€erent values}

6.4. Recurrent incidence peaks

In the model calculation of Woolhouse et al. [21], it was found that in an open ¯ock the ap-proach to endemic equilibrium was oscillatory, resulting in recurrent incidence peaks. In order to understand the origin of such an oscillatory behavior, and the e€ect of parameter choices on this phenomenon, we here consider a simpli®ed model within which a linear stability analysis can be done analytically. In this model we consider only two di€erent genotypes, one resistant and one susceptible, and we assume that when one of the parents is resistant and one susceptible, the o€spring has equal probability of becoming either susceptible or resistant. The model equations read

where X1…X2† denotes the number of animals of resistant (susceptible) genotype. The basic

re-production number is given by

R0 ˆ

b…1ÿp† ‡l…1ÿp

2†

l‡m :

From a linear stability analysis, we ®nd for R0 >1 that the damping characteristics of endemic

equilibrium is determined by the sign of the expression

8p2…R0ÿ1†u3ÿ …R0

Here um=land in case of a positive (negative) sign endemic equilibrium is approached in an oscillatory (non-oscillatory) manner. The heuristic interpretation of this condition is as follows: As in a standard SIR model, the oscillations occur as a result of a di€erence in timescales: the timescale of epidemic growth (here inversely proportional to l‡m) is `faster' than the timescale 1=lof replenishment of susceptible individuals. How fast the timescale 1=mneeds to be depends on bothR0andp, as illustrated in Fig. 5. IfR0is large, the equilibrium fraction of susceptibles will be

small, making it harder to have `excess' depletion of the susceptible pool, thus leading to a high threshold value for u. For small R0 (i.e. close to 1), the threshold is high because of the small

equilibrium fraction of infective individuals. Generally, the threshold is increasing with decreasing

p(when keepingR0®xed). This is because decreasingpmeans decreasing the fraction of resistant

animals, thus reducing the `over¯ow compartment' needed for excess depletion of the susceptible pool. For the present model there is a criticalpvalue of approximatelypˆ0:15. Above this value, the value for R0 with the lowest threshold for u is always at R0 ˆ3=2. Ifp is smaller than this

critical value, we ®nd oscillatory behavior for u above a threshold value and, for a range of R0

7. Conclusions

In this paper we have presented a theoretical framework for the study of the transmission dynamics of the aetiological agent of scrapie in a sheep ¯ock. Our aim has been to provide basic

Fig. 5. (a) Boundaries between overdamped ()) and underdamped (+) endemic states in theu;R0 plane for di€erent

results that will be useful as background and tools for more data-oriented modeling work in the future. Our framework comprises most of the aspects that are of relevance (potentially or in reality) to the transmission dynamics of scrapie. The basic theoretical results concern the interplay between horizontal and vertical transmission, the timescale of disease invasion in a ¯ock, the e€ect of disease invasion on the population genetics of a ¯ock, and the characteristics of endemic states. These results were obtained using deterministic descriptions of the population dynamics of scrapie in a sheep ¯ock. If and how they are changed when taking into account demographic stochasticity will be discussed in a forthcoming paper [31]. This stochasticity can be important when the number of infected animals is small. In endemic situations, small numbers of infected animals can result from a small ¯ock size, low frequencies of susceptible genotypes, or a basic reproduction number close to 1. For a given basic reproduction number, e€ects of transmission stochasticity will be most pronounced for long latent but short infectious periods. An environmental infectivity reservoir acts to suppress stochastic ¯uctuations in endemic infection prevalence, i.e. reduce the probability of endemic fade-out.

Acknowledgements

T.J.H., C.A.D. and R.M.A. thank the Wellcome Trust and MAFF for research grant support. NMF thanks the Royal Society and MAFF for research grant support.

Appendix A. The relationship between the real-time growth rate and_R0 for structured but separably

mixing populations

As explained in [27], there is in general no direct relationship between the real-time growth rate

rof the epidemic andR0, other than the equivalenceR0ÿ1>0()r >0. Here, we will formally

expandrin powers of ln…R0†for a structured but homogeneously mixing population (see also [32],

and for closely related results in the context of demography, see [33]). For speci®c situations, truncations of this expansion may provide useful approximate (or even exact) relations betweenr

and R0. It follows from Eq. (25) in [26] that the real-time growth rater for separable mixing is

given by the implicit equation

1ˆB…0†

Z

S…a†g…a†

Z

exp…ÿrs†A…s;a†dsda: …A:1†

Using the de®nition of R0 given in Eq. (18), we may write

hexp…ÿrs†i_K ˆ 1

R0

; …A:2†

wherehi_K denotes a normalized average obtained by the integration oversagainst a `generation' kernel K…s†

hf…s†i_K

R1

0 f…s†K…s†ds

R1

0 K…s†ds

where

K…s†

Z

S…a†g…a†A…s;a†da: …A:4†

The averaging in the left-hand side of (A.2) can be interchanged with the exponentiation

hexp…ÿrs†i_K ˆexp X

whereCn are the cumulants de®ned by

Cn

These cumulants characterise the distribution of the generation time; e.g. C1 is the average

gen-eration timeTg (as de®ned in Eq. (33)), andC2 is the variance. After taking the logarithm of Eq.

(A.2), we arrive at

X1

nˆ1

…ÿr†n

n_! Cn ˆ ÿlnR0: …A:7†

If we now formally expandr in powers of lnR0 as well,

rˆX

1

iˆ1

q_i…lnR0†i: …A:8†

Eq. (A.7) enables us to express the coecientsq_iin terms of the cumulantsCn. For the ®rst three

coecients, we ®nd

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