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Mathematical Biosciences 168 (2000) 117±135
www.elsevier.com/locate/mbs

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 number R0 and the mean generation
time Tg . The timescale of epidemic outbreaks is expressed in terms of R0 and 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-

*

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.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 4 8 - 1

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T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

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 encephalopathy 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 ¯ocklevel dynamics of scrapie can be studied for a wide range of underlying between-animal transmission scenarios. We characterise di€erent transmission 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, including 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.
Here we concentrate on the case of a single ¯ock, assuming for simplicity that there is no
recruitment of animals from outside and that there are no infectious contacts with animals from
outside ¯ocks. Fig. 1 provides a graphical representation of our model framework. In mathematical terms, a general population-level formulation is given by the following integral equations:

T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

119

Fig. 1. Flow diagram of the proposed model structure.

oX c
oX c
…t; a† ‡
…t; a† ˆ d…a†s…t†Bc …t† ÿ ic …t; a† ÿ l…a†X c …t; a†;
ot
oa
Z a
oN c
oN c
…t; a† ‡
…t; a† ˆ d…a†s…t†Bc …t† ÿ
f c …s†ic …t ÿ s; a ÿ s† ds ÿ l…a†N c …t; a†;
ot
oa
0

…1†
…2†

dI
…t† ˆ kdh …t† ÿ gI…t†;
dt

…3†

ic …t; a† ˆ d…a†BcY …t† ‡ kc …a; t†X c …t; a†;

…4†

kc …t; a† ˆ ghc …a†…vI…t† ‡ kdh …t††;

…5†

kdh …t† ˆ

XZ Z

a0

0

0

Acdh …t; s; a0 †ic …t ÿ s; a0 † ds da0 ;

B0 bc …t†
Bc …t† ˆ P c0 ;
c0 b …t†
c

b …t† ˆ

…6†

0

c0

XZ

…7†

0

0

a…a0 †Gcc N c …t; a0 † da0 ;

…8†

c0

BcY …t†

ˆ

gvc

XZ Z
c0

a0

0

0

0 c
0
0
Acc
v …t; s; a †i …t ÿ s; a † ds da ;
0

…9†

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T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

Here Eqs. (1) and (2) describe the change in X c …t; a†, the number of susceptible (or resistant)
animals with genotype c of age a, and N c …t; a†, the total number of animals with genotype c of 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 genotype c lambs
(corresponding to an increase in the number of animals of age a ˆ 0, as denoted by the `delta
function' d…a†), B0 a constant overall birth rate, BcY …t† the birth rate of infected lambs with genotype c, ic …t; a† the incidence of new infections in genotype c animals of age a, l…a† the rate of
routine culling of animals of age a, resulting in an equilibrium survival probability S…a† in absence
of disease, and f c …s† the incubation period distribution for genotype c animals. kc …t; a† denotes the
force of infection for the horizontal transmission route (including both direct transmission, corresponding to the infection risk kdh , and transmission via the environment, proportional to vI…t†),
v a parameter that moderates the transmissibility of scrapie via the environment (relative to the
direct horizontal transmissibility), and 1=g the characteristic time of decay for the infectivity of the
environmental reservoir. Furthermore, ghc …a† is the relative susceptibility/exposure to horizontal
infection of an animal of age a and genotype c, gvc the relative susceptibility to vertical infection of
an animal of genotype c, Acdh …t; s; a† the expected force of direct horizontal infection experienced at
time t by an animal with unit susceptibility caused by an animal with genotype c that was infected
0

s units of time ago at age a, and Acc
v …t; s; a† the expected birth rate (after taking out the seasonality
s…t†) of infected genotype c o€spring, assuming unit susceptibility, born at time t to an animal with
genotype c0 that was infected s units of time ago at age a. In Eq. (8), a…a† is a lambing rate, and
0
Gcc the proportion of genotype c lambs born to genotype c0 ewes. For a single locus-two allele
model we have c 2 fRR; RS; SSg (where R is the resistance and Sthe susceptibility allele), and for
0
random mating, the proportions Gcc are elements of the matrix (see [18])
0

p
G ˆ …G † ˆ @ 1 ÿ p
0
cc0

p=2
0:5
…1 ÿ p†=2

1
0
p A;
1ÿp

…10†

where p is the frequency of the R allele in the rams.
We can write Acdh …t; s; a† as the product of the probability S…a ‡ s†=S…a† that the primary infective survives from routine culling at least until age a ‡ s conditional on its survival to age a, an
expected net infectiousness bc …s† (`net' because this infectiousness is weighed by the probability of
not yet having died from the disease a time s after infection), and, assuming mass-action scaling of
the contact rate with population size, a factor 1=N …t† (with N …t† the total number of animals in the
¯ock at time t)
Acdh …t; s; a† ˆ

S…a ‡ s† c
b …s†:
S…a†N …t†

…11†

In stochastic implementations of the model framework, N …t† is replaced by N…t† ÿ 1 in the above
0
expression (see [19]). Similarly, Acc
v …t; s; a† can be written as the product of the probability S…a ‡
s†=S…a† that the primary infective survives at least until age a ‡ s conditional on its survival to age
a,Pits expected net vertical infectiousness c …s†, the age-dependent lambing rate B0 a…a ‡ s†=
0
0
… c0 bc …t††, and the proportion Gcc :

T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

121

0

0

Acc
v …t; s; a† ˆ

S…a ‡ s†
B
a…a ‡ s†Gcc c0
P 0c0
 …s†:
bc …t†
S…a†
c0 b …t†

…12†

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 frequency p, 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 SItype 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
oY0c oY0c
‡
ˆ d…a†BcY …t† ‡ kc …t; a†X c ÿ mc0 Y0c ÿ l…a†Ykc ;
…13†
ot
oa
oYkc oYkc
c
‡
ˆ mckÿ1 Ykÿ1
ÿ mck Ykc ÿ l…a†Ykc
ot
oa
Z
0
1 X c0
bk
Ykc …t; a0 † da0 ;
kdh …t† ˆ
N…t† c0 ;k
N c …t† ˆ X c …t† ‡

X

…1 < k < kf †

Ykc …t†;

…14†
…15†
…16†

k

BcY …t†

ˆ

s…t†gvc

XXZ
0
0
B0
0
P c00
a…a0 †Gcc ck Ykc …t; a0 † da0 :
c00 b …t† c0
k

…17†

Eqs. (13) and (14) describe the change in the number Ykc …t; a† of infected sheep of age a and genotype c in incubation stage k, with mck the transition rate from incubation stage k to …k ‡ 1† for
genotype c. Furthermore we have introduced bck , the horizontal infectiousness of an animal genotype c in incubation stage k, and ck , the vertical infectiousness of an animal genotype c in

122

T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

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 stage k, we obtain a gamma distribution for
the incubation period, given by the probability density function
…mc †n …nÿ1†
t
exp…ÿ…mc †t†:
…n ÿ 1†!
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 infections 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 R0 P 1. In more
realistic stochastic models (in which demographic and infection events are modeled as random
processes that occur with a certain rate), if R0 < 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 calculation of R0 was presented before for the partial-di€erential equation model for scrapie employed
in [21,23,20,24]. For a derivation of R0 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 for R0 reduces to [27, Eq. (20)], and translates to
Z
Z
X
c
c
B …0†g …a† Ach …0; s; a† ds da
…18†
R0 ˆ S…a†
c

in our notation. We recall that Bc …0† is the number of new susceptible individuals of genotype c
born per time unit, S…a† is the survival probability to age a, and gc …a† is the age-dependent susceptibility; the product S…a†Bc …0†gc …a† represents the distribution of primary infections over age
and genotype. We have introduced Ach …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 genotype c that was infected s units 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




Z 1
v
c
c
0
0
Acdh …t; s; a†:
…19†
v expf ÿ g…s ÿ s†g ds Adh …t; s; a† ˆ 1 ‡
Ah …t; s; a† ˆ 1 ‡
g
s
In the expression for R0 resulting from Eqs. (18) and (11), the S…a† factors cancel out.

T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

123

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 factor Bc …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 infectiousness, 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 complicated 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 matrix Mg , and R0 is 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 6  6 matrix, since six different types of infected animals need to be considered (three genotypes times two transmission
routes). Ordering these types as …yhc ; yvc †, where the subscripts h and v denote `horizontally infected'
and `vertically infected', respectively, this matrix has the form


Mhh Mhv
;
…20†
Mg ˆ
Mvh Mvv
where the elements of the four 3  3 submatrices read:
Z Z
0
0
cc0
c0
c0
Mhh ˆ B …0†
S…a†ghc …a†Ach …0; s; a† da ds  Mhh
;
Z
Z
0
0
c0
cc0
;
ˆ Bc …0† S…a†ghc …a† da Ach …0; s; 0† ds  Mhv
Mhv

…21†
…22†

124

T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135
cc0
Mvh
cc0
Mvv

ˆ

gvc

RR

ˆ

gvc

Z

0

0

S…a†ghc …a†Acc
v …0; s; a† da ds;
R
0
S…a†ghc …a† da
0

Acc
v …0; s; 0† ds:

…23†
…24†

Generally, one needs to resort to numerical root-®nding methods for obtaining the largest eigenvalue of the matrix Mg (see [24]). If we assume that the genotype dependency of the susceptibility is the same for vertical and horizontal transmission, Mhh is proportional to Mhv and Mvh is
proportional to Mvv . In this case the characteristic equation for Mg for a single-locus two-allele
system is e€ectively of fourth order (two zero eigenvalues). If the RR genotype is truly resistant
against infection, the problem simpli®es further into a cubic eigenvalue equation, so that R0 can 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 infection. In this case the generation matrix is (essentially) a 2  2 matrix with elements
Mhh ; Mhv ; Mvh ; Mvv , and
q
Mhh ‡ Mvv 1
2
‡
…Mhh ‡ Mvv † ‡ 4Mhv Mvh ÿ 4Mhh Mvv :
R0 ˆ
…25†
2
2
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 for R0 , 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 4Mhv Mvh term). We note that in a nonexpanding sheep ¯ock, vertically infected animals will on average infect at most 1±p animals
vertically each, i.e. Mvv 6 1 ÿ p. 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 to R0 of transmission pathways not purely
horizontal, R0 ÿ Mhh , is largest in the limit, where Mhh  Mvv (e.g. when the horizontal transmission coecient is large), where it is entirely due to the alternating transmission pathway and
given by R0 ÿ Mhh ˆ Mhv Mvh =Mhh . In fact, this limit is relevant under many scenarios, since it will
be approached already for `modest' values of R0 even when Mvv is close to unity. As is illustrated
in Fig. 2, for the examples of Mvv ˆ 0:2 and 0.4, the limit is reached or almost reached already
once R0 > 1.
During the initial phase of disease invasion, the infectives are distributed over transmission
routes as yh =yv ˆ …R0 ÿ Mvv †=Mvh ˆ Mhv =…R0 ÿ Mhh †, where yh …yv † is the fraction of infected animals that acquired infection via the horizontal (vertical) route. In endemic equilibrium, the disease
propagation is described by an e€ective generation matrix Mg (taking into account, amongst other
things, the disease-induced reduction in the number of susceptible individuals), of which the

T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

125

Fig. 2. (a) Contribution to R0 of the vertical transmission route (R0 ÿ Mhh ), shown in units of j  Mhv Mvh =Mhh ) as a
function of the horizontal transmission Mhh , for Mvv ˆ 0:2 and di€erent values of j. The arrows indicate where R0 ˆ 1.
(b) Same as (a), but with Mvv ˆ 0:4.

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



:
P p =Mvh
†=Mvh
yh =yv ˆ …1 ÿ Mvv

…26†

126

T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

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 Mvv
=Mvv and Mvh
=Mvh


are generally bigger than their horizontally transmitting counterparts. Whether yh =yv < 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
< Mvv ); a lowering
induced mortality) will reduce the fraction of vertically acquired infections (Mvv
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


$ Mvh ). We note that for p ˆ 0, one has Mvv
ˆ Mvv . If in this
horizontally infected animals (Mvh

case vertical transmission is perfect (Mvv ˆ 1), Eq. (26) gives yh ˆ 0. In this case, all animals are


ˆ Mvv and the ratio Mvh
=Mvh is increasing with overall horizontal
infected; see also [30]. If Mvv
transmission coecient (for example when the lowering of the average age at infection of horizontally infected animals has the e€ect of enhancing the number of births while infectious), it
follows from Eq. (26) that yh =yv decreases with increasing horizontal transmission eciency. This
explains the ®nding in [30] that the highest equilibrium vertical transmission rates are seen when

=Mvh in fact arises
horizontal transmission is very ecient (In [30], the increase with R0 of Mvh
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
infection in a naive population
RR
 vmax s ds
M…s†
;
Tg ˆ R R 
M…s†
vmax ds


mhh …s† mhv …s†

;
…27†
M…s†
ˆ
mvh …s† mvv …s†;
where

0
v
ˆ B …0†
Acdh …0; s; a†;
g…g ‡ v†s
a


Z
0
v
cc0
c
c0
mhv …s† ˆ B …0† S…a†gh …a† da 1 ‡
Acdh …0; s; 0†;
g…g ‡ v†s
P
c0
c0
cc0
0
c
a S…a†n …0†gh …a†Av …0; s; a†
mcc
;
R
0
vh …s† ˆ gv
S…a†ghc …a† da
0
mcc
hh …s†

0

c0

0

X

0

0
S…a†ghc …a†

c cc
mcc
vv …s† ˆ gv Av …0; s; 0†;


1‡

…28†
…29†
…30†
…31†

T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

127

and vmax is an eigenvector corresponding to the maximum eigenvalue of M. In the absence of
vertical transmission, this reduces to
RR
C…s; a†s ds da
v
‡
;
…32†
Tg ˆ R R
C…s; a† ds da g…g ‡ v†
where

C…s; a† ˆ S…a ‡ s†

X

Bc …0†gc …a†bc …s†:

…33†

c

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 of S…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
rate r during the initial phase of invasion is given by r ˆ ln…R0 †=Tg . In the more interesting case of
a distributed generation time, corrections to this expression for r can be obtained using a cumulant expansion of the generation time distribution, as shown in Appendix A. To second order in
ln…R0 †, we obtain
!
ln…R0 †
r2 …Tg †
rˆ
1‡
ln…R0 † ;
…34†
Tg
2Tg2
where r2 …Tg † 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 prevalence 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
We simplify our PDE model of Section 6.1 by leaving out the age structuring and the strati®cation into latent classes from the general framework discussed in Section 2, and concentrate on
the situation where the only susceptible animals are those homozygous for the susceptibility allele.
This translates into the following ordinary di€erential equation model:

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T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

dX1
dt
dX2
dt
dX3
dt
dY
dt

lp
X2 ‡ mp2 Y ÿ lX1 ;
2
l
ˆ l…1 ÿ p†X1 ‡ X2 ‡ lp…X3 ‡ Y † ‡ m2p…1 ÿ p†Y ÿ lX2 ;
2
l…1 ÿ p†
Y
ˆ
X2 ÿ l…1 ÿ p†…X3 ‡ …1 ÿ †Y † ‡ m…1 ÿ p†2 Y ÿ b X3 ÿ lX3 ;
2
N
Y
ˆ l…1 ÿ p†Y ‡ b X3 ÿ lY ÿ mY ;
N
ˆ lpX1 ‡

…35†
…36†
…37†
…38†

where the subscripts 1,2,3 denote the genotypes RR; RS; SS, respectively, l is the birth and death
rate in absence of scrapie,  the vertical transmission rate, and m is 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 to mY 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 ˆ Rh0 ‡ Rv0 , where Rh0 ˆ b…1 ÿ p†2 =…l ‡ m† and
Rv0 ˆ l…1 ÿ p†=…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 equilibrium analytically. In absence of vertical transmission ( ˆ 0), the equilibrium fraction of susceptible animals is a factor 1=R0 smaller than the initial fraction …1 ÿ p†2 . Maternal transmission
further reduces X3 , however in a linear manner
X3 =N ˆ

l ‡ m ÿ l…1 ÿ p†
1 ÿ Rv0
ˆ …1 ÿ p†2
:
b
Rh0

…39†

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


l
…1 ÿ p†2
2

…40†
=N
ˆ
ÿ
X
…R0 ÿ 1†:
…1
ÿ
p†
Y  =N ˆ
3
l ‡ m…4p ÿ 3p2 †
Rh0 …1 ‡ m=l…4p ÿ 3p2 ††

In Fig. 3, we study the dependence of Y  =N on p and m=l. If there is no resistant genotype (p ˆ 0),
disease-induced mortality has no e€ect on endemic prevalence. For p 6ˆ 0, the endemic prevalence
is reduced when increasing the degree of disease-induced mortality (measured by here by m=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
p ÿ p ˆ

2pm…l ‡ m†
2p…1 ÿ p†2 m=l
…R0 ÿ 1†:
ÿ
1†
ˆ
…R
0
b…l ‡ m…4p ÿ 3p2 ††
Rh0 …1 ‡ m=l…4p ÿ 3p2 ††

…41†

T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

129

Fig. 3. Relationship between endemic prevalence Y  =N and initial frequency p of the resistant allele, for di€erent values
of u  m=l.

The increase depends linearly on the vertical transmission coecient . We illustrate how it depends on p and u  m=l in 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 of u  m=l.

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T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

6.4. Recurrent incidence peaks
In the model calculation of Woolhouse et al. [21], it was found that in an open ¯ock the approach 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
dX1
l…p ÿ 1†
lp
X1 ‡ …X2 ‡ Y † ‡ mpY ;
ˆ
…42†
2
2
dt

dX2
l…1 ÿ p†
lp
p
Y
X1 ÿ X2 ‡ l 1 ÿ …1 ÿ †Y ÿ b X2 ‡ m…1 ÿ p†Y ;
ˆ
…43†
2
2
2
N
dt
dY
Y
p
ˆ b X2 ÿ lY ‡ l…1 ÿ †Y ÿ mY ;
…44†
dt
N
2
where X1 …X2 † denotes the number of animals of resistant (susceptible) genotype. The basic reproduction number is given by
R0 ˆ

b…1 ÿ p† ‡ l…1 ÿ p2†
:
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 …R0 ÿ 1†u3 ÿ …R0 ÿ 1†2 ÿ …8p2 ÿ 6p†…R0 ÿ 1† ÿ p2 u2
2



3
2
:
…45†
ÿ 2…R0 ÿ 1† ÿ …1 ‡ 6p†…R0 ÿ 1† ‡ p u ÿ R0 ÿ
2

Here u  m=l and 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=l of replenishment of susceptible individuals. How fast the timescale 1=m needs to be depends on
both R0 and p, as illustrated in Fig. 5. If R0 is 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 keeping R0 ®xed). This is because decreasing p means 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 critical p value of approximately p ˆ 0:15. Above this value,
the value for R0 with the lowest threshold for u is always at R0 ˆ 3=2. If p is smaller than this
critical value, we ®nd oscillatory behavior for u above a threshold value and, for a range of R0
values just below R0 ˆ 3=2, also for a small bounded region of low u values.

T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

131

Fig. 5. (a) Boundaries between overdamped ()) and underdamped (+) endemic states in the u; R0 plane for di€erent
values of p. (b) Boundaries between overdamped and underdamped endemic states in the u; p plane for di€erent values
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

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T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

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
r of the epidemic and R0 , other than the equivalence R0 ÿ 1 > 0 () r > 0. Here, we will formally
expand r in 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 between r
and R0 . It follows from Eq. (25) in [26] that the real-time growth rate r for separable mixing is
given by the implicit equation
Z
Z
1 ˆ B…0† S…a†g…a† exp…ÿrs†A…s; a† ds da:
…A:1†
Using the de®nition of R0 given in Eq. (18), we may write
hexp…ÿrs†iK ˆ

1
;
R0

…A:2†

where h
iK denotes a normalized average obtained by the integration over s against a `generation'
kernel K…s†
R1
f …s†K…s† ds
hf …s†iK  0 R 1
;
…A:3†
K…s† ds
0

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T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

where
K…s† 

Z

…A:4†

S…a†g…a†A…s; a† da:

The averaging in the left-hand side of (A.2) can be interchanged with the exponentiation
!
1
X
…ÿr†n
…A:5†
hexp…ÿrs†iK ˆ exp
Cn ;
n!
nˆ1
where Cn are the cumulants de®ned by

 n

o
lnhexp…s†iK  :
Cn 
o
ˆ0

…A:6†

These cumulants characterise the distribution of the generation time; e.g. C1 is the average generation time Tg (as de®ned in Eq. (33)), and C2 is the variance. After taking the logarithm of Eq.
(A.2), we arrive at
1
X
…ÿr†n
Cn ˆ ÿ ln R0 :
n!
nˆ1

…A:7†

If we now formally expand r in powers of ln R0 as well,
rˆ

1
X

qi …ln R0 †i :

…A:8†

iˆ1

Eq. (A.7) enables us to express the coecients qi in terms of the cumulants Cn . For the ®rst three
coecients, we ®nd
1
;
C1
C2
;
q2 ˆ
2C13
C2
C3
q3 ˆ 25 ÿ 4 :
2C1 6C1

…A:9†

q1 ˆ

…A:10†
…A:11†

For a gamma-distributed (with parameters l and h) generation time, the coecients qn satisfy the
relationship
qn ˆ

q1
1
h
ˆ …nÿ1† ˆ n ;
…nÿ1†
n!l
n!l
Tg n!l
1=l

and the series (A.8) can be seen to converge to the analytical result r ˆ h…R0 ÿ 1†. With increasing n, the values of coecients qn become increasingly sensitive to the detailed form of the
long-time tail of the generation time distribution. Therefore, in practice, when estimating the
growth rate r from approximate knowledge of the generation time distribution, it will not be
useful to take into account more than the ®rst two or three terms in (A.8).

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T.J. Hagenaars et al. / Mathematical Biosciences 168 (2000) 117±135

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