Page 1

A modelling approach to vaccination and

contraception programmes for rabies control

in fox populations

Christelle Suppo1, Jean-Marc Naulin2*, Michel Langlais2and Marc Artois3

1IRBI-UMR CNRS 6035, Universite¨ deT ours, 37200 T ours, France (suppo@univ-tours.fr)

2UMR CNRS 5466,`Mathe¨matiques Applique¨es de Bordeaux’ , BP 26, Universite¨ Victor Segalen Bordeaux 2,

33076 Bordeaux Cedex, France (jean-marc.naulin@mi2s.u-bordeaux2.fr, michel.langlais@mi2s.u-bordeaux2.fr)

3AFSSA Nancy, Laboratoire d’Etudes sur la Rage et la Pathologie, des Animaux Sauvages, B.P. 9, 54220 Malzeville, France

(marc.artois@nancy.afssa.fr)

In a previous study, three of the authors designed a one-dimensional model to simulate the propagation

of rabies within a growing fox population; the in£uence of various parameters on the epidemic model

was studied, including oral-vaccination programmes. In this work, a two-dimensional model of a fox

population having either an exponential or a logistic growth pattern was considered. Using numerical

simulations, the e¤ciencies of two prophylactic methods (fox contraception and vaccination against

rabies) were assessed, used either separately or jointly. It was concluded that far lower rates of administra-

tion are necessary to eradicate rabies, and that the undesirable side-e¡ects of each programme disappear,

when both are used together.

Keywords: discrete modelling; rabies; foxes; oral vaccination; contraception

1. INTRODUCTION

Fox rabies is a major veterinary public-health problem in

several countries of the world (Blancou et al. 1991). Oral

vaccination of foxes carried out by distribution of vaccine

baits has had a clear impact on the prevalence of the

virus in Western Europe (StÎhr & Meslin 1997; Pastoret

& Brochier1999).

Data from fox-hunting records indicate that the

European fox population tends to increase (Artois 1997).

This has been observed in both rabies-free (Great Britain,

Tapper 1992; J.-A. Reynolds, personal communication)

and rabies-infected areas (Belgium, De Combrugghe

1994; Germany, MÏller 1995). This does not mean that

fox populations are not regulated over the long term, but

simply that over the short term the population income^

outcome ratio is not balanced for unknown reasons

(increase of resources and/or decrease in mortality).

Whatever the cause of fox-population increase, it could

eventually impede the success of further oral-vaccination

campaigns when the number of non-immunized foxes

becomeshigh enoughto

(Breitenmoser et al. 1995; Vuillaume et al. 1997). A su¤-

cient level of culling to achieve a sustainable control of

the population is di¤cult to obtain if the rabies threshold

density is much lower than that of the population

carrying capacity (Anderson et al. 1981). The combination

of culling and vaccination is still a matter of debate

(Smith 1995; Barlow 1996). A promising solution could be

the limitation of recruitment of healthy foxes through

fertility control. Increasing e¡orts have been focused on

this technique for red fox predation control in Australia

(Bradley1994).

In a previous model, vaccination by the oral route

alone was examined as a way of controlling rabies in

carryon the infection

high fox-population density areas (Artois et al. 1997). This

model emphasized that a vaccination rate lower than 70%

will allow the epidemic to persist, a ¢gure already

described by Smith (1995). In this study, the focus was on

fertility controlthroughthe use ofbaits¢lledwitha contra-

ceptive vaccine in conjunction with a rabies vaccine as a

possible method of controlling rabies when vaccination

aloneis not su¤cient fordiseaseeradication(Smith1995).

2. DESCRIPTION OF THE MODEL

The present model was based on the one-dimensional

discrete deterministic model of Artois et al. (1997). The

fox population has been structured in space (a two-

dimensional model in this paper with N home ranges), in

age (young and adult, i.e. dispersing foxes or residents

one year old and more), in sex (female and male) and in

disease state (healthy, exposed and vaccinated). This gave

12 classes of foxes per cell through which rabies propa-

gated (¢gure 1). The density of healthy young females in

cell n at time t has been denoted by HYF(n,t), with analo-

gous notations for the 11other fox classes.

The time-step, ¢t ˆ10 days, chosen in the simulations

is longer than the life expectancy of clinically ill indivi-

duals (1^4 days) (Blancou et al. 1991). Thus no speci¢c

class of infectious individuals has been considered.

Instead, the number of infectious individuals in the time

interval from t to t+¢t is proportional to the number of

exposed individuals; the proportionality coe¤cient ¼(t) is

the inverse of the latency period.

(a) Demography

As a further contrast with Artois et al. (1997), this

paper simulates the demography of the fox population as

either exponentially increasing or density dependent.

In a density-dependent fox population the natural

mortality is di¡erent for young foxes and adult ones,

Proc. R. Soc. Lond. B (2000) 267, 1575^1582

Received 6 March 2000

1575

© 2000 The Royal Society

Accepted 12 April 2000

doi10.1098/rspb.2000.1180

*Author for correspondence.

Page 2

according to season. Survival is therefore also density

dependent: for adult female foxes it has been determined

as

saf(t) ˆ s^ af(t) £

1

1 ‡ ¯(n,t)P(n,t);

(1)

where sa ª f(t) is the natural survival rate of adult female

foxes, P(n,t) is the total density of foxes living at time t in

home range n and ¯(n,t) is a non-negative parameter.

When ¯ ˆ 0 there is no density dependence, while a posi-

tive ¯ will yield a logistic e¡ect. Similar formulae have

been used for other age and sex classes. In simulations, ¯

is a constant, numerically evaluated to supply an average

fox density of 0.01ha71(Artois 1989). Only healthy and

vaccinated females were able to reproduce as incubation

period was shorter than gestation and weaning duration;

hence infected cubs had no chance of survival. The

density of healthy young females in cell n is

HYF(n,t+¢t) ˆ b(t)£saf(t)£HAF(n,t),(2)

where the birth function, b(t), satis¢es b(t) ˆ b0on 1 April

and b(t) ˆ 0 otherwise (Artois 1989), with 2b0being the

average number of cubs per litter per female and b0

referred to as the half birth rate.

(b) Two-dimensional spatial structure

and dispersion

A rectangular domain is subdivided into cells having a

hexagonal shape, each cell corresponding to the size of an

average fox’s home range. Cells have been numbered

from 1 to N; the hexagon lying at the intercept of line i

and column j is numbered

n ˆ (i71)£nmax+ j,

where nmaxis the maximal number of cells on a line

(¢gure 2).

Conversely, the location (i, j) on the grid of a hexagon

having number n could quickly be found from

(3)

i ˆ I

n ¡ 1

nmax

‡ 1,I ‰rŠ

is the integerpart of the realnumber r,(4)

j ˆ n ¡ (i ¡ 1) £ nmax.

During the dispersal process, young foxes leave their

parental home range to become territorial. In our model,

we assumed that a young fox disperses one way along a

straight path and crosses at most L home ranges before

settling down (Lloyd 1980; Macdonald & Bacon 1982;

Trewhella et al. 1988). Thus it can reach 1+3L(L+1)

di¡erent cells (¢gure 2). In this model D(n,L) was de¢ned

as the set of cells that a young fox living in cell n can

reach through dispersal and as the set of cells from which

a young fox arriving in cell n started from. Finally, the

radial distance between two cells was determined through

a simple algorithm.

The probability of a young fox settling in a given home

range was assumed to depend only on the number of cells

it crossed, i.e. the radial distance between the end-points

of its path. As a model we took a linearly decreasing

function of the distance travelled: the probability of

reaching a cell located at a radial distance d is

(5)

C(d) ˆ? £ ‰(L‡ 1)¡ dŠ

6 £ d

, with ? ˆ

2

(L ‡ 1)(L ‡ 2),

for d ˆ 1,....

(6)

As an example, in a rabies-free situation, the densities of

healthy young and adult females are given by the

following equations:

HYF(n,t+¢t) ˆ (17F(t))£syf(t)£HYF(n,t),(7)

HAF(n,t+¢t) ˆ F(t)£hydf(n,t)+saf (t)£HAF(n,t), (8)

1576C. Suppoandothers

Rabiesvaccination and sterilization

Proc. R. Soc. Lond. B (2000)

vaccinated

adult

males

reproduction

mr

mn

mnmnmnmn

mr

mn

mn

mn

mnmn mnmnmr mr

d

d

d

d

d

dvv

vv

cc

cc

healthy

adult

males

healthy

young

males

infected

young

males

infected

adult

males

healthy

young

females

healthy

adult

females

vaccinated

young

males

vaccinated

adult

females

infected

adult

females

infected

young

males

vaccinated

young

females

Figure 1. Interaction between the 12 classes of foxes. mn,

natural mortality; mr, mortality induced by rabies; v, vacci-

nation; d, dispersal; c, contamination.

mmax

nmax

Figure 2. Structure of the two-dimensional domain. The

shaded area represents the set of cells that a young fox living

in cell n can reach through dispersal.

Page 3

where F(t) is the proportion of young foxes that disperse,

syf(t) is the survival rate of young females, saf(t) is the

survival rate of adult females and hydf(n,t) is the number

of healthy young females that arrive in cell n.

At time t

X

where ’(k,n) is the probability of a fox located in cell k

coming into cell n at a radial distance d from cell k:

hydf(n,t) ˆ

k2D(n,1)

’(k,n) £ syf(t) £ HYF(k,t),(9)

’(k,n) ˆ Á(d).

As in the one-dimensional model, there was a problem

for young foxes leaving a home range close to the

boundary of the domain. Here we considered that young

foxes that would have left the domain through dispersal

remained in their parental territory, therefore no fox

crossed the boundary. Under this assumption, numerical

simulations show that in a rabies-free situation, the global

dynamics is that of a large isolated population. Numerical

simulations in a disease-free environment show no

density increase at the edges. Since the aim of this paper

was to compare the e¤ciency of di¡erent control

strategies within the centre of the domain, the edge e¡ect

at the boundary could be neglected.

(10)

(c) Transmission

Transmission of rabies occurs through bites and licking,

as the rabies virus is transmitted via the saliva (Blancou

et al. 1991). It is thought that rabies can be propagated by

two modes:

(i) Outside the dispersal of juveniles, between foxes

living in the same or adjacent home ranges (Artois

1989). For a given cell, n, these home ranges have

numbers

vj(n) ˆ n7nmax, n7nmax+1, n71, n, n+1,

n+ nmax, n+ nmax+1,

(11)

the density of infected young females being deter-

mined as

IYF(n,t+¢t) ˆ syf(t)£IYF(n,t)+?j(t)

£(syf(t)£HYF(n,t))£I(n,t);

where ?j(t) is the transmission rate from an infectious

fox to a healthy young fox at time t, and I(n,t) is the

number of infectious foxes living in cell n and in the

six surrounding cells determined as

X

‡ IAM(vj,t)Š.

As indicated earlier ¼(t) is the inverse of the latency

period.

(ii) During dispersal (October and November) (Artois

1989), infected young individuals carry the infection

further than one home range (see ¢gure 2). The

density IAF(n,t +¢t) of infected adult females in cell

n at time t+¢t is given by

(12)

I(n,t) ˆ

vj(n)

¼(t)£‰IYF(vj,t)‡ IAF(vj,t)‡ IYM(vj,t)

(13)

saf(t)£IAF(n,t)+?j(t)+(syf(t)£HAF(t))£I(n,t)

+ F(t)£iydf(n,t),

where iydf(n,t) is the number of infectious young

females that could arrive in cell n at time t:

X

(14)

iydf(n,t) ˆ ¼(t)

k2D(n,l)

’(k,n)£ syf(t)£ IYF(k,t). (15)

(d) Vaccination and sterilization programmes

Two vaccination campaigns per year were simulated in

our model (Aubert 1995): one in spring to target adult

animals, and one in autumn to target all age classes. The

number of healthy and vaccinated young females in a cell

n at time t+¢t has been determined as

HYF(n,t+¢t) ˆ (17vay(t))£syf(n,t)£HYF(n,t),(16)

VYF(n,t+¢t) ˆ vay(t)£syf(n,t)£HYF(n,t),

where vay(t) is the vaccination rate at time t.

In fertility-control campaigns, each autumn only

females are concerned and contraception is only e¡ective

during one breeding season. T o take contraception into

account the equations of the previous model were modi-

¢ed: the number of healthy young females, or males, in a

cell n at time t+¢t was determined as

(17)

HYF(n,t+¢t) ˆ HYM(n,t+¢t)

ˆ b(t)£(17st(t))£HAF(n,t)£saf(n,t),

where st(t) is the sterilization rate at time t.

(18)

3. SIMULATION RESULTS

Numerical simulations were performed on a worksta-

tion using a Fortran 77 code on the data shown in table1.

Various numerical values of the birth and transmission

rates have been used. More precisely values for ? and b0

were determined that were consistent with either an

endemic state or a disease-free state. The e¤ciency of fox

contraception, dependent on or independent of vaccina-

tion against rabies was then easier to analyse.

In Artois et al. (1997) birth rate, b0, was 2.5. Here b0

varied from 1.02 (see ½3(a)) to a maximum of 3.5, which

corresponded to seven cubs per litter per female, with a

balanced sex ratio (Artois1989).

In Garnerin et al. (1986) and Artois et al. (1997) trans-

mission rate, ?, was 0.18. Here ? varied from 0.04 to a

maximum of 0.20 (see ½ 3(b)).

Rabies vaccination and sterilization

C. Suppoandothers 1577

Proc. R. Soc. Lond. B (2000)

Table 1. Data used in simulations

studied area

dispersaldistance

survivalrate

(Artois etal. 1997)

summer

winter

latencyperiod

(Blancou etal. 1991)

birth rate

transmissionrate

nmaxˆ 61; N ˆ nmax£ mmaxˆ 3721

l ˆ 5, or91 reachablehome ranges

adult

0.99

0.98

young

0.97

0.98

21 daysor ¼ ˆ 0:48

b(t)ˆ b0on 1 April; b(t)ˆ 0 otherwise

?(t) ˆ ?a(t) ˆ ?

Page 4

Actually, ? is unknown, but b0varied within a narrow

range (Voigt & Macdonald 1984).

(a) Population equilibrium

Simulations were carried out using one healthy pair of

foxes per cell and no young as initial distribution levels.

First, for a constant survival rate (¯ ˆ 0), there was a

thresholdvaluebmin(0)closeto1.02;ifb05 bmin(0)the popu-

lationgoes extinct and if b04 bmin(0)the populationfollows

a Malthusian growth pattern. For di¡erent birth rates, the

corresponding Malthusian parameters have been deter-

mined (table 2). As in the one-dimensional model (Suppo

1996),forabirthrateb0ˆ 2.9thisparameteris0.016.

Second, with a density-dependent survival rate (¯4 0),

after a transient period a maximum yearly periodic

distributionof individuals

b04 bmin(¯), while the population will become extinct for

b05 bmin(¯). Eventually, the maximum number of foxes

will be achieved on 1 April and can be determined with a

suitably designed ¯. Typically, a fox group on 1 April will

be composed of one male, two fertile females and their

litters, i.e. an average of 13 individuals for b0ˆ 2.5 (Artois

1989). In the model this will occur when ¯ ˆ 0.0025 and

b0ˆ 2.5. For ¯ ˆ 0.003 the minimum threshold birth rate,

bmin(0.003), needed to prevent the population from going

extinct is close to 1.3. For birth rates varying up to

b0ˆ 3.5, the number of foxes per home range was deter-

mined for di¡erent values of ¯ (¢gure 3). A total of 13

foxes per home range can be obtained with di¡erent

combinations of ¯ and b0.

From a numerical point of view, we proceeded along

two lines in order to get a prescribed maximum number

willbe observedfor

of foxes on 1 April. Assuming this maximum to be 13 and

b0ˆ 2.5, we ¢rst put a pair of healthy adults and no

young in each cell and ran the program until a yearly

periodic distribution of individuals was achieved. Using a

dichotomy method we estimated ¯ ˆ 0.0025, the average

transient time being12 years.We next modi¢ed the initial

distribution of healthy foxes and re-ran the program with

¯ ˆ 0.0025 until a yearly periodic distribution of indivi-

duals was achieved; we again found 13 to be the

maximum number, with variable transient times.

(b) Rabies-endemic equilibrium

In this section, we introduced a pair of exposed adult

foxes in a singlecell located at the centre ofthe domainand

assumed thateachcellcontainedonepairof healthy foxes.

Assuming a Malthusian growth trend (¯ ˆ 0), we ¢rst

determined the set of pairs (?,b0) needed to yield an

endemic state. Results show that for each ¢xed birth rate,

b0, rabies will not remain if the transmission rate is larger

than a maximum threshold ?max(b0,0) (¢gure 4).

Furthermore, the transmission rate must also be larger

than a more or less constant value to further an endemic

state, ?min(b0,0) ˆ 0.04. For b0ˆ 2.5 and ? ˆ 0.08 numerical

simulations show similar results to those obtained in the

1578C. Suppoandothers

Rabiesvaccination and sterilization

Proc. R. Soc. Lond. B (2000)

Table 2. Malthusian parameters for di¡erent birth rates

birth rateMalthusianparameter

1.02

1.5

2.0

2.5

3.0

3.5

0.0001

0.005

0.0095

0.0133

0.0166

0.0182

50

40

30

20

10

0

0.0090.0070.0050.003

d

number of foxes per home range

0.001

Figure 3. Number of young and adult foxes per cell as a

function of the density-dependent parameter ¯ for b0ˆ 3.5

(top curve), 3, 2.5 and 2 (bottom curve).

0.2

0

3.532.5

birth rate

2

transmission rate

1.5

0.15

0.1

0.05

Figure 4. Maximum transmission rate ?max(b0,¯) for ¯ ˆ 0.003

(open diamonds), 0.002 (dashed line), 0.001 (asterisks) and 0

(open squares).

35 000

30 000

25 000

20 000

15 000

10 000

5000

0

3630 24 18

time (years)

126

number of foxes

Figure 5. Dynamics of rabies for b0ˆ 2.5, ? ˆ 0.08 and

¯ ˆ 0.003: healthy individuals (top curve) and infected

individuals (bottom curve).

Page 5

one-dimensional model (Suppo1996): between two waves,

the population resumes a Malthusian growth trend as in a

disease-free situation; during the ¢rst ten years the occur-

rence of four successive waves with high prevalence of

infection was observed; during the next 30years, the

number of successive waves of rabies increased, the growth

of the healthy population was regulated by rabies and a

periodicendemic state emerged (¢gure 5).

For a logistic situation, the same pairs (?,b0) were

determined for di¡erent values of ¯ (see ¢gure 4). For

b05 1.7 rabies could be sustained with higher transmis-

sion rates when ¯4 0 than when ¯ ˆ 0; this threshold

?max(b0,¯) increased with ¯ and b0. Again, the transmis-

sion rate had a minimum threshold ?min(b0,¯) to further

an endemic state, but the latter was strictly larger than

0.04, increased with ¯ and decreased with b0. Finally,

there existed an optimum transmission rate ?opt(b0,¯),

?min(¯)5 ?opt(b0,¯)5 ?max(¯), at which the prevalence

was maximal (see ½ 3(d)). These results show that for a

given transmission rate, if the birth rate decreased below

1.7, an endemic state could not be obtained and rabies

disappeared. Thus, depending on the size of the birth

rate, a sterilization method could decrease this rate and

lead to eradication of rabies. In addition, for a birth rate

b0ˆ 2.5 and a transmission rate ? ˆ 0.07 rabies waves will

occur every six or seven years.

If ¯ is close to zero, results would be similar to the

Malthusian growth model.

(c) E¤ciency of fertility control

The e¤ciency of sterilization programmes could be

deduced from the previous computations. We can draw

di¡erent conclusions from Malthusian and logistic growth

trends.

First, we will consider a Malthusian growth trend (see

¢gure 4). For a ¢xed transmission rate lying between 0.04

and 0.12, in order to eradicate rabies an initial birth rate

in the range 2^3.5 must be decreased to a value close to

bmin(0). Thus, it would be di¤cult to employ a steriliza-

tion method alone because a healthy population would go

extinct before rabies was eradicated.

Let us also consider a logistic growth with ¯4 0, (see

¢gure 4). For a transmission rate lying between ?min(b0,¯)

and ?max(b0,¯), for rabies to disappear an initial birth rate

in the range 2^3.5 must be decreased to a minimum value

bopt(¯,?) 4 bmin(¯). Consequently, an e¤cient sterilization

rate could be determined for di¡erent birth rates and a

¢xed transmission rate (¢gure 6): for ¯ ˆ 0.0025, b0ˆ 2.5

and ? ˆ 0.08 a sterilization rate close to 35% is required.

(d) E¤ciency of vaccination against rabies

In our computations, vaccination programmes would

begin after three years, corresponding to the control of an

unexpected outbreak of rabies spreading quickly across

the spatial domain. The vaccination rate was considered

as e¤cient when rabies was totally eradicated; numeri-

cally this means the total number infected in the whole

domain equal to zero for at least 20years.

First, for a Malthusian growth trend, a minimum

e¤cient vaccination rate was determined for various pairs

(?,b0) in order to eradicate rabies (¢gure 7). This

minimum rate was larger in high-density populations and

decreased when ? increased. For b0ˆ 2.5 and ? ˆ 0.06,

simulations gave the minimum e¤cient rate of vaccination

necessary to eradicate rabies as 70%, which is close to the

upper limit achieved in the ¢eld during actual vaccination

campaigns (Aubert 1995). In other words, both numerical

simulation and ¢eld ¢ndings show that there is a density at

whichvaccinationfails to eradicate rabies.

Second, for a logistic growth trend, the same computa-

tions were carried out to emphasize the di¡erence for

lower-density populations; see ¢gure 8 for ¯ ˆ 0.003.

According to the results, in order for a vaccination to be

e¤cient the rate must be higher for high birth rates

(corresponding to a larger population), with maximum

values at the optimum transmission rate ?opt(b0,¯).

It is worth noting that in both cases (logistic and

Malthusian) the dynamics of rabies was modi¢ed by low

vaccination rates. For a vaccination rate between 20 and

30% (¢gure 9), the ¢rst wave of rabies was delayed, but

afterward the number of successive waves increased and a

new one could occur every year with the same preva-

lence. For a rate lower than, and close to, the e¤cient

Rabiesvaccination and sterilization

C. Suppoandothers1579

Proc. R. Soc. Lond. B (2000)

60

0

3.4 2.8 3.23 2.62.4

birth rate

2.2

transmission rate

2

50

40

30

20

10

Figure 6. Sterilization rate required to eradicate rabies for

¯ ˆ 0.003, transmission rates ? ˆ 0.06 (top curve), 0.08, 0.10,

0.12 and 0.14 (bottom curve); variable birth rates.

80

0

0.04

birth rate

vaccination rate (%)

60

40

20

0.06

0.08

0.10

0.11

transmission rate

2

2.5

2.9

Figure 7. Vaccination rate required to eradicate rabies for

di¡erent pairs of ? and b0for ¯ ˆ 0.

Page 6

rate, the ¢rst wave appeared later but the prevalence of

following waves increased continuously.

Now we come to the key point of our analysis. For some

pairs of b0and ?, the vaccination rate needed to eradicate

rabies had to be larger than 70%, which is di¤cult to

achieve in the ¢eld (Breitenmoser et al. 1995; Vuillaume

et al. 1997). In these cases contraception is required to

improve the e¤ciency of anti-rabies vaccination.

(e) V accination and fertility control combined

For a Malthusian growth trend, we saw that contra-

ception alone could lead to extinction of foxes (see ½3(c)).

A combination of both methods could be e¤cient if birth

rates were decreased to a value requiring a lower vaccina-

tion rate. It is straightforward to observe from ¢gure 7

that coupling vaccination and sterilization could be

successful. Thus, we could ¢x several vaccination rates

less than or equal to that needed to be successful when

vaccination alone was used. We could then deduce the

minimum sterilization rate required to eradicate rabies.

Figure 10showstherequired

?min(3,0)ˆ 0.04 and ?max(3,0)ˆ 0.11.

combinationsfor

For a logistic growth trend, we saw that a successful

sterilization rate could be determined but would be

di¤cult to obtain. A combination of both control methods

is also bene¢cial (see ¢gure 8). For ¯ ˆ 0.003, ¢gure 10

shows the successful combinations of sterilization and

vaccination for ?min(3.5,0.003)ˆ 0.08 and ?max(3.5,0.003)

ˆ 0.14. We observed a linear relationship between the

vaccination and sterilization rates; linear regression

yields a slope equal to 70.76 for ?min(b0,0.003).

4. DISCUSSION

Over time, more than 15 models have been devoted to

fox rabies (reviews in Barlow (1995) and Pech & Hone

(1992)).The main value of the one herein presented lies in

the use of recent and actual data from fox baiting in

France (Aubert 1995). Additionally, the use of contra-

ception to manage rabies in fox populations is considered

(Artois & Bradley 1995). As with many models of the

same type, ours is oversimpli¢ed in several regards and

some of the results obtained could be consequences of

these oversimpli¢ed choices.

We have omitted di¡erences between the dispersal

modes of male and female animals. Also we do not take

into account the e¡ect of culling, because its e¤ciency has

not been fully demonstrated on the European continent

(Aubert 1994). Therefore, it was considered that, to a

large extent, fox control by various methods constituted a

part of the density-dependent mortality. Additionally,

1580 C. Suppoandothers

Rabiesvaccination and sterilization

Proc. R. Soc. Lond. B (2000)

70

60

50

40

30

20

10

0

birth rate

vaccination rate (%)

0.08

0.10

0.12

transmission rate

0.14

2

2.5

3

3.5

Figure 8. Vaccination rate required to eradicate rabies for

di¡erent pairs of ? and b0for ¯ ˆ 0.003.

3000

2500

2000

1500

1000

500

0

108642

time (years)

number of foxes

0

Figure 9. Dynamics of rabies for a vaccination rate close to

25%: healthy individuals (dotted line) and infected indivi-

duals (solid line).

60

(b)

(a)

50

40

30

20

10

0

80

vaccination rate (%)

sterilization rate (%)

0 10 20 3040506070

60

50

40

30

20

10

0

sterilization rate (%)

Figure 10. The shaded area contains combinations of

vaccination and sterilization rates required, that could be

achieved in the ¢eld, to eradicate rabies for (a) ¯ ˆ 0 and

?min(3,0) ˆ 0.04 (dashed line)4?40.11 ˆ ?max(3,0) (solid

line) and (b) ¯ ˆ 0.003 and ?min(3.5,0.003)ˆ 0.08 (dashed

line)4?40.14 ˆ ?max(3.5,0.003) (solid line).

Page 7

density dependence in this study acted only on survival

and not on reproduction. This is close to what has been

observed in nature: a lack of variability in the fertility

rate within a wide range of natural conditions suggests

that fecundity is a stable demographic parameter in

Europe. Finally, the dispersal mode used in this model

enabled us to estimate the population size after yearling

dispersal, but did not simulate a preferred settlement of

young in less densely occupied areas. Knowledge about

dispersal patterns that include this behaviour is currently

so limited (see Lloyd 1980; Harris 1981; Macdonald &

Bacon 1982; Trewhella et al. 1988; Allen & Sargeant 1993)

that this simpli¢cation is worth keeping.

Obvious trends in fox demographic indices suggest a

steady population increase over the long term (MÏller

1995; Artois et al.1997). Ecological reasons for this popula-

tion increase remain unclear, but links with a decrease in

human control seem the most likely explanation (Aubert

1994; Szemethy & Heltai 1997). These trends were simu-

lated in this model through a Malthusian growth process

obtained by a constant that ensured reproduction greater

than mortality. Trends in the model are similar but not

precisely adjustedto those observedunder ¢eldconditions.

Socio-spatial adjustment with increasing density was not

considered in this model.This could have an in£uence in a

spatial modelof rabies di¡usion: inbrief, foxes are regarded

as `contractors’ (Kruuk & Macdonald 1984) maintaining

the smallest economically defensible home range. Addi-

tional residents would be tolerated as long as su¤cient

resources are available (compatible with the resources

dispersion hypothesis, see Carr & Macdonald (1986)).

According to ¢eld observations, the number of adult indivi-

duals within a social group is nevertheless limited to four or

¢ve (constant territory size hypothesis (CTH) versus terri-

tory inheritance hypothesis (TIH), see LindstrÎm et al.

1982;Von Schantz1984; LindstrÎm1986). Few behavioural

studies have been recently devoted to this aspect of the

spatial behaviour of foxes. Therefore, the response to a

decrease in mortality within a situation of stable accessi-

bility to resources is unknown.The model herein presented

accepts the unveri¢ed hypothesis that under these condi-

tions the number of individuals within a social group could

reach a limit transgressing the CTH^TIH hypotheses.

Further ¢eld research is needed to clarify this aspect.

Concerning the propagation of the virus, a uniform trans-

mission rate was used; there is then no variation due to sex

or age (dispersers could be less exposed than resident

adults, see Artois & Aubert (1985)), and no variation in

contactratebetween foxesliving withinthe same orin adja-

cent territories (seeArtois & Aubert1985).

Our model is considered in a constant environment,

unlike that of Pech et al. (1997), who have studied the e¡ect

of environmental variability on the use of fertility control

of foxes in arid Australia. No compensatory phenomena

(Hone 1994) to contraception, such as an increase in the

birth rate of non-sterilized females (Newsome 1995), an

increase in the survival rate of foxes, or immigration

(Seagle & Close 1996), were introduced in our model.

Therefore, no side e¡ects or retarded e¡ects could be

expected in our short-term analysis; with the purpose of

this project being the fast control of an outbreak of rabies,

long-term e¡ects did not need to be considered.

In the conditions described by our model of an isolated

host population, one observes that a stable endemic equi-

librium emerges with rabies regulating the fox population

in both demographic settings, i.e. logistic and Malthusian.

This occurs when demographic and epidemiological

parameters lie within a reasonably realistic range. Under

our assumptions this stable equilibrium between the virus

and the host requires a fast turn-over of the healthy foxes.

Since survival of the population is assured by dispersal

(October) and reproduction (April), it is understandable

that for a small transmission rate the virus does not

propagate at a su¤cient speed to survive, while for a

large transmission rate the mortality due to rabies cannot

be compensated for in time. Still under our assumptions,

it follows that the propagation of the epidemic disease is

not very sensitive to dispersal, while being more sensitive

to birth rate. This should moderate biological considera-

tions that could be drawn from our model. Nevertheless,

provided that these simpli¢cations can be accepted, the

model suggests that sterilization turns out to be a strong

complement for controlling fox rabies. Additionally, our

model suggests that density dependence smoothes out

£uctuations at equilibrium between the host and the

virus. In contrast with intuitive predictions, the successful

rate of rabies eradication is higher when the host popula-

tion is not regulated, i.e. Malthusian growth.

Nevertheless, for a fox population experiencing a

Malthusian growth curve, vaccination alone would be

ine¤cient to eradicate rabies, as expected, so sterilization

turns out to be speci¢cally helpful here.

Finally, as an alternative approach we compared our

results to a deterministic and time-continuous model,

given in an electronic appendix (http://durandal.mass.u-

bordeaux2.fr/~naulin/appendix/appendix.html), based on

a system of ordinary di¡erential equations such as that

used in Anderson et al. (1981) and Barlow (1996). As a

¢rst di¡erence this continuous model does not predict

self-eradication of rabies when the transmission rate is

large. Also, for a Malthusian population growth trend,

the vaccination e¤ciency cannot be predicted by the

model and depends on the parameters de¢ning the initial

state.Nevertheless, similar

sterilization can be drawn from both continuous and

discrete models. Discrete-time modelling appears, then,

to be more appropriate for our purpose. Predictions

obtained from both models are encouraging in consid-

ering immunocontraception as a possible method of

controlling a re-emerging outbreak of rabies in highly

dense fox populations. Nevertheless, additional biological

hypotheses that need to be taken into account in further

studies include fox culling considered as a non-density-

dependent mortality factor, changes in spacing strategies

when density increases, and the in£uence of dispersal in

the recovery of healthy populations.

conclusionsconcerning

Supported by the CNRS under the grant `Mode ¨ lisation de la

circulation de parasites dans des populations structure ¨ es’.

REFERENCES

Allen, S. H. & Sargeant, A. B. 1993 Dispersal patterns of

red foxes relative to population density. J. Wildl. Mgmt 57,

526^533.

Rabiesvaccination and sterilization

C. Suppoandothers 1581

Proc. R. Soc. Lond. B (2000)

Page 8

Anderson, R. M., Jackson, H. C., May, R. M. & Smith,

A. D. M. 1981 Population dynamics of fox rabies in Europe.

Nature 289,765^770.

Artois, M. 1989 Le renard roux. Encyclope ¨die des Carnivoresde France.

3. Puceul, France: Socie ¨ te ¨

Protection des Mammife © res.

Artois, M. 1997 Managing problem wildlife in the `OldWorld’: a

veterinary perspective. Reprod. Fertil. Dev. 9, 17^25.

Artois, M. & Aubert, M. 1985 Behaviour of rabid foxes. Rev.

Ecol. (T erre etVie) 5,171^176.

Artois, M. & Bradley, M. 1995 Un vaccin contre les renards.

Pour enrayer la prolife ¨ ration des animaux inde ¨ sirables, un

appa ª t contraceptif. La Recherche 281, 40^41.

Artois, M., Langlais, M. & Suppo, C. 1997 Simulation of rabies

control within an increasing fox population. Ecol. Model. 97,

23^34.

Aubert, M. 1994 Control of rabies in foxes: what are the

appropriate measures? Vet. Rec. 134, 55^59.

Aubert, M. 1995 Epide ¨ miologie et lutte contre la rage en France

et en Europe. Bull. Acad. Nat. Me¨d. 179,1033^1054.

Barlow, N. D.1995 Critical evaluation of wildlife disease models.

In Ecology of infectious diseases in natural populations (ed. B. T.

Grenfell & A. P. Dobson), pp.230^259. Cambridge University

Press.

Barlow, N. D. 1996 The ecology of wildlife disease control:

simple models revisited. J. Appl. Ecol. 33, 303^314.

Blancou, J., Aubert, M. F. A. & Artois, M. 1991 Fox rabies. In

The natural history of rabies, 2nd edn (ed. G. M. Baer),

pp. 257^290. Boca Raton, FL: CRC Press.

Bradley, M. P.1994 Experimental strategies for the development

of an immunocontraceptive vaccine for the European fox

Vulpes vulpes. Reprod. Fertil. Dev. 6, 307^317.

Breitenmoser, U., Kaphegyi, T., Kappeler, A. & Zanoni, R.

1995 Signi¢cance of young foxes for the persistence of rabies

in northwestern Switzerland. In Proceedings of theThird Congress

of the European Society of VeterinaryVirology, pp.391^396. France:

Fondation Me ¨ rieux.

Carr, G. M. & Macdonald, D. W. 1986 The sociality of solitary

foragers: a model based on resource dispersion. Anim. Behav.

34,1540^1549.

De Combrugghe, S. A. 1994 Statut des mammife © res sauvages en

Wallonie. Annls Me¨d.Ve¨t. 138, 229^235.

Garnerin, P., Hazout, S. & Valleron, A. J. 1986 Estimation of

two epidemiological parameters of fox rabies: the length of

incubation period and the dispersal distance of cubs. Ecol.

Model. 33,123^135.

Harris, S. 1981 An estimation of the number of foxes (Vulpes

vulpes) in the city of Bristol, and some possible factors a¡ecting

their distribution. J. Appl. Ecol. 18, 455^465.

Hone, J. 1994 Analysis of vertebrate pest control. Cambridge

University Press.

Kruuk, H. & Macdonald, D.W.1984 Group territories of carni-

vores: empires and enclaves. In Behavioural ecology: ecological

consequences of adaptative behaviour (ed. R. M. Sibly & R. H.

Smith),pp.521^536.Oxford,

Publications.

LindstrÎm, E. 1986 T erritory inheritance and the evolution of

group-living in carnivores. Anim. Behav. 34,1825^1835.

Franc ° aise pour l’Etude et la

UK:BlackwellScienti¢c

LindstrÎm, E., Poulsen, O. & Von Schantz, T. 1982 Spacing of

the red fox Vulpes vulpes L. in relation to food supply. In

Population ecology of the red fox in relation to food supply (ed. E.

LindstrÎm), pp.82^107. PhD thesis, University of Stockholm,

Sweden.

Lloyd, H. G.1980 The redfox. London, UK: B.T. Batsford Ltd.

Macdonald, D. W. & Bacon, P. J. 1982 Fox society, contact rate

and rabies epizootiology . Comp. Immunol. Microbiol. Infect. Dis.

5, 247^256.

MÏller, W.W. 1995 Oral vaccination and high density fox popu-

lations. Rabies Bull. Eur. 19,14^15.

Newsome, A. E. 1995 Socio-ecological models for the red fox

populations subject to fertility control in Australia. Ann. Zool.

Fenn. 32, 99^110.

Pastoret, P. P. & Brochier, B. 1999 Epidemiology and control of

rabies in Europe.Vaccine 17,1750^1 754.

Pech, R. P. & Hone, J. 1992 Models of wildlife rabies. In

Wildlife rabies contingency planning in Australia (ed. P. H.

O’Brien & G. Berry), pp.147^157. National Wildife Rabies

Workshop,12^16March

Government Publishing Service.

Pech, R., Hood, G. M., McIlroy, J. & Saunders, G. 1997 Can

foxes be controlled by reducing their fertility? Reprod. Fertil.

Dev. 9, 41^50.

Seagle, S. W. & Close, J. D. 1996 Modeling white-tailed deer

(Odocoileus virginianus) population control by contraception.

Biol. Conserv.76, 87^91.

Smith, G. C. 1995 Modelling rabies control in the UK: the

inclusion of vaccination. Mammalia 59, 629^637 .

StÎhr, K. & Meslin, F. X. 1997 Oral vaccination of wildlife in

Europe. In Rabies control in Asia (ed. B. Dodet & F. X. Meslin),

pp.27^34. Amsterdam,The Netherlands: Elsevier.

Suppo, C. 1996 Mode ¨ lisation et analyse mathe ¨ matique de la

propagation des viroses dans les populations de carnivores.

Thesis in Mathematics, Universite ¨ Bordeaux I, France.

Szemethy, L. & Heltai, M. 1997 E¡ects of per-oral vaccination

against rabies on red fox population dynamics in Hungary. In

23rd Congress of the International Union of Game Biologists, Lyon,

France,1 September1997. Re ¨ sume ¨ s des posters.

Tapper, S. 1992 Game heritage. An ecological review from shooting and

gamekeepingrecords.Fordingbridge,UK:GameConservancyLtd.

Trewhella, W. J., Harris, S. & Macallister, F. E. 1988 Dispersal

distance, home-range size and population density in the red

fox (Vulpes vulpes): a quantitative analysis. J. Appl. Ecol. 25,

423^434.

Voigt, D. R. & Macdonald, D. W. 1984 Variation in the spatial

and social behaviour of the red fox, Vulpes vulpes. Acta Zool.

Fenn. 171, 261^265.

Von Schantz, T. 1984 Carnivore social behaviourödoes it need

patches? Nature 307, 388^390.

Vuillaume, P., Aubert, M., Demerson, J. M., Cliquet, F., Barrat,

J. & Breitenmoser, U. 1997 Vaccination des renards contre la

rage par de ¨ po ª t d’appa ª ts vaccinaux a © l’entre ¨ e des terriers. Ann.

Med.Vet. 141, 55^62.

1990.Canberra: Australian

As this paper exceeds the maximum length normally permitted,

the authors have agreed to contribute to productioncosts.

1582 C. Suppo andothers

Rabies vaccination and sterilization

Proc. R. Soc. Lond. B (2000)