Page 1

Journal of Applied

Ecology 2001

38, 1362–1370

© 2001 British

Ecological Society

Blackwell Science Ltd

Estimating disease transmission in wildlife, with emphasis

on leptospirosis and bovine tuberculosis in possums, and

effects of fertility control

PETER CALEY*† and DAVE RAMSEY*‡

*Landcare Research and ‡Marsupial Cooperative Research Centre at Landcare Research, Private Bag 11052,

Palmerston North 5301, New Zealand; and †Applied Ecology Group, University of Canberra, ACT 2601, Australia

Summary

1. We present methods for estimating disease transmission coefficients in wildlife, using

Leptospira interrogans infection (a bacterial disease transmitted predominantly during

social contacts) in brushtail possums Trichosurus vulpecula as a model system.

2. Using data from a field experiment conducted on a naturally infected possum

population, we estimated disease transmission coefficients assuming either ‘density-

dependent’ or ‘frequency-dependent’ transmission.

3. A model-selection approach determined that density-dependent transmission was

the most appropriate form of the transmission of L. interrogans infection in brushtail

possums.

4. We used the chosen model of transmission to examine experimentally the effect of

tubally ligating female brushtail possums on the epidemiology of L. interrogans. The

estimated transmission coefficient was 28% higher (P = 0·16) in populations subject to

tubal ligation, raising the possibility that fertility control of this type may increase dis-

ease transmission rates.

5. Altering mating behaviour through fertility control may have the potential to control

diseases such as bovine tuberculosis in brushtail possums, although the potential of

fertility control techniques to change disease transmission coefficients and disease

epidemiology requires further investigation. This would require models that exam-

ine the combined effects of fertility control on population dynamics, social behaviour

and disease transmission coefficients simultaneously.

Key-words: epidemiology, Leptospira interrogans, modelling, Mycobacterium bovis,

Trichosurus vulpecula.

Journal of Applied Ecology (2001) 38, 1362–1370

Introduction

Based on the projections of host–pathogen models,

reducing susceptible host abundance is often proposed

as a strategy for eradicating diseases from wildlife

populations, such as Mycobacterium bovis (Karlson &

Lessel) infection (bovine tuberculosis) in brushtail

possums Trichosurus vulpecula (Kerr) (Barlow 1991b,

1996; Roberts 1996) and badgers Meles meles (L.)

(White & Harris 1995), and Brucella abortus infection

(brucellosis) in bison Bison bison (L.) (Dobson &

Meagher 1996). Indeed, reducing the population den-

sity of animals is one of the most frequently attempted

management strategies for controlling disease in wild

animals (Wobeser 1994). This logically follows from

the paradigm of threshold density for the establish-

ment and persistence of disease (Kermack & McKen-

drick 1927; Anderson & May 1979; May & Anderson

1979) and the implicit assumption underlying this par-

adigm that disease transmission scales positively with

abundance. However, the projections of these host–

pathogen models are greatly affected by the way in

which transmission between infected and susceptible

hosts is modelled (McCallum, Barlow & Hone 2001).

Estimating disease transmission coefficients is con-

sidered to be a very difficult parameter estimation

problem (Anderson & May 1991) and remains a great

challenge in field ecology today (McCallum, Barlow &

Hone 2001). Disease transmission coefficients are

model-dependent, and an important issue is the form

of the model for the scaling between host population

density and parasite transmission rate (McCallum,

Correspondence: P. Caley, Landcare Research, Private Bag

11052, Palmerston North 5301, New Zealand (fax + 64 6355

9230; e-mail caleyp@landcare.cri.nz).

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Estimating disease

transmission rates

© 2001 British

Ecological Society,

Journal of Applied

Ecology, 38,

1362–1370

Barlow & Hone 2001; Grenfell & Bolker 1998 and ref-

erences therein). Resolving this issue can be considered

a model-selection problem for which Akaike’s infor-

mation criterion (AIC) provides an approach for

choosing between competing models of transmission

(Burnham & Anderson 1998), providing data sets exist.

Studies that estimate transmission coefficients for

diseases of free-living vertebrates, let alone estimate

the effect of management on disease transmission, are

uncommon in both the laboratory (Bouma, De Jong &

Kimman 1995) and the field (Hone, Pech & Yip 1992;

Swinton et al. 1997; Begon et al. 1999). A result of this

is a general paucity of data on transmission rates (De

Leo & Dobson 1996).

Reducing susceptible host abundance and/or popu-

lation density may be achieved by a variety of means,

such as lethal control, vaccination or reducing fertility

(fertility control). Fertility control has been proposed

as an alternative non-lethal tactic for reducing the

abundance of species such as brushtail possums (Bar-

low 1996; Cowan 1996; Barlow, Kean & Briggs 1997)

and badgers (Swinton et al. 1997; Tuyttens & Macdon-

ald 1998) below the threshold for disease (M. bovis)

persistence. Here, fertility control is broadly defined as

a reduction in the birth rate that should decrease the

rate of population increase, assuming no compensa-

tory change occurs in the death rate (Hone 1992). The

possibility that population control may cause a change

in transmission coefficients through effects on social

behaviour is receiving increasing theoretical interest.

For example, social perturbation arising from lethal

control of badgers may act to promote transmission of

M. bovis (Swinton et al. 1997). In contrast, Tuyttens &

Macdonald (1998) considered that fertility control

(sterilization) of badgers could reduce vertical trans-

mission of M. bovis, and transmission of M. bovis

during mating through changed behaviour.

Endemic M. bovis infection in New Zealand possum

populations is the single biggest threat to the nation’s

livestock industry, with M. bovis-infected possums

occurring over about 24% of the land mass (Coleman &

Caley 2000). The presence of this wildlife reservoir of

M. bovis infection has hampered efforts at controlling

the disease in livestock (O’Neil & Pharo 1995; Coleman

& Caley 2000), similar to the problem encountered in

England (Zuckerman 1981) and Ireland (OMairtin

et al. 1998a,b) arising from M. bovis-infected badgers.

While reducing abundance by non-selective culling is

presently the primary strategy for controlling M. bovis

infection in brushtail possum populations in New Zea-

land (Barlow 1991b; Caley et al. 1999), fertility control

is being pursued as an alternative method of reducing

abundance (Cowan 1996, 2000). Methods of fertility

control that block fertilization, such as immunocontra-

ception, may leave the endocrine system intact. Thus

sterile but hormonally competent females may have

an increased frequency of mating contacts due to an

increased frequency of oestrus, as observed in white-

tailed deer Odoicoileus virginianus (Miller) (McShea

et al. 1997) and elk Cervus elaphus (Bailey) (Heilmann

et al. 1998) subjected to this type of contraception.

This prediction is supported for brushtail possums by a

field trial by Ji, Clout & Sarre (2000). Whilst increased

frequency of mating could enhance the transmission

of a hypothetical transmissible biocontrol vector in

brushtail possums (Barlow 1994), it could possibly also

increase the transmission coefficient of M. bovis, thus

negating some of the benefits of reduced abundance

resulting from fertility control. Alternatively, methods

of fertility control that target endocrine control of

reproduction may result in behavioural changes, in-

cluding inhibition of mating behaviour. For brushtail

possums this could mean reduced sexual contacts and

possibly also reduced agonistic contacts, which would

be associated with a reduction in disease transmission.

Clearly, altered behaviour arising from fertility control

techniques potentially may help or hinder disease man-

agement in wildlife, although little attention has been

given to altering high risk behaviour of wildlife to

reduce disease transmission. This is in contrast to the

management of disease in humans, where behaviour

modification (e.g. changing sexual habits in the case of

sexually transmitted diseases) is one of the most com-

monly used methods of management of public health

(Anderson & May 1991; Morris 1996).

In this paper, we present methods for estimating

disease transmission coefficients in wildlife, using

Leptospira interrogans serovar balcanica (Kmety &

Dikken 1993) (hereafter L. balcanica) infection in

brushtail possums as a model system. We compared

two models of transmission (density-dependent and

frequency-dependent) and used the selected best

model to estimate from a field experiment the effect of

behavioural changes induced by fertility control (here

tubally ligating female possums) on the transmission of

disease. We then examined the theoretical implications

of attempting to use fertility control to manage M. bovis

infection in brushtail possums.

Materials and methods

The ability of a pathogen to establish and persist in

animal populations is largely determined by the basic

reproductive rate of the disease (Ro). This is defined as

the expected number of secondary infections caused in

an entirely susceptible population by a typical infected

host. By definition, if Ro is greater than or equal to

unity the disease will establish and, conversely, if Ro is

less than unity the disease will fail to establish (Ander-

son & May 1991). We use a simplified version of the

compartment model for a directly transmitted disease,

as presented by Anderson & May (1979), to illustrate

the possible effects of fertility control relative to that of

culling and vaccination. We choose a model with hor-

izontal transmission, a negligible latent period and no

Page 3

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P. Caley &

D. Ramsey

© 2001 British

Ecological Society,

Journal of Applied

Ecology, 38,

1362–1370

life-long immunity. Host population dynamics assume

exponential population growth, with the exponential

rate of increase r = a – b, where a and b are the instan-

taneous per capita birth and death rates, respectively.

Note that this is an illustrative example, rather than an

exact description of the dynamics of infection. Under

this model, mixing is assumed to be homogenous and

hence transmission is ‘density-dependent’, occurring

at a rate βSI, where β is the disease transmission coef-

ficient and S and I are the abundance of susceptibles

and infectives, respectively. Ro is estimated as:

eqn 1

where α is the per capita disease-induced death rate, b is

the per capita natural death rate, v is recovery rate from

disease, and N is the initial size of the susceptible pop-

ulation. Equation 1 can be interpreted as an infected

animal making β infectious contacts per unit time with

N susceptibles for its life expectancy. Life expectancy is

the reciprocal of the combined mortality rate due to

natural causes and disease. By setting Ro equal to unity

in equation 1, the threshold density (KT) for disease

establishment is found:

eqn 2

The approximation of density-dependent transmission

is reasonable for many directly transmitted diseases,

but may not be adequate for sexually transmitted dis-

eases, for example, where the number of sexual partners

(η) is independent of the absolute population size and

hence no threshold abundance exists (May & Anderson

1979). In this situation, the transmission rate may

be approximated by βηSI/N (May & Anderson 1979),

or here by β′SI/N (replacing βη with β′). This

model of transmission (commonly termed ‘frequency-

dependent’ transmission; Begon et al. 1999; McCallum,

Barlow & Hone 2001) is most appropriate for diseases

transmitted through contacts that are largely density-

independent (e.g. mating). However, De Jong, Diek-

mann & Heesterbeek (1995) and Bouma, De Jong &

Kimman (1995) suggest that frequency-dependent

transmission may have wider application to host–

pathogen systems where disease contact rates are

not necessarily density-independent. For frequency-

dependent approximation of the transmission process,

the maintenance of disease is independent of the pop-

ulation size, and occurs when β′ > (b + α + v) (May &

Anderson 1979). It follows that the basic reproductive

rate may be calculated (Roberts & Heesterbeek 1993;

Heesterbeek & Roberts 1995) as:

eqn 3

which is independent of population size or density.

We use equations 1, 2 and 3 to investigate qualitat-

ively the possible effects of fertility control-induced

behaviour change on disease. Under density-dependent

transmission, Ro is decreased by either reducing

the numerator (βS), increasing the denominator

(α + b + v) or reducing βS by proportionally more than

(α + b + v) is reduced. For example, culling of possums

aims to increase the mortality rate (b) and reduce the

abundance of susceptible possums (S), both of which

act to lower Ro. Vaccination also acts to reduce the

abundance of susceptible possums; in effect, vaccinated

possums are removed from the susceptible population,

thus reducing S. Equation 3 makes the important

prediction that for frequency-dependent transmission,

disease may only be controlled by increasing the

mortality rate (assuming β′, α and v are constants).

Equations 1, 2 and 3 clearly illustrate that wildlife

management could target the actual mechanisms of

disease transmission that make up β as an alternative

to targeting hosts per se. Historically, β is considered

difficult to measure, as it subsumes many processes

involved in initiating infection (Anderson & May

1991). Additionally, β is model-specific. So, given a

model, how do we estimate the transmission coefficient

from field data? For simplicity, we consider our simple

disease–host model where the transmission of disease

is either frequency-dependent or density-dependent.

Under density-dependent transmission, susceptibles

are infected at rate βSI, hence the per capita instanta-

neous incidence of disease (λ), termed the force of

infection (Muench 1959), equals βI, and β can be sim-

ply expressed in terms of the abundance of infectives

and the force of infection:

eqn 4

A similar argument gains an expression for β′ in terms

of prevalence (p):

eqn 5

Importantly, equations 4 and 5 show that under the

simple models chosen, β and β′ may be calculated from

the force of infection and the abundance of infectives.

This is done below for L. balcanica infection in brush-

tail possums.

: L. BALCANICA

To determine if methods of fertility control that block

fertilization (e.g. immunocontraception) could result

in behavioural changes affecting the transmission coef-

ficient β, we undertook an experiment to estimate the

transmission rate of L. balcanica in wild populations

of the brushtail possum subject to fertility control

(tubal ligation of females). Leptospira balcanica is a

commonly occurring disease in possums thought to be

transmitted predominantly by sexual contact (Durfee

& Presidente 1979; Day et al. 1997, 1998). Clinical

disease due to L. interrogans infection in most wild

animals is rare (Bender & Hall 1996) and no clinical

R

N

bv

ο

=

++

β

α

K

b

β

v

T

=

++α

R

bv

ο

=

+

′

+

β

α

β

λ

=I

′ =

=β

λλ

p

N

I

Page 4

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Estimating disease

transmission rates

© 2001 British

Ecological Society,

Journal of Applied

Ecology, 38,

1362–1370

symptoms were observed in possums infected with

L. balcanica in a study by Hathaway (1981). The

hypothesis tested in the present study was that im-

munocontraception as modelled by tubal ligation

increased frequency of oestrus in sterilized females,

leading to increased mating contacts with males, result-

ing in a higher transmission coefficient for L. balcanica.

Increased frequency of oestrus in tubally ligated

captive female possums has been demonstrated ex-

perimentally in both captive (Rekha 1997) and

free-ranging (Ji, Clout & Sarre 2000) possums.

The experiment formed part of a larger study to

determine the effects of various levels of fertility con-

trol on the population dynamics of brushtail possums

(Ramsey 2000). Four live-trapping grids were estab-

lished in podocarp/broadleaf forest in the lower North

Island of New Zealand, two in the Orongorongo Valley

(separated by c. 500 m), east of Wellington (174°58′ E,

41°21′ S), and two in the Turitea catchment (separated

by c. 1 km), east of Palmerston North (175°41′ E,

40°26′ S). Each grid consisted of 150 cage traps set at

30 m spacing. Sites were established in October 1995

and trapped three times yearly in January, June and

September. Experimental treatments, consisting of

either 0% (although with 80% of females subject to

sham operations) or 80% of mature females surgically

sterilized by tubal ligation, were applied randomly to

the two blocks in the Orongorongo Valley and the two

blocks in the Turitea catchment between January–

April 1996. Each year the number of sterilized females

was adjusted to maintain the 80% sterility level. Move-

ment of animals between grids at each location was

negligible.

On first capture, possums were anaesthetized with

ether and given a unique tattoo and ear tag for identi-

fication. During the June trapping sessions from years

1996–99, c. 5 ml of blood was collected from 60–100

possums on each of the experimental control (0%

sterilized) and 80% treatment grids. The June trapping

period was selected for sampling as it occurs directly

after the main autumn breeding period of March–June

(Fletcher & Selwood 2000). The serum was separ-

ated by centrifugation and submitted to the Central

Animal Health laboratory, Wallaceville, New Zealand,

for serology. The serological micro-agglutination test

(MAT), using doubling dilutions of serum beginning at

1 : 50, was used to detect the presence of leptospiral

antigens (Horner, Heath & Cowan 1996). If no reaction

was seen in the 1 : 50 dilution, the result was scored as

negative. A positive result indicated the presence of

active leptospiral infection (Cowan, Blackmore &

Marshall 1991).

For testing the effect of sterilization treatment on the

transmission coefficient for L. balcanica infection of

possums, we assume that the force of infection is con-

stant within years (although not necessarily between

years) with no disease-induced mortality (α = 0). This

allows us to use the very tractable exponential dis-

tribution (Lee 1992) to model the force of infection

observed in the experiments. Over n intervals of length

ti, each having a force of infection λi and abundance of

infectives Ii, the probability of an individual acquiring

infection (p) is given by:

eqn 6

The force of infection is the parameter of interest in

many studies of disease (McCallum 2000); however,

here it is the transmission coefficient that is of intrinsic

interest. Substituting for λ from equation 4 into equa-

tion 6 and rearranging equation 6 yields an expres-

sion for the observed probability of infection (p) under

density-dependent transmission in terms of the trans-

mission coefficient (β) and the abundance of infectives

(Ii) in each time interval (ti):

i

eqn 7

Likewise, for frequency-dependent transmission, sub-

stituting for λ from equation 5 into equation 6 gives an

expression for the observed individual disease preva-

lence in terms of transmission coefficient (β′) and the

prevalence of infectives (Zi) in the population during

each time interval (ti):

i

eqn 8

Equations 7 and 8 are generalized linear models

(GLM) that enable direct estimation of the transmis-

sion coefficients, and testing of the effect of sterility

control treatment on them. They are somewhat similar

in structure to the GLM presented by Becker (1989) for

estimating β from an outbreak with complete observ-

ability (i.e. a notifiable disease with all cases reported

and total population known). The key difference is

that, rather than having a known absolute number of

cases during the experiment, in our experiment we

have a sampled proportion of new-case individuals,

hence our data are binomially rather than normally

distributed.

Equations 7 and 8 were fitted to the incident cases of

L. balcanica that occurred during the experiment using

a GLM utilizing a complementary log–log link, and

the logarithmic term on the right-hand side of each

equation fitted as an offset (equivalent to fixing the

slope of the regression to 1; Collett 1991). The response

variable was the number of incident cases of disease

during the experiment with the binomial denominator

equal to the number of individuals tested. Individuals

who were seropositive on their first sample were dis-

carded from the analysis, except when calculating the

overall prevalence of disease (Zi). The exceptions were

animals that showed a greater than twofold increase in

antibody titre between serum samples. In studies on

captive animals, L. balcanica infection in possums was

pe

i i

t

i

n

=

1

−

∑

=

−

1

λ

ln( ln(

−

)) ln ln

=−+

=∑

1

1

pI t

i i

n

β

ln( ln(

−

)) ln ln

=β−

′ +

=∑

1

1

p Zt

i i

n

Page 5

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P. Caley &

D. Ramsey

© 2001 British

Ecological Society,

Journal of Applied

Ecology, 38,

1362–1370

followed by a peak in antibody titres within a month of

infection (Hathaway 1981). Antibody titres then slowly

decreased over the following 13 months. The most

likely explanation for a large (> 2 doubling dilutions)

increase in antibody titres between annual serum sam-

ples, therefore, is that the individual had been recently

re-exposed to infection. The treatment 0% or 80% ste-

rility was fitted as a factor. The abundance of infectives

on each grid in each year (I) was estimated from the

sampled prevalence of disease, and the abundance of

possums estimated using the jack-knife estimator

(Burnham & Overton 1978). Time at risk (t) for an indi-

vidual was defined as the period from the date of first

capture or the date of application of treatments for pos-

sums resident before the start of the study to the date of

last serum sampling. Locality (Orongorongo Valley or

Turitea) was fitted as a fixed block effect in all models.

The competing hypotheses of density-dependent vs.

frequency-dependent transmission, as represented by

equations 7 and 8, were compared by calculating AIC

scores for each, with the best model selected being that

with the lowest AIC score (Burnham & Anderson

1998). All statistical analysis was performed in S-plus

version 2000 (Mathsoft, Seattle, WA).

Results

: L. BALCANICA

A summarized form of the data used to estimate dis-

ease transmission coefficients is shown in Table 1. The

point prevalence of L. balcanica infection was high

although variable (mean = 55·2%, range 38·8–70%),

as was the abundance of possums on each trapping grid

(mean = 148, range 63–257). The incidence of L. bal-

canica was estimated from 213 possums with more than

one serum sample. Of these 213 possums, 30 of 103

possums in experimental control sites, and 39 of 110

possums in the sterility sites, sero-converted during

the course of the study (Table 1).

The model assuming density-dependent transmis-

sion was most strongly supported by the data, based on

having the lowest AIC (Table 2). However, we present

the results of both models as, based on a ∆AIC of 3, the

model assuming frequency-dependent transmission is

bordering on having substantial support in compar-

ison (Burnham & Anderson 1998). There was little

evidence from either model that the transmission

coefficient differed between geographical locality (density-

dependent P = 0·77, frequency-dependent P = 0·22),

so this variable was removed from the models. Assum-

ing density-dependent transmission, β was estimated

to be 1·3 times higher in the 80% sterility sites than the

experimental control sites, although there remained

doubt as to the probability that this was a true treat-

ment effect (P = 0·16; Table 2). Estimates of β′ under

the frequency-dependent transmission model were

1·24 times higher for possums in the 80% sterility sites

than for possums in the experimental control sites

(Table 2).

With negligible disease-induced mortality (α = 0),

no disease recovery (v = 0) and natural instantaneous

mortality rate (b) for adult possums of 0·24 year–1

(Efford 1998), the threshold abundance for the estab-

lishment for L. balcanica infection in brushtail pos-

sums is estimated to be 96 possums for unmanipulated

populations, and 75 possums with 80% tubal ligation

of females, a reduction of 16%. For the grid size used

in our study with an effective trapping area of c. 22 ha

(D. Ramsey, unpublished data), the corresponding

population densities for disease establishment are

4·4 possum ha–1 for unmanipulated populations and

3·4 possum ha–1 for populations subject to 80% tubal

ligation of females.

Table 2. Parameter estimates from generalized linear models fitted to incident cases of Leptospira balcanica in possums subject

to 0% or 80% sterilization of females, assuming density-dependent (transmission type βSI) or frequency-dependent (transmission

type β′SI/N) disease transmission. The parameter estimate for ‘+ Sterility’ is the contribution of the 80% sterility treatment to

ln(β). P-values are for one-tailed tests. ∆AIC gives the comparative AIC scores for the two models

Transmission typeModel parameters Parameter estimateSE

Z*

P

∆AIC

βSI

ln(β)

+ Sterility

ln(β′)

+ Sterility

–8·49

0·25

–3·45

0·21

0·19

0·26

0·19

0·26

–

1

–

0·8

–

0·16

–

0·21

–

0

–

3

β′SI/N

*Standard normal deviate.

Table 1. Prevalence of Leptospira balcanica (Zˆ ), estimated

possum abundance (N), number of possums at risk of

infection that were sampled (S), and the number of incident

cases of Leptospira balcanica (I) for different levels of sterility

treatment

Treatment SiteYear

Zˆ (%)

N

SI

0% sterility Turitea

Turitea

Turitea

Orongorongo

Orongorongo

Orongorongo

1996

1997

1998

1996

1997

1998

53·4

51·6

63·2

56·1

59·6

54·1

257

226

142

136

110

123

19

13

23

30

22

17

5

1

4

11

5

4

80% sterilityTuritea

Turitea

Turitea

Orongorongo

Orongorongo

Orongorongo

1996

1997

1998

1996

1997

1998

59

43·9

38·8

70

54·7

57·9

63

106

90

190

171

163

24

24

17

27

17

16

10

7

3

12

4

3