Ecological Modelling 167 (2003) 139–157
Investigating the spatial dynamics of bovine tuberculosis in badger
populations: evaluating an individual-based simulation model
Mark D.F. Shirley a,∗ , Steve P. Rushton a , Graham C. Smith b ,
Andrew B. South a,1 , Peter W.W. Lurz a
a
Centre for Life Sciences Modelling, University of Newcastle, Newcastle upon Tyne NE1 7RU, UK
b Central Science Laboratory, Sand Hutton, York YO41 1LZ, UK
Received 8 July 2002; received in revised form 11 February 2003; accepted 22 April 2003
Abstract
We describe an individual-based spatially-explicit model designed to investigate the dynamics of badger populations and TB
epidemiology in a real landscape. We develop a methodology for evaluating the sensitivity of the model to its input parameters
through the use of power analysis, partial correlation coefficients and binary logistic regression. This novel approach to sensitivity
analysis provides a formal statement of confidence in our findings based on statistical power, and a solution for analysing sparse
data sets of disease prevalence. The sensitivity analysis revealed that the simulated badger population size after 20 years was
most dependent on five parameters affecting female recruitment (probability of breeding, mortality of adult females in the
first half of the year, mortality of juvenile females in the second half of the year and mortality of female cubs in the both
halves of the year). The simulated prevalence of TB was most affected by the population size, the rate at which infectious
badgers transmit the disease to other members of their social group, and the rate at which the disease is spread outside of the
social group.
The spatial and temporal predictions of the model were tested against badger demography and TB prevalence data derived
from the field. When validated in space, the model generated population sizes and disease incidence that were consistent with
the observed field population. We conclude that modelling TB dynamics must include spatial and temporal heterogeneity in life
history parameters, social behaviour and the landscape.
Based on parameter sensitivity and data availability, we suggest priorities for future empirical research on badgers and bovine
tuberculosis.
© 2003 Published by Elsevier Science B.V.
Keywords: Spatially-explicit; Sensitivity analysis; Power analysis; Mantel tests; Latin hypercube sampling; Logistic regression; Meles meles;
TB
1. Introduction
∗ Corresponding author.
E-mail address: [email protected] (M.D.F. Shirley).
1 Present address: Anatrack Ltd., EGI, Department of Zoology,
University of Oxford, Oxford OX1 3PS, UK.
The European badger (Meles meles) is a natural
reservoir of Mycobacterium bovis and has been implicated in outbreaks of bovine tuberculosis (bovine TB)
in cattle in the British Isles (Muirhead et al., 1974;
Zuckerman, 1980; Krebs, 1997). The disease can
cause lesions in the respiratory and urinary systems
0304-3800/$ – see front matter © 2003 Published by Elsevier Science B.V.
doi:10.1016/S0304-3800(03)00167-4
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in badgers (Clifton-Hadley et al., 1993) which may
lead to contamination of the agricultural environment
and create a potential risk of disease transmission
to cattle. Understanding the spread of bovine TB by
a wildlife reservoir is, therefore, of importance to
agriculture in the UK.
In the UK there is considerable geographical variation in badger population densities and a range of
social systems, from solitary animals to large social groups with carrying capacity determined by the
availability of food and suitable sett sites (Kruuk and
Parish, 1987; Rogers et al., 1997, 1999). In southern
England and Wales, where bovine TB is most prevalent, badger populations typically consist of social
groups occupying mutually exclusive territories. Badger social group sizes in this region may vary between
3 and 10 individuals (Neal and Cheeseman, 1996),
with records of up to 27 individuals (Rogers et al.,
1997). The territorial behaviour and social structure of
badger populations has implications for understanding
the spread of bovine TB within badger populations.
The incidence of disease transmission may be influenced by badger social organisation and excretory
behaviour (e.g. Brown, 1993) or by disease-induced
changes in badger behaviour (Cheeseman and
Mallinson, 1981). It has been suggested that the
spatial perturbation of badger social groups during
culling operations to reduce the incidence of TB in
cattle may facilitate disease transmission between
groups (Tuyttens et al., 2000). Landscape structure
and heterogeneity, together with farm management
practices, are therefore important factors in understanding the prevalence of TB in badger populations
and the incidence of TB breakdowns in cattle herds.
Since experimentation in field populations is largely
impractical, modelling has been used extensively to
analyse TB dynamics in badger populations. Models
have been of two major types: analytical modelling
and simulation modelling. Deterministic analytical
approaches, such as Anderson and Trewhella (1985),
Swinton et al. (1997) and Smith and Cheeseman
(2002) are primarily strategic models, giving insights into the factors which control changes in the
population structure. These models are useful for understanding principles of the disease dynamics, such
as cyclicity and threshold population densities, but
ignore badger social structure by assuming homogeneous mixing of populations. Furthermore, these
models do not include stochasticity and environmental heterogeneity that are likely to be important in real
landscapes in which badger-TB is a problem.
The second approach has been based on stochastic
simulation modelling, (Smith et al., 1995, 2001a,b;
White and Harris, 1995). These are tactical models,
attempting to deal with the detailed practicalities
of TB control, and were spatially-explicit in that
the dynamics of the badger population and disease
were simulated in a landscape of grid cells, between
which animals were allowed to disperse. These authors analysed the conditions under which TB could
remain endemic within badger populations and concluded that group size was important in determining
spread. Smith et al. (1995, 2001a,b) went on to investigate the role of heterogeneity in group structure on
the dynamics of the disease, and demonstrated that
epidemiological models for homogeneously mixing
populations were inappropriate because of the social
structuring of badger social groups.
While these models have been used to predict disease dynamics of bovine TB in badgers and to investigate different control strategies (e.g. Smith et al., 1997,
2001a,b; White et al., 1997), none of them have considered the disease problem in real landscapes. Dispersal and disease epidemiology is non-homogeneous in
natural populations. The social and territorial organisation of badgers (Kruuk and Parish, 1987; Rogers
et al., 1997, 1999; Delahay et al., 2000) results in
differential rates of contact between individuals of
different social groups, and reduced rates of contact
between territories separated by distance or landscape
features. Since contact rates are dependent on spatial
organisation of badger territories, there is clearly a
need for a spatially-explicit model which examines
the relationship between the behaviour of individuals
and the spread of TB (Ruxton, 1996).
In this paper, we describe a spatially-explicit model
to investigate the dynamics of TB and badger populations, based on the approach by Smith et al. (1997).
Spatially articulated models have been used to investigate the effects of disease transmission between
sympatric species of squirrels in the UK (Rushton
et al., 2000a,b), and control strategies such as immunocontraception to control grey squirrels (Rushton
et al., 2002) and possums (Barlow, 1994). In contrast
to previous approaches, the model relies on observed
social group boundaries to simulate population and
M.D.F. Shirley et al. / Ecological Modelling 167 (2003) 139–157
disease dynamics rather than a grid-based representation of social structure.
There is no accepted procedure for testing the sensitivity of a stochastic model. Swartzman and Kaluzny
(1987) recommend that a successful method meets
the following criteria: (a) it must be clearly defined,
straightforward and specify the number of model runs
required; (b) it must account for the effects of interactions between parameters; (c) it must include information on the variability associated with parameter
estimates; and (d) it must allow interpretation for several output variables. This paper describes a protocol
for a sensitivity analysis of an individual-based simulation model which includes all of these features. We
use Latin hypercube sampling (LHS) to sample the
data range of each input variable, using the restricted
pairing technique of Iman and Conover (1982) to
eliminate correlation between input variables. In addition, the calculation of partial correlation coefficients
for each input variable takes into account the variance in model results caused by other input variables
and calculates the proportion of the variance in the
output which is uniquely accounted for by each input
variable. Power analysis of the partial correlation coefficients allowed for the specification of the number
of model runs needed to detect sensitivity. Binary
logistic regression was used to analyse the effect of
the input variables on disease persistence, where the
sparse data set makes partial correlation analysis less
powerful. A combined partial correlation and GLM
approach of the type presented here gives a full picture
of the interaction between model inputs and model
outputs.
A model should not just be an experiment in
thought—to justify modelling studies to the funding
bodies, models should also have an application in a
predictive or management role. If a model is to be
‘believable’ and, therefore, of any use in an applied
role, it must undergo a preliminary stage of testing,
broadly termed ‘validation’. Validation (Rykiel, 1996)
is “. . . a demonstration that a model within its domain
of applicability possesses a satisfactory range of accuracy consistent with the intended application of the
model”. Assessment of the predictive efficacy of models has concentrated on comparing disease prevalence
to various sets of field data (Bentil and Murray, 1993;
Smith et al., 1995, 1997, 2001a,b; Ruxton, 1996) or on
determining the disease transmission parameters that
141
gave rise to a good fit of the model results to the field
data (White and Harris, 1995; White et al., 1997).
The first stage of this procedure was to derive transmission rates for the model that gave the best fit to the
field data for the landscape as a whole. This procedure
ignored spatial trends and, therefore, looked at change
in badger populations and disease dynamics through
time. The second validation stage was to look at the
effects of the parameters derived in the first stage on
the spatial spread within the landscape.
This study describes the testing of the predictions
of the model with regards to population size and TB
prevalence on a temporal and spatial scale using data
from a long term study at Woodchester Park, Gloucestershire.
2. Model description
2.1. General
A spatially-explicit simulation model was written in
the programming language ‘C’ (Fig. 1). This was an
individual-based model with the age, TB status and sex
of each badger in each social group as the state variables. The model interrogated each badger at 6-month
time step to determine stochastically the life history of
the individual. Life history ‘decisions’ were made using a probabilistic approach, each individual having a
particular chance that it will become infected or pass
into the next stage of the disease (depending on TB
status), change social groups and die (both based on
age and sex), and, if female, breed.
The first 6-month season each year was the spring/
summer season, and included subroutines governing
reproduction, TB transmission, TB-induced mortality, natural mortality and movement (Smith et al.,
1995, 1997). The second season, representing autumn/winter, included all these subroutines except for
reproduction, which in badgers only occurs once a
year. Values for each of the life history and disease
transmission parameters (Table 1) were obtained from
Smith et al. (2001a,b), and all operated on a 6-month
time step. A difficulty with this simulation model was
that while there were reliable data on the life history
parameters of badgers at Woodchester Park (Table 1);
the information on TB transmission between infected
and healthy animals is very limited. The sensitivity
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Fig. 1. Structure diagram (Dent and Blackie, 1979) of the simulation model. The diagram is read as a sequence of steps executed from
left-to-right and top-to-bottom (within each step).
analysis described in Section 3.1 investigated the impacts of variations in all of these input parameters at
each stage in the model.
2.2. Reproduction
Female badgers produce only one litter each year
(usually one to five cubs), with the majority giving
birth between mid-January and mid-March (Neal and
Cheeseman, 1996). In high density populations usually only one female per social group produces a
litter, although others may also breed if sufficient setts
and resources are available (Cresswell et al., 1992;
Woodroffe and Macdonald, 1995). To this end, the process of reproduction was modelled as a sequential
Markov chain: there is a base probability that a female
will produce a litter and if she does, the model checks
to see if a second female (if present) produces a litter,
and so on. This allows for variation between years
based on the assumption that there is a fluctuating resource to support multiple breeding females in a social
group. The number of cubs in each litter was determined by a cumulative probability distribution, based on field data collected at Woodchester Park (unpublished).
2.3. Dynamics of bovine tuberculosis
Amongst badgers, bovine tuberculosis is thought to
be spread in four major ways: as a respiratory aerosol;
during aggressive encounters involving bite wounds;
at latrine sites and by vertical (or pseudo-vertical)
transmission between mother and cubs (Anderson and
Trewhella, 1985; Cheeseman et al., 1988a,b). When
healthy badgers become infected, they are assumed
M.D.F. Shirley et al. / Ecological Modelling 167 (2003) 139–157
143
Table 1
Badger life history and TB parameters used in the simulation
model
Parameter
Value
Natural mortality
Female cub mortality season 1
Female juvenile mortality season 1
Female adult mortality season 1
Female cub mortality season 2
Female juvenile mortality season 2
Female adult mortality season 2
Male cub mortality season 1
Male juvenile mortality season 1
Male adult mortality season 1
Male cub mortality season 2
Male juvenile mortality season 2
Male adult mortality season 2
0.240
0.122
0.122
0.229
0.122
0.122
0.240
0.161
0.161
0.296
0.161
0.161
Movement
Female movement probability
Male movement probability
0.0025
0.0279
Fecundity and fertility
Probability of producing first litter
Probability of producing second litter
Probability of producing third litter
Probability of producing fourth litter
Probability of litter size of 1
Probability of litter size of 2
Probability of litter size of 3
Probability of litter size of 4
Probability of litter size of 5
0.74
0.37
0.30
0.30
0.05
0.21
0.51
0.18
0.05
TB related parameters
Super-excretor infection rate (HLw )
Excretor infection rate (HLw /2)
Between-social group transmission (HLb )
Excretor females to latent (ELf )
Excretor males to latent (Elm )
Latent to super-excretor (LS)
Latent to excretor (LE)
Additional super-excretor mortality, females
Additional super-excretor mortality, males
Varies
Half above
Varies
0.52
0.12
0.08
0.09
0.175
0.308
All values are given as a 6-month probability; and are derived
from Smith et al. (2001b).
to undergo a latent period before becoming infectious
themselves. A tracheal aspirate taken from an infectious badger has a positive reaction in an ELISA test
for M. bovis; although no other bodily fluids contain
the bacillus. A small proportion of badgers (particularly adult males) become ‘super-excretors’ (Smith
et al., 1995); who test positive for the TB bacillus in
saliva, urine, faeces, semen and in open bite wounds,
in addition to the tracheal aspirate; and therefore, have
a higher transmission rate than normal excretors.
Fig. 2. TB status and transmission rates used in the badger model.
The probability of within- and between-social group infection from
a excretor is half that of the probability from a super-excretor.
Following Smith et al. (2001a,b) all badgers in the
model were assumed to be in one of four health stages:
healthy (i.e. uninfected), latent (i.e. infected but not
infectious), excretor, or super-excretor (Fig. 2). Only
the last two stages could infect other badgers. Since
super-excreting badgers shed such a high amount of
M. bovis in all bodily fluids, it was assumed that badgers had twice the probability of becoming infected
with TB from a super-excretor than from an excretor
(Smith et al., 2001a,b). Healthy badgers became latent
by one of two transmission mechanisms, one governing within social group infection (where an excretor
or super-excretor could infect any healthy member
of the same social group, with frequency HLwithin /2
and HLwithin , respectively) and the other governing
between social group infection (where an excretor
or super-excretor could infect any healthy member
of any neighbouring social group, with frequency
HLbetween /2 and HLbetween , respectively). Latent badgers could become excretors or super-excretors (at
frequencies LE and LS); and excretors could become
latents or super-excretors (with frequency EL and ES).
2.4. Mortality
Natural mortality data for the model were taken
from Smith et al. (2001a,b) and varied according to
age, sex and season (Table 1). Only super-excreting
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badgers suffer life history consequences from TB infection (Wilkinson et al., 2000). Mortality due to TB
infection was assumed to be additive and was assessed
before natural mortality.
2.5. Movement
High density badger populations such as the one at
Woodchester Park are typified by a low level of movement of individuals between social groups (Cheeseman
et al., 1988a,b; Woodroffe and Macdonald, 1993).
Rogers et al. (1998) found that 73.1% of badgers
(n = 208) that changed social groups between 1978
and 1995 were ‘occasional movers’ and 22.1% were
‘permanent movers’. Of the 703 recorded movements of badgers between social groups in the study
area, the majority occurred between neighbouring
groups.
The spatial arrangement of the intensively studied core of 21 badger social groups at Woodchester
Park (1981–1996) was preserved in the simulation
model, such that badgers could only interact (through
dispersal or TB transmission) with animals from social groups whose territories were adjacent to their
own. Movement was simulated at the end of each
season in the model. Each badger had a probability
of dispersing by moving to a randomly-determined
neighbouring social group chosen from all contiguous
badger social groups. If the badger was female, then
it moved to the neighbouring social group if it was
smaller than the badger’s current social group. If the
social group was larger than the current social group,
then the model chose another neighbour at random,
for up to four attempts. Male badgers dispersed to
a randomly-determined neighbouring social group regardless of its size.
3. Results
3.1. Sensitivity analysis
The sensitivity of the population dynamics model
to the input parameters was investigated by analysing
the response of total population size and total number of infected individuals in simulated badger populations over 20 years to variations in the model inputs.
These input parameters were randomly sampled from
within known ranges using a LHS regime, and then relationships between input parameters and model output were analysed using partial correlation and binary
logistic regression.
A LHS strategy following the methods of Vose
(1996, chapter 4) was used to select input parameters
for the model from the known or estimated ranges of
the different variables in the model. The aim was to
provide a range of input values for each variable that
could potentially occur under field conditions. In other
words the model would be run a sufficiently large
number of times to encompass the potential range
of conditions that occur naturally rather than simply
worst and best case scenarios (sensu Bart, 1995). In
this method, sample values of the input parameters
were selected by a randomisation procedure subject
to constraints on the extent of correlation of input
variables that were imposed by the modeller. There
were insufficient data available to identify the distribution function for all parameters; furthermore there
were no data available to assess the extent to which
each of the life history parameters was correlated
with the others. A uniform distribution was assumed
for each variable with upper and lower limits derived
from the literature. Variables were also assumed to be
independent of each other. This approach will lead to
an overestimate of the size of the likely universe of
possible values that each life history parameter could
take, since firstly, it is likely to lead to the selection
of values for parameters that are near the extremes
of their distributions more frequently than would
be expected in reality. Secondly, the assumption of
non-independence between the life history variables
will lead to variable pairs being selected in the model
that are unlikely to occur in the field (e.g. high mortality and high fecundity). On the other hand it also
ensures that all potential values (within the known
range of observed behaviours for each variable) are
sampled. In other words, whilst we know that the
hyperspace of possible values for each parameter in
the model will be larger than reality, we know that
reality lies somewhere in that space and not outside it
(Rushton et al., 2000a,b).
3.1.1. Sensitivity analysis: number of simulations
There is a trade-off to consider when choosing the
number of simulations to perform in a sensitivity analysis. In assessing the effects of individual parameters
M.D.F. Shirley et al. / Ecological Modelling 167 (2003) 139–157
on model output it is critical not only to be able to
accept the alternate hypothesis of an effect with confidence (i.e. significance, α); but also to have sufficient
confidence in the predictions to avoid mistakenly rejecting the null hypothesis (i.e. power, 1 − β). The
power of a statistical test is reliant on the effect size
looked for (that is, the posited difference between the
sampled test statistic and the true test statistic) and the
number of samples (Cohen, 1988). Thus, the number
of input parameter sets generated by the LHS can be
chosen to achieve the required criteria for significance
and power (i.e. minimise Type I and Type II errors).
In an ideal world, millions of replicates would
be performed, producing high statistical power and,
therefore, high confidence in the results. On the other
hand, computer run-time dictates the maximum number of replicates possible, as does the capacity of
statistical programs to analyse the data. With an LHS
procedure, there is a maximum of (n!k−1 ) parameter
sets, where n is the number of simulations and k is the
number of variables. Iman and Helton (1985) suggest
n > 4/3k as a minimum number of simulations, however, this number was reached from experience with
their models, and is not necessarily a portable rule.
Therefore, to investigate the effect of number of
simulations on the sensitivity analysis, a heuristic approach was used. The LHS procedure was used to generate 50 sets of the 28 life history parameters in the
model (Table 1). The restricted pairing technique of
Iman and Conover (1982) was used, rejecting parameter sets with significant correlations. The model was
then run 50 times, once for each parameter set. Each
model run consisted of 100 replicates of a 20-year
simulation, using as the initial population the data collected at Woodchester Part in 1981. Another 50 sets of
input parameters were then generated using the LHS
procedure and these were added to the previous 50
parameter sets, and the model runs and analysis were
repeated using these 100 parameter sets. This process was repeated to 1000 model runs (i.e. 20 × 50
samples).
For each model run the total number of badgers in
the population were output at the end of the 20 years.
The total population size was then correlated with the
input variables and partial correlation coefficients calculated to assess the impact of the individual life history parameters on the dynamics of the population as
simulated by the model. Partial correlation coefficients
145
reflect the contribution of that parameter to the outputs
of the model, having partialled out the effects of the
other variables in the model (Cohen, 1988). However,
when modelling events of low probability such as TB
incidence it is likely that the frequency of animals predicted to have disease will have been low and there
will have been many incidences with no disease. The
presence of many zeros in these output data rendered
the partial correlation analysis less powerful at testing
the effects of disease parameters on disease epidemiology. In contrast, binary logistic regression analysis
examines disease incidence events rather than the magnitudes of such events; and in this particular example,
where disease transmission was a comparatively rare
event, provided a better analytical approach for these
data. For this reason, binary logistic regression analysis was undertaken for the disease incidence data.
3.1.2. Sensitivity analysis: partial correlation
analysis
Partial correlation coefficients between each model
parameter and the badger population size after 16
years were calculated after 1000 simulations (see
Table 2). The power of the partial correlation coefficients was calculated exactly using the method of
Cohen and Cohen (1983).
To ensure that 1000 model runs were sufficient,
power analysis was also used to calculate the number
of simulations required to give sufficient power to detect a given effect size. Using the method of Cohen
and Cohen (1983), the number of simulations needed
to achieve a power of 0.80 was calculated for all sixteen variables found to be significant; these values are
given in Table 2. It can be seen that five of these variables had sufficient power after 1000 simulations, and
another two parameters would have achieved a power
of 0.80 after another 800 simulations. The remaining
eight parameters would have required at least an order
of magnitude more simulations to achieve the required
statistical power.
The data in Table 2 show that those parameters that
have a large effect on the model output (those with
relatively high partial correlation coefficients) can be
easily identified as key driving parameters at relatively
low numbers of simulations (less than 100 in these two
cases). However, those parameters that have a small
effect on the model output, while having a low probability of making a Type I error (i.e. high significance)
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M.D.F. Shirley et al. / Ecological Modelling 167 (2003) 139–157
Table 2
Partial correlation coefficients and associated F-values after 1000 simulations, relating the predicted total number of badgers to the different
life history parameters used in each model run
Variable
Partial correlation
coefficient
F-value
P-value
Power at
α = 0.01
Probability of producing first litter
Probability of producing second litter
Probability of producing third litter
Probability of producing fourth litter
Female cub mortality season 1
Female juvenile mortality season 1
Female adult mortality season 1
Female cub mortality season 2
Female juvenile mortality season 2
Female adult mortality season 2
Male cub mortality season 1
Male juvenile mortality season 1
Male adult mortality season 1
Male cub mortality season 2
Male juvenile mortality season 2
Male adult mortality season 2
Female movement probability
Male movement probability
Super-excretor infection rate
Excretor infection rate
Between-social group
transmission, females
Between-social group
transmission, males
Excretor females to latent
Excretor males to latent
Latent to super-excretor
Latent to excretor
Additional super-excretor
mortality, females
Additional super-excretor
mortality, males
0.690
−0.006
−0.004
0.022
−0.678
−0.507
−0.528
−0.816
−0.612
−0.600
−0.229
−0.125
−0.088
−0.252
−0.079
−0.060
−0.030
−0.250
−0.275
−0.067
−0.079
881.671
0.031
0.019
0.456
824.895
335.974
375.063
1942.241
582.876
548.151
53.872
15.437
7.572
66.012
6.062
3.553
0.865
64.752
79.718
4.349
6.058
0.000∗∗∗
0.860
0.889
0.500
0.000∗∗∗
0.000∗∗∗
0.000∗∗∗
0.000∗∗∗
0.000∗∗∗
0.000∗∗∗
0.000∗∗∗
0.000∗∗∗
0.006∗∗∗
0.000∗∗∗
0.014∗
0.060
0.353
0.000∗∗∗
0.000∗∗∗
0.037∗
0.014∗
1.000
0.008
0.008
0.008
1.000
0.373
0.519
1.000
0.984
0.963
0.010
0.008
0.008
0.011
0.008
0.008
0.008
0.011
0.012
0.008
0.008
0.001
0.001
0.973
0.008
0.046
−0.031
−0.046
−0.048
−0.031
2.104
0.933
2.106
2.268
0.937
0.147
0.334
0.147
0.132
0.333
0.008
0.008
0.008
0.008
0.008
0.001
0.001
0.970
0.008
Number of simulations to
achieve power = 0.8 at α = 0.01
291
323
1797
1442
83
614
707
68597
854078
3534422
46211
5441545
47770
31850
10572572
5451653
The F-values have 1 and 972 degrees of freedom. Power is calculated assuming a significance level for the F-distribution at 0.01, and
predicted number of simulations to achieve a power of 0.8 is given for the significant life history parameters.
∗ P < 0.05.
∗∗∗ P < 0.001.
have a high probability of making a Type II error (i.e.
low power).
The results can be divided into three categories. First
there are those partial correlation coefficients which
have a high significance (thus, a low probability that
the null hypothesis is true; α ≤ 0.01) and a high power
(thus, a high probability that the null hypothesis is
false; 1 − β ≥ 0.80). We can say with a high degree of
certainty that variables in this category are important
drivers in the model. Five parameters fall clearly into
this category (Table 2): female cub mortality in season 2, probability of producing the first litter, female
cub mortality in season 1, female juvenile mortality in
season 2 and female adult mortality in season 2. Two
other parameters would enter this category after about
800 more data sets (Table 2): female juvenile mortality in season 1 and female adult mortality in season 1.
Thus, all female mortality rates were important drivers
in determining total population size after 20 years, reinforcing the effects of baseline recruitment due to cub
production.
Secondly, there are partial correlation coefficients
which have high significance but low power; indicating that there is a low probability that there is no
M.D.F. Shirley et al. / Ecological Modelling 167 (2003) 139–157
correlation between the variable and the response, but
also a low probability that there is a real correlation.
There are insufficient model runs for these parameters
to reject the null hypothesis as false with confidence.
However, their high significance indicates that they
have an effect on the total number of badgers after 20
years; but the size of this effect is small. Nine model
parameters fall into this category (Table 2): all male
mortality factors except for male adult badgers in
season 2, male movement probability, super-excretor
infection rate, excretor infection rate and rate of
transmission of TB between social groups by female
badgers.
Finally, there are partial correlation coefficients
which have low significance and low power. These
variables have no detectable effect on population size
in the model. The remaining 12 parameters fall into
this category: the probability of producing a litter after the first, male adult mortality in season 2, female
movement probability, TB transmission rates between
social groups by males, all disease state transition
probabilities and the additional mortality imposed on
super-excreting badgers of both sexes. It should be
noted that because of the way in which fecundity is
modelled, if the first female doesn’t breed, then the
social group does not produce any cubs that year,
explaining why the other fecundity probabilities were
not important in the model.
3.1.3. Sensitivity analysis: binary logistic
regression analysis
The data resulting from the sensitivity analysis were
transformed into presence/absence of TB in a modelled badger social group after 20 years. Binary logistic regressions were performed on the LHS variable
set to identify variables contributing significantly to
the presence of infected badgers after 20 years.
Of the 28 variables described in Table 1, only 6
were not significant predictors at the 5% level of the
persistence of TB after 20 years in the simulation
model—probability of the second, third and fourth female breeding, male juvenile mortality in season 1,
infection due to excretor badgers and the rate at which
males in the excretor stage reverted to latent. There
was a 93% concordance between persistence of TB in
the simulation model and the regression line fitted to it
based on the 22 significant predictors. The coefficients
for the significant predictors are given in Table 3.
147
These results suggest that long term persistence of
TB in the modelled badger populations is affected
by population size (indicated by negative coefficients
for female mortality and positive coefficients for the
probability of producing a litter), transmission from
super-excreting badgers, and transmission between social groups by female badgers. The additional mortality imposed upon super-excreting badgers of either
sex reduces the probability of TB persistence.
3.2. Force of infection
To investigate further the differing roles of male and
female badgers in the model, the mean within-group
force of infection was calculated by simulating the
life histories of 1000 male and 1000 female badgers
in the simulation model. Each of these badgers was
assumed to be latent for TB infection, and the number of months that each badger spent in the excreting
and super-excreting state was recorded (Table 4). The
mean force of infection was calculated as the sum of
the mean time spent in each infectious stage multiplied by the within-group transmission rate of that
infectious stage. As the actual values of within-group
transmission rate are not known (just the values derived in Section 3.3.1), the calculated forces of infection for males and females can only be used relative
to each other. Table 4 shows that the force of infection
calculated for female badgers is 1.5 times greater than
the male force of infection. Even though male badgers are more likely to become super-excretors, they
suffer higher mortality than female badgers, and so
their overall effect on disease dynamics is predicted
to be less.
3.3. Comparing temporal and spatial trends in the
model and field data
The results of the simulation model were compared
with the observed changes in the demography and
prevalence of infection in the core 22 badger social
groups of the population at Woodchester Park between
1981 and 1996 (Rogers et al., 1997, 1998; Delahay
et al., 2000).
3.3.1. Temporal trends in the model and field data
While reliable information was available for badger life history parameters for the model, few data
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Table 3
The 22 significant predictors of TB persistence after 20 years from a binary logistic regression, arranged in order of importance (Z-value)
Predictor
Coefficient
Standard deviation
Z-value
P
Female cub mortality season 2
Female cub mortality season 1
Additional super-excretor mortality, females
Female juvenile mortality season 2
Additional super-excretor mortality, males
Between-social group transmission, females
Super-excretor infection rate
Female adult mortality season 1
Female adult mortality season 2
Probability of producing first litter
Female juvenile mortality season 1
Male cub mortality season 2
Between-social group transmission, males
Latent to super-excretor
Male cub mortality season 1
Male movement probability
Male juvenile mortality season 2
Latent to excretor
Male adult mortality season 2
Male adult mortality season 1
Excretor females to latent
Female movement probability
−0.0090
−0.0086
−0.0074
−0.0093
−0.0068
0.0141
0.0045
−0.0062
−0.0059
0.0431
−0.006
−0.004
0.010
0.012
−0.004
−0.004
−0.004
0.009
−0.004
−0.003
−0.001
−0.008
0.0008
0.0008
0.0009
0.0011
0.0008
0.0018
0.0007
0.0010
0.0010
0.0073
0.001
0.001
0.002
0.002
0.001
0.001
0.001
0.002
0.001
0.001
0.000
0.004
−10.67
−10.25
−8.47
−8.17
−7.98
7.64
6.17
−5.96
−5.93
5.90
−5.85
−5.74
5.40
5.14
−5.10
−4.60
−4.00
3.92
−3.63
−2.95
−2.78
−2.24
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.003
0.005
0.025
92.7% of the pairs of predicted and observed values were classified correctly.
are available on the rates of TB transmission. For
this reason, multiple simulations of the model were
performed, varying the within-social group and
between-social group TB transmission parameters,
with replicate simulations for each output. In total,
21 values of within-social group transmission and
21 values of between-social group transmission were
used in combination, resulting in 441 parameter sets.
One hundred replications of each parameter set were
performed, and, after 20 years of simulation for each
replicate, the total population size and TB prevalence
Table 4
Force of infection as calculated from the model
Sex
Male
Female
Mean time in state
(months)
Excretor
Super-excretor
6.0
2.9
5.4
10.8
Mean force of infection
per infected animal
0.42
0.62
The mean within-group force of infection is the mean time in
months that a badger remained in an infective state multiplied by
the probability of disease transmission of that infective state. These
values were calculated from 1000 modelled badgers of each sex.
was output. In order to compare the model outputs
with the observed data, a ranking system was developed which determined the position of the observed
data within the range of outputs of these replicates.
Preliminary outputs of badger demography from the
simulation model indicated that the results were not
normally distributed. The position of the observed
outputs relative to the replicated simulations was,
therefore, used as a measure of goodness of fit. The
rank of the field data within the 100 replicates was
calculated; and the (absolute) difference between this
rank and the median of the replicates was used as a
prediction accuracy score. Under this transformation,
when an observed output was identical to the median
of the replicates, the ranking criterion was zero, and
the goodness of fit criterion was assumed to be maximal (Fig. 3). This method avoids the problem of making assumptions about the distribution of the model
output, and was used in this case to compare the
predictions of the model under different transmission
rates of TB between and within badger social groups.
There was a good match between the simulation
model and the field data for both population dynamics
M.D.F. Shirley et al. / Ecological Modelling 167 (2003) 139–157
149
Fig. 3. The prediction accuracy (rank deviance) of the badger-TB simulation model when two TB transmission rates are varied in tandem.
The figure on the left shows the results for total badger population size after 20 years, the figure on the right the results for total infected
population size after 20 years. Low values represent high prediction accuracy.
and TB prevalence under a number of parameter values for within- and between-social group transmission
(black areas in Fig. 3). However, there was little overlap between the areas where the model closely predicts both badger population size and TB prevalence
(i.e. the black areas on the two graphs in Fig. 3 do not
show much coincidence).
The best fit to both badger population dynamics
and TB prevalence was given by a within-social group
transmission probability of 0.05 and a between-social
group transmission probability of 0.2. These values are for badgers in the excretor stage, rather
than super-excretors, and compare to values of
within-group transmission probabilities of 0.065 and
between-group probabilities of 0.045 for excretor
badgers derived from simulations by Smith et al.
(2001a,b).
The TB parameters best fitted the demographics
of badgers and the epidemiology of TB observed
at Woodchester (discovered in the section described
above) were used to produce spatial output of TB distribution. One hundred simulations of the model were
performed, using the badger population and disease
status of badgers in Woodchester Park in 1981 to initialise the model. Values for the number of years that
an infected badger was present (termed TB prevalence), the number of times that a group becomes
infected (termed TB incidence) and the average duration of infection (prevalence divided by incidence)
were calculated for each social group in the 100
simulations. The means of all the simulations were
then compared to values of prevalence, incidence and
average duration derived from the field data.
The number of years that social groups were infected in the model and in the field is shown in Fig. 4.
For all but four of the social groups, the observed
incidence of number of years in which the group was
infected was within the 95% confidence intervals of
the model predictions. This indicates that in general
the model was able to predict the heterogenous pattern
of disease incidence. The four social groups outside
the confidence interval (Colepark, Old Oak, Nettle
and Wood Farm) were all adjacent to each other, and
at the eastern end of the study site, suggesting that
some other spatial process was responsible for this
variation.
The average duration of infection events per social
group (expressed as the total number of infected years
divided by the number of infection events) is shown in
Fig. 5. In both the field and the model simulations, TB
persistence was greater in the badger social groups to
the west of the study site, however, the model produced
results that were more homogeneous than in the field
data.
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Fig. 4. Total number of years that each social group in the Woodchester population had at least one TB-infected badger. The lines show
the means and 95% confidence intervals for the model (arranged in descending order); the diamonds show the field data.
3.3.2. Spatial trends in the model and field data
Mantel tests were used to investigate the extent to
which the predicted spatial pattern of disease incidence and social group demographics matched those
observed in the field. Mantel tests compare two (or
more) distance matrices. One matrix typically contains
a measure of geographical distance separating the
points of interest (e.g. Sanderson et al., 1995), whilst
the second contains measures of ecological distance as
calculated using the Minkowski metric (Everitt, 1980).
The Mantel statistic r is then calculated as the sum of
cross-products of the two matrices and its significance
is evaluated by Monte Carlo permutation of the matrices (Manly, 1998) or normal approximation (Cliff
and Ord, 1981). Comparisons were made between the
predicted and observed patterns of disease and badger demographic information summarised for the 16
years of the model run. For each run of the model the
number of times a social group became infected (from
a disease-free state) and the duration of disease were
recorded and the mean duration of disease in each
social group was calculated. In addition, the mean reproductive rate of each social group over the duration
of the 16 years of the model run was also calculated.
These data were then used to calculate Euclidean
distance matrices (Everitt, 1980), for subsequent
analysis.
The g-statistics derived from Mantel tests comparing the spatial pattern of disease incidence and duration in the model to those observed in the field are
Fig. 5. The mean duration of infection (in years) in each of the 21 badger social groups at Woodchester Park: (a) shows the field data and
(b) shows the mean of 10 simulations of the same time period.
M.D.F. Shirley et al. / Ecological Modelling 167 (2003) 139–157
151
Table 5
Standardised normal deviates derived from Mantel tests comparing the distance matrices derived from the predicted social group disease
characteristics with those observed in the field at Woodchester Park
Variable comparison
g-statistic
P-value
Pperm
Mean duration of disease
Frequency of social group becoming infected
Mean reproductive rate of social groups
Population growth after 20 years
Mean population size of social groups
1.683
4.068
2.789
5.072
2.370
0.046
<0.001
0.003
<0.001
0.009
0.067
0.004
0.018
0.001
0.025
P-value estimated normal deviate, Pperm probability derived from 1000 permutations.
shown in Table 5. The analyses can be split into two
types. Firstly, analyses comparing the spatial patterns
in the demography of the badger social groups and
secondly, analyses comparing the spatial dynamics of
TB amongst social groups. The null hypotheses that
there were no associations between the distance matrices derived from model predictions of average growth
rate, average population size and the mean reproductive rates of each social group and their respective matrices derived from the field data were rejected. This
indicates that the spatial patterns of growth, reproduction and population size amongst the 21 social groups
were similar in the model output and the field. The
null hypotheses that there were no associations between distance matrices describing the differences in
the duration of disease amongst social groups and the
frequency with which social groups became infected
were also rejected. This indicates that the spatial pat-
tern of disease incidence and the duration predicted by
the model were similar to those observed in the field.
The g-statistics derived from Mantel tests comparing the distance matrices summarising differences
between predicted total and observed total populations of badgers in each social group are shown in
Fig. 6a.
The change in g-statistic comparing the predicted
numbers of TB-infected badgers in each social group
with those observed in the field from 1981 to 1996
(Fig. 6b) shows that in only four out of 16 years was
the hypothesis that there was no association between
the patterns of disease incidence in the model and the
field rejected. For the remaining 12 years the predicted
patterns of disease incidence were not similar to those
observed in the field. As with the badger population
size analyses the pattern of association declined with
time and most consistently after year 11.
Fig. 6. The change in g-statistic of a Mantel test associating field data with model results over time (a) average population size and
(b) average number of infected badgers. The dotted line indicates the critical value of the g-statistic at the 5% level.
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4. Discussion
“Models are constructions of knowledge and caricatures of reality” (Beissinger and Westphal, 1998),
and as such, the heuristic aspects of modelling are often of more interest than the model results. That is,
models may be used to highlight the important drivers
of processes; and these results in themselves may be
as valuable as using models predictively. One of the
best approaches to identifying these heuristic aspects
is the use of sensitivity analysis.
In this paper we present a combined methodology for testing the sensitivity of our stochastic model
which meets all the criteria of Swartzman and Kaluzny
(1987, chapter 8). In this analysis, parameters that
have the strongest effect on the model output will
be significant at comparatively low numbers of simulations. Conversely, parameters with a weak influence will tend to become significant only after large
numbers of simulations. This is particularly important
when the parameter space is so large, such as in this
model with 28 life history parameters. By following
the present method, a significant variable allows us to
distinguish between false positives and true positives
with high confidence. Thus, if we observe an effect of
a variable with high significance we can be confident
that the effect is real. A powerful variable allows us
to distinguish between false negatives and true negatives. If we do not observe an effect of a variable with
low power, we cannot be certain that it is truly absent.
It is therefore important to determine the minimum
number of simulations necessary to achieve adequate
power for the partial correlation analysis.
In the absence of power calculations, the sensitivity
analysis would have identified 14 out of the 16 significant variables as significant at α = 0.001. Using the
analysis presented here, this list has been reduced to
five variables that are statistically powerful as well as
being statistically significant. The probability of producing at least one litter and female mortality of all
age groups were the prime drivers of population size.
This is in overall agreement with empirical studies,
e.g. Anderson and Trewhella (1985), report up to 70%
mortality of badgers in the first year of life, although
adult death rate is fairly low and constant with age.
Of the TB-related parameters in the model, three had
a significant impact upon badger population sizes: the
infection rate of both excretor and super-excretors; and
between-group transmission rate by females. The fact
that TB-induced mortality was not a significant predictor of badger population size was due to interdependence of this parameter with the infection rates. In
simple terms, the more infected badgers there are, the
more there are to kill. The variation introduced to the
system through TB-induced mortality was trivial compared to that introduced by the transmission rates. As
we did not know precise values for the transmission
rates, this highlights the importance of this sensitivity
analysis; in that transmission rates are the weak point
within our knowledge of TB epidemiology.
The main reason for modelling TB and badgers was
the issue of the spread of TB in the wider landscape.
Since this is a spatial phenomenon, a key point is
to what extent spatial features were important drivers
of the model output. The spatially-explicit information used in the model was the distribution of badger
social groups (and their composition) and the extent
to which animals could move from one social group
to another. Therefore, the key spatially-explicit component of the model affecting disease transmission
was the rate of movement between social groups. To
what extent were the model outputs sensitive to this
dispersal parameter? Male movement probability was
found to be a significant driver of both population size
and disease prevalence; whereas changes in female
movement probability only significantly affected TB
prevalence.
Long term persistence of TB in the modelled badger population was most greatly affected by badger
population size (in terms of low mortality and high
probability of producing a litter), infection from
super-excreting badgers and transmission between
social groups by female badgers. Transmission by
females was shown to be important because they
have a higher force of infection. Increased mortality
of super-excreting badgers reduced the probability of
TB persistence. In field studies, disease incidence has
been related to movement of badgers between social
groups (Rogers et al., 1998), and the binary logistic
regression showed both male and female movement
rate to be significant predictors of TB persistence.
Between social group transmission of TB can either
occur through dispersal behaviour, that is individuals
moving to another social group; by social interactions
or by environmental contamination (e.g. at latrines,
outlying setts, etc.). Currently, social mobility in the
M.D.F. Shirley et al. / Ecological Modelling 167 (2003) 139–157
model does not include observed behaviours such
as the tendency for males to move to groups with a
higher proportion of resident females (Rogers et al.,
1998). Permanent dispersal movements of individuals
between social groups at Woodchester Park are relatively rare (Cheeseman et al., 1988a,b; Rogers et al.,
1998) with fewer than 10% of animals making permanent movements between social groups over the 20
years of study. The probability of dispersal between
badger social groups was parameterised from observations made at Woodchester Park and, therefore,
incorporate this low level of permanent movement.
The comparison of model outputs with field data
as a validation exercise was complicated by the requirement that the comparisons had to be made in
both space and time. Simple comparisons of maps
can be undertaken using coefficients such as the simple matching coefficient (SMC) and the Czekanowski
coefficient (Krebs, 1999, chapter 11); however, these
treat each point in space as independent and they often provide very crude estimates of goodness of fit
(Rushton et al., 2000a,b). A better method is to create a measure of how the points in space differ in
each map and then compare the internal differences in
maps between maps. This at least allows for the fact
that the points in the map are not independent. Mantel
tests have been used extensively to analyse ecological
patterns (Legendre and Legendre, 1998). When used
here it was obvious that the spatial pattern of fit was
poorer than the overall estimate provided by comparing population and disease prevalence of the system
as a whole.
Why didn’t the model predict the spatial distribution
of TB incidence amongst social groups accurately?
There are two possibilities for this, either the current
representation of disease transmission in the model is
incorrect or there are other processes which are currently not included in the model. As the model predictions for the spatial distribution of the duration of
infection per social group were more homogeneous
than the field data, this would suggest that the model
is missing some other source of spatial heterogeneity.
One aspect of the model which may have had considerable impact on the goodness of fit of prediction
to the observed population was the role of spatial
and temporal environmental heterogeneity. The model
assumed that social group territories were homogeneous. An overlay of social group boundaries with
153
a map of different habitats present in the study area
showed that there was considerable variation in habitat composition in social groups. In addition, there was
considerable variation in topography, with some social groups on north-facing and others on south-facing
slopes. None of this spatial variation was included in
the model, because there is little understanding of the
role of landscape composition in determining badger
demographics and the epidemiology of TB. There is
some evidence that habitat quality influences the structure and dynamics of badger social groups (e.g. Kruuk,
1978; Hofer, 1988; da Silva et al., 1993; Feore and
Montgomery, 1999). Social group size is smaller in areas of poorer quality habitat and fewer females breed
in each social group (Kruuk and Parish, 1982).
The model did not include explicitly social interactions facilitating opportunities for disease spread; like
previous approaches, all these processes were subsumed into an overall measure of inter-group disease
transmission rate (e.g. Smith et al., 2001a,b). This
may have had considerable impact on the success
in modelling disease spread. Firstly, the distribution
of landscape structures such as linear features influence badger marking behaviour (White et al., 1993)
and the spatial and temporal distribution of latrine
locations along social group boundaries is not homogeneous (Rogers et al., 2000). Thus, it is unlikely that
the opportunities for disease transmission between
social groups was spatially homogeneous. Secondly,
the badger population at Woodchester Park grew in
size substantially since 1978 (Rogers et al., 1997) and
represents a population subject to limited perturbation
that is thought to be near carrying capacity, with limited opportunity for between social group dispersal
(Rogers et al., 1998, 1999). In addition, occasional
movements of individuals from one social group to
another were relatively common (44%) and of those
that moved, more than 73% were classified as ‘occasional movers’ (Rogers et al., 1998). It is probable
that disease transmission was modelled too simplistically. The counter-intuitive result that between-social
group transmission had to be four times higher than
the within-social group transmission rate to explain
the temporal trends in the model is perhaps explained
by the absence of temporary movements in the model.
The correspondence between predicted and observed total population size is greatest at low withingroup TB transmission and is relatively unaffected
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M.D.F. Shirley et al. / Ecological Modelling 167 (2003) 139–157
by between-group transmission. Within-social group
transmission affects the total population size, whereas
between-social group transmission affects the spatial
distribution of TB within the landscape. This contrasts with the correspondence between observed and
predicted disease prevalence, which is greatest at intermediate levels of the two disease parameters. If
TB transmission is too low in the model, the disease
never becomes as prevalent as observed in the field;
if it is too high, then too many badgers die. Whilst
the spatial pattern of model predictions of population
size was similar to that observed in the field for the
first 10–11 years of the model run, there was a rapid
divergence after 1992 when patterns of badger demography and, to a lesser extent, disease incidence
changed. Delahay et al. (2000) noted that disease incidence became significantly more aggregated after
1992; it is possible that the deviation of the model
Table 6
Research priorities for the life history parameters used in this model based on the sensitivity analyses for population size and disease
persistence
PCR
BLR
Sensitivity
Existing LH data
Research priority
3
3
3
3
3
3
3
3
2
2
6
6
6
5
5
P
G
G
G
G
M
L
L
L
L
Variables with moderate importance in both outputs
Female juvenile mortality season 1
2
Female adult mortality season 1
2
2
2
4
4
G
G
L
L
Variables with high importance in disease persistence only
Between-social group transmission, females
1
Male cub mortality season 1
1
Male cub mortality season 2
1
Male dispersal probability
1
Super-excretor infection rate
1
Additional super-excretor mortality, females
0
Additional super-excretor mortality, males
0
3
2
2
2
2
3
3
4
3
3
3
3
3
3
X
P
G
M
X
M
M
H
L/M
L
L
H
M
M
Variables with low importance in both outputs
Between-social group transmission, males
Latent to super-excretor
Male adult mortality season 1
Male juvenile mortality season 2
Male juvenile mortality season 1
Excretor infection rate
Male adult mortality season 2
Female dispersal probability
Excretor females to latent
Latent to excretor
0
0
1
1
1
1
0
0
0
0
2
2
1
1
0
0
1
1
1
1
2
2
2
2
1
1
1
1
1
1
X
P
G
G
G
X
G
M
P
P
H
M
L
L
L
M
L
L
M
M
Variables with no importance
Probability of producing second litter
Probability of producing third litter
Probability of producing fourth litter
Excretor males to latent
0
0
0
0
0
0
0
0
0
0
0
0
P
P
P
P
L
L
L
L
Variables with high importance in both outputs
Female cub mortality season 1
Female cub mortality season 2
Female juvenile mortality season 2
Female adult mortality season 2
Probability of producing first litter
The results of the sensitivity analyses are ranked as follows: Partial correlation rank (PCR) based on significance and power. 3: (P ≤ 0.01,
power ≥0.8). 2: (P ≤ 0.01, power ≥0.5). 1: (P ≤ 0.05). 0: (P > 0.05). Binary logistic rank (BLR) based on significance and Z. 3:
(|Z| ≥ 7.0). 2: (|Z| ≥ 4.0). 1: (|Z| < 4.0, P ≤ 0.05). 0: (P > 0.05). Sensitivity = PCR + BLR. Existing life history (LH) data: G, good;
M, moderate; P, poor; X, no information. Research priorities: H, high; M, moderate; L, low.
M.D.F. Shirley et al. / Ecological Modelling 167 (2003) 139–157
from the observed pattern reflects this. The fact that
the increased difference between model and field data
was also observed between field data and the starting conditions in the field suggests that there was a
change in the demography of the badgers in the field
that was spatially heterogenous in 1992. This was
observed in the field, since the proportion of breeding
females declined in the field population from 1992
onwards (Delahay, unpublished data).
So what does the sensitivity analysis and validation tell us about the overall utility of the model?
For this purpose models are treated as hypotheses or
experiments rather than an accurate or faithful representation of reality (Starfield, 1997). The sensitivity
analyses described here inform us of which parameters are important drivers of either population size or
TB persistence. These results can be divided quantitatively into categories according to the results of the
analysis. Thus, for the partial correlation analysis, the
most important drivers are both statistically powerful
and statistically significant. The next most important
parameters are those that were significant, but with
unacceptable statistical power. Likewise, the logistic
regression analysis can rank parameters according to
their Z-values, for the parameter that has the highest
value for Z is a more important driver than one with a
lower value, even if both parameters are highly significant. Swartzman and Kaluzny (1987, p. 218) present
a qualitative sensitivity analysis used to direct future
research by setting research priorities based on parameter sensitivity and data availability. An analysis of
this type is shown in Table 6. Some parameters (such
as juvenile and adult mortality) are important drivers
in the model but have a low research priority as they
are well understood. Particularly deserving of more
focussed empirical research are parameters such as
TB transmission rates (both within and between social
groups), which are poorly understood, and have a large
impact on the predictions of the model. The results
of these studies are likely to lead not only to better
parameterisation, but also to more realistic modelling.
For example, the between-social group rate of transmission for males has been given a higher priority than
the sensitivity analysis might otherwise suggest; this is
because the model does not include sex-biased transmission. The more aggressive males are more likely
to transmit the disease to badgers from other social
groups than the model might suggest. This qualitative
155
analysis of simulation performance demonstrates that
models are more than just their results; the means of
getting to those results are of equal importance.
This paper describes the testing of a complex model
of badger population dynamics and TB incidence
with historical data. Nonetheless, it is clear from the
results that the model was not capable of generating
all of the observed spatial variation in badger and TB
demographics. Key features that were missing were
environmental heterogeneity arising from spatial variations in habitat composition and temporal variations
in weather. Since badgers rely on a food resource
whose availability is strongly dependent on weather
(Neal and Cheeseman, 1991) and TB persistence in
the environment has a strong environmental correlate
(King et al., 1999), these need to be included in the
next phase of modelling. Very little is known about
disease transmission parameters within badger populations, and this approach will further the understanding of TB epidemiology. More explicit modelling
of social interactions between individuals and their
impact on TB transmission rates within and between
social groups is likely to enhance the accuracy of
predictions and the overall utility of the model.
Acknowledgements
Data from the Woodchester Park study were collected by the CSL Wildlife Disease Ecology field
team, with the kind permission of local farmers and
landowners.
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