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Keywords:

  • control;
  • individual-based model;
  • TB;
  • wildlife disease

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix
  • 1
     An individual-based stochastic simulation model was used to investigate the control of bovine tuberculosis (TB) in the European badger Meles meles by using a live test to determine the presence of infection. The model was an extension of earlier models, and nearly all population and epidemiological parameters were derived from one study site.
  • 2
     This is the first TB model to examine sex differences in disease epidemiology, and the transmission of TB from badgers to cattle. The latter is an essential step if reactive badger control strategies are to be modelled.
  • 3
     Heterogeneity was introduced to the simulation model by the use of a carrying capacity, which defined the maximum number of breeding females per social group.
  • 4
     The prevalence of TB, and the number of simulated cattle herd breakdowns, was reduced for all control strategies using a live test, namely localised culling, ring culling and proactive culling. However, only proactive culling resulted in a marked reduction in these values within a few years.
  • 5
     If trapping efficacy was increased above its current value (80%), this did not improve the effectiveness of these culling strategies.
  • 6
     If the number of individual badgers caught and tested per social group was doubled from two to four animals per group, then the overall level of effectiveness of these strategies could be doubled.
  • 7
     The effectiveness could be improved if the sensitivity of the live test was increased, but did not continue to show an improvement above a sensitivity of about 70%.
  • 8
     Given the constraints of the current live test sensitivity (41%) and a trapping efficacy of 80%, proactive culling, following the testing of four individuals per group, led to an average of three cattle herd breakdowns per year in the simulation, compared with an average of 31 per year when simulating the live test trial as used between 1994 and 1996.

Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

Mycobacterium bovis (Karlson & Lessel 1970), the cause of bovine tuberculosis (TB), remains a serious disease of cattle in the UK. There is strong circumstantial evidence that infectious badgers Meles meles (Lin. 1758) cause a significant proportion of the total number of cattle herd breakdowns, particularly in the south-west of England (Krebs et al. 1997). From 1975 to 1996 the Ministry of Agriculture, Fisheries and Food (MAFF) has attempted to control TB in the badger population through culling.

An independent review (Dunnet, Jones & McInerney 1986) recommended the development of a diagnostic test to identify infected animals, so that healthy animals could be released during culling exercises. This live test, based on an enzyme-linked immunosorbent assay (ELISA), first became available in 1994 (Goodger et al. 1994; Clifton-Hadley, Sayers & Stock 1995). Although it has a good specificity, its low sensitivity (41%) has meant that it has been restricted to determining the presence of disease in badger groups rather than in individual badgers (Clifton-Hadley, Sayers & Stock 1995; Clifton-Hadley & Cheeseman 1997).

There have now been a number of attempts to model bovine TB in the badger (Anderson & Trewhella 1985; Bentil & Murray 1993; Smith et al. 1995; White & Harris 1995a,b; Ruxton 1996a,b; Smith, Cheeseman & Clifton-Hadley 1997; Swinton et al. 1997; White, Lewis & Harris 1997). The early approaches have, in general, been limited by a lack of accurate data on the epidemiology of bovine TB in the badger, and none of them has made any attempt to simulate the transmission of TB from badgers to cattle. The aim of this study was to construct a model capable of correctly simulating the prevalence and spatial distribution of TB in badgers and use it to examine the effect of different badger control strategies where the live test is utilised. In order to be able to simulate any reactive badger control policy that is invoked in response to detection of M. bovis in cattle, it is also necessary to simulate the transmission of TB from badgers to cattle.

In Britain the badger generally lives in territorial social groups, where each territory contains one main sett, which is the focus of activity, and usually a number of other setts. Some of these may be quite large and can be used for breeding when more than one sow reproduces in a territory (Neal & Cheeseman 1996). The status of a sett is not fixed so that different setts may be the main sett in different years (Neal & Cheeseman 1996). In an undisturbed population, the badger social group size averaged 8·8 adults in 1993, and may reach over 20 (Rogers et al. 1997b). Permanent movement between territories was rare, but temporary movement and short-term forays were much more common (Rogers et al. 1998). The distribution of animals between setts within a territory, and the frequency of changes between setts, is not well documented. Males and females are less active in winter and spring and tend not to travel to alternative setts within their territory, although young males may travel to setts in other territories in search of mating opportunities (Brown 1993). During summer and autumn males are more active than females and are more likely to traverse their territory and visit alternative setts (Brown 1993).

Between 1994 and 1996 the live test was used to identify setts with infected badgers within 24 areas. Each area was defined around herd incidents believed, after investigation, to have had a badger origin. Each area, of average size 12·25 km2, was surveyed for badgers and the location, size and activity of each sett was recorded. Baited cage traps were used at each active sett for a period of 1 week and the infection status of all badgers caught was determined. If one or more badgers gave a positive ELISA result, badgers continued to be trapped at that sett and all badgers destroyed, until there were no further signs of activity. For setts where no badgers that were trapped during the first week gave a positive blood test, all animals were released and trapping discontinued. No attempt was made in the field to group badger setts into territories. This can be done by standard bait marking techniques (Kruuk 1978) but is most reliable during spring and autumn.

The prevalence of infection in badgers caught under the live test trial was similar to that under the interim strategy, where badgers were trapped only on the farm in which the cattle breakdown occurred (Krebs et al. 1997). The live test trial was suspended in September 1996 before sufficient data had been collected to determine whether there was a significant effect on herd breakdown rate. An analysis of the available data concluded that the live test, as implemented, would be unlikely to reduce the overall prevalence of TB in badgers and thus the risk to cattle (Woodroffe, Frost & Clifton-Hadley 1999).

The low sensitivity of the ELISA test was seen as very disadvantageous. As a result, the current badger control strategy involves a large-scale experimental trial, with a scientific control, to determine the total level of effect of TB in the badger population on the number of cattle herd breakdowns (as recommended by Krebs et al. 1997). Only after determining the level of this effect can optimal control strategies be designed. Such strategies may, or may not, involve the use of a live test, but it is important to ascertain the optimal use of a live test so that it can be considered against other strategies.

Methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

Badger population data

Wherever possible all data were derived from a long-term study site at Woodchester Park, south-west England. The badger has a flexible social structure (Kruuk 1978; Kruuk & Parish 1982; Evans, Macdonald & Cheeseman 1989) and the ability to adjust its fecundity (Cresswell et al. 1992), so the use of a single study site to derive population and epidemiological data is beneficial. At Woodchester Park, data on badger population dynamics and epidemiology have been collected systematically since 1981 (for a description of the study site and protocol see Rogers et al. 1997b). Badgers in this population live in clearly defined and temporally static social groups (Rogers et al. 2000).

Analysis of capture–mark–recapture data from this study site has shown that cubs have a higher mortality than adults and that males have a higher mortality rate than females (Cheeseman et al. 1987; Rogers et al. 1997b; Wilkinson et al. 2000). The higher rate of mortality in cubs is due to pre-emergent mortality, previously estimated at 24% (Rogers et al. 1997b). All pre-emergent mortality was assumed to occur in the first 6 months of life, and all post-emergent mortality in the second 6 months of the first year.

Fecundity within badger social groups is determined by average litter size and the number of females that give birth. Because breeding success appears to be related to social status (Woodroffe & Macdonald 1995), we ignored the possible effect of the female’s age on litter size and assumed that the older females breed preferentially (Rogers et al. 1997b). Litter size was modelled probabilistically from the distribution of known litter sizes (Neal & Cheeseman 1996), with a mean of 2·94 cubs litter−1.

Density-dependence was modelled by putting a limit on the maximum number of breeding females, k. To determine this figure for each social group, the number of cubs produced each year was divided by three (mean litter size), and the maximum value across all years is k. The relative frequency of one, or more, litters could then be used to calculate the probability of one, two, three or four females successfully breeding. The maximum number of litters in a social group, max(k), was thus determined to be four, which is in agreement with other studies (Woodroffe & Macdonald 1995). The probability of each female breeding is given in Table 1. The average limit on the number of breeding females per group, k, at Woodchester was 2·58. The distribution of k-values at Woodchester Park was used for input into the model for all simulations using a heterogeneous carrying capacity, unless otherwise stated.

Table 1.  Badger population demography parameters as used in the model. All values are given as a 6-monthly probability
Parameter Value
Male cub mortality February–Julydmc0·240
Male cub mortality August–Januarydmc0·296
Female cub mortality February–Julydfc0·240
Female cub mortality August–Januarydfc0·229
Male adult mortality February–Julydma0·161
Male adult mortality August–Januarydma0·161
Female adult mortality February–Julydfa0·122
Female adult mortality August–Januarydfa0·122
Probability of producing first litterp10·74
Probability of producing second litterp20·37
Probability of producing third litterp30·30
Probability of producing fourth litterp40·30
Male dispersal probabilityδm0·0279
Female dispersal probabilityδf0·0025
Disease transmission probabilities  
Latent to excretortle0·09
Latent to super-excretortls0·08
Excretor males to latenttmel0·12
Excretor females to latenttfel0·52
Excretor (combined) to latenttes0·42
Excretor males to super-excretortmes0·88
Excretor females to super-excretortfes0·48
Excretor (combined) to super-excretortes0·58
Additional super-excretor mortality, malesσms0·308
Additional super-excretor mortality, femalesσfs0·175
Additional super-excretor mortality, combinedσs0·224

Permanent movement between social groups is relatively uncommon for the badger (Cheeseman et al. 1988; Woodroffe, Macdonald & Da Silva 1995; Rogers et al. 1998). Temporary movement between social groups can be simulated by increasing intergroup disease transmission rates, and does not have to be modelled independently in an individual-based model. For this model, 6-month permanent dispersal rates were derived from Rogers et al. (1998) (Table 1).

Badger disease data

In line with previous models (Smith et al. 1995; Smith, Cheeseman & Clifton-Hadley 1997), we assumed four health states: healthy; latent (animals reacting to an ELISA test and negative for M. bovis culture tests); excretors (animals where M. bovis can be cultured, but excluding animals with a single positive result from faeces unless substantiated by a later result; R.S. Clifton-Hadley, unpublished data); and super-excretors (animals with at least one positive culture result on each of two or more consecutive captures, or animals with two positive culture results from different body locations on the last capture occasion). An analysis of badger health states showed that in 94% of cases, only one change of health state occurred within any 6-month period, so individual simulated badgers only need to be examined for the possibility of changing disease status every 6 months. This was therefore used to define the time step used in the model. All excretors return to being latent, or become super-excretors, at the end of a 6-month period. However, as male and female badgers show different probabilities of becoming super-excretors (Wilkinson et al. 2000), rates were calculated for males and females separately and for both sexes combined, in order to determine whether this has any effect on disease dynamics. The exact probabilities of transition between states is given in Table 1.

A recent analysis of mortality rates (Wilkinson et al. 2000) showed that mortality of diseased animals was significantly different from that of healthy animals only among the super-excretors. These had mortality rates approximately double those of healthy animals. Again there was a significant difference between male and female mortality rates, so combined and separate 6-monthly rates were calculated (Table 1). This additional mortality was applied subsequent to natural mortality. In addition to the above sex bias in disease progression and disease-induced mortality, there was a sex bias in the types of positive culture results obtained. Males gave significantly more frequent culture-positive bite wound swabs, probably as a result of being bitten by infectious badgers, whereas females had significantly more frequent culture-positive results from tracheal aspirate samples (Wilkinson et al. 2000). This implies a difference in the means of disease spread between male and female badgers. Of 12 male super-excretors, M. bovis was cultured from bite wounds in 10 animals and from tracheal aspirate in seven animals. Of 13 female super-excretors, M. bovis was cultured from bite wounds in three animals and from tracheal aspirate in 12 animals. We could therefore examine the assumption that infected bite wounds were caused by aggression during intergroup disputes, and M. bovis isolation in tracheal aspirate was caused by inhalation during intragroup below-ground activity. Thus male between-group transmission occurs in 10/12 cases, while female between-group transmission occurs in 3/13 cases. Therefore the relative rate of female between-group transmission is 0·277 that of male between-group transmission (3/13 × 12/10). A similar calculation gives the within-group transmission rate of males as 0·632 compared with females.

If we assume that all TB infection that began with a culture-positive bite wound was a result of intergroup aggression, then 10 out of 33 cases at Woodchester Park were caused by intergroup infection. Each social group at Woodchester has an average of five neighbouring social groups. Therefore intergroup infection probability would approximate 6% of the intragroup infection probability.

The probability of individuals becoming infected cannot be measured easily in the field, so following previous models (Smith et al. 1995; Smith, Cheeseman & Clifton-Hadley 1997) we sought to find intergroup and intragroup transmission rates that simulated the prevalence and spatial spread of the disease at the study site. This approach will also result in decreasing the effect of uncertainty in the other parameters (Wallach & Genard 1998). In the Woodchester Park badger population disease prevalence levels have varied between 10% and 18% (Delahay et al. 2000) with, on average, one-third of social groups being infected in any one year. Over the period 1981–95 the percentage of social groups with infected animals present varied between 14% and 52%.

Badger to cattle transmission data

A further parameter, the rate of disease transmission from badgers to cattle, cannot easily be measured. However, first approximations of this rate can be made by a number of different means. Two national badger surveys have now been conducted. The badger population in the south-west of England was estimated at 62 250 adults in 1988 (Harris et al. 1995). The number of badger social groups in this region had increased by 23% by 1997, very close to the national average (Wilson, Harris & McLaren 1997), and thus we will assume that the total number of badgers in the south-west has also increased by the national average of 77% to about 110 000 adults. During the period 1981–94, of 868 badgers caught at Woodchester Park 25 animals were considered to be super-excretors (2·9%). Extrapolation to the entire south-west region would result in 1805 super-excretors in 1988 and 3190 in 1997, assuming no dramatic changes in disease prevalence over this time. In 1988 there were 62 cattle herd breakdowns attributed to badgers in the south-west (MAFF 1989), and in 1996 there were 192 breakdowns attributed to badgers in the same area (MAFF 1997). Thus the probability of a super-excretor causing a cattle herd breakdown has varied from 3·4% to 6·0% per annum. This is equivalent to 1·7–3·1% 6 months−1. While these figures rely on numerous important assumptions, slightly different methods of calculation produce estimates in the same order of magnitude. We therefore used a mean ‘detection’ probability of 0·025, investigated the effect of changing this probability between 0·01 and 0·05 per 6-month period, but give the results in terms of the number of cattle herd breakdowns per annum.

Model

A stochastic spatial individual-based simulation model was constructed on a 10 × 10 grid where each cell represents one badger social group. The grid therefore represents an isolated non-torus population without emigration or immigration, and runs on 6-monthly time steps starting with the birth of cubs in early spring. Following this, subroutines were called to simulate disease transmission and progression, natural mortality, dispersal, any applied mortality, and social perturbation in each 6-month period. These routines are described below. The effect of different model structures and longer time steps have been examined elsewhere (Smith et al. 1995; Smith, Cheeseman & Clifton-Hadley 1997; G. Smith, unpublished data) and the 6-monthly time step is also justified above. All events were independently and probabilistically applied to each individual. All spatial changes to the array were performed through secondary arrays so that the order of cell examination in the grid had no effect (Ruxton 1996c; Smith & Bull 1997). Following Smith and colleagues (Smith et al. 1995; Smith, Cheeseman & Clifton-Hadley 1997), two alternative assumptions were made regarding pseudovertical transmission of TB from a mother to her young. Either 100% of cubs were infected by infectious (excretor and super-excretor) mothers, or there was no preferential transmission between mothers and young. Three age categories were used in the model: cubs, yearlings and adults. Breeding females were chosen randomly from the adult category, and assigned a litter with a 50 : 50 sex ratio.

Each excretor or super-excretor badger was then given an opportunity to infect healthy animals within its own social group (Fig. 1 and the Appendix) and within any of the eight immediate neighbouring groups on the simulated grid. Each of these original animals (i.e. not including any that have just been infected) was given the chance to change disease status, and the additional disease-induced mortality was applied to existing super-excretors. These super-excretors were then examined for possible transmission of infection to a cattle herd. For simplicity each herd was considered to map exactly with one badger social group and assumed to contain the same number, or density, of cows. Infected cows can be detected during the same 6-month time period, and any reaction (badger culling) would occur during the subsequent time step. The movement of cattle between herds was not modelled.

image

Figure 1. A diagrammatic representation of the simulation model. The broken line represents pseudovertical transmission, which can be switched on or off.

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Natural mortality rates were then applied to each animal and they were given the opportunity to disperse to a neighbouring social group. Age and season were assumed to have no effect on dispersal rates. Methods of controlling TB in badgers was the next subroutine (see the next section), and strategies not involving the live test are considered elsewhere (Smith et al. 2001).

The social perturbation caused by culling part of a social group may influence disease dynamics (Swinton et al. 1997). A subroutine was therefore constructed to allow for perturbation as a result of social groups with very few or no badgers. Each cell was checked for male and female badgers. If the number of badgers of a particular sex was zero, each of its eight neighbours was examined. Any neighbouring group with more than one adult animal of the opposite sex moved a single ‘surplus’ animal into the empty group. Thus in normal circumstances no social group declined to zero animals, as replacement was instantaneous, and there was always at least one neighbouring social group with surplus badgers.

Badger social group size appears to be limited ultimately by prey biomass (Kruuk & Parish 1982), and proximately by breeding success (Woodroffe & Macdonald 1995; Rogers, Cheeseman & Langton 1997a). Carrying capacity in the model was expressed in terms of the maximum number of breeding females per group, k. This could be adjusted from one to four breeding females per social group, and the effect of this variation on population size and disease prevalence is examined in the model.

The model therefore incorporates two forms of density-dependence that will come into effect when badgers are culled: an increase in productivity and a breakdown of social structure that promotes interterritorial movement (social perturbation). Both of these effects occur naturally in the model as it stands. The decrease in number of females alive does not change the probability of litter production in each social group (except when the number of females drops below k) so the same number of females will breed, which leads to an increase in overall productivity, given the smaller population size. An increase in the movement of animals between territories occurs whenever a decreasing social group size results in no animals or only animals of one sex in a territory. This effect occurs much more frequently following badger culling (Smith et al. 2001) and in the absence of quantifiable data the above seems a logical and conservative way to model social perturbation.

As in all epidemiological models the parameters with least certainty are those concerning disease transmission. Following Smith and colleagues (Smith et al. 1995; Smith, Cheeseman & Clifton-Hadley 1997), the disease transmission probability for excretors was set to half that of super-excretors, and the intergroup infection probability was calculated by using a multiplier of the intragroup probabilities, thus resulting in only one unknown parameter.

The model was seeded with different numbers of badgers in each social group, dependent on the carrying capacity of that territory, and each group was seeded with one infectious animal. The model was then run for 50 years to achieve an average steady-state population in terms of both disease prevalence and spatial organisation, before manipulations were imposed and results compared.

Badger control

Four types of badger control are considered here: no control; local reactive culling (one social group following a cattle herd breakdown); ring reactive culling (the badger group that caused a cattle herd breakdown and its eight immediate neighbours); and proactive culling (all groups tested once per year and any groups in which a badger gives a positive reaction to the ELISA test culled). Local culling is therefore most similar to the interim culling strategy adopted by MAFF from 1986 to 1996, and ring culling most similar to the live test strategy implemented from 1994 to 1996, in terms of the number of badger social groups subjected to control. For reactive culling we conservatively assumed that any infection passed from badgers to cattle in one 6-month period was detected, and culling completed by the end of the next 6-month period. We are therefore simulating areas with at least annual testing of cattle for M. bovis. For proactive culling we assumed that the human resources were only sufficient to test each social group once per year, therefore half of the simulated population is tested in each 6-month period. Following earlier models (White & Harris 1995b; Smith, Cheeseman & Clifton-Hadley 1997) we assumed a trapping efficacy of 80% of the social group as a default. This means that each individual badger has an 0·80 probability of being trapped.

In practice, badger culling was performed by setting traps at each active sett. Therefore infected social groups with more than one active sett will have a lower trapping efficacy if badgers positive to the ELISA test are not caught at all the active setts. During the live test trial an average of two badgers was caught and sampled at each sett (Woodroffe, Frost & Clifton-Hadley 1999). If setts are allocated to a social group, for example by using Dirichlet tessellations (essentially assigning all setts to the nearest main sett; Doncaster & Woodroffe 1993) or bait marking (Kruuk 1978), the effective number of badgers sampled per group is increased, and badgers at all setts within the group can then be trapped and tested to increase efficacy. As a default we assumed that two badgers are caught and tested in each social group subject to trapping. We therefore examined the effect of increasing the number of badgers sampled, and adjusting the trapping efficacy.

It is clear that the ELISA test is more sensitive to animals in the later stages of TB progression (Clifton-Hadley, Sayers & Stock 1995; Clifton-Hadley & Cheeseman 1997). It is therefore important to investigate the effect of modelling this differential sensitivity. If badgers are classified as healthy or infected then the sensitivity of the ELISA is 41% (Clifton-Hadley, Sayers & Stock 1995). If the animals are classified as healthy, infected (no visible lesions postmortem) and excretors (visible lesions), then infected animals have a sensitivity of 37% and infectious animals 62% (Clifton-Hadley, Sayers & Stock 1995). We assumed here, conservatively, that the 37% sensitivity applies to the infected and excretor classes and the 62% to the super-excretor class.

If the current live test is not sufficiently sensitive to control TB in badger populations, it may be that a more sensitive test, or a combination of tests, would be. We therefore investigated the effect of varying the level of live test sensitivity without explicitly determining the means of such increased sensitivity. It has been shown previously that pseudovertical transmission of TB from mother to offspring would result in a higher disease prevalence (Smith, Cheeseman & Clifton-Hadley 1997), and some badger control policies have released any lactating females caught. We examined both of these options to determine their possible effects. Lactating females were presumed to be present only during the first 6-month period of the year.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

Model structure

A series of runs was performed without the introduction of TB. Once a steady state had been reached after the first 50 years, the population size was dependent upon the carrying capacity. Simulations with a heterogeneous carrying capacity and a distribution of k-values identical to those at Woodchester Park gave a mean group size of 8·6 adults and yearlings. When compared with a homogeneous carrying capacity, a heterogeneous carrying capacity tended to result in smaller social groups than expected, although this effect was less noticeable at high density. Subsequent simulations were therefore performed with a heterogeneous carrying capacity averaging 2·58 breeding females per group to simulate the Woodchester Park population.

A number of within- and between-group infection probabilities produced similar disease prevalence and spatial characteristics. We chose median values for when k = 2·58 and examined their sensitivity to change by varying each one by 20% and examining disease prevalence and the mean number of infected groups. The results showed that the model was most sensitive to changes in the super-excretor infection probability, and least sensitive to changes in the excretor infection probability. The latter was to be expected as it was a transitory phase.

The results above were obtained under the assumption that disease transmission and progression were independent of the sex of the animals involved. Four sex-biased factors have been determined from the data: (i) male super-excretors have a higher mortality than females; (ii) males are more likely to become super-excretors per unit time; (iii) intergroup transmission may be more commonly caused by males, and within-group transmission by females; and (iv) females may exhibit pseudovertical transmission of disease to their young. The inclusion of each factor meant that small changes were required in the transmission probabilities in order to maintain a similar prevalence. The greatest change was for the inclusion of sex-biased transmission, which resulted in a mean disease prevalence of just 5% and required higher infection probabilities to compensate (0·13 for super-excretors and 0·065 for excretors). All of these changes were too small to use a field estimate of transmission in order to determine whether they occur in real life. For the remaining simulations we included the sex-biased disease-induced mortality because reliable estimates have been calculated for this parameter. In order to give a mean disease prevalence of about 16% over the second 50 years of the simulation, infection probabilities were set to 0·10 for super-excretors, 0·05 for infectious and 0·045 for relative between-group infection, which also resulted in 34–36% of social groups infected in any year.

If we assume that these infection probabilities do not change with badger population density, then we can examine the effect of spatial heterogeneity and social group size. Table 2 shows the effect of carrying capacity and heterogeneity in badger social group size on disease epidemiology. At similar densities, a heterogeneous carrying capacity resulted in a lower disease prevalence and less social groups infected. This can be explained partly by the lower population size in the heterogeneous simulations, as the degree of population depression resulting from TB was fairly similar between homogeneous and heterogeneous populations at the same carrying capacity.

Table 2.  The effect of carrying capacity on disease dynamics in the badger. Each figure is the mean of at least 10 runs
Carrying capacity (k)Population sizeDisease prevalenceNumber of infected groupsPercentage cases showing diseas extinctionPopulation depression
Homogenous carrying capacity
1436·70·000 0·0100 0
2649·20·13632·0  015·0
3740·60·24849·3  024·7
4689·00·25752·0  038·8
Heterogeneous carrying capacity
1·5568·40·025 5·8 60 3·6
2627·10·09220·8 2013·4
2·58657·40·16536·3  023·6
3682·30·18740·2  028·8
3·5690·60·24348·3  036·5

A correlation of population size with disease prevalence, using a heterogeneous carrying capacity similar to Woodchester Park, indicated no clear relationship. During 20 50-year simulations, seven had significantly positive correlations, seven significantly negative and six were not significant. Subsampling showed that in some simulations positive correlations over the first 20 years changed to negative over the next 20 years, and vice versa. The percentage of social groups with infection varied between 14% and 58%, with a mean of 35·9%, over 20 runs each of 20 years.

If the perturbation subroutine was excluded from the model, a total of 0·4% of social groups declined to zero animals per year. If included, the minimum group size was always at least one animal. The average maximum social group size over 10 runs of 50 years each was 20·9. The absolute maximum social group size was 33 badgers. In each year between 24% and 52% of adult females bred, with an overall mean of 38·8%.

The effect of carrying capacity and transmission of TB to cattle within the predefined limits were then examined. The most important factor in explaining the number of cattle herd breakdowns was the prevalence of disease in the badger population (r2 = 0·93), and in a stepwise regression it was the only significant factor. However, in reality information on TB prevalence in the badger is rare, but even without this (assuming that TB is present in the population) then the carrying capacity (k) accounted for 65% of the variation in herd breakdowns. This relationship was not apparent when the badger carrying capacity was 1·0, as relatively few simulations maintained TB in the badger population for the initial 50-year settling-down period, and in those that did the disease died out within 3–27 years.

Control with live test

For each combination of variables a minimum of 40 simulations was performed. This gave a sufficiently smooth average value for the output variables to compare results visually. The differential ELISA sensitivity to badgers at different stages of disease progression was examined for trapping efficacies of 20%, 40%, 60%, 80%, 85%, 90% and 95%. Regardless of whether a standard ELISA sensitivity of 41% was used, or whether a differential sensitivity of 0·37 for infected and 0·62 for super-excretors was used, or whether this differential was increased to give 0·34 and 0·85, respectively, no differences could be seen in the average results. This demonstrated that the model was not sensitive to changes in the method of applying the live test to the simulated population, therefore all subsequent simulations were performed assuming a single sensitivity of 41% to all diseased animals.

The effect of the different control policies (no control, local culling, ring culling and proactive culling) for a generalised ELISA sensitivity of 41% and a trapping efficacy of 80% is shown in Figs 2 and 3. For clarity the standard deviation of each line is not shown. An effect can be seen on the prevalence of TB in the badger (Fig. 2) and the number of cattle herd breakdowns (Fig. 3) of each culling policy when compared with no control. Neither local culling nor ring culling resulted in a reduction in the size of the badger population of more than 10%. Proactive culling reduced the population size by 20%, but the subsequent reduction in disease prevalence allowed the population to recover to 116% of its original size.

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Figure 2. Mean prevalence of TB in a simulated badger population subjected to 50 years of no control, local culling, ring culling and proactive culling. Live test sensitivity is 41% and trapping efficacy 80%.

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Figure 3. Mean number of simulated cattle herd breakdowns per year caused by a badger population subjected to 50 years of no control, local culling, ring culling and proactive culling. Live test sensitivity is 41% and trapping efficacy 80%.

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However, these effects were the means of 40 simulations, and may take a number of years to become apparent. Scatter plots produced to demonstrate the level of variation in disease prevalence (Fig. 4a,b) clearly showed the difficulty of evaluating the success of any single control policy against the background prevalence of disease. For local culling it would not be possible to demonstrate that any single disease prevalence projection is significantly different from no control. With ring culling this differentiation becomes more and more likely after some 20 years. Only with proactive culling (Fig. 4b) is there a clear difference between all simulations after only a few years. The variation in cattle herd breakdown rate is even more confusing because, over the relatively small area simulated, there is only a small number of such breakdowns per year. For the remainder of this paper we will therefore make the assumption that a halving of the prevalence of TB or cattle herd breakdown rate could be detected in the field, and use this as a nominal standard of success.

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Figure 4. A scatter plot of TB prevalence in 40 simulated badger populations subject to (a) no control and (b) proactive culling.

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For the prevalence of TB in badgers, our definition for success shows that local culling is never successful, ring culling is successful after 19 years and proactive culling from year 4. For evaluating the success of badger control on cattle herd breakdowns, local culling is never successful, ring culling after year 13, and proactive culling from year 3. There is, not surprisingly, a close correlation between a reduction in disease prevalence in the badger and a reduction in cattle herd breakdown rate.

By using the above definition of success, the ring culling strategy is successful at the halfway point of the simulation (Fig. 2). Figure 5 shows how this result depends on the efficacy of trapping. Disease prevalence can be reduced slightly even with a low trapping efficacy. However, there appears to be no great benefit of increasing this trapping efficacy much above 60%. None of these policies has much chance of eradicating the disease from the simulated population within a 50-year period. One possible reason for this is the low number of badgers (two) sampled in each social group to determine whether the social groups are infected.

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Figure 5. Mean prevalence of TB in a simulated badger population subjected to 50 years of ring culling with a trapping efficacy of 20%, 40%, 60% (broken line), 80%, and 95% (broken line).

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The effect on the reactive ring culling strategy of different numbers of badgers sampled in each social group is shown in Fig. 6. The maximum number of badgers sampled was five, as this is approximately the average social group size in Britain and would thus represent the majority of all badgers caught and tested. A similar result to trapping efficacy is seen, with the top two lines (representing one or two badgers caught) being less effective than the other scenarios.

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Figure 6. Mean prevalence of TB in a simulated badger population subjected to 50 years of ring culling with one, two, three or five badgers caught and live tested in each social group.

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Thus a reactive ring culling strategy with a trapping efficacy in excess of 60% and an average of three or more badgers sampled per social group should give the best results. Because trapping efficacy in the field is already about 80% (if the whole social group is trapped), a comparison has been made between two or four badgers sampled per social group (see Fig. 7 for ring culling). Increasing the number of badgers caught and tested decreased the time taken to reduce the disease prevalence by half.

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Figure 7. Mean prevalence of TB in a simulated badger population subjected to 50 years of ring culling. The upper solid line represents two badgers live tested and lactating females released, the dotted line represents two badgers live tested and lactating females killed, and the lower solid line represents four badgers live tested and lactating females killed.

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If lactating females were assumed to be released when caught (during the first 6 months of the year), this had little effect on controlling TB (see Fig. 7 for ring culling). Only for proactive culling did the released lactating females consistently result in a slightly higher disease prevalence than if they were killed.

By holding the trapping efficacy and number of badgers tested constant (at 80% and two individuals), we could examine the effect of varying the sensitivity of the ELISA live test in a reactive ring culling strategy (Fig. 8). As before, the nominal success rate improved with slight increases in the efficacy, but failed to improve further above a live test sensitivity of about 70%.

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Figure 8. Mean prevalence of TB in a simulated badger population subjected to 50 years of ring culling with increasing live test sensitivity, from 41%, 60% (broken line), 80% (broken line) and 100%.

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An alternative way to look at the success of these strategies is to consider the total number of cattle herd breakdowns that occur in the 50-year simulation period. Where no badger control was used there was, on average, a total of 65 simulated cattle herd breakdowns. Under the closest comparison with the live test as used in reality (a ring cull of 41% sensitivity, 80% trapping efficacy and two badgers tested) there was, on average, a total of 31 breakdowns. However, under the optimum strategy (proactive culling, testing of four badgers, but still assuming a 41% sensitivity and 80% trapping efficacy) there were, on average, only three breakdowns.

So far consideration has not be given to the possible effect of pseudovertical transmission or releasing lactating female badgers. With 100% pseudovertical transmission the mean disease prevalence in the uncontrolled population increased from 0·149 to 0·226. This had little overall effect on the nominal success rate.

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

A stochastic simulation model of bovine TB was produced to examine badger population dynamics, disease epidemiology, transmission to cattle and the effect of different control strategies using a live test. Simulations without TB, using a carrying capacity dependent upon the maximum number of breeding females per social group, gave an average group size of 8·6 adults and yearlings. This is in close agreement with the average social group size at Woodchester Park, which was 8·8 adults in 1993 (Rogers et al. 1997b). Infection rates were adjusted to give a mean disease prevalence of about 16% for a carrying capacity similar to that seen at Woodchester Park. This resulted in a decrease in mean social group size to 6·6 adult and yearling badgers: equivalent to a population depression of 24%. Although this is higher than an earlier approach with a similar model in a microsimulation (Smith, Cheeseman & Clifton-Hadley 1997), it is still much lower than predicted by other models (Anderson & Trewhella 1985; White & Harris 1995a). If sex-biased transmission rates are included in the model, then the various infection parameters need to be increased in order to maintain a similar disease prevalence and spatial distribution. This results in a population depression of 20%, with 6·8 adults and yearlings per group.

The maximum social group size in the simulation, using a similar carrying capacity to Woodchester Park, was 33 animals. The largest recorded social group at Woodchester Park contained 27 known animals (Rogers et al. 1997b). The percentage of adult females breeding at Woodchester Park has usually stayed between 20% and 40%. In the simulation it varied between 24% and 52%. The simulated figures would be slightly higher than expected when the badger population size is depressed.

The above results give a very close representation to those seen in the field data, and support the accuracy of this simulation modelling approach. They cannot be taken as a formal validation of the model, because the data are being compared with the population used to generate the model input, albeit through indirect measurements. For a comparison of population recovery rates with field data following badger removal see Smith et al. (2001).

The level of population depression in the present model was slightly greater than in earlier models (Smith, Cheeseman & Clifton-Hadley 1997). This was due to the wider categorisation of the super-excretor class, as more animals would reach this class and die prematurely than in the earlier more restrictive super-infectious class. However, it is interesting to note that reducing the additional mortality rate of animals in this class does not affect the level of population depression until this additional mortality is almost totally removed, although any reduction in the mortality does slowly increase the disease prevalence. Because the transmission probabilities are adjusted to produce realistic prevalence levels, any inaccuracy in the calculation of the disease-induced mortality will be compensated for (Wallach & Genard 1998). The insensitivity of population depression to the level of disease-induced mortality therefore means that the model is very robust to the inaccuracy of this calculation. No evidence has been found in the field to support a significant level of population depression, but without knowing the true carrying capacity at Woodchester Park (i.e. removing TB from the population) it is probably not possible to measure a depression of some 20%.

Although this model was closely based on an earlier version (Smith et al. 1995; Smith, Cheeseman & Clifton-Hadley 1997), there are some important differences. In the current model the time step was decreased from 1 year to 6 months to more closely simulate the occasional rapid progression of TB. Also, the highly infectious state super-excretor was a slightly broader category than the previous super-infectious state. Despite this broader categorisation the ELISA test is 85% sensitive to animals in this category (R.S. Clifton-Hadley, unpublished data). Other minor changes in the present model include the removal of a low level of disease-induced mortality applied to infectious animals.

White & Harris (1995a) proposed a disease-free threshold group size of about six adults and yearlings in order to sustain an enzootic infection, and Smith et al. (1995) calculated a figure of eight animals per group in a diseased population simulation model. The results of the present study agree with these estimates, as when social group size drops below about 6·3 adults and yearlings (k ≈ 2) disease extinction may occur within 50 years. White & Harris (1995a) used a homogeneous group size to evaluate disease persistence. In this model it is interesting to note that if the carrying capacity is heterogeneous, then disease prevalence is reduced. However, these models have not considered density-dependent changes in contact rates between animals either within or between social groups, and this could also affect disease persistence and prevalence.

Non-spatial models are often criticised for a lack of realism due to the assumption of homogeneous mixing throughout the population. Spatial models go some way towards addressing this issue by generally only allowing homogeneous mixing within small units of the model: badger social groups in this case. Homogenous mixing within badger social groups would be likely if all animals shared one sett, but would be less likely where more than one sett was occupied within a territory. Also, if there is behavioural segregation of infected or infectious badgers this assumption would no longer hold. Analysis of badger removal operations throughout the south-west has shown that disease prevalence may be greater in outlying setts (Woodroffe, Frost & Clifton-Hadley 1999), and analysis of trapping data at Woodchester has shown that social groups with more outlying setts tend to have a higher disease prevalence (Delahay et al. 2000). This heterogeneous mixing would affect disease transmission within social groups, and may exacerbate the effect of sex-biased transmission, particularly by emphasising the importance of pseudovertical transmission to cubs.

A second aspect not modelled previously is the interspecies transfer of infection to cattle. Previous attempts to simulate spatial control of TB in badgers have used an arbitrary approach by initiating local infection at particular prevalence levels and attempting immediate disease control (White & Harris 1995b). The current model is capable of initiating badger control dependent on the detection of infection in cattle, and because the initial population is stored in arrays various control strategies can be performed on identical disease patterns.

In the present model cattle herd breakdowns are highly correlated with disease prevalence and badger population density (measured as carrying capacity in the model). However, using this model disease prevalence in badgers is not always positively correlated with badger population density. This lack of a strong positive relationship may explain why, in the present model, disease prevalence in badgers is a much better predictor of herd breakdown than carrying capacity. In reality the interspecies transmission probability may vary from farm to farm because of factors relating to cattle management practice, and exposure to badgers and their excretory products. This variation between farms is likely to be much greater than the variation used in the present model, but herd breakdown rates at parish or county level may show less variation, due to averaging, and may therefore be a good predictor of disease prevalence in the badger.

Only seven of 20 simulations gave a significant positive correlation between disease prevalence and badger carrying capacity. A strong relationship is normally expected in conventional disease models, and the weak relationship found in this model is backed up by the difficulty of finding such a relationship in the field data collected at Woodchester Park (Smith et al. 1995). Thus, both empirical data and modelling demonstrate that this population is spatially heterogeneous with respect to disease, and this can impact on disease control strategies (May & Anderson 1984).

The inclusion of cattle into the model, albeit in a simple form, represents an important first step in realising the utility of models, and expanding them to include the species of economic interest. Despite the inclusion of cattle we have continued to examine the prevalence of disease in the badger as the primary goal, due to the noise associated with the low frequency of cattle herd breakdown (Fig. 3). This facilitates comparisons of control strategies.

During the live test trial an average of two badgers was caught and tested at each sett (Woodroffe, Frost & Clifton-Hadley 1999). If these setts were allocated to social groups by using the Dirichlet tessellation method, then an average of just over three badgers would have been sampled per social group, giving an average of 1·8 setts per social group (Woodroffe, Frost & Clifton-Hadley 1999). This partial trapping of social groups must have reduced the overall trapping efficacy, and this could have had a small negative effect. However, the pooling of setts into groups, even if errors occur, increases the number of badgers sampled, and this had a dramatic effect on the ability of the control strategy to reduce the prevalence of disease.

In order to have an effect similar to testing four individuals per social group, either the trapping efficacy must be increased above 95%, or the overall sensitivity of the live test must be increased to 70% or more. While both of these are possible, it is much easier and less time consuming to pool badger setts into social groups. We suggest that Dirichlet tessellations are used to investigate the proportion of badger setts that would be correctly allocated to a social group. To date, Dirichlet tessellations have been used to try and identify territorial boundaries (Doncaster & Woodroffe 1993).

Both reactive strategies were capable of reducing the prevalence of TB in the badger, and thus the number of cattle herd breakdowns. The strategy that tested and culled more infected social groups (ring culling) was more effective in reducing prevalence, although even this strategy took 10–20 simulated years to reduce disease prevalence in the badger by half. Woodroffe, Frost & Clifton-Hadley (1999) concluded that it is unlikely that the live test trial, as implemented, would be effective in reducing the prevalence of TB. The results of these simulations contradict their conclusion, but because many social groups were only partially trapped, trapping efficacy in the field will have been below 80%, and the time taken to noticeably reduce disease prevalence will have been too long to be useful.

However, of greater interest is the effect of changing from a reactive strategy to a proactive strategy using the live test. Issues of human resources have not been considered here as it has been assumed that all social groups within the control area can be tested and trapped in each year. Given the large difference in the effect of the reactive and proactive strategies, it seems very likely that this effect would be seen in reality. Each reactive culling area is a distinct patch within a diseased badger population, whereas the proactive culling occurs over the entire simulated area, resulting in no possibility of immigration of infected animals. Simulations with proactive trapping occurring for 5 years out of every 10 still gave a very strong reduction in disease prevalence. During non-culling years the badger population began to recover, but the disease prevalence did not immediately increase.

If a proactive control strategy were to be used, it would be necessary to determine which areas should be subjected to culling, and how much culling is cost-effective. The current UK badger control field trial involves proactive culling, but without any live test. The areas for this experiment were determined by analysis of the cattle herd breakdown rates. It would probably be more effective to demarcate these areas based upon systematic collection of data on the prevalence on TB in the badger population, but to date no easy method of determining this has been found and validated. A live test offers at least a partial solution, and could be combined with a cost–benefit analysis, with proactive badger culling, using the live test, performed at different intensities (e.g. every year, every second year, etc.).

The reactive culling strategies, in particular, depend upon a number of assumptions. First, TB must be detected very early in cattle and the subsequent control must be performed within an average of 6–12 months after the infection occurred. Changing the super-excretor badger to cattle transmission rate (0·025 per 6 months) has a significant effect on the success of the reactive control strategies. When this transmission rate exceeds about 0·1 6 months−1 then the ring culling strategy becomes almost as effective as the proactive strategy.

We have made no attempt to simulate the behaviour of individual badgers. There is evidence that TB infection is more prevalent in individuals residing in outlier setts (Woodroffe, Frost & Clifton-Hadley 1999) or at least in social groups with more outlying setts (Delahay et al. 2000). This may have some effect on their probability of being trapped, and also any outlier setts that are wrongly assigned to a social group may negatively impact on the success of the control. Similarly, the mapping of one herd of cattle to each badger social group is simplistic and a more accurate mapping system is required in order to investigate fully the spatial consequences of control. Such an approach is now under investigation using a Graphical Information System (GIS) version of the model. In addition, the majority of the data used in the model is derived from a single population.

Mycobacterium bovis is capable of surviving in the environment (King, Lovell & Harris 1999) and this aspect has not been considered here. Environmental survival of M.bovis would lead to a reduced correlation between prevalence of TB in badgers and the number of cattle herd breakdowns and an increased ability for the disease to survive in a post-culled population. This would make the disease more difficult to control, and increase the negative effect of releasing lactating sows.

Because of the simplistic representation of cattle herds in the model so far, cattle-to-cattle and cattle-to-badger transmission of TB has not been considered, nor has the movement of infected cattle. Cattle-to-cattle transmission of the disease seems to be low even under conditions of isolation (Costello et al. 1998) and would only be relevant to a model of badger control if the cattle were moved between herds. Cattle-to-badger transmission is therefore likely to be even lower, but may be important where infectious cattle are placed in herds with access to relatively high-density, disease-free, badgers. Cattle movement, the importation of infected cows and immigration of infected badgers should all be considered once optimum control strategies have been identified. These may not involve the use of a live test, and so these aspects are not considered here.

The consequences of culling on the badger population size, growth rate and recovery have not been investigated here as we have been concerned with use of a live test. This area is reported elsewhere (Smith et al. 2001), where control strategies not using a live test are also examined.

In conclusion, the model suggests that proactive culling is much more efficient than reactive culling. It is therefore more effective to increase the number of animals tested than to increase the trapping efficacy. Also, a live test with increased sensitivity could significantly reduce the time taken to decrease disease prevalence in the badger. Analysis of results obtained for the current culling trial (Krebs et al. 1997; Bourne et al. 1999) should also help to provide further data to evaluate the suggested options.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

The authors thank the Animal Health and Veterinary Group of the Ministry of Agriculture, Fisheries and Food for funding this project. Thanks also go to the many people who have helped collect and analyse the Woodchester Park database, to Drs L. Rogers and R. Delahay and to the referees for valuable comments on an earlier draft.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix
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Received 28 May 1999; revision received 5 September 2000

Appendix

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix

The spatial stochastic model can be described by the following equations, which apply to every social group of badgers in the simulation grid. Variables given in Table 1, or elsewhere, are not described here. Lower case letters represent probabilities, while upper case represent numbers. The average number of healthy males cubs in a social group is described by:

inline image

where 1 ≤ k ≤ 4, and pseudovertical transmission is set to 0. When pseudovertical transmission is set to 1, all offspring of the ith litter are born into the infected class. The number of males per litter, B, is defined by a binomial distribution from the total litter size, and the probability of the ith litter being produced is pi. In the first time step per year, pi is set to the values in Table 1, otherwise, pi = 0. Ψ is the number of neighbouring badgers in the same age/health category; τ is the probability of a healthy badger moving to an adjacent social group as a result social perturbation; wg is the probability of healthy animals becoming infected due to within-group infection; bg the probability due to between-group infection; and mt is the applied mortality at time t. The average number of healthy male adults is described by:

inline image

the average number of latent male adults by:

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and similar equations exists for excretor male adults. The average number of super-excretor male adults is described by:

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Ageing occurs instantaneously at the start of the year when each age category is set to the total of the previous age group (adults are the sum of the current and previous age groups). Similar sets of equations exists for yearlings, infected cubs and females.