MRC Centre for Outbreak Analysis and Modelling, Department of Infectious Disease Epidemiology, Division of Epidemiology, Public Health and Primary Care, Faculty of Medicine, Imperial College London, St Mary's Campus, Norfolk Place, London W2 1PG UK
1Bovine tuberculosis (TB) occurs in cattle and badgers in the UK and control efforts are undertaken to reduce the spread of the disease.
2This study evaluates relationships predicted by nine epidemiological two-host models of disease spread generated by various combinations of density-dependent, frequency-dependent and environmental pathogen transmission. The relationships of interest are between measures of TB in cattle and in badgers from 10 sites which were randomly selected to be proactive badger culling sites in the UK Randomized Badger Culling Trial. The data are from the initial badger cull only.
3There was most support (Akaike weight = 0·562, R2 = 0·869) for models that predicted a positive linear relationship between density of infectious cattle per square kilometre and the density index of infectious badgers. There was less support (Akaike weight = 0·060) for a model that predicted a positive linear relationship between density of infectious cattle per square kilometre and the proportion of badgers infectious with Mycobacterium bovis. A correction to reduce effects of badger carcase storage and an examination of effects of the 2001 foot-and-mouth disease epidemic had little impact on estimated relationships.
4Synthesis and applications. The results provide support for two-host disease models of TB in cattle and wildlife such as badgers, although the form of disease transmission cannot be identified clearly by these analyses. The implication of the results is that the best-fitting models predict that, in the absence of intervention-related changes in badger behaviour, a reduction in density of infectious badgers should reduce the density of infectious cattle. However, analysis of bait-marking data collected following experimental badger culls indicated that culling badgers profoundly alters their spatial organization as well as their population density, potentially influencing contact rates. Effective vaccination of badgers, were it to become available, would be expected to reduce the density of infectious badgers without directly affecting their behaviour.
There are a variety of models of TB in wildlife such as those for badgers in the UK (Smith et al. 2001) and brushtail possums Trichosurus vulpecula in New Zealand (Barlow 1991, 2000), and for cattle and badgers (Cox et al. 2005) and cattle and possums (Kean et al. 1999). Generic modelling has shown how multiple hosts can complicate disease dynamics and control (Woolhouse, Taylor & Haydon 2001; Grenfell & Keeling 2007). Single-host models will predict that the disease status of each host will be independent, but two-host models could predict relationships between disease status. The need to evaluate assumptions and predictions of models has been well described (Caley 2006).
The aim of this deductive study is to evaluate relationships predicted by a priori two-host models of relationships between cattle TB and M. bovis infection in badger using data from a randomized field experiment. Two-host models are used as there is evidence of association of TB infection between cattle and badgers, and badgers have been culled in efforts to reduce TB in cattle populations (Krebs et al. 1997). The implications for disease modelling and analysis are discussed.
Predictions of two-host disease models
The predictions of nine two-host disease models (Ia, Ib to VIII) are described, reflecting different assumptions about the form of disease transmission. There are many ways of modelling disease transmission (McCallum et al. 2001; Begon et al. 2002; Turner et al. 2003) and the specific form appropriate for bovine TB is unclear. As a result, three forms of transmission are investigated, density-dependent, frequency-dependent, environmental transmission, and various combinations thereof, to assess evidence for each. Other forms of transmission, such as vertical (Anderson & Trewhella 1985), are not investigated to restrict the size of the study. The models are forms of multiple working hypotheses in the sense of Chamberlin (1965).
Disease dynamics are described here by compartment models using deterministic differential equations as used by Anderson & Trewhella (1985) and Barlow (1991, 2000). It is assumed cattle are susceptible (Sc), latent (Lc) or infectious (Ic), and wildlife are susceptible (Sw), latent (Lw) or infectious (Iw). The total density (km−2) of cattle (Nc) equals the sum of susceptibles, latents and infectious. Similarly, this applies for wildlife density (Nw). The model assumes exponential growth of cattle with per capita rates of natural (non-TB and non-culling) death rates (d) and birth rates (b), no immunity, no recovery from infection, no vertical transmission, no restocking of cattle and no immigration or emigration of wildlife. There is culling of wildlife (gw) and infectious cattle (hc), disease-induced mortality (α) and the inverse of the latent period (σ) at per capita rates. These assumptions are based on knowledge of the disease as described in the various models of Anderson & Trewhella (1985) and Barlow (1991, 2000). Growth of the badger population is modelled as density-dependent using the generalized logistic equation (Anderson & Trewhella 1985) with the intrinsic growth rate rm, carrying capacity K and the shape parameter θ. The models assume that animal behaviour, and hence, transmission coefficients and other parameter values do not change in response to wildlife control. These are recognized as simplifying assumptions.
Solving the equations for the equilibrium situation and rearranging it can be shown that
Equation 7 describes a linear relationship (model Ia) between the equilibrium density of infectious cattle () and the density of infectious wildlife (), with the intercept being the origin as shown in Fig. 1a. The relationship can be positive if, and negative if Equation 7 as a fitted regression has the form with the slope The equation predicts that as the culling rate of infectious cattle (hc) increases, then the equilibrium density of infectious cattle () decreases.
No estimate of the variable was available; thus, it is combined with other parameters into the estimated slope. Such an approach has been used previously in ecological modelling (Hone & Clutton-Brock 2007). The two terms and are not independent; however, it can be shown that with a very low TB-induced mortality rate in cattle (αc) and a low proportion of cattle being infectious with TB and hence, a low culling rate of infectious cattle (hc), then and are only weakly related as the proportionality parameter between them is approximately zero. For them to be independent would require use of an orthogonal experimental design, which was not possible. The effect of the lack of independence is that the estimated regression slope will be a function of ; higher values of will give higher estimated slopes and vice versa. If is very large, then the relationship could be negative. Hence, the estimated slope is conditional on the specific data used. The focus here is the level of evidence of a relationship between and not on the specific estimated slope.
The cattle–badger model of Cox et al. (2005, equation 4), hereafter called model Ib, is similar to model Ia and describes a positive linear relationship between the density of infectious cattle and density of infectious badgers. The relationship in the Cox model can have an intercept greater than 0 if restocking with infected cattle occurs as shown in Fig. 1a. The fitted regression for model Ib has the form Model Ib predicts no intercept with no restocking and in that situation makes the same predictions as model Ia.
The approach can be used for other forms of pathogen transmission, namely frequency-dependent and environmental transmission and various combinations thereof. Environmental transmission of bovine TB has been suggested as possibly occurring (Anderson & Trewhella 1985; Swinton et al. 2002). The detailed results are described in the Supporting Information Appendix S1 with a summary of the relevant fitted equilibrium solutions given in Table 1 and shown in Fig. 1(b–d). The nine models describe six different equilibrium equations for the density of infectious cattle () (Table 1).
Table 1. Equilibrium solutions, shown in the form of the fitted regressions, for two-host disease models. Details of forms of pathogen transmission and symbols are described in the text and in the Supporting Information Appendix S1
Density-dependent transmission within and between species (model Ia)
Frequency-dependent transmission within and between species (model II)
Density-dependent transmission from cattle (to cattle and to badgers) and frequency-dependent transmission from badgers (to badgers and to cattle) (model III)
Frequency-dependent transmission from cattle (to cattle and to badgers), and density-dependent transmission from badgers (to badgers and to cattle) (model IV)
Density-dependent transmission from cattle (to cattle and to badgers) and from badgers to cattle and frequency-dependent transmission within badgers (model V)
Environmental transmission (model VI)
Environmental transmission and density-dependent transmission within and between species (model VII)
Environmental transmission and frequency-dependent transmission within and between species (model VIII)
The equilibrium solutions can be interpreted in several ways. First is that the cattle–badger-pathogen system is in equilibrium. This is unlikely, given variable badger control and variable cattle testing and culling. Second is that the equations estimate the equilibrium levels of infection towards which the system would move if undisturbed. The second interpretation is what we used here. Such an interpretation is analogous to that for plant-herbivore systems that show high environmental variation and strong feedback effects resulting in the equilibrium state being rarely achieved (Caughley 1987). However, the equilibrium state can be described although it is rarely observed.
Materials and methods
The data are from the Randomized Badger Culling Trial (RBCT) which has been described elsewhere, such as by Bourne et al. (1999) and Donnelly et al. (2003, 2006, 2007). In brief, the trial comprises 10 triplets with each triplet comprising a proactive badger control treatment, a reactive badger control treatment and an experimental control (no badger control) area, with random allocation of treatments within each triplet. The collection of such data on M. bovis in badgers from a randomized experiment with independent populations (areas) overcomes a concern expressed in a previous study (Krebs et al. 1997) of non-random sampling when using road-kill badger data. Each area in the RBCT is approximately 100 km2. Areas are located in western and south-western England (Woodroffe et al. 2005). The present data are from the 10 proactive badger cull areas and comprise data from badgers removed during the initial cull, including data from Woodroffe et al. (2005, Table 2). The culling procedures are described in detail by Woodroffe et al. (2005) and occurred across sites from December 1998 to December 2002 (Woodroffe et al. 2006a). The cattle TB data are for the 12-month period prior to each cull. Data on M. bovis infection consisted of results of skin tests, culture tests and TB lesion examinations (Woodroffe et al. 2005). The analyses of cattle data from culture tests and TB lesion examinations showed no significant relationships so are not reported further. The data on M. bovis infection in badgers were from tissue culture (Woodroffe et al. 2005). An index of badger density per square kilometre was estimated from the number of badgers removed in the initial cull. It was assumed that this was a linear index.
Table 2. Parameter estimates for models of bovine TB in cattle and badgers. The models are described in the text, Table 1 and in the Supporting Information Appendix S1. The response variable in each model is density of infectious cattle. RSS, residual sums of squares; K, parameters; wi, Akaike weights for models i = I to VIII; and R2, coefficient of determination for the best model. NE, not estimated. To reduce effects of badger carcase storage, data from triplets A, C and E were deleted in the second set of analyses. The best-fitting models are shown in bold
a1 = 2·099
V, VI, VII
a1 = 1·764
a2 = 0·223
a3 = 15·722
a4 = –0·480
a5 = –295·100
a6 = 5·378
a7 = 6·848
a8 = 1·500
a9 = –139·200
a12 = 0·016
a13 = 0·021
Data for triplets A, C and E deleted
a1 = 2·106
V, VI, VII
a1 = 1·929
a2 = 0·128
a3 = 1·331
a4 = –0·508
a5 = –264·000
a6 = 5·441
a7 = 7·200
a8 = 1·496
a9 = –151·000
a12 = 0·021
a13 = 0·006
For disease modelling and management, it was assumed that cattle infection as shown by reaction on skin test was equivalent to the animal being infectious, and that there is no carrier state in cattle or badgers. Use of cattle reactors as a measure of infectiousness may be biased. Cattle can become reactors before being infectious (Kean et al. 1999). The prevalence of M. bovis in juvenile and adult badgers were highly correlated (R2 = 0·84, d.f. = 7, P = 0·0005); thus, age classes are not discriminated in the models and analyses. There is an attempt to reduce an effect of badger carcase storage on diagnosis of M. bovis infection: storage for more than 1 week (which was accompanied by freezing) can result in underestimation of rates of M. bovis infection (Woodroffe et al. 2006a). An initial analysis used data from all triplets and then data from triplets A, C and E were deleted as all carcases from triplets A and C and most from E were stored for more than a week (Woodroffe et al. 2006a). The foot-and-mouth disease (FMD) epidemic of 2001 delayed TB testing of cattle and badger culling, and elevated the badger M. bovis infection levels (Woodroffe et al. 2006a). Initial proactive culling of badgers in triplets D, I and J occurred after the FMD epidemic; hence, analyses examined whether the cattle TB rates in those three triplets were higher than expected.
The study did not aim to fully parameterize each model but to evaluate evidence for the predicted regressions. The study did not aim to run simulations, in contrast to studies such as those of Anderson & Trewhella (1985) and Barlow (1991, 2000), which would have required values for all model parameters. The univariate associations between TB in cattle and badgers, as relationships described above from models I, III, V, VI and VII were analysed by least-squares linear regression. The multiple regression relationships of models II, IV and VIII were estimated by non-linear least-squares analysis using sas version 8·2 (Freund & Little 1986). Powerful tests of regression assumptions were not possible because of the small sample size (n = 10). All data had equal weighting in analyses. The analysis of relationships assumes that the independent variables are estimated without error. That may not be correct and would lower the numerical estimate of regression coefficients (Snedecor & Cochran 1967); however, if the error is small, then the effect on parameter estimates will be small (McArdle 2003). Model selection was based on identifying the model with the lowest Akaike Information Criterion (AIC) corrected for small sample size (AICc), and Akaike weights (wi) were used as weight of evidence (Burnham & Anderson 2002). Each Akaike weight is the probability that model i is the best model of the candidate set, given the data (Anderson 2008). Use of AIC analysis has become more common in ecology and widely discussed and its use evaluated (Johnson & Omland 2004; Stephens et al. 2005, 2007). AICc was estimated from the residual sums of squares for regression, as outlined by Burnham & Anderson (2002) and Anderson (2008).
A variety of farm features and cattle movements have been shown to be associated with herd-level TB breakdowns (Johnston et al. 2005). Such farm features were not investigated here. Johnston et al. (2005), however, did not investigate any association between cattle TB and M. bovis infection in badger, which was done here. Farm environmental features may be correlated with badger density, and thus, the results between studies may not be independent. Hence, interpretation of the present results and analyses is conditional on the data used.
evaluation of model predictions
As predicted by models Ia, V, VI and VII, there was a positive relationship between density of infectious cattle per square kilometre, as estimated from skin test data, and the density index of badgers infectious with M. bovis (Fig. 2, Table 2). The Akaike weights (wi) strongly support models Ia, V, VI and VII (Akaike weight = 0·562, Table 2). The model with the second lowest AICc was model VIII (ΔAICc = 2·653, Akaike weight = 0·149, Table 2) although model Ib had similar support (Table 2). In model Ib, the estimated intercept (0·223) had a 95% confidence of –0·217 to 0·662 and thus included 0, consistent with no restocking with infectious cattle. The regressions for models Ia, V, VI and VII (F1,8 = 59·85, P < 0·001) and for model Ib were highly significant (F1,8 = 20·32, P = 0·002). Models II, III and IV had ΔAICc > 4 (Table 2). As predicted by model III there was a significant (F1,8 = 35·00, P < 0·001) positive relationship between the density of infectious cattle and the proportion of badgers infectious with M. bovis (Fig. 3).
The non-linear least-squares regression analyses of models II and IV estimated standard errors and 95% confidence intervals of only two of the three coefficients in each model, presumably associated with the small data set (n = 10) and high variability. AICc values were considerably higher (model II ΔAICc = 4·138, and model IV ΔAICc = 7·506) for those two models than for model I (Table 1) but are difficult to interpret because of the lack of precision for one parameter in each model. The high ΔAICc values suggested the models had little comparative support. On this basis, excluding models II and IV and recalculating Akaike weights gave weights of 0·614 (models Ia, V, VI and VII), 0·163 (model VIII), 0·158 (model Ib) and 0·066 (model III), and thus showed strongest comparative support for models Ia, V, VI and VII.
effects of carcase storage and the fmd epidemic
The above analyses were repeated after deleting data from triplets A, C and E to reduce effects of badger carcase storage. In general terms, the effects were minor. The AICc ranking of models was very similar and again only two of three parameters in models II and IV were estimated with standard errors and 95% confidence intervals. On this basis, excluding models II and IV and recalculating Akaike weights gave weights of 0·850 (models Ia, V, VI and VII), 0·071 (model III), 0·050 (model VIII) and 0·030 (model Ib), and thus showed strongest comparative support for models Ia, V, VI and VII. Analyses that were previously statistically significant and non-significant remained so, with one exception (Table 1). The relationship between the density of infectious cattle and the proportion of badgers infectious was significant (P = 0·013) in the full analysis but marginally non-significant in the reduced analysis (P = 0·057). In all analyses the data, from triplets D, I and J showed relatively high levels of TB in cattle. However, those data showed a scatter above and below the fitted lines (Figs 2 and 3), implying no consistent trend of increased or decreased TB at those sites as a result of delayed cattle testing and badger culling because of the FMD epidemic.
The model prediction with the most support (lowest AICc, Table 2) was the positive linear relationship, through the origin between the density of infectious cattle and the density index of badgers infectious with M. bovis (models Ia, V, VI and VII, Fig. 2). Models Ia, V, VI and VII make the same prediction, as shown in Fig. 1a (dotted line), yet are derived from different assumptions about transmission. Clearly, the common pattern can be generated by different transmission processes. There is a need to better differentiate between the predictions of the four models and that is a topic of future research. Model Ib of Cox et al. (2005) had slightly less support. However, it is recognized that in Fig. 2 (and Fig. 3), there are apparent clusters of data points in the bottom left and towards the top right of each graph. A more uniform spread of data across the range of density of infectious badgers would have been desirable. Hence, the conclusions reached from those analyses are tentative. The model predictions with the least empirical support are the predicted negative relationships between densities of infectious cattle and measures of M. bovis infection in badgers (Supporting Information Appendix S1; equations 7, 14, 21, 28, 35, 43, 51 and 59 if the denominators were negative). The conclusions are conditional on the small set. A larger data set would presumably allow estimation of more parameters with greater precision, such as in models II and IV.
The equilibrium relationship (equation 14) in model II (Table 1), which assumes frequency-dependent transmission, can be rearranged to predict a positive linear relationship, through the origin, between the equilibrium proportion of cattle infectious () and the proportion of badgers infectious (). Such a relationship does not explicitly include the term for cattle density () in the denominator although it is in the right-hand side of the equation. The relationship is highly significant (F1,8 = 49·36, P < 0·001, R2 = 0·85). Such positive relationships have also been reported in New Zealand between the proportions of cattle and brushtail possums with bovine TB (Barlow 1991, Fig. 3), and the proportions of red deer Cervus elaphus and brushtail possums with bovine TB (Nugent 2005, Figure 3.8c). Such associations suggest control of TB in cattle in New Zealand may require control of relevant wildlife, and that occurs (Barlow 1991, 2000).
Evidence of an association between TB in cattle and badgers is presented here. The study does not attempt to describe the specific effects of badger control. The relationships in Figs 2 and 3 imply that transmission of TB occurs from cattle to cattle and badgers to cattle. Woodroffe et al. (2005, 2006b) concluded that their results were consistent with transmission between cattle and badgers, but could not evaluate the relative importance of transmission in each direction. Confirmation of identical genotypes (spoligotypes) of TB in both species (Woodroffe et al. 2005) is consistent with between-host transmission. However, this is not strong evidence of an effect of badger control on TB in cattle. Such evidence is provided by more experimental studies as described by Donnelly et al. (2003, 2006, 2007), Griffin et al. (2005), Woodroffe et al. (2006b) and Jenkins, Woodroffe & Donnelly (2008).
The Randomized Badger Culling Trial (RBCT) was designed, overseen and analysed by the Independent Scientific Group on Cattle TB (John Bourne, Christl Donnelly, David Cox, George Gettinby, John McInerney, Ivan Morrison and Rosie Woodroffe). The RBCT was funded by the Department of Environment, Food and Rural Affairs (Defra) with the cooperation of the many farmers and land occupiers in the trial areas who allowed the experimental treatments to operate on their land. Thanks also to D. R. Cox and P. Caley for comments on a draft manuscript. J.H. acknowledges support from the University of Canberra and C.A.D. acknowledges the MRC for Centre funding support.