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Keywords:

  • bovine tuberculosis;
  • ferret;
  • force of infection

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  • 1
    The force of Mycobacterium bovis infection (λ) in feral ferret Mustela furo populations in New Zealand was estimated, by fitting candidate models to age-specific disease-prevalence data. The candidate models were constructed from a set of a priori hypotheses of how M. bovis infection is transmitted to ferrets, and model selection used to assess the degree of support for each hypothesis.
  • 2
    The estimated force of M. bovis infection ranged between five sites from 0·14 year−1 to 5·8 year−1, and was twofold higher in males than in females.
  • 3
    The data most strongly supported the hypothesis that transmission of M. bovis to ferrets occurs from the ingestion of M. bovis-infected material from the age of weaning, as modelled by the force of infection being zero up to the age of weaning, and constant thereafter. Other candidate transmission hypotheses (e.g. mating, suckling, routine social interaction) and combinations thereof were unsupported in comparison, and hence it was concluded that transmission from these postulated mechanisms must be insignificant compared with dietary-related transmission.
  • 4
    The preferred transmission hypothesis was nearly equally supported regardless of whether disease-induced mortality was included or not, although omitting disease-induced mortality resulted in a lower force of infection estimate. The dietary transmission hypothesis (omitting disease-induced mortality) could be easily represented by a generalized linear model, enabling simple analysis of critical experiments designed to identify the source of M. bovis infection in feral ferrets.

Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Estimating rates of disease spread and associated model parameters of microparasites is an essential component of understanding host–pathogen dynamics, and remains a great challenge in field ecology (McCallum, Barlow & Hone 2001). A result of this is a general paucity of data on transmission rates (De Leo & Dobson 1996). Estimating ‘the force of infection, λ’ (Muench 1959) and relating it via a model to host density and the relative abundance of susceptibles and infecteds, in combination with other demographic parameters, is one practical approach for estimating transmission rates (McCallum et al. 2001). In New Zealand, Mycobacterium bovis (Karlson & Lessel 1970) infection is prevalent in many feral ferret Mustela furo Lin. 1758 populations (Caley 1998). The brushtail possum Trichosurus vulpecula Kerr 1792 is a known maintenance host for the disease in New Zealand (Coleman & Caley 2000), in a similar role to that played by the Eurasian badger Meles meles Lin. 1758 in Britain (Cheeseman, Wilesmith & Stuart 1989). However, it is unclear whether ferrets are maintenance hosts for the disease [i.e. the disease is capable of cycling independently in ferret populations in the absence of external (non-ferret) sources of infection] or whether the observed disease is simply a spillover from brushtail possum populations. Determining the host status of ferrets requires estimating disease-infection rates. Hence methods for estimating the force of M. bovis infection in ferrets are needed.

This study was concerned with estimating the force of M. bovis infection in feral ferret populations in New Zealand from age-prevalence data, and using the data to infer the likely pattern of disease transmission. Previous inference regarding M. bovis infection in feral ferrets has been based on estimates of point prevalence, which is simply the proportion of animals diagnosed as being infected at the time of survey. For example, Caley (1998) observed the prevalence of macroscopic M. bovis infection in ferrets from 11 sites in New Zealand to range from 0 to 31·6%, and reported a positive correlation between the prevalence of infection in ferrets and possum abundance but no correlation with ferret abundance. However, there are limitations to the utility of point prevalence estimates alone for making epidemiological inference. The prevalence of M. bovis infection in ferrets is highly age-specific, with a higher proportion of adults infected than juveniles (Lugton et al. 1997). Mycobacterium bovis infection in ferret populations can be better quantified by using age-prevalence data to estimate the instantaneous rate at which feral ferrets acquire M. bovis infection. In epidemiological terms, λ is the per capita rate of acquiring disease (Anderson & May 1991). In survival analysis, λ is analogous to the hazard rate (the hazard being becoming infected). Observing how the prevalence of disease changes with increasing age provides a starting point for estimating the rate of disease transmission, as age can be used as a surrogate for time (Grenfell & Anderson 1985). Indeed, age-prevalence data are considered of great value in the determination of the net force of infection within host communities (Anderson & Trewhella 1985). Furthermore, different mechanisms of transmission may result in the prevalence of infection changing with age in different ways (different shaped curves), which can be related to different underlying hazard models. Thus, for some diseases it should be possible to use observed age-specific prevalence data to screen for adequate candidate models of disease transmission.

The development of an a priori set of candidate models before undertaking any data analysis and model fitting has been recommended (Burnham & Anderson 1998). Following this approach, this study developed an a priori set of hypotheses of how M. bovis is transmitted to ferrets. Transmission of M. bovis to ferrets has been postulated to occur by a number of routes, including pseudovertical through suckling (as opposed to true vertical transmission across the placenta) (Lugton et al. 1997), horizontal-direct through routine social activities (den-sharing, etc.) (Ragg 1998b), horizontal-direct through fighting (Lugton et al. 1997) and scavenging on M. bovis-infected carcasses (Lugton et al. 1997). These possible routes of transmission can be thought of as a priori hypotheses of the underlying transmission mechanisms of M. bovis among ferrets. Not explicitly stated by any author, but one commonly considered in the transmission of disease, is that of environmental contamination. The hypotheses, none of which are mutually exclusive, are spelt out as follows.

Hypothesis 1 (H1): transmission occurs from mother to offspring (pseudovertically) during suckling until the age of weaning, which occurs at 1·5–2·0 months of age (Lavers & Clapperton 1990).

Hypothesis 2 (H2): transmission occurs during mating and fighting activities associated with it, from the age of 10 months when the breeding season starts (Lavers & Clapperton 1990).

Hypothesis 3 (H3): transmission occurs during routine social activities from the age of independence, estimated to be at 2·0–3·0 months (Lavers & Clapperton 1990), such as sharing dens simultaneously.

Hypothesis 4 (H4): transmission occurs through scavenging/killing tuberculous carrion/prey from the age of weaning (1·5–2·0 months of age).

Hypothesis 5 (H5): transmission occurs from birth because of environmental contamination.

These hypotheses correspond to various hazard functions, where the hazard represents the instantaneous probability of becoming infected (schematically shown in Fig. 1) and equate directly to λ. The possible combinations of five hypothesized underlying hazards (assumed to be additive as they are not mutually exclusive) yields many possible hypotheses for how the force of M. bovis infection may vary with age (Fig. 1). Note that H5 may also potentially arise through combinations of either H1 and H3 or H1 and H4. Just how many hypotheses there are depends on the relative size of the different baseline hazards (Note that when plotting the combined hazard functions, we have not distinguished between H3 and H4). For example, depending on whether the hazard for H1 is higher or lower than H3 or H4 determines whether the resulting hypothesized hazard function takes on the shape of H5, H7 or H8. These various hypotheses may be represented by different mathematical models of infection that fulfil the requirement of Burnham & Anderson (1998), that to be included as a candidate model for making inference, a particular model must make biological sense. The models can be fitted to data as a ‘test’ of the competing hypotheses, although this is not hypothesis testing in the usual falsification sense. It is an exercise in determining which model is most likely, given the data and the number of parameters fitted, as opposed to which model is truly correct, given that there can never be a fully correct model, only a best approximating model (Burnham & Anderson 1998). This model-fitting approach for making inference has been termed ‘post-Popperian’ science, and contrasts with classical reductionist experiments with emphasis on falsification, which can seem too constraining for some workers in the biological sciences (Walker 1998). We then used the best model as a means of comparing the force of M. bovis infection in ferrets between sites and sexes, as a precursor to using the model to estimate transmission rates of M. bovis between ferrets from manipulative studies. The results of the latter will be reported separately.

image

Figure 1. Schematic representation of possible baseline hazard functions for transmission of Mycobacterium bovis infection to feral ferrets for hypotheses 1–5 (H1–H5), and possible composite hazard functions (H6–H12) arising from these five baseline hazard functions. The scaling of the y-axis is arbitrary.

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Materials and methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

SELECTION OF STUDY SITES

This study utilized data collected from cross-sectional surveys of M. bovis infection in feral ferrets at five sites in New Zealand (Fig. 2). Sites were primarily selected for survey on the basis that M. bovis occurred in wildlife, as inferred either from previous wildlife surveys or from tuberculin testing of cattle herds. Sites were deliberately chosen to sample a range of possum and ferret densities. In the case of possum density, this ranged from low at Lake Ohau in the Mackenzie Basin, which has a naturally sparse population of possums, to moderate at Scargill Valley in North Canterbury, to high at Awatere Valley in Marlborough and the Castlepoint and Cape Palliser study sites in the coastal Wairarapa. Possum and ferret abundance are in general inversely related (Caley 1998). In New Zealand, ferrets occur at highest densities in semi-arid regions, where their principal prey species (European rabbits Oryctolagus cuniculus Lin. 1758) are most abundant, whereas possums tend to be more abundant in areas of higher rainfall. For sites that were subjected to repeated surveys (e.g. Castlepoint), the number of ferrets removed in each survey was considered insignificant, hence data from all surveys were included for analysis. For the purpose of analysis, the magnitude of unmeasured factors considered possibly to influence the force of M. bovis infection in ferrets (specifically site) was assumed to be constant over time. Becker (1989) points out that, from a single cross-sectional survey, any effects of age and time on λ are confounded, so whether λ is age-dependent [i.e. λ(a)], time-dependent [i.e. λ(t)] or both [i.e. λ(a,t)] can not be determined.

image

Figure 2. Sites of cross-sectional surveys of Mycobacterium bovis infection in feral ferrets in New Zealand. Sample sizes are shown in parentheses.

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FERRET SAMPLING

Ferrets were captured in Victor Soft-Catch® leg-hold traps (size ½) baited with fresh rabbit, hare Lepus europaeus occidentalis de Winton 1898 or domestic chicken meat. Traps were set at approximately 200-m intervals, usually over 5–10 nights, and checked daily. Animals were humanely killed at the trap site where they were captured. Traps were located in all areas of each study site thought most likely to be frequented by ferrets, particularly areas of highest rabbit density. All fieldwork procedures were approved by the Landcare Research Animal Ethics Committee (Approval Project No. 98/10/4).

DIAGNOSIS OF m. bovis INFECTION

From each ferret caught, the jejunal (mesenteric), both caudal cervical (prescapular) and both retropharyngeal lymph nodes were collected. All other major lymph nodes and organs were also examined, and a portion of any potentially tuberculous lesion added to the lymph-node pool, which was stored frozen. Diagnosis of M. bovis infection in ferrets was made from bacterial culture of the pooled lymph-node samples for each animal, whatever its apparent disease status. There is an unknown period between infection and positive diagnosis based on the mycobacterial culture of pooled lymph nodes. However, because of the high sensitivity of modern mycobacterial culture techniques, and the collection of the entire lymph nodes considered to be the sites of predilection, this period can be considered negligible (G. de Lisle, unpublished data).

ESTIMATING FERRET AGE

Each winter, ferrets lay down a dense cementum layer that becomes visible as a distinct band from early spring onwards. Ferret age was initially estimated to the nearest year by counting cementum annuli in sections of a lower canine tooth (Grue & Jensen 1979). The age of each animal was then calculated to the nearest month, from the date of capture and seasonality of breeding, with all ferrets assumed to have been born on 30 October. This date was arrived at by estimating the median birth date of juveniles caught during February trapping sessions using the growth curve for European polecats Mustela putorius (Shump & Shump 1978).

MODEL SPECIFICATION

Mathematical models were used to represent the various hypotheses of disease transmission among ferrets, and model fit used as a method of choosing the best hypothesis (or best working model) (Burnham & Anderson 1998). For each hypothesis, we consider candidate mathematical models first with, and then without, disease-induced mortality (α). Few data exist on the disease-induced mortality rate of M. bovis infection in feral ferrets. Lugton et al. (1997) document a radio-collared feral ferret surviving at least 1 year with tuberculosis infection, and suggest that the time of survival after infection probably ranges from several months in a few cases, to in excess of a year in many cases. For badgers, the most closely related species (also Family Mustelidae) for which data are available, disease-induced mortality arising from M. bovis infection is moderate, although highly variable (Wilkinson et al. 2000).

For α equal to zero, H1 may be modelled by the exponential model (Lee 1992), by only allowing transmission during the suckling period (s) (hazard function 1 and model 1.1; Table 1). For α non-zero, H1 may be modelled based on the model of Cohen (1973; see below) (model 1.2; Table 1).

Table 1.  Details of each hazard function (hazard) in terms of the age-specific force of infection (λ(a)) for various age classes (age), and the age-specific disease-prevalence model without (α = 0) and with (α > 0) disease-induced mortality. The suckling period is s, and the guarantee time g. Model numbers are given to the right of brackets Thumbnail image of

Hypotheses H2, H3, H4 and H5 may be modelled by the exponential model, modified to allow for a period when ferrets are not exposed to infection, here termed g (hazard function 2; Table 1). This is analogous to the concept of a guarantee time in survival analysis (Lee 1992). In epidemiological studies it commonly arises when individuals are protected from disease for a period after birth due to the presence of maternal antibodies (for mycobacterial infections such as M. bovis, immunity is cell-mediated only, hence there is no maternally derived immunity). The value of g was set to specify each relevant hypothesis (10, 2·5, 1·75 or 0 months for H2, H3, H4 and H5, respectively). For α equal to zero, the age-prevalence solution is model 2.1 (Table 1). For non-zero α, the age-specific prevalence for hypotheses H2–H5 can be obtained from the solution of Cohen (1973), although modified as before to include the term g and omitting the disease latent period term (model 2.2; Table 1).

To represent the candidate hazard functions H6–H12 (Fig. 1), the hazard function needs be able to take different values (not just 0 or λ) for anything up to 3 age classes. For hypotheses with a single step in the hazard function at g1 (H7–H9) this is represented by hazard function 3 (Table 1). For α equal to zero, the age-specific prevalence for H7–H9 is modelled as model 3.1 (Table 1). For non-zero α, the resulting age-specific prevalence for hypotheses H7–H9 can be obtained from the solution below (model 4.2) with g1 set to zero.

Hypotheses H6, H10, H11 and H12, which have two steps in the hazard function (say at g1 and g2), are modelled by hazard function 4 (Table 1). For α = 0, the age-specific prevalence for H6 (setting λ1 = 0), H10, H11 and H12 (setting λ2 = 0) is model 4.1 (Table 1). For non-zero α, there are considerable complications in finding solutions of the age-specific prevalence. For reasons made clear in the Results, solutions with a non-zero force of infection up until the age of weaning were not needed. For the piece-wise constant exponential model with λ1 = 0 (H6), the age-specific prevalence including disease-induced mortality (G. Fulford, personal communication) is given by model 4.2 (Table 1).

As an alternative to the piece-wise smooth exponential models being used to account for λ varying with age, the exponential models can be generalized to the Weibull model (Lee 1992). The Weibull model contains an additional shape parameter γ, with λ now termed a scale parameter. Lambda, now age-dependent and including a guarantee time, g, is given by hazard function 5 (Table 1), and increases with age when γ > 1 and decreases with age when γ < 1; hence the Weibull hazard may model an increasing, decreasing or constant λ. The age-specific solution is given by model 5 (Table 1). Setting γ equal to 1 simplifies the hazard function to the exponential case [λ(a) = λ]. The flexibility of the Weibull model, modified to include a guarantee time, enables it to represent broadly all the hypotheses except those that are U-shaped (H11 and H12). It does not, however, represent any of the hypotheses explicitly. There is no explicit solution for the Weibull model with a non-zero disease-induced mortality rate (G. Fulford, personal communication).

Finally, we fitted the polynomial hazard function (of order k) (model 6; Table 1) of Grenfell & Anderson (1985), which has the flexibility to fit many shaped curves, including those that the Weibull model is unable to fit. Grenfell & Anderson (1985) allowed λ(a) to be zero below a lower threshold age, which is analogous to the guarantee time used in this study. As for the Weibull model, analytical solutions for the polynomial models including disease-induced mortality do not exist (G. Fulford, personal communication), except for the case with k = 0 (equivalent to the model 2 version of the exponential model).

MODEL FITTING

All models were fitted by maximum likelihood. The likelihood (L) to be maximized was the binomial likelihood (equation 1), where pi is the modelled probability of infection, and yi is the number of M. bovis-infected individuals out of a total ni in each age class i (m in total).

  • image(eqn 1)

Maximizing L was achieved by numerically minimizing the negative log-likelihood [ln(L)] with respect to inline image (a separate estimate for each site), gender effect (single multiplicative factor) and inline image (if applicable), after substituting for pi from the relevant model. The gender effect acted only on the elevation (cf. slope) of the hazard function in question. The slope was considered a constant for all sites and sexes (e.g. γ held constant for Weibull models, and sex and site factors multiplied on b0 only for the polynomial hazard model). When undertaking numerical minimization, biological ( inline image and inline image were constrained to be positive for all models) and hypothesis-generated bounds were placed on the values for parameters. Hypothesis-generated bounds were: H6, inline image2 ≤ inline image3; H7, inline image1 ≥ inline image2; H8, inline image1 ≤ inline image2; H9, inline image1 ≤ inline image2; H10, inline image1≤ inline image2 ≤ inline image3; H11, inline image1 ≥ inline image2 ≤ inline image3. S-PLUS (Data Analysis Products Division, MathSoft, Seattle, WA) was used for numerical minimization.

MODEL SELECTION

Akaike’s information criterion corrected for sample size (AICc) (Burnham & Anderson 1998) was used to compare models. Burnham & Anderson (2001) suggest that models having ΔAICc (difference in AICc scores) within 1–2 of the best model have substantial support. Models within about 4–7 of the best model have considerably less support, while models with ΔAICc > 10 have essentially no support. Plots of Pearson residuals (Collett 1991) were used to assess model fit further. For the chosen model, confidence intervals for inline image were calculated by profile likelihood (McCallum 2000). Confidence intervals for estimates of λ were not estimated here. Rather, model 2.1 was used to test for the relative differences between sites and sexes in inline image, as this model can be fitted as a generalized linear model (GLM), making estimates of standard errors for parameters relatively straightforward (see below).

As this study aimed to estimate the absolute rate at which ferrets encounter M. bovis infection, Cox’s proportional hazards model (Cox 1972) was not considered, despite its popularity for many epidemiological investigations. Cox’s model is primarily concerned with estimating the proportional effects of different factors on the hazard rate, rather than the baseline hazard function, which in the current study is the variable of intrinsic interest.

HYPOTHESIS TESTING FOR EFFECTS OF SEX AND SITE

The Weibull model may be linearized into the form of a GLM (equation 2), and this provides a convenient method for testing the relative effects of gender and site on λ. The simplest exponential model (model 2.1) is nested within this model by setting γ to 1.

  • ln(−ln(1 − p(a))) = lnλ + γ ln(a − g)(eqn 2)

Equation 2 was fitted to the data using the computer software package glim4 (Francis, Green & Payne 1993). Mycobacterium bovis prevalence data were classified by sex, site and age. The error structure of p(a) was specified as binomial, with the response variable the number of animals infected, and the binomial denominator the total number of individuals in that age group. The link function was specified as complementary log-log. The term ln(a – g) was fitted as an offset (Collett 1991). The significance of sex and site on the force of M. bovis infection was assessed using the deletion test (Crawley 1993), which assesses the change in model deviance (Collett 1991) arising from the removal of a parameter from the model. The interaction between sex and site was not examined, as there was no a priori reason for doing so. The adequacy of the fit of the chosen model was examined by testing the significance of the residual model deviance (Collett 1991).

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

COMPARISON OF CANDIDATE AGE-SPECIFIC PREVALENCE MODELS

H4 (dietary-related transmission), as represented by model 2.2 (exponential model including disease-induced mortality with g = 1·75 months), had the lowest AICc of all the models fitted (Tables 2 and 3). It also had the highest likelihood (best fit to the data). As neither the Weibull nor the polynomial hazard models represent any one hypothesis explicitly, their ranks were not calculated; however, their ΔAICc scores are shown in Table 3. Great difficulty was had in finding starting parameters for the polynomial model that led to the likelihood function convergence for k greater than or equal to 2. As the AICc scores all worsened (got larger) going from k = 0 to k = 1, the process was abandoned for k = 2.

Table 2.  Akaike’s information criterion (AICc) scores and differences in AICc (ΔAICc) scores of candidate hypotheses for the transmission of Mycobacterium bovis infection to feral ferrets, as represented by various models fitted to age-specific M. bovis infection prevalence data. Steps in the hazard functions are given by g1 and g2. Disease-induced mortality rate =α. All models have sex and site fitted as factors (assumed multiplicative)
HypothesisModelg1 or s (months)g2 (months)αAICcΔAICcRank
  • *

    inline image1 = inline image2 (i.e. hit bound);

  • inline image2 = inline image3 (i.e. hit bound);

  • inline image1 = inline image2 = inline image3; NF, not fitted.

H11·1s = 1·75   0 154·4 51·422
 1·2s = 1·75≥ 0NF
H22·110   01987·21884·223
 2·210≥ 01987·41887·424
H32·1 2·5   0 142·1 39·121
 2·2 2·5≥ 0 138·2 35·219
H42·1 1·75   0 103·6  0·6 2
 2·2 1·75≥ 0 103  0 1
H52·1 0   0 105·9  2·9 7
 2·2 0≥ 0 108·3  5·310
H64·1 1·7510   0 105·6  2·6 3
 4·2 1·7510≥ 0 105·7  2·7 4
 4·2 2·510≥ 0 140·9 37·920
H73·1 0 1·75   0 108·5  5·512
 4·2 0 1·75≥ 0 111·0  815
H83·1 0 1·75   0 105·8  2·8 5
 3·1 0 2·5   0 106·6  3·6 9
 4·2 0 1·75≥ 0 106  3 8
 4·2 0 2·5≥ 0 105·8  2·8 5
H93·1 010   0 108·5  5·512
 4·2 010≥ 0 111·0  815
H104·1 1·7510   0 108·4*  5·411
 4·1 2·510   0 109·2*  6·214
H114·1 1·7510   0 111·2  8·217
 4·2 1·7510≥ 0NF
H124·1 1·7510   0 132·8 29·818
 4·2 1·7510= 0NF
Table 3.  Akaike’s information criterion (AICc) scores and differences in AICc (ΔAICc) scores for the Weibull and polynomial hazard models (model 5 and model 6) with respect to the best-performing exponential model (model 2.2)
Modelg1AICcΔAICc
Weibull 0 108·2  5·2
  1·75 103·7  0·7
  2·5 138·7 35·7
 101985·81882·8
Polynomial (k = 1) 0 108·3  5·3
  1·75 106·1  3·1
  2·5 144·6 41·6
 101989·51886·5

Two additional models have substantial support: H4, as represented by model 2.1 (exponential with no disease-induced mortality and g = 1·75 months) (ΔAICc= 0·6; Table 2), and the Weibull model (model 5 with g = 1·75 months) (ΔAICc= 0·7; Table 3). This Weibull model estimated γ to be 0·94, which equates with inline image decreasing slightly with increasing age. This is possibly an artefact of disease-induced mortality causing a lower prevalence for a given age than expected if γ was equal to unity. The hypothesis that matched this best performing Weibull hazard function was H4. Two other hypotheses (H6 and H8), as represented by models, had ΔAICc < 3. However, in both cases there was either no support for a step (increase) in the hazard function (H6, inline image2 = inline image3) or, in the case of H8, inline image in the second time interval (a > g) was 10-fold greater than that in the first time interval (a ≤ g). So essentially both hypotheses were trying to ‘emulate’ H4 (hence with similar likelihoods) and the observed difference (c. 2) in the AICc score was simply caused by the addition of a single parameter ‘penalizing’ the AICc score by approximately twice the number of additional parameters (one in this case), as opposed to any difference in the statistical likelihood. Hence it can be concluded that H4 has considerably more support than the other candidate models.

Model 2.2 estimated α to be 1·4 year−1 (95% CI –1·1–4·4 year−1), the wide confidence interval (including biologically unrealistic negative numbers) showing that the likelihood function with respect to inline image must be very flat indeed. Note that α is an instantaneous rate so can be greater than 1. Indeed, H4 as represented by either model 2.1 or model 2.2 appeared to fit the data well over the range of ages sampled, with the residuals reasonably evenly spread when plotted against ferret age (Fig. 3). This demonstrates that the hazard function of this simple model has captured the key components of the disease-transmission processes that shape the age-specific prevalence of disease.

image

Figure 3. Pearson residuals for (a) model 2.1 with g = 1·75 months and α = 0 year−1; and (b) model 2.2 with g = 1·75 months and inline image = 1·4 year−1, plotted against age of ferrets.

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FORCE OF INFECTION

Models 2.1 and 2.2 differed in their estimates of λ. The effect of ignoring disease-induced mortality was to lower inline image significantly for model 2.1 relative to model 2.2, with this discrepancy becoming more pronounced as the estimated force of infection decreased. Despite the wide confidence interval around inline image, we choose H4 as represented by model 2.2 as a working model for estimating the force of M. bovis infection in ferrets, as it seems most biologically plausible that some disease-induced mortality should occur. For this model, inline image in males was 2·2 times that in females. Ferrets at Castlepoint encountered M. bovis infection at about six times the rate of ferrets at Scargill Valley, and about 40 times the rate of ferrets at Lake Ohau (Table 4), resulting in a major difference in the age-specific disease prevalence (Fig. 4).

Table 4.  The estimated force of Mycobacterium bovis infection ( inline image) in feral ferrets from five sites in New Zealand, as determined from modelling age-specific disease prevalence using a modified exponential model including disease-induced mortality at 1·4 year−1 and a guarantee time of 1·75 months (model 2.2, see text for details)
SiteSexNo. examinedNo. infectedinline image(year−1)
Lake OhauMale 57 30·19
 Female 54 20·09
 Sexes combined111 50·14
Scargill ValleyMale 37 51·40
 Female 39 80·65
 Sexes combined 76131·02
Cape PalliserMale 15112·69
 Female 23101·24
 Sexes combined 38211·97
CastlepointMale 27217·90
 Female 21103·65
 Sexes combined 48315·77
Awatere ValleyMale 24164·64
 Female 22122·15
 Sexes combined 46283·40
image

Figure 4. Observed age-specific prevalence of Mycobacterium bovis infection in ferrets from (a) Castlepoint (•) and Lake Ohau (▴); (b) Cape Palliser; (c) Awatere Valley; and (d) Scargill Valley. Fitted lines are for the exponential model including disease-induced mortality (model 2.2) using the mean estimates of inline imagefor males and females from Table 4. Data are pooled over age classes (cf. Figure 3) for illustrative purposes; for an assessment of model fit see text and Fig. 3.

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TESTING SITE AND GENDER EFFECTS ON FORCE OF INFECTION

Because of its far superior ease for undertaking hypothesis testing, H4 as represented by model 2.1 was used to test formally for differences (as opposed to estimating differences) between the sites and sexes. This model explained 74% of the overall binomial deviance of the data, and the model fit was very good (χ2 = 51·0, d.f. = 53, P = 0·55), which was not surprising given the even spread of residuals (Fig. 3a). Lambda was significantly higher for males compared with female ferrets (χ2 = 8·5, d.f. = 1, P < 0·001) and differed significantly among sites (χ2 = 153·7, d.f. = 4, P < 0·001) (Table 5).

Table 5.  Estimates of the effect of sex and site on the natural logarithm of the yearly force of Mycobacterium bovis infection in feral ferret populations estimated using model 2.1 (see text for details). The estimate of the intercept is the value of the natural logarithm of inline image for male ferrets from the Scargill Valley site. Each additional parameter estimate represents its added contribution to the estimate of loge( inline image) for the intercept
ParameterEstimateStandard errort*P
  • *

    The observed t-values are distributed with infinite degrees of freedom (Z-scores).

  • Significance of the difference of each parameter estimate from that of the intercept (sex = male, site = Scargill Valley).

Intercept (Male, Scargill Valley)  0·0870·1
Female−1·00·26−3·8< 0·001
Castlepoint  2·00·38  5·2< 0·001
Awatere Valley  1·30·38  3·6< 0·001
Cape Palliser  0·50·39  1·3   0·10
Lake Ohau−2·50·53−4·6< 0·001

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

The use of age-prevalence curves for estimating the rate of disease spread in vertebrate wildlife has been limited. Anderson & Trewhella (1985) estimated the force of M. bovis infection in badger populations by fitting an age-structured version of a deterministic model to prevalence data. Dobson & Meagher (1996) presented age-specific prevalence data for Brucella abortus infection in bison Bison bison Lin. 1758, although they did not use it to estimate any parameters. They did, however, make some qualitative inferences from the data, concluding that the absence of any tendency for the prevalence to increase with age suggests that there is no significant mortality associated with brucellosis infection in bison. This is not strictly correct, as disease-induced mortality may only cause a plateau in prevalence as the animals age, as inferred by Anderson & Trewhella (1985). The general lack of use of age-prevalence data for studying disease in wildlife is in contrast to the study of disease incidence in humans, where analysis of age-prevalence data is used routinely (Farrington 1990). The difficulties in obtaining adequate sample sizes in studies of disease in wildlife is one reason for this disparity, along with difficulties in the diagnosis of infection and ageing animals accurately. This study has demonstrated that estimating the force of infection from age-prevalence data is possible, and this can assist in discriminating between alternative hypotheses about routes of disease transmission.

The model-selection exercise undertaken here has identified that the hazard function underlying the observed age-specific prevalence of M. bovis infection in ferrets is adequately modelled by a constant λ from the age of weaning, supporting H4 (dietary-related transmission). Other candidate hypotheses, of which there were a reasonably exhaustive number representing all hypothesized or combinations of hypothesized transmission mechanisms, were unsupported in comparison. Notably, the data gave no support for transmission occurring in the suckling period before weaning (H1). Neither did the data support an increase in λ once ferrets became socially independent (H3), sexually mature (H2), nor λ being a constant from birth due to environmental contamination (H5). Hence it is concluded that transmission during mating, suckling and routine social activities must be insignificant compared with dietary-related transmission, in agreement with the observations of Lugton et al. (1997).

Utilizing a model-selection approach can increase knowledge of the underlying processes of disease transmission, knowledge that otherwise would not be obtained by fitting a single model only. The similar AICc scores of the exponential model used to represent H4 (dietary-related transmission) with (model 2.2) and without (model 2.1) disease-induced mortality provide an example of the difference between model selection as opposed to hypothesis testing of parameter estimates. The similar AICc scores cannot be taken to mean that there is no disease-induced mortality arising from M. bovis infection (especially when the parameter estimate is large). Rather, they can be interpreted as meaning that the age-specific prevalence of M. bovis infection in ferrets can be adequately modelled using either model. Whether or not disease-induced mortality occurs would be considered an estimation rather than a model-selection problem (Burnham & Anderson 1998). Because of its ease of fitting within a statistical framework, model 2.1 provides a logical starting point for comparing the incidence of M. bovis infection in feral ferret populations, although the estimates are probably biased downwards. Model 2.2, however, probably provides more accurate estimates of λ, and hence is more useful for obtaining parameter estimates needed for subsequent modelling of disease transmission. There is a need to obtain independently a more reliable estimate of the rate of disease-induced mortality caused by M. bovis infection in ferrets.

A dietary-related working hypothesis for M. bovis transmission to ferrets must account for λ to be twofold higher in males than females. Possible causes that remain consistent with dietary-related transmission supported by H4 include dietary composition (male ferrets being more prone to scavenge tuberculous carcasses than females), immunological (males being more susceptible to becoming infected) and ecological (larger male home range having a greater probability of including a source of M. bovis) reasons (Lugton et al. 1997). Ragg (1998a) reported no intraspecific differences in diet in the species postulated to be the main source of infection for ferrets, making the dietary composition hypothesis unlikely. However, while no intersexual differences may exist in the composition of ferret diet, due to their pronounced sexual dimorphism (Lavers & Clapperton 1990; male mean weight = 1187 g, female mean weight = 627 g), male ferrets need to consume significantly more food than females, and hence could be exposed to a greater risk of encountering M. bovis-infected carcasses simply through greater dietary intake. Gender differences in the susceptibility of ferrets to M. bovis infection have not been evaluated; however, in other species there appear to be gender differences in susceptibility. For example, male badgers appear more susceptible than females to disease progression and have a higher rate of disease-induced mortality (Wilkinson et al. 2000). Hence the immunological hypothesis should not be ruled out. Home ranges of male ferrets are consistently larger than females, and as the distribution of M. bovis-infected possums is typically highly spatially aggregated (Caley 1996), differences in home range size could elevate λ in males.

Our study has identified the consumption of tuberculous carrion/prey as the most strongly supported hypothesis for the transmission of M. bovis infection to feral ferrets, and has identified a suitable model for estimating the force of M. bovis infection in feral ferrets. However, we have not identified what the source of this infection is. As well as accounting for the difference in λ between the sexes, a dietary-related working hypothesis for M. bovis transmission to ferrets must also allow for λ to differ by an order of magnitude between sites. Although the diet of ferrets consists mainly of lagomorphs (Ragg 1998a), they also scavenge extensively, and will readily eat possum and ferret carcasses (Ragg, Mackintosch & Moller 2000). Mycobacterium bovis infection has been recorded, although at a very low prevalence, in common prey items of ferrets, including rabbit (Gill & Jackson 1993), hare (Cooke, Jackson & Coleman 1993) hedgehog (Lugton, Johnstone & Morris 1995), and of course ferrets themselves. For all these species other than for ferrets, M. bovis-infected possums are considered the underlying reservoir of infection. The highest λ occurred at the sites with the highest density of possums, hence the hypothesis that M. bovis infection in ferrets is simply a spill-over from possum populations is an obvious candidate hypothesis for critical testing. However, the hypothesis of dietary-related transmission is not inconsistent with ferret-to-ferret transmission through ferrets scavenging on M. bovis-infected ferret carcasses. If this occurs at a high enough rate, it could enable M. bovis to cycle independently in ferret populations, irrespective of the contribution from possums. The critical experiments required to answer this are underway, and involve manipulating the density of M. bovis-infected possums and/or ferrets, to examine their influence on the force of M. bovis infection in feral ferrets, estimated using the methods presented here.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Funding was provided by the Foundation for Research, Science and Technology (contracts C09301, C09603 and C09801) and the Animal Health Board (Projects 44-90 and R-10481). Assistance with trapping was given by Scott Akins-Sellar, Warwick Baldwin, Chris Bee, Chris Brausch, Nyree Fea, Richard Heyward, Darren Lindsay, Gary McElrea, Lisa McElrea, Gert Vermeer, Ivor Yockney and Jim Young. Glenn Fulford generously provided an analytical solution for model 4.2 and commented on a draft manuscript that was further improved by the comments of Graham Smith and an anonymous referee. Bacteriology was undertaken by Gary Yates and staff from AgResearch (Wallaceville). Lisa McElrea sectioned ferret teeth.

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  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
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