## Introduction

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.