Ethics statement

Badgers were trapped as part of a wider ongoing study at Woodchester Park under a Natural England Science and Conservation licence [11, 32]. Clinical samples were collected by experienced personnel holding relevant Personal Home Office licences and acting under a Home Office Project licence. All animals were examined and sampled under the supervision of a Named Veterinary Surgeon and a Named Animal Care and Welfare Officer. All work with live animals at Woodchester Park is subject to approval and periodic review by the Animal and Plant Health Agency Animal Welfare and Ethical Review Board.

Data

The data are collected and curated by the Animal and Plant Health Agency on behalf of the Department for the Environment, Food and Rural Affairs, and are described in more detail in Section A of S1 Appendix. Since there are some heterogeneities in trapping effort with respect to different social groups over time, we define a monitoring period for each social group as a range of dates during which all the setts within certain social groups would have been checked for activity and trapped if signs of activity were found. We restrict attention to a core set of 34 social groups that were monitored from 1980 onwards. As such, the full dataset contained 2, 751 individual badgers, but we removed 360 individuals (13%). Of these 165 (6%) were never captured in any of the core social groups, 63 (2%) were initially captured in a non-core social group and moved to a core social group afterwards, and 132 (4.8%) were first captured in undefined social groups. From the social group monitoring periods we could therefore define a monitoring period for each animal, which we assume ends if the animal leaves the core area. We conduct inference on the epidemiological states of the animals only in their corresponding monitoring periods. However, trapping events occurring outside of the monitoring period and core groups are still used in the mortality component of the model.

Model

We model transmission through a discrete-time compartmental model, defined over quarterly periods, where at any point in time individuals are either susceptible-to-infection (S), infected but not yet infectious (i.e. ‘exposed’—E), infected and infectious (I) or dead (D). This structure is similar to other models of bTB infection in animal hosts [7, 33, 34], except that since the Woodchester Park badger population is not artificially managed—and the study period extends over several generations of badgers—we also incorporate a natural mortality process that models individuals moving to the D state from any other state. Since the observed data consist of the trapping effort, the capture history and diagnostic test results for individual badgers, the underlying epidemiological states are not directly observed. Furthermore, we must include an additional stochastic observation process that relates the hidden states to the observed data. This is important because the capture process and diagnostic test accuracy are both imperfect. As such this model can be considered a coupled hidden Markov model, governed by transition probabilities between the hidden states and the probabilities of observing the data given the hidden states.

Individuals born during the study period are assumed to be susceptible at birth, and those born before the monitoring period begins in any social group are given a non-zero probability of being in either the S, E or I states once monitoring commenced. As is usual with discrete-time models, we assume that the state transition probabilities are fixed in each time interval [t − 1, t), that events are conditionally independent given the transition probabilities, and that states are updated at the end of each time period.

Once monitoring has started we assume there are two sources of infection—a background infectious pressure (corresponding to indirect sources-of-infection and infection from other sources such as cattle), and direct badger-to-badger transmission. Environmental contamination (e.g. from bacteria shed in urine and faeces) may be a major source of bTB infection in badgers and cattle [35], and studies have shown that external sources of infection can contribute to bTB persistence in badger populations [36]. Several studies analyse the spatial organisation of badger populations and its implications for disease transmission at different spatial scales [32, 37–41]. As such we consider a set of constant social-group specific background rates-of-infection, but do not consider a direct between-social group badger-to-badger transmission process, outside of the explicit movements of individual animals between groups captured in the data. Therefore we model the probability of infection (S → E) for a susceptible individual i in the period [t − 1, t) as (1) where g_i(t−1) is the social group that individual i belongs to in [t − 1, t), is the background rate of infection of group g_i(t−1), β is the transmission rate, with and the number of infectious individuals and the total number of individuals respectively in g_i(t−1). The parameter q measures the strength of frequency dependence [33, 42], and K is a rescaling constant that makes the units of β independent of q [42]. The value of K that we used is the median social group size across all social groups and all time. We assume that the social group structure is known [43], and that each badger had only moved to a new social group from the first time it was captured there.

The probability of an infected individual becoming infectious (E → I) in the period [t − 1, t) is given by (2) where τ is the mean length of the latent period.

The mortality process in any mammal population is age-dependent, and conceptually a bathtub-shaped hazard function would be appropriate [44], where high early-life mortality risk drops to a constant level during mid-life and then increases exponentially in later-life. In wild badgers early-life mortality is often hidden, since deaths in newborns occur underground and so are never observed [13], so instead we model the conditional mortality distribution given survival to first capture as a Gompertz-Makeham distribution [45, 46]. Therefore the conditional probability of an individual i dying in the period [t − 1, t), given they are alive at t − 1, is (3) where is the age of individual i at time t, and (4) is the continuous-time Gompertz-Makeham survival function. Thus the mortality function is age-dependent, but since the birth times of badgers are assumed known then the mortality process is still Markovian.

Therefore to model the observation process given the hidden states we assume that badgers surviving this period of early-life mortality can be captured according to seasonally-varying (quarterly) capture probabilities given that they are alive when trapping occurs [12]. Since carcasses of dead animals are rarely recovered, mortality information is typically right-censored at the last capture time for most badgers (this can be treated as interval-censored if a carcass is recovered). The capture process is thus very important for identifying the mortality parameters, since despite rarely recovering dead badgers the probability that an animal is alive but evades capture at successive time points quickly becomes negligible unless the capture probabilities are very small [13].

We assume that the different diagnostic tests are conditionally independent given the infection status of each animal, and thus the observation process can be governed by the sensitivities and specificities of the different tests, which we wish to estimate. This work extends ideas in [12] to incorporate a fully mechanistic model of disease transmission and progression, meaning that we are able to infer key quantities-of-interest that result from the mechanistic model, such as individual effective reproduction numbers. Since there is no recovered state in this system, the mortality process limits the length of the infectious periods for individual badgers. We also fit to a much longer time period (1980–2020); incorporating two additional types of test (Brock and DPP) and we integrated a change-point for the Brock test sensitivities and specificities (based on finding some evidence of model mis-specification when this was not included—the justification is described in more detail in the Results section).

More comprehensive mathematical details of the model and parameters are given in Section B of S1 Appendix. We used vague or weakly informative prior distributions for the parameters (see Section B3 in S1 Appendix).

Here the iFFBS algorithm acts as a Gibbs sampler over the hidden states for each individual, conditioned on the states of all other individuals. As such it can be readily embedded within a data-augmented Markov chain Monte Carlo (MCMC) routine, where it provides an efficient and generalisable way to sample the hidden states of a CHMM. These update steps only require transition and observation probabilities to be specified, and so could also be a good candidate for embedding in general-purpose inference software in the future. Standard FFBS approaches scale exponentially in the number of individuals, but the iFFBS algorithm (in certain situations) can be made to scale linearly with the number of individuals, substantially reducing computational overheads. It is also possible to sample semi-Markov processes through the inclusion of Metropolis-Hastings steps, though this impacts the efficiency of the algorithm. For full details of the method, see [31]. For updating the parameters in the MCMC, we employed a combination of Hamiltonian Monte Carlo, Gibbs Sampling and Metropolis-Hastings updates. For full details, see Section D in S1 Appendix.

Posterior predictive samples for the test results from individual badgers can be easily obtained and aggregated to any spatio-temporal level, enabling a visual assessment of model fit against the observed data. Similarly, posterior distributions for the relative contribution of direct badger-to-badger transmission on any new infection can be calculated through the ratio of the badger-to-badger transmission rate over the total transmission rate in the corresponding social group at each given infection time, which can again be aggregated to any desired spatio-temporal resolution (see Section F in S1 Appendix).

To quantify superspreading, we estimate individual effective reproduction numbers, R_i, from which a population-level effective reproduction number, R, can be derived [16–18]. We define R_i to be the expected number of secondary infections caused by an infected individual i across its lifetime. [16] derive a likelihood-based estimator for the individual effective reproduction number, R_i, for an individual i as: (5) where p_{(k, i)} is the relative likelihood of individual k being infected by individual i, with and the infection times for individuals k and i respectively. Hence R_i is the sum of these relative likelihoods over all possible future infections . In [16], an estimator for p_{(k, i)} was derived based on the generation-time distribution and a fixed estimate for . The generation-time interval is non-trivial to define for our model, since the infectious period is governed by the age-dependent mortality distribution, and we don’t have a simple proxy for . However, since the iFFBS algorithm generates random samples for the unobserved event times at each iteration of the MCMC algorithm, we can condition on these directly to calculate (6) and thus generate posterior samples of R_i based on (5). Taken over the whole MCMC chain, these can be used to estimate the posterior distribution for R_i which numerically integrates over the relevant hidden states. In addition, (6) adjusts for the fact that we have additional sources of infectious pressure through the background rates-of-infection, and therefore our estimates of R_i should account for this.

There are various ways in which population-level effective reproduction numbers can be defined, see for example [16–18], however here we do something more akin to the approach of [16] and obtain a population-level estimate by simply averaging across individuals, giving: (7) where R_i is defined in (5) and is the number of badgers that have died up to time t. We choose to calculate R_i only for animals that have died, and so each estimate adjusts for the latent and infectivity processes across the lifetime of each animal, and automatically accounts for the social-group specific background infection rates and changes in group sizes over time (and also any movements of badgers between social groups). For full derivations of these quantities, please see Section E in S1 Appendix.

Reproduction numbers

Posterior summaries for the individual- and population-level reproduction numbers (Fig 2) reveal substantial heterogeneity in reproduction numbers between individuals: some badgers have R_i ≈ 8, whereas most have R_i < 1. Despite some animals having high individual reproduction numbers, the population effective R is on average < 1 (Fig 2B) with a posterior mean estimate of 0.66 (95% credible interval [CI]: 0.59–0.72). To explore key factors that might contribute to the estimated heterogeneity in R_i, we plot the posterior mean R_i values against a set of other individual-level summary measures: the posterior mean infectious period, the average number of social groups a badger belonged to, and the average size of social groups a badger belonged to (Fig 3). Heterogeneity in the duration of the infectious period due to variations in infection time and survival seems to be the main (although not sole) driver of these differences (Fig 3A). Since R_i values do not associate strongly with the average number of social groups that a badger has been associated with, or with average group sizes (Fig 3B–3D), it is most likely that on average badgers infected at a younger age simply have more opportunity for onwards transmission. We also compare the posterior mean probability of infection against the posterior mean R_i values across individuals and show that superspreading is not just a function of the probability of infection (S1 Fig).

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Fig 2. Posterior summaries of reproduction numbers.

(A) Posterior means and credible intervals (50% and 95%) for the individual reproduction number for all individuals, ranked in increasing order of posterior means, and (B) posterior distribution for the population effective reproduction number.

https://doi.org/10.1371/journal.pcbi.1012592.g002

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Fig 3. Posterior mean estimates of individual reproduction numbers against other factors.

Plots show the posterior mean R_i against (A) posterior mean duration of the infectious period; (B) average number of social groups an individual belonged to; (C) average number of social groups an individual belonged to divided by the duration of the infectious period; (D) the average size of social groups that the individual belonged to.

https://doi.org/10.1371/journal.pcbi.1012592.g003

Hidden burden of infection

We also estimate the probability of infection/infectivity of any badger at any given time, and the corresponding event times (e.g. S2 Fig), accounting not only for the diagnostic test history for each animal, but also information from other badgers as captured through the transmission model. These estimates can be aggregated up to higher spatial levels, such as the social-group or population levels. For example, Fig 4A shows the predicted number of individuals in each epidemiological state across the whole population over the study period, giving an estimate of the overall hidden burden of infection [33]. Analogous plots for each social group are shown in S3 and S4 Figs. The model predicts relatively low levels of underlying infection in the population over the study period, however, there is a steep decline in population size from about 1993 onwards, with the proportion of infected badgers increasing towards the end of the study period. We predict that on average 72% (95% CI: 65%–78%) of new infections can be attributed to direct badger-to-badger transmission, with the rest attributed to the background rate-of-infection terms (Fig 4B). In addition to transmission arising from other species (e.g. cattle), the background infection risk can also encompass indirect transmission from badgers, such as from environmental contamination, and hence the 72% estimate represents a lower bound for the relative rates of all badger-to-badger transmission in WP.

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Fig 4. Posterior predictive summaries for aspects of the underlying epidemic.

(A) Posterior predictive means and 95% prediction intervals for the number of individuals in each epidemiological state over time; (B) posterior predictive means and credible intervals (50% and 95%) for the relative contribution of badger-to-badger transmission vs. background rates-of-infection on new infections over time (the solid black line is the average contribution over all time points, with the dashed lines giving the corresponding 95% credible interval); (C) the observed proportion of positive test results over time for each test.

https://doi.org/10.1371/journal.pcbi.1012592.g004

Epidemiological parameters

Posterior summaries for all epidemiological, CMR, mortality and diagnostic test parameters are shown in Table 1, and Figs D–E and Table A in S1 Appendix. We obtain an estimate of the average (discrete-time) latent period (the time between infection and infectiousness) of 3.7 (95% CI: 2.9–4.7) years. We also obtain estimates for the infectious period distribution, which is derived from the age-dependent mortality curves and the inferred infection times. To understand the transmission potential of individual badgers, we define an effective infectious period distribution as the time between when an animal becomes exposed (i.e. when they enter the E class) and when they die. We then define the infectious period as the time between when an animal becomes infectious (i.e. when they enter the I class) and when they die, here equivalent to the conditional survival distribution given survival beyond infectiousness. Since we simulate the time-of-infectiousness and the time-of-death for each animal, we can generate empirical estimates for these distributions that average over the unobserved event times (Fig 5). It is clear that a large proportion of badgers die before becoming infectious, and so have an effective infectious period of zero (Fig 5A); shown even more clearly if averaged over the population (Fig 5B). Considering the infectious period distribution conditional on being infectious, then we obtain a posterior mean of 1.4 (95% CI: 0.25–4) years averaged across the population (Fig 5C).

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Fig 5. Posterior summaries for the length of the infectious periods.

(A) Posterior means and credible intervals (50% and 95%) for the individual effective infectious period distribution across all individuals, ranked in increasing order of posterior means; (B) posterior distribution for the population-averaged effective infectious period distribution; (C) posterior distribution for the population-averaged infectious period distribution.

https://doi.org/10.1371/journal.pcbi.1012592.g005

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Table 1. Posterior means and 95% credible intervals for key model parameters.

Results given to 2 significant figures (4 s.f. for changepoint). For full details of the model structures and parameters, see Section B in S1 Appendix.

https://doi.org/10.1371/journal.pcbi.1012592.t001

Transmission can be considered as lying on a continuum between density- and frequency-dependent, captured here by a parameter q, which mediates the extent to which contact rates are governed by badger density [42]. We infer the posterior mean for q to be 0.59 (95% CI: 0.23–0.91), which provides evidence consistent with the transmission process being somewhere in-between frequency- and density-dependent (defined when q = 1 and q = 0 respectively; [33, 42]). Following [33], our choice of a U(0, 1) prior distribution stops q from being exactly 0 or 1, but the area of high posterior density is nevertheless away from either of these two extremes.

Mortality and capture processes

We estimate a median survival time of 3.6 (95% CI: 3.5–3.8) years, with posterior predictive plots of the survival and hazard functions shown in S5 Fig. We estimate seasonal differences in capture probabilities, ranging from 0.18 (95% CI:0.17–0.18) in the winter, to 0.46 (95% CI: 0.45–0.47) in the summer (Table 1).

Diagnostic test performance

We estimate diagnostic test sensitivities and specificities relative to infection, rather than to a proxy measure such as visible lesions [12, 33]. In the first version of our model we assumed that each of the five tests had constant sensitivity and specificity values over time. We obtained posterior mean estimates for the sensitivities (for infectious animals) of: Brock 0.88 (95% CI: 0.84–0.91), Culture 0.32 (0.29–0.35), StatPak 0.93 (0.89–0.96), γ-IFN 0.57 (0.53–0.62) and DPP 0.63 (0.53–0.73). The sensitivity of the γ-IFN test was much lower than expected from cattle studies [47], although previous estimates in badgers are more variable: [12] estimated a γ-IFN sensitivity of about 0.44, whereas [10] estimated a value of 0.799; although the former did not differentiate between latently infected and infectious badgers (which may drive the estimate to be lower if sensitivity is lower during the latent period), and the latter used a different method but had a more informative prior distribution with a median sensitivity of 0.796.

This original low sensitivity estimate for the γ-IFN test led us to explore other possible explanations. Plotting the proportion of test positive results over time for the five tests, we see that from about 2005 onwards the proportion of test-positives for the Brock test increases rapidly, whilst the other tests (notably culture) drop or remain constant (Fig 4C). The exception is the StatPak (an antibody test like the Brock), and similar inconsistencies with the other tests have previously been noted [43]. Consequently, we developed our current model, where the Brock test sensitivity and specificity were permitted to change at some changepoint. Following this modification the posterior distribution for the changepoint is estimated to be between the final quarter of 2004 and the first two quarters of 2005 (Table 1), consistent with the data in Fig 4C. Consequently the pre- and post-changepoint Brock test sensitivities (denoted Brock1 and Brock2 respectively) are both high (0.91 and 0.93), but the specificity of the Brock post-changepoint drops to 0.78 (from 0.98 pre-changepoint). The inclusion of the changepoint also results in the γ-IFN sensitivity estimate rising to about 0.85, which is more consistent with earlier literature (except [12] as discussed above). Model fits, parameter estimates and predictive plots for the initial (non-changepoint) model are shown in the Section G4 in S1 Appendix. We note that the model fit was not as good for this model (Fig F in S1 Appendix).

Conclusions and future work

To summarise, we have developed an individual-level stochastic meta-population compartmental model of bTB in badgers and fitted this to a dataset derived from an intensive long-term capture-mark-recapture study using an iFFBS algorithm. The model fitted in a few hours, enabling us to conduct efficient Bayesian inference simultaneously on all epidemiological parameters, as well as those relating to age-dependent mortality, the CMR surveillance process and diagnostic test performance.

The model provides novel estimates of individual effective reproduction numbers, suggesting that a small proportion of infected badgers, or superspreaders, contribute disproportionately to the transmission potential of the disease. This is despite the population-level effective reproduction number being low, suggesting that management strategies could be improved by targeting these animals or associated social groups with interventions or more intensive surveillance. Although our current model cannot directly identify animals with superspreading potential in advance, given systematic surveillance data the model can produce probabilistic predictions for the hidden epidemiological states of individual badgers over time, which can be aggregated to different spatio-temporal resolutions as required. These kinds of models could be used to help target surveillance or management strategies since we can estimate the likelihood of individuals or social groups harbouring hidden infection, and produce estimates of the expected infectious periods for infected badgers, accounting for the age-dependent mortality process and the infection times. These estimates are not just based on the diagnostic test history for individual animals, but also borrow weight from other animals through the transmission model structure. Of course few wildlife systems have systematic surveillance at the resolution of Woodchester Park, and since the predictive resolution of any compartmental infectious disease model is constrained by the resolution of the available data, the way these models might be employed in practice would have to be informed by the available data (for example, it may be feasible to estimate social-group specific infection risks, but not individual-level risks in more low resolution settings).

Nonetheless, if detailed data are available then the iFFBS algorithm has demonstrated its value in modelling such data efficiently, and could therefore be employed in other settings; for example the modelling of within-herd spread of bTB in livestock in the UK, where detailed surveillance data is collected. It also highlights additional benefits that could be generated if more detailed surveillance was employed in a specific wildlife setting. Nevertheless, there are various areas that could be developed further. For example, exploring more nuanced infectivity-profile dependent sensitivity and specificity processes, rather than the coarse step-functions that we currently employ could be insightful about disease progression [57]. It may also be valuable to include individual-level covariate dependencies in different components of the model, for example, to explore sex-specific differences in mortality [13]. Also of interest is investigating the effect of infection on mortality risk [13, 48], or sex-/age-specific differences in transmission/susceptibility. With suitable data it would be of interest to integrate different choices of spatial connectivity structure and/or environmental transmission, or to deal with multiple species [9, 34]. We also propose further exploration of the characteristics of superspreader badgers, which may inform operational targeting. The insights described above have been gleaned from a single plausible model, but a key area of future work is to extend this framework to allow for systematic model comparison and/or model averaging, which could provide even more robust predictions and help disentangle complex model interactions [30, 58].