Abstract
Contact-tracing data (CTD) collected from disease outbreaks has received relatively little attention in the epidemic modeling literature because it is thought to be unreliable: infection sources might be wrongly attributed, or data might be missing due to resource constraints in the questionnaire exercise. Nevertheless, these data might provide a rich source of information on the disease transmission rate. This paper presents a novel methodology for combining CTD with rate-based contact network data to improve posterior precision, and therefore predictive accuracy. We present an advancement in Bayesian inference for epidemics that assimilates these data and is robust to partial contact tracing. Using a simulation study based on the British poultry industry, we show how the presence of CTD improves posterior predictive accuracy and can directly inform a more effective control strategy.
1. Introduction
In a world in which people and animals move with increasing frequency and distance, authorities must respond to disease outbreaks at maximum efficiency according to economic, social, and political pressures. Field epidemiologists are typically faced with making decisions based on imperfect and heterogeneous data sources. European Economic Community (1992) Council Directive 92/119/EEC requires member states to identify “other holdings … which may have become infected or contaminated”; this is achieved through contact tracing, performed by the competent authority—for instance, Defra's Framework Plan for Exotic Animal Diseases (Defra, 2009). Though resource limitations might prevent the implementation of efficient case detection and removal based on contact-tracing data (CTD), the presence of CTD might provide a rich source of information, both on contact frequency and probability of infection given a contact. In a practical setting, however, rigorous collection of contact tracing information from all detected cases is difficult to achieve; more typically, it will be collected from as many individuals as resources allow. Therefore, for a given epidemic the amount of CTD available for analysis is not guaranteed in advance, the challenge for the field epidemiologist being how best to use these data to investigate disease dynamics, and inform disease control decisions.
This paper introduces a Bayesian approach to the assimilation of CTD into model-based inference and control for infectious diseases based on spatiotemporal case data. This can be applied to general classes of continuous-time epidemic models, and we provide illustration using a flexible epidemic modeling framework as described in O'Neill and Roberts (1999) and Jewell, Kypraios, Christley, and others (2009); Jewell, Kypraios, Neal, and others (2009). Our methodology is particularly useful since it automatically adapts to the information contained in CTD, over and above that contained in the adjunct timeseries.
In advance of an outbreak, network and spatiotemporal dynamical models of disease have become a standard for evaluating the likely effect of various control measures and are now routinely used for contingency planning. Using demographic data as well as case time series data, previous work has shown that real-time estimation of epidemic model parameters can potentially be highly effective in informing decision-making (Cauchemez and others, 2006; Jewell and others, 2009). In the early stages of an epidemic case data are often sparse, though this is when choosing the right course of control strategy is most critical (Anderson and May, 1991). Therefore, decisions must often be made in the face of considerable uncertainty. For such a statistical decision support tool, assimilating different sources of data are often helpful in maximizing statistical information, leading to more accurate model predictions and improved decision-making throughout the epidemic (Presanis and others, 2010). Nevertheless, incorporating diverse data sources into a tractable likelihood function commonly presents a methodological problem (see, e.g. Diggle and Elliott, 1995).
Much has been written on how contact tracing may be used to decrease the time between infection and detection (notification) during epidemics. However, this focuses on the theoretical aspects of how contact tracing efficiency is related to both epidemic dynamics and population structure. It has been shown that, providing the efficiency of following up any contacts to look for signs of disease is high, this is a highly effective method of slowing the spread of an epidemic, and finally containing it (see, e.g. Eames and Keeling, 2003; Kiss and others, 2005; Klinkenberg and others, 2006). In contrast, the use of CTD in inferring epidemic dynamics does not appear to have been well exploited. During the UK outbreak of foot and mouth disease in 2001, the Ministry of Agriculture, Farming, and Fisheries (now Defra) inferred a spatial risk kernel by assuming that the source of infection was correctly identified by the field investigators, giving an empirical estimate of the probability of infection as a function of distance (Ferguson and others, 2001; Keeling and others, 2001; Savill and others, 2006). Strikingly, this shows a high degree of similarity to spatial kernel estimates based on the statistical techniques of Diggle (2006) and Kypraios (2007) without using contact-tracing information. However, Cauchemez and others (2006) make the point that the analysis of imperfect CTD requires more complex statistical approaches, although they abandon contact-tracing information altogether in their analysis of the 2003 SARS epidemic in China. In human health, Blum and Tran (2010) devise a model that incorporates contact tracing for HIV–AIDS by dividing cases into those detected by random sampling or contact tracing. This allows them to estimate the proportion of undetected cases in the population at a particular observation time, although the construction of the model does not allow them to estimate the probability of infection given a contact.
Addressing this, we present a model which uses contact-tracing information between pairs of individuals for time periods when it is available and falls back on a Poisson process likelihood for time periods when it is not. We illustrate our methodology using an example from the UK agricultural industry.
In Section 2, we describe a motivating example of a potential outbreak of avian influenza in the British poultry industry. In Section 3, the existing approach to continuous time epidemic models is reviewed. Section 4 then presents our new class of embedded model for incorporating CTD. In Section 5, we develop a reversible jump Markov Chain Monte Carlo (MCMC) algorithm to estimate the posterior density, showing the feasibility of exact Bayesian computation for problems of this type. Section 6 then applies this methodology to the avian influenza example, showing how this methodology might be used in the event of an outbreak of high pathogenicity avian influenza.
2. Motivating example
One motivation for this paper stems from the example of a potential avian influenza outbreak in the British poultry industry (Jewell, Kypraios, Christley, and others, 2009), using data from Defra's Great Britain Poultry Register as well as contact network data (Dent and others, 2008; Sharkey and others, 2008). We identify three contact networks contained in the data with different levels of information. The company association network is “static”: an edge is present if and only if two poultry holding belong to the same production company. The feed mill delivery and slaughterhouse networks are “dynamic” meaning they contain edge frequency information—for example, we know not only that two holdings are connected by a feed mill delivery, but how often a feed lorry runs between them. Geographic coordinates for each holding allow us to consider non-network spatially related transmission. We also consider “background” infection sources, accounting for infections unexplained by the other transmission modes.
The presence of explicit contact networks within the poultry industry presents an interesting opportunity to investigate the possibility of including CTD into the SIR-type epidemic model (Kermack and McKendrick, 1927). Where contact frequency data are available, we can view the networks as dynamic with contacts occurring according to a Poisson process with intensity equal to the frequency along a directed edge of the network. Normally, this process is not observed, and we have only a mean contact intensity between individuals to use as covariate data in a statistical analysis. However, the collection of CTD presents the possibility for making inference from direct observations of the contact process, albeit for a select period of time leading up to a case detection.
In the typical livestock setting, CTD is gathered in response to the notification (i.e. case detection) of an infectious premises (IP). The resulting data are a list of contacts (with time, source, destination, and type) that have been made in and out of the IP during a period prior to the notification. The length of this period—the “contact tracing window” —during which contact tracing is gathered is stipulated by policy, and is typically longer than the expected infection to detection time for the disease in question.
Though somewhat idealized here, these data provide us a platform from which to develop the methodology required to make use of CTD for inference on epidemic models. In livestock epidemics, CTD is not routinely passed to the modeling community, and it is hoped that the results presented in this paper will encourage the authorities to do so.
3. A review of continuous time epidemic models
To begin, we provide an overview of the SINR epidemic model for heterogeneously mixing populations (see also Jewell and others 2009). The SIR model is extended by considering the population to be composed of individuals who, at any time t, exist in one of four states: susceptible, infected, notified, and removed. Progression through these states is assumed to be serial: individuals begin as susceptible, become infected, are notified (i.e. disease is detected), and are finally removed from the population (either by death, or by life-long immunity). The term “individual” refers to an epidemiologically discrete unit, which might be a person or animal, or might be of higher order, such as a household or farm (as is the case for our HPAI example). It is then assumed that individuals become infected via transmission modes k = 1,…, K which might be contact networks or spatial proximity to infected individuals. Independent “background” sources of infection are also included to account for infection sources other than those explicitly modeled.
Conditioning on the initial infective κ, individuals j are assumed to become infected according to a time-inhomogeneous Poisson process with instantaneous rate λj (t) equal to the sum of the infection rates across all transmission modes from all infected and notified individuals present in the population. Let 𝒮(t), ℐ(t), 𝒩(t), and ℛ(t) be the sets of susceptible, infected, notified, and removed individuals at time t, with the restriction that the entire population 𝒫 = 𝒮(t) ∪ ℐ(t) ∪ 𝒩(t) ∪ ℛ(t). Furthermore, let I, N, and R the corresponding vectors of individuals' infection, notification, and removal times. Infection times are, of course, never directly observed and we therefore treat ℐ as missing data. Since the aim of the methodology is to make inference on an epidemic in progress observed at time Tobs, infected individuals are partitioned into j:Nj > Tobs occult (i.e. undetected) infections with right censored notification and removal times, and j:Nj ≤ Tobs known (i.e. detected) infections. Given the disease transmission parameters θ = {ϵ, p, β, γ, ψ, η, ξ} (see below), the likelihood function (here denoted by LA (·)) for the epidemic process up to the analysis time Tobs is
In these examples, the function 0 ≤ q(i;ξ, t) ≤ 1 represents the infectivity of individual i at time t, and 0 ≤ s(j;η) ≤ 1 the susceptibility of individual j which is assumed constant. The rate of spatial (or environmental) transmission is given by Ψk (i,j;ψ) and would typically be a function of Euclidean distance between i and j. Network rates of transmission are represented by βkcij(k) and pkrij(k). The important distinction between the latter two terms is that cij(k) represents network associations (0 or 1) between i and j with associated infection rate βk, whereas rij(k) represents a vector of (potentially infectious) network contact rates with associated probability pk that a contact results in an infection.
We remark that, if contacts are assumed to occur according to some underlying Poisson process with rate rij(k), infections occur according to a thinned version of this Poisson process, the thinning being governed by pk. We assume that the contact rate is a priori independent of the probability of the infection being transmitted. Thus, a contact between infected individual i and susceptible j will not necessarily result in j being infected, just that there is some positive probability of infection being transmitted. Considering a putative full model containing several networks, the information contained in p therefore gives a direct measure of risk for each contact network in rij.
4. Incorporating CTD
CTD, 𝒞, represents a list of all known contacts, both incoming and outgoing, that have occurred between newly detected cases j and all other individuals during a contact-tracing window defined as the interval [Tjc, Nj). In other words, these data provide a means to observe the underlying contact process driving the infectious process as discussed in Section 3.
Let Cijk (t) be a right continuous counting process describing the number of contacts that have occurred between infected individuals i and j along network k up to time t, with ΔCijk (t) = limδ↓0 [Cijk (t) − Cijk (t − δ]. If it were possible to obtain CTD for all time, as well as perfectly observing individuals' infection states, then inference on p could be made using a geometric model with likelihood denoted by LΩ(·). For example,
The limitation to this approach is, of course, that CTD is available only for the contact tracing window and so the naive application of such a likelihood will give a biased estimate of p.
To address this, we divide the epidemic into periods of time for which CTD is observed, and periods for which it is not. For each individual j in the population, let 𝒲j be the set of times for which contact tracing is observed (i.e. the contact tracing window), and 𝒲jc the set of times for which it is not. Then, the σ-algebras ℱ = σ(ΔCijk (t);t ∈ 𝒲j) and 𝒢 = σ(Δ Cijk (t);t∈ 𝒲jc) represent the information contained in CTD, and that lost by not having CTD, respectively.
Theorem 1 Where CTD is available only for contact tracing windows 𝒲j, if the underlying contact rates rij(k), k = 1, …, K are known, the likelihood with respect to ℱ is obtained by taking an expectation with respect to G such that
See supplementary material A available at Biostatistics online for the proof.
Theorem 1 relies on knowing the underlying contact rate for a particular network. For associative and spatial transmission modes (and indeed background pressure), the contact rate between individuals, rij(k), is not known and these cases do not therefore fit into our contact-tracing framework. However, since the overall infection rate βk = pkrij(k), where it is assumed that rij(k) is constant with respect to both i and j, we can regard infections via these modes as occurring due to an unobserved contact process ( equation (3.3)). The likelihood function in equation (4.2) is therefore sufficiently flexible to include these transmission modes in the 𝒲jc related terms.
Finally, we highlight that the likelihood function is written in terms of the data I, N, and R, given the transmission parameters θ. However, the epidemic process means that infection times are never directly observed. We therefore treat I as missing data, using Bayesian data augmentation techniques described in the next section.
5. Bayesian inference
A Bayesian approach permits the incorporation of prior information early in the epidemic and provides a natural way to deal with I, the vector of censored infection times.
Independent priors are assigned to the model parameters θ = {ε, p, β, γ, ψ, η, ξ}. Typically, gamma distributions are used for rate parameters (i.e. ϵ, β, γ, ϕ) with beta distributions used for probability parameters (i.e. p), as demonstrated in Section 6.
The main features of the adaptive reversible jump MCMC algorithm used to fit the model are outlined here, for the full implementational details see supplementary material B available at Biostatistics online. An adaptive multisite update (Haario and others, 2001) is used for the disease transmission parameters θ, which avoids the necessity for time-consuming tuning runs. This is useful to speed up the algorithm usage in a real-time setting. The implementation of reversible jump (Green, 1995) is 2-fold. First, the dimension of I is allowed to increase or decrease according to either an addition or a deletion update step for occult infections from the parameter space (Jewell and others, 2009). This allows the algorithm to explore the possibility of occult infections present at the time of analysis, consistent with the model. Secondly, infection events (including occults) are attributed to either a traced contact related to the 𝒲j components of transmission, or spatial/untraced infections related to the 𝒲jc components. The switching in and out of the set 𝒲j comprises a reversible jump move, since it results in the infection appearing in different parts of the likelihood. The update step for an infection time therefore proposes the reversible jump with probability 0.5, and a contact-to-contact, or non-contact-to-non-contact move otherwise.
Shared memory parallel computing (GNU C ++ using OpenMP) is used to calculate the likelihood, ensuring that the algorithm runs in an overnight timeframe (Jewell and others, 2009).
6. Case study
To illustrate how this contact-tracing methodology might be used in practice, we return to the dataset presented in Section 2. First, we describe the special case of HPAI within the British poultry industry. Since no epidemic outbreak of HPAI has yet occurred in Britain, we provide two simulation studies to demonstrate our approach for the given model. We then demonstrate how posterior information changes in response to the amount of data available from an epidemic, both in terms of the length of the timeseries and the presence or absence of CTD. Finally, we present a simulation study of four outbreak scenarios in which different surveillance strategies are employed for the early detection of new infections.
6.1. The model
For this example, four modes of contact (Feedmill, Slaughterhouse, Company Networks, and Spatial [environmental] transmission) contribute to the infection rate between infected or notified individual (i.e. farm) i and susceptible j. For the Feedmill and Slaughterhouse networks, contact frequency information is available, whereas for the Company network only the presence or the absence of a business association is known. The spatial transmission rate is then parameterized as a function of the Euclidean distance between the two individuals, centered at 5 km to improve MCMC mixing and aid parameter interpretability. The transmission modes are therefore
The infectivity function q(i;ξ, t) is defined as
Transmission parameter “true” values used to simulate the test epidemic for contact tracing
| Parameter | True value | Prior |
|---|---|---|
| ε | 1−6 day−1 | Γ(0.15,5000) |
| p1 | 0.3 | Beta(1,1) |
| p2 | 0.9 | Beta(1,1) |
| β1 | 0.008 day−1 | Gamma(2.048,256) |
| β2 | 0.009 day−1 | Gamma(2,111) |
| γ | 0.5 | Gamma(1.5,3) |
| ψ | 0.2 km−1 | Gamma(10,50) |
| η1 | 1 | fixed |
| η2 | 0.6 | Gamma(1,10) |
| η3,…,10 | 0.3 | Gamma(1,10) |
| Parameter | True value | Prior |
|---|---|---|
| ε | 1−6 day−1 | Γ(0.15,5000) |
| p1 | 0.3 | Beta(1,1) |
| p2 | 0.9 | Beta(1,1) |
| β1 | 0.008 day−1 | Gamma(2.048,256) |
| β2 | 0.009 day−1 | Gamma(2,111) |
| γ | 0.5 | Gamma(1.5,3) |
| ψ | 0.2 km−1 | Gamma(10,50) |
| η1 | 1 | fixed |
| η2 | 0.6 | Gamma(1,10) |
| η3,…,10 | 0.3 | Gamma(1,10) |
6.2. Posterior information
To illustrate how the acquisition of CTD can improve posterior precision, we use a simulated epidemic on the GBPR dataset as described in Section 2 using the model described in Section 3. The epidemic is realized using a stochastic simulation based on the Doob–Gillespie algorithm (Gillespie, 1976) applied at the level of individual contacts (to generate CTD), with the extension that retrospective sampling is used to determine whether a contact results in an infection (Jewell and others, 2009).
Here we investigate how the assimilation of improves inference by increasing posterior precision. We base our test analysis on a typical simulated epidemic lasting 109 days in which 350 farms become infected, using the parameter values presented in Table 1. (See supplementary material C, Figure S1, available at Biostatistics online.) Two timepoints were analyzed, day 40 (incomplete epidemic) and day 109 (complete epidemic), as shown in Table 2. CTD for a 21-day period preceding a notification event was recorded for each notified individual. The algorithm was then run for the two timepoints during the epidemic, with and without the associated CTD.
The state of the epidemic at each observation time
| Contacts resulting in infection | Contacts not resulting in infection | |||||
|---|---|---|---|---|---|---|
| Time/days | Notified infections | Occult infections | Feedmill | Slaughterhouse | Feedmill | Slaughterhouse |
| 40 | 159 | 19 | 3 | 1 | 25 | 3 |
| 109 | 350 | 0 | 7 | 5 | 78 | 12 |
| Contacts resulting in infection | Contacts not resulting in infection | |||||
|---|---|---|---|---|---|---|
| Time/days | Notified infections | Occult infections | Feedmill | Slaughterhouse | Feedmill | Slaughterhouse |
| 40 | 159 | 19 | 3 | 1 | 25 | 3 |
| 109 | 350 | 0 | 7 | 5 | 78 | 12 |
True number of epidemiologically relevant (i.e. originating at infectious individuals) contacts are known from the simulation algorithm.
The density plots for parameters ϵ, p1, p2, β1, β2, γ, and ψ are shown in Figure 1. Of particular interest are the plots for the probability parameters p1 and p2, which are, of course, directly informed by the contact tracing information. It is immediately apparent that the addition of CTD affects the marginal posteriors of these parameters, with an increase in precision in each case. The other parameters are far less affected, with only minor differences apparent in the complete epidemic analysis. For the density plots for the components of η and histograms of the posterior number of occults present on day 40 see supplementary material C, Figures S2–S4, available at Biostatistics online.
Kernel density estimates of the marginal posterior distributions of ϵ, p1, p2, β1, β2, γ, ψ for (a) day 40 and (b) day 109.
Kernel density estimates of the marginal posterior distributions of ϵ, p1, p2, β1, β2, γ, ψ for (a) day 40 and (b) day 109.
6.3. Practical application
To test a prospective application of our methodology, we focus on the statistical detection of occult infections. We postulate that this provides a method to target limited surveillance resource to the most likely infected individuals. A simulation study was performed in which four surveillance strategies were tested. In the “Reactive” strategy, no active surveillance is performed, and cases are notionally detected and reported by the farmer. The three remaining strategies use active (pre-emptive) surveillance to look for disease as well as a case detection by the farmer, with the strategy being implemented at day 14, mimicking the application of such a strategy once it has become apparent that an epidemic is taking off. “Random” surveillance represents a strategy in which a random sample of size z holdings within the statutory 10 km surveillance zone are visited and tested daily (Defra, 2009). The “Bayes” targeted strategy uses our algorithm (without CTD) on a daily basis to rank holdings in terms of occult probability; the top z holdings are then visited and tested for disease. The “ Bayes–CT” strategy then adds in CTD to target surveillance similarly. For the purposes of our simulation study, we assume a surveillance resource of z = 15. We also assume that each surveillance team has at their disposal a perfect test for the disease—i.e. 100% sensitivity and specificity—and that if positive, the farm is culled immediately. For each strategy, 500 epidemic realizations with random index cases were generated to integrate over the stochasticity in the epidemic process and to study the efficacy of each surveillance strategy. These were further conditioned to involve at least two individuals and last > 14 days, such that the epidemic could be said to have “taken off”. For a flow diagram of the simulation study see supplementary material C, Figure S5, available at Biostatistics online.
As metrics for comparing the performance of the four surveillance strategies, the number of culled holdings (both as a result of Reactive and surveillance detection) and epidemic duration (i.e. time from the first infection to the last cull with no further infected or notified holdings) were used. These results are summarized in Table 3 and Figure 2, and indicate that active surveillance as implemented here is effective in reducing the probability of large-scale outbreaks. The Random strategy, whilst successful in reducing the mean size of the outbreak, does not appreciably affect the probability of a large epidemic. Using Bayes targeting has the effect of both dramatically reducing the mean epidemic size and the probability of a large epidemic. The addition of CTD improves the efficacy of surveillance further: the Bayes strategy reduces the mean epidemic size by 4-fold when compared with the Reactive strategy, the Bayes–CT strategy reduces it by 10-fold. A similar picture is reflected in the results for epidemic duration, with the presence of a surveillance strategy greatly reducing the mean. As before, the effect of Bayes targeting is profound, with the presence of CTD reducing the mean variance of the duration distribution. These results suggest that Bayes targeting of surveillance using case incidence data and contact tracing may have much to offer in disease control resource prioritization.
Histograms of total number of farms culled and epidemic duration under the four control strategies in the simulation study. (a) The distribution of the logarithm of total number of culled premises (reactive culling plus active surveillance culls), log (mean number culled) shown by dashed lines. (b) The distribution of epidemic duration, mean duration shown by dashed lines.
Histograms of total number of farms culled and epidemic duration under the four control strategies in the simulation study. (a) The distribution of the logarithm of total number of culled premises (reactive culling plus active surveillance culls), log (mean number culled) shown by dashed lines. (b) The distribution of epidemic duration, mean duration shown by dashed lines.
Mean number of holdings culled and mean epidemic duration (time from first infection to last removal) for each surveillance strategy, conditional on the epidemic lasting longer than 14 days
| Strategy | Mean # culled (95% CI) | Mean duration (95% CI) |
|---|---|---|
| SOS | 203.5(2,727) | 73.2(14.5,147.2) |
| Random | 120.1(2,709) | 60.0(16.8,122.4) |
| Bayes | 48.1(2,521) | 23.8(14.3,50.5) |
| Bayes–CT | 19.3(2,204) | 27.3(14.2,64.8) |
| Strategy | Mean # culled (95% CI) | Mean duration (95% CI) |
|---|---|---|
| SOS | 203.5(2,727) | 73.2(14.5,147.2) |
| Random | 120.1(2,709) | 60.0(16.8,122.4) |
| Bayes | 48.1(2,521) | 23.8(14.3,50.5) |
| Bayes–CT | 19.3(2,204) | 27.3(14.2,64.8) |
CI credible interval.
7. Discussion
The results presented in the previous section show that CTD is a useful addition for inference and prediction on SIR-type epidemic models. The purpose of this methodological innovation is demonstrated well in Figure 1, showing an increase in posterior precision in response to the acquisition of the contact-tracing information in parameters p1 and p2. Of particular interest is the behavior of the posterior in response to differing amounts of data, and for this reason the density plots should be considered in conjunction with Table 2. For day 40, only one out of four slaughterhouse contacts resulted in infection, compared with 3 out of 28 for Feedmill contacts. Importantly, the low contact frequency of the slaughterhouse network (see supplementary material C, Figure S6 and material D available at Biostatistics online.) means that the marginal posterior for p2 is poorly informed without the addition of CTD. Conversely, the full epidemic dataset on day 109 contains more contact observations, reflected in the narrower posterior distributions. Here, there is perhaps little difference in the marginal posteriors for p1 with and without CTD, though the effect on p2 remains marked.
Despite its simplistic setup, the results of the surveillance strategy simulation study are encouraging, lending evidence to Bayes-guided surveillance being highly effective for reducing infection to detection time, and hence reducing the propensity for epidemic spread. Importantly, this methodology provides a way of immediately linking targeted surveillance to changes in outbreak dynamics due to unexpected behavior of a new strain of the disease, changes in control strategy, or changes in underlying population behavior—changes that may well not be apparent to the naked eye in an evolving epidemic time series.
Given the marked influence that CTD has on the posterior parameter estimates, care must be taken to avoid biasing the results through the use of biased datasets. The structure of the likelihood assumes independence between the data contained in individual contact tracing questionnaires (i.e. between individuals), and therefore the model is robust to the absence of contact tracing from random cases. This is important as the analysis is unbiased in the case that the speed of the disease process exceeds the capacity to collect CTD. Source of bias, however, may well arise through a case's reluctance to declare certain contacts, and therefore carefully designing tactful contact tracing questionnaires should be of high priority. Nevertheless, in our Bayesian setup, the model extends naturally to include expert opinion on the level of such reporting bias, so an attempt to correct for it can be made.
In a UK livestock setting, although CTD is collected and used informally by field operatives to identify at risk farms, there currently appears to be no formal dissemination of these data to analysts. However, given the strong evidence for its use presented here, we strongly recommend that it should be made routinely available together with the case data. Given reliable sources of such data, therefore, we envisage that real time inference and risk prediction for epidemics will become commonplace in reactive disease control strategy.
Supplementary material
Supplementary material is available at http://biostatistics.oxfordjournals.org.
Funding
This research was supported by BBSRC.
Acknowledgments
We thank Professors Laura Green and Matt Keeling for helpful discussions. We thank Defra for supplying demographic data, and also for their valuable discussions on the nature of CTD. Conflict of Interest: None declared.
