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Edited by: Xiaonan Lu, University of British Columbia, Canada

Reviewed by: Abhinav Mishra, University of Georgia, United States; Alessandra De Cesare, University of Bologna, Italy; Thomas Alter, Freie Universität Berlin, Germany

This article was submitted to Food Microbiology, a section of the journal Frontiers in Microbiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Despite continued efforts to improve biosecurity protocols,

Poultry meat has been decisively implicated as a leading infection route for campylobacteriosis in humans (EFSA Panel on Biological Hazards,

Current attempts at tackling outbreaks of

Understanding of the spread of

An area of more recent study is the role played by “super shedders,” birds who consistently shed high amounts of

Some factors affecting transmission are well-reported, if incompletely understood. The effect of seasonal variation on both the carriage rate, and number of

This study explores the impact of multiple factors on the transmission of multiple sequence types (STs) of

We use a robust data set from Colles et al. (

A Bayesian approach is considered for this study due to the methodology's innate strengths in analyzing incomplete data (Dorazio,

The field data used for this study were originally presented in Colles et al. (

As such we build a dataset providing information on real-time evolution of

Histogram showing the count of positive and negative samples from a breeder flock for different species of

Within each species, multiple STs are recorded. In

The 5-weeks rolling average number of positive samples for

The 5-weeks rolling average number of positive samples for

We notice from

In this section we discuss the general methodology behind all of our models. A general step-by-step process to model formulation is also presented in

Choose how data should be classified, and construct matrix

Choose how model will define transition probabilities and dependencies.

Define prior probability distributions for model parameters. Program and run Bayesian model using JAGS, to acquire a posterior probability distribution for all model parameters defined in step 2.

Investigate model output to assure posterior distribution is well-constructed and has converged.

Plot the transition probabilities, π_{i, j}, and interpret the results.

Once the matrix is defined, each model uses a Bayesian process to find the transition probabilities between these states. Formally we seek the matrix π, where π_{i, j} = _{i, j} is the probability that a chicken moves from state _{1} ∈ [0, 1] and α_{2} ∈ [0, 1] that best fit the data _{i, j} between 0 and 1, as each value represents a probability. Likewise each row of π must sum to 1, as these probabilities cover all transition possibilities. In the example of Equation (1) above, when starting from state 1, one can transition to state 2 (π_{1,2}), or remain in state 1 (π_{1,1}), hence π_{1,1} + π_{1,2} = 1. Different models below will use more complex definitions for π to explore the impact of time, density dependence, and chicken health on transitions between different states.

A Bayesian statistical model provides a way to iteratively deduce parameters of interest in regards to given data. The process is derived from Bayes' theorem:
_{1} and α_{2} in the example above. The data,

Below we present a series of case studies presenting our different models and their results. All models were run using JAGS (Plummer,

Our first model investigates how time affects the transition probabilities between states. Following the process outlined in

To assess how the transition probabilities vary through time we must ensure that we define our transition probabilities such that they depend on time. One way would be to adapt Equation (1) above such that π_{1,1} was a function of α + β_{i,j,t} remain correctly bounded. The underlying theory is that we assume there is some mean probability for π_{i,j,t} across all _{1} and α_{2}. We then assume that, for each _{1}[_{2}[

Now that we have decided on our model formulation, we move to step 3 and run the model to find the posterior distributions for α_{1}, α_{2}, _{1}, and _{2}. First we define our prior probability distributions for each of the model parameters. This distribution represents our initial assumptions on what value our variables may take, and is often informed by expert opinion. Since we do not have any initial assumptions on what values our variables may take, we use wide non-informative priors. For α_{1} and α_{2} we choose a prior distribution of _{1} and _{2}, we wish each element of these vectors to be a small perturbation away from the mean of α_{1} or α_{2}. As such, we would ideally have these elements drawn from a normal distribution with mean 0, and some, yet to be determined, standard deviation. This represents a hierarchical model formulation (discussed further in _{1} and _{2}. Following the advice of Gelman (

Convergence was considered well-achieved via investigation of the trace plots of the chains, the effective sample size (ESS) and Monte Carlo Standard Error (MCSE) of the variables. The Gelman-Rubin statistic, or “shrink factor,” is the most commonly used metric for convergence, with a value close to 1 signifying effective convergence. Heuristically, any shrink factor below 1.1 is considered by Kruschke (

The results for this model are presented below in _{1,1,t}, (4B) π_{1,2,t}, (4C) π_{2,1,t}, (4D) π_{2,2,t} are plotted, and a linear regression is fit to these outputs using the

Transition probabilities between two states, “uncolonized” and “colonized.” Plots show _{1,1,t}, _{1,2,t}, _{2,1,t}, and _{2,2,t} against time. Each point is the calculated transition probability for that time point. Also plotted is a linear regression against these points in blue, with a shaded region depicting the 95% confidence interval of the regression.

These findings suggest that, as time progresses, colonized chickens become more likely to remain colonized, and similarly become less likely to clear such a colonization.

For the next model we investigate transition differences between the two species present in the study:

Transition probabilities between three states, “uncolonized,” “colonized by

We now combine the previous two models together, to investigate how the transitions between species alter across time. We once again therefore classify our data into three categories, as per the previous model.

We will be constructing a three-dimensional array once again for our transition probabilities, with each time period being described by a separate 3 × 3 transition matrix. To ensure each row of these matrices sums to 1, we start by framing the transition probabilities as an unbounded array _{i}[_{i} terms to all be drawn from a six-variable multivariate normal distribution, with mean (0, 0, 0, 0, 0, 0) and a covariance matrix as our parameter to be defined. JAGS requires the input of a precision matrix (the inverse of the covariance matrix) for its formulation of the multivariate normal distribution, so we set a prior distribution on the precision matrix of Wishart(_{6}, 6), where _{6} is the 6 × 6 identity matrix.

The model was run with two chains for an initial burn-in period of 5,000 iterations, and then a posterior distribution was built from a sample of 250,000 iterations, thinned at a rate of 1 in 5, meaning only 1 in every 5 iterations was used for the posterior distribution so as to reduce autocorrelation. Results are plotted below in

Transition probabilities between three states, “uncolonized,” “colonized by _{1,1,t}, _{1,2,t}, _{1,3,t}, _{2,1,t}, _{2,2,t}, _{2,3,t}, _{3,1,t}, _{3,2,t}, and _{3,3,t} against time. Each point is the calculated transition probability for that time point. Also plotted is a linear regression against these points in blue, with a shaded region depicting the 95% confidence interval of the regression. Five transition probabilities were found to be statistically significant for correlation against time: π_{1,3,t}, π_{2,1,t}, π_{2,3,t}, π_{3,1,t}, and π_{3,3,t} (

Of the nine transition probabilities presented, five were found to be statistically significant for correlation against time when a linear regression was applied: π_{1,3,t}, π_{2,1,t}, π_{2,3,t}, π_{3,1,t}, and π_{3,3,t} (_{2,3,t} (

Transition probabilities between state 2 “colonized by

For this model, we extend model 2 to now capture species-specific ST perseverance within a chicken. To do this, we re-classify the data into five different states: “S1: uncolonized,” “S2: new _{1,3} = π_{4,3} = π_{5,3} = 0, and likewise for transitions to state 5: π_{1,5} = π_{2,5} = π_{3,5} = 0. The non-zero transition probabilities can then be calculated by drawing each row from a 3 or 4 variable Dirichlet distribution. Formally we set a prior on each row of,

Transition probabilities between five states, “uncolonized,” “colonized by a new

The most notable difference is seen in the perseverance of

Whereas model 1 considered how transition probabilities vary across time, we now consider how transition probabilities vary across different chickens. We follow a very similar framework to model 1, beginning by classifying all data as one of two states: “S1: uncolonized” or “S2: colonized.” We then, like model 1, consider some average transition probability that each chicken is close to, and then consider some small “correction term” unique to each chicken, which may make them more or less likely to transition to a certain state. Formally, we write,
_{1} and α_{2} of _{1}[_{2}[_{2}, 2), where _{2} is the 2 × 2 identity matrix.

The model was run with two chains for an initial burn-in period of 20,000 iterations, before posteriors were then constructed from a sample of 50,000 iterations. Convergence was well-achieved, with all chains well-mixed and all parameters sampled with a high ESS and MCSE < 0.01. The mpsrf was unable to be calculated due to the high number of stochastic nodes, however there were no signs to suggest invalid convergence.

Upon calculating our transition probabilities for each bird, we plot the values for π_{1,2} against the value of π_{2,1} for each bird and investigate the correlation.

Transition probabilities for each bird in the flock from a state of being colonized to uncolonized (y-axis) against the transition probability from uncolonized to colonized (x-axis). Contours show the fit of a multivariate normal distribution to the output.

The strong linear relation observed reveals the presence of distinct sub-groups within the flock of birds who are colonized often, and those who are colonized very rarely.

We now alter the previous model to consider the differences in transition between species of _{i}[_{i} parameter, and the six chicken correction terms, _{i}[_{6}, 6), where _{6} is the 6 × 6 identity matrix.

The model was run with two chains for an initial burn-in period of 10,000 iterations, before posterior distributions were constructed from a sample of 50,000 iterations, thinned at a rate of 1 in 25, meaning only one iteration was kept in every 25.

The idea of this model is to assess how bird variation affects the transition of each species of _{1,1,c}, the transition from uncolonized to uncolonized as the y-axis. This acts as a rough metric for “bird resilience to colonization,” as the more resistant birds are more likely to continue being uncolonized. As such plots 10A–C depict how transitions related to each species vary according to host bird susceptibility. Plot 10D uses π_{3,3,c}, the transition from

Transition probabilities for a three state system. In these plots, “U” refers to being uncolonized, “J” refers to colonization by

Interestingly, the gradients of all the shifting transition probabilities are different between species, confirming that, indeed, the transition probabilities of each species varies differently across chickens. We see that the probability of a species persisting, unsurprisingly increases as bird susceptibility increases, but curiously our linear regressions for each species overlap. This result indicates that, in the more resilient birds,

It is interesting to note that the gradient of the lines in each plot are distinctly different from one another, highlighting how each species responds differently to variations in host bird health.

This model builds on model 5 by now considering how transition probabilities are affected by the number of total colonizations in the previous week.

The model formulation is then as follows,
_{t} is the number of birds that data is available for at time _{i} represents some mean transition probability that all birds are clustered around, and _{i}[_{t} as, for most weeks 75 birds are recorded for every _{i} are parameters signifying the strength of the density dependent effect.

The model was initialized with prior distributions of _{i} and β_{i} parameters. The chicken corrections terms _{i}[_{2}, 2) where _{2} is the 2 × 2 identity matrix. The model was run with two chains for an initial burn-in period of 6,000 iterations and then posterior distributions built from a sample of 25,000 iterations. This was done twice with two variations of the model. One where density dependence is calculated from provided data, and one with the addition of imputed data. The posterior distributions of our model parameters were used to simulate the transition probabilities for each flock across a full range of total flock prevalences, i.e., using the median values for α_{i}, β_{i}, _{i} and the precision matrix, we are able to build the functions

Transition probabilities from a state of uncolonization (by

Importantly

An improved understanding of the transmission dynamics of

A number of the analyses were consistent with the existence of a “

The models further indicated that individual bird status was more important than

Having confirmed the existence of variation in bird transition probabilities, the possible impact of such variation on the proliferation of

With respect to the two species, there was evidence that

Human incidence of campylobacteriosis has been shown to vary in a repeated pattern each year (Nylen et al.,

The age of the flock has been shown to be an important factor associated with increasing

We also stress the importance of our final model in not just investigating the weekly bacterial prevalence turnover, but how it further substantiates the importance of bird-to-bird transmission. Without this one could argue from our earlier results that more resilient birds were simply the ones who did not ingest a more invasive ST.

In conclusion, these analyses have highlighted the diversity of individual bird response to bacterial challenge, and how this range of responses can be a key driver of

The original contributions presented in the study are publicly available. This data can be found here:

FC collected the data. TR, RP, MM, and MB conceived the study. TR and RP built the models and wrote all associated code. TR wrote the manuscript. MD, FC, and MB supervised the project. All authors reviewed the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This manuscript has been released as a pre-print at Rawson et al. (

The Supplementary Material for this article can be found online at: