Probabilistic graphical models

The PGM presented in the present paper has been designed using the HUGIN tool, a commercial software package from Hugin Expert (http://www.hugin.com/). Briefly, PGMs include prior knowledge and uncertainties using a number of variables and probabilistic dependence relationships between the variables (Madsen et al. Reference Madsen, Karlsen, Barker, Garcia, Hoorfar and Jensen2012; Garcia et al. Reference Garcia, Madsen and Vigre2013). The PGMs formed by a set of variables have two components: a qualitative and a quantitative part. The qualitative part includes diverse ‘nodes’ such as variables, decision nodes and utility functions and arcs representing relationships between them (as exemplified in Fig. 1). A chance node (an oval in Fig. 1) represents a random variable, while a decision node (a rectangle in Fig. 1) defines decision alternatives at a specific point in time and a utility node (a diamond in Fig. 1) represents a reward or cost function. The relationships of dependence or independence between the variables included in the models can be represented and the strength of the relationships can be defined using conditional probability distributions (Heckerman et al. Reference Heckerman, Mamdani and Wellman1995). The quantitative part of the models includes the mathematical expressions and probability distributions. The results from the models comprise posterior probability distributions (for every strategy) related to expected Campylobacter numbers in the flock (in logs) before and after the implementation of the decision(s) and the expected cost–benefit balance associated with each decision.

Fig. 1.

The Commercial Broilers in Denmark (ComBDK) model designed based on epidemiological and financial data from Denmark.

Prior knowledge related to poultry production management factors

The developed decision support model (Fig. 1) integrates epidemiological data from a study on risk factors potentially associated with Campylobacter infection in Danish broiler flocks (Chowdhury et al. Reference Chowdhury, Sandberg, Themudo and Ersbøll2012). These authors investigated >1700 variables and found the following risk factors to be significantly associated with Campylobacter positive flocks: season, increasing age of the birds, the age of the poultry houses, previous Campylobacter positive flocks in the same houses and the number of persons entering the poultry houses (Chowdhury et al. Reference Chowdhury, Sandberg, Themudo and Ersbøll2012). Odds ratio (OR) values were used to represent the concept of ‘increased risk’ due to the factor(s) considered in their research. To integrate this information into the PGM, the significant risk factors were included in the model (Fig. 1) and the OR values were transformed to a probability of infection with Campylobacter at varying states of the risk factor (P(s)) using the following formula:

Table 1.

Significant risk factors, frequency of occurrence and associated probability of Campylobacter introduction in broiler flocks in Denmark (Chowdhury et al. Reference Chowdhury, Sandberg, Themudo and Ersbøll2012)

* Based on a basic level of Campylobacter of 14% (Chowdhury et al. Reference Chowdhury, Sandberg, Themudo and Ersbøll2012).

† Reference category.

$$P{\rm (}s{\rm )} = \displaystyle{{\exp (\ln ({b_{\rm p}}/1 - {b_{\rm p}}) + \ln ({\rm ORs}))} \over {1 + \exp (\ln ({b_{\rm p}}{\rm /}1 - {b_{\rm p}}) + \ln ({\rm ORs}))}}$$

where b _p is the baseline Campylobacter flock prevalence and ORs is the odds ratios (Table 1). In addition, Campylobacter introduction in the flock due to the presence of risk factors was assumed to be conditional to a baseline Campylobacter flock prevalence (b _p: lowest level of Campylobacter in broilers close to slaughter time found in the literature) of 0·14 (Chowdhury et al. Reference Chowdhury, Sandberg, Themudo and Ersbøll2012).

The ‘Commercial Broilers in Denmark’ (ComBDK) model: design and assumptions

The ComBDK model (Fig. 1) was developed to assist poultry producers in Denmark to make decisions regarding Campylobacter controls in commercial broilers based on the following assumptions:

1. (i) A microbiological detection level of Campylobacter was considered to be 2 logs colony forming units (CFU)/g or ml of sample while the maximum colonization level was assumed to be 10 logs CFU/g or ml of sample. Thus, in this model, a Campylobacter level of 0–2 logs will produce a negative result while a positive result suggests Campylobacter numbers present in the samples from 2 to 10 logs. Intervals for bacterial concentration with two log widths (e.g. 0–2 logs, 2–4 logs, 4–6 logs, 6–8 logs and 8–10 logs) were considered.

2. (ii) The contributions from different risk factors to the level of Campylobacter are independent.

3. (iii) The ‘measured Campylobacter numbers at slaughter’ will depend on the microbiological quantitative methods used and the ‘true numbers’ of Campylobacter in chickens. In the model, a hypothetical nearly perfect quantitative method has been considered.

4. (iv) The Campylobacter controls included in the model are: the use of fly screens and hypothetical Campylobacter vaccines A and B. Fly screens are used during the summer but the costs associated with the use of fly screens are always considered (Lawson et al. Reference Lawson, Jensen and Lund2009).

In this model, data related to the influence of the risk factors on the expected Campylobacter status of the flock were based on a positive (2–10 logs)/negative (0–2 logs) result while data related to the effect of interventions were based on a log-scale because the objective was to develop a control strategy (e.g. vaccination strategy) able to reduce the numbers of Campylobacter in commercial broilers. Data must be on the same scale to obtain reliable results from the model. For this reason, a mathematical transformation from positive/negative to the diverse Campylobacter levels (in logs)-scale was performed. Figure 2 illustrates the mathematical transformation used to translate results from a positive/negative format (‘Campylobacter slaughter time without vaccine’) to a log scale format (‘Campylobacter slaughter time logs without vaccine’). A flat distribution was selected to transform a general expected posterior probability of a Campylobacter flock (e.g. 0·72) into a distribution of equal probabilities (0·18) for the different levels of Campylobacter considered in the model (Fig. 2).

Fig. 2.

Illustration showing the part of the model where a mathematical transformation is performed to translate results from a positive/negative format (‘Campylobacter slaughter time w/o vaccine’) to a log scale format (‘Campylobacter slaughter time logs w/o vaccine’).

The effectiveness of Campylobacter controls and cost–benefit functions included in the model

The use of a cost-effective vaccine against Campylobacter in chickens is desirable; however, no commercial vaccine is presently available. Based on prices of other vaccines used to control diseases in poultry (Dianova 2013), a hypothetical price can be considered to be around 0·15–0·50 Danish Kroners (DKK), equivalent to €0·02–0·07. Two hypothetical vaccines (A and B) were incorporated in the model with different vaccine effectiveness and costs (Table 2). The use of fly screens in poultry houses in order to control Campylobacter has been recommended (Hald et al. Reference Hald, Skovgard, Bang, Pedersen, Dybdahl, Jespersen and Madsen2004, Reference Hald, Sommer and Skovgard2007; Bahrndorff et al. Reference Bahrndorff, Rangstrup-Christensen, Nordentoft and Hald2013). The average cost (including capital costs that cover long-term investments and variable expenses) of implementing fly screens on broiler farms was calculated to be 0·13 DKK/chicken (Lawson et al. Reference Lawson, Jensen and Lund2009), equivalent to approximately €0·02, and included in the model (Table 2). The impact of interventions and the cost–benefit functions were based on a log-scale because the objective was to develop a control strategy that could reduce the numbers of Campylobacter in already infected chickens. The use of fly screens can reduce the prevalence of Campylobacter in broiler flocks significantly; therefore, it was assumed in the model that the effectiveness of fly screens was 0·90 (expected reduction of Campylobacter numbers in more than 2 logs).

Table 2.

Costs and effectiveness related to control measures included in the model

DKK, Danish Kroners.

Two reward systems (Table 3) were designed based on an ‘average’ gross profit (for Danish farmers producing chickens carrying an ‘average’ number of Campylobacter) considered to be DKK 2·92/chicken (equivalent to €0·39/chicken) based on financial data from 2013 (Farmtal Online 2013). It was assumed that an average broiler chicken from a positive flock in Denmark will carry Campylobacter in a concentration of 4–6 logs CFU/g or ml of sample (from the digestive tract). The reward system 1 was designed based on a system implemented in Denmark where poultry producers get an extra payment when the flock is identified as Campylobacter negative before slaughter (Marianne Sandberg, personal communication, 2013). In the model, an extra payment (between DKK 0·02 and 0·04) is given to farmers producing chickens carrying Campylobacter in numbers <4 logs and a reduced payment (between 0·04 and 0·08 less) is established for farmers producing poultry flocks that carry Campylobacter in numbers >6 logs before slaughter (Table 3). A different reward system 2 was designed based on an extra payment of 2·5 times the normal price for chickens testing negative for Campylobacter before slaughter in order to test the sensitivity of the reward variable in the model.

Table 3.

Reward systems 1 and 2

DKK, Danish Kroners.

Sensitivity analyses

Three different types of sensitivity analyses were performed on the model. Firstly, parameter sensitivity analyses were conducted (using the Hugin Expert software) in order to determine which parameters in the model had the strongest influence on final outcomes (Nielsen & Jensen Reference Nielsen and Jensen2003), for example, the effect of vaccination impact (in logs reduction) on measured Campylobacter at slaughter (Fig. 3). Furthermore, the impact of changes in parameters related to the reward system was investigated such as the effect of changes in ‘p’ (increased/decreased payment based on numbers of Campylobacter in commercial chickens) on the cost–benefit balance (or expected utility) of different control measures (Fig. 4). Secondly, an evidence- (or scenario-) driven sensitivity analysis was performed in order to determine the sensitivity of the decisions considered in the model under diverse evidence scenarios (Jensen Reference Jensen1996). For this purpose, different scenarios (e.g. best/worst case scenarios) and the two different reward systems previously described were used. Thirdly, a sensitivity analysis using a number of control measures including a third hypothetical vaccine C (with an effectiveness level of 0·90 and a cost of DKK 0·26/ chicken equivalent to €0·04 was conducted.

Fig. 3.

Results from sensitivity analyses performed to investigate the effect of vaccination impact (in logs reduction when using Vaccine A) on measured Campylobacter at slaughter.

Fig. 4.

Results from sensitivity analyses in terms of changes of the expected utility (cost–benefit balance) when selecting some decisions (no controls and using vaccines A or B) as a function of ‘p’ (decreased/increased payment considered in reward system 1).