A Quantitative Risk Assessment Model for Listeria monocytogenes in Ready-to-Eat Smoked and Gravad Fish

Abstract

This study introduces a quantitative risk assessment (QRA) model aimed at evaluating the risk of invasive listeriosis linked to the consumption of ready-to-eat (RTE) smoked and gravad fish. The QRA model, based on published data, simulates the production process from fish harvest through to consumer intake, specifically focusing on smoked brine-injected, smoked dry-salted, and gravad fish. In a reference scenario, model predictions reveal substantial probabilities of lot and pack contamination at the end of processing (38.7% and 8.14% for smoked brined fish, 34.4% and 6.49% for smoked dry-salted fish, and 52.2% and 11.1% for gravad fish), although the concentrations of L. monocytogenes are very low, with virtually no packs exceeding 10 CFU/g at the point of sale. The risk of listeriosis for an elderly consumer per serving is also quantified. The lot-level mean risk of listeriosis per serving in the elderly population was 9.751 × 10⁻⁸ for smoked brined fish, 9.634 × 10⁻⁸ for smoked dry-salted fish, and 2.086 × 10⁻⁷ for gravad fish. Risk reduction strategies were then analyzed, indicating that the application of protective cultures and maintaining lower cold storage temperatures significantly mitigate listeriosis risk compared to reducing incoming fish lot contamination. The model also addresses the effectiveness of control measures during processing, such as minimizing cross-contamination. The comprehensive QRA model has been made available as a fully documented qraLm R package. This facilitates its adaptation for risk assessment of other RTE seafood, making it a valuable tool for public health officials to evaluate and manage food safety risks more effectively.

1. Introduction

Cold-smoked and gravad fish are products with considerable public health implications in regards to listeriosis since they are not heat-treated, are generally eaten with no prior heating, and have a long shelf life. Over more than a decade, many reports and surveys have highlighted that ready-to-eat (RTE) seafood products are prone to contamination with L. monocytogenes [1,2,3,4,5,6]. Data have shown that fish used as raw material can be contaminated with L. monocytogenes cells before harvesting, which then multiply previous to smoking or marination when in the processing plant [7]. Implementation of whole genome sequencing (WGS) has revealed that different genotypes dominate in the environment, processing, and in different countries, both in sea and on land [8]. According to the latest European Union (EU) zoonoses report [9], in 2022 RTE fish and fishery products was the food category with the highest occurrence of L. monocytogenes, with an overall mean prevalence of 7.1% (N = 9727), varying from 0.0 to 20.0% between Member States (MS).

While qualitative methods provide valuable insights for the risk ranking of pathogens in food chains, a quantitative approach is essential for precisely quantifying the efficacy of control measures, enabling the assessment of risk levels and the mathematical modeling necessary to effectively predict and manage public health outcomes. Quantitative risk assessment (QRA) is a systematic approach used to estimate the risk of infection and illness from exposure to pathogens through various environmental mediums like food, water, or air [10].

Recently, using a “generic quantitative risk assessment” model, EFSA [11] contrasted the risk of listeriosis in the EU elderly population linked to foods products such as RTE fish, pâté, cooked meats, sausages, soft and semi-soft cheeses, and blanched frozen vegetables and determined that gravad fish in normal atmosphere packaging and hot-/cold-smoked fish in reduced-oxygen packaging were the top-most high-risk products. In terms of reported outbreaks, according to EU surveillance data [12], in the period between 2010 and 2020, fish and fish products (namely, crab meat, crustaceans, shellfish, mollusks, smoked fish, and non-specified seafood) caused 23% of the 53 strong-evidence total outbreaks in the EU, whereas, in 2022, EFSA [9] ranked L. monocytogenes in fish and fish products as one of the top ten pathogen/food vehicle pairs causing the highest number of deaths in strong-evidence outbreaks in the reporting EU MS. The high share of RTE seafood as a causative agent of listeriosis was also purported by a recent genomic-based epidemiological study [13], which estimated that 27% of 228 listeriosis cases in Germany between 2010 and 2020 were most likely caused by smoked or gravad salmon products.

More recently, ECDC and EFSA [14] investigated a prolonged cross-border outbreak of L. monocytogenes ST173 that, between 2017 and 2024, caused 73 cases of listeriosis, including 14 deaths, in Belgium, the Czech Republic, Germany, Finland, the Netherlands, Italy, and the UK. WGS cluster analysis and tracing evidence indicated that the strain spread in Europe originated from a past single source in the fish production chain. Contrary to Europe, where seafood like gravad and smoked fish in certain packaging types pose high risks, the attribution of outbreaks of listeriosis to seafood in the USA appears to be lower [15].

In view of the fact that most of the listeriosis cases occur sporadically [16], it is also pertinent to point out the outcomes of a meta-analysis on case-control studies of sporadic listeriosis [17]. These authors found that, among the RTE food categories, seafood, processed meats, cheese, and composite foods, RTE seafood presented the highest association with sporadic listeriosis, with pooled odd ratios of 10.75 (p < 0.001) for non-perinatal populations (immunocompromised and the elderly) and 6.273 (p < 0.001) for all susceptible populations (perinatal/non-perinatal, immunocompromised, and the elderly). In order to provide valuable insights on practices and strategies to reduce the current risk of listeriosis associated with RTE seafood products, various QRA models have been developed. A recent critical review on listeriosis QRA models [18] revealed that, although 12 out of the 13 seafood models retrieved (published between 1998 and 2022) investigated the food products specifically considered as high risk—cold-smoked or gravad fish—all of them represented short supply chains only, either from end processing/retail to table [16,19,20,21,22,23,24] or the sole consumption module [25,26,27,28,29,30]. None of the available QRA models included a processing module nor at least a qualitative assessment of the most relevant opportunities of cross-contamination during processing, probably due to the insufficiency of data at the time of their development. In 2022, the Joint FAO/WHO Expert meeting on microbiological risk assessment of Listeria monocytogenes in foods recommended the development of a full primary production (harvest and farming) to consumption risk assessment for the risk of L. monocytogenes for RTE seafood (hot- and cold-smoked fish and gravad fish) [31].

In light of new data and predictive microbiology models, the objective of this study was to build a QRA model of longer scope for RTE smoked and gravad fish capable of representing the growth, inactivation, and potential cross-contamination from processing to consumption as well as the retarding effect of background microbiota on the development of L. monocytogenes.

This study addresses the critical need for an updated quantitative risk assessment model that incorporates the latest data and technological advancements in predictive microbiology. By developing a comprehensive model that evaluates the entire chain of production from primary processing to consumption, this paper aims to provide actionable insights and strategies to significantly reduce the health risks associated with the consumption of RTE smoked and gravad fish.

The model was designed to be able to assess the contribution of the initial contamination of the incoming fish, the contribution of cross-contamination during the filleting of fish and slicing prior to packaging, the effect of the sampling schemes at the end of processing, the effect of time and temperature throughout the logistics of the end-product, the effect of improved practices throughout the supply chain, and the effect of lactic acid bacteria (LAB) cultures added for the biocontrol of L. monocytogenes. The model’s structure for both RTE smoked and gravad fish was developed according to the recent Expert Panel recommendation of the Joint FAO/WHO Expert meeting on microbiological risk assessment [31]. The present article aims to describe in detail the QRA model and subsequently illustrate its functionality by reference and what-if scenario analysis.

2. Materials and Methods

2.1. Exposure Assessment

The exposure assessment model was developed for two RTE seafood products, smoked fish and gravad fish, and both are presented in this article since they share most of the processing operation units. The processing stages are schematized in Figure 1. The model starts from whole fish units and considers the storage before filleting, which can represent the transport to the primary processing facility or waiting time at the processing facility. After filleting, there is a holding-off time at the facility before preparation, yet it can alternatively correspond to the transport of fish fillets to a secondary processing facility.

At this point, there is a differentiation between smoked fish and gravad fish. For the processing of smoked fish, the stages modelled are salting/brining and smoking/maturation, whereas for the gravad fish the stages are the smearing of fillets with condiments, followed by maceration. The stages that follow are common to both products—the slicing of fillets, packaging, within-lot testing, cold chain distribution (i.e., transport to retail, display at retail, and transport from retail to home), and home storage and portioning—although the parameters feeding the model are product-specific.

Each of the stages shown in Figure 1 was coded as a function stochastically estimating the microbial prevalence and numbers after a process of microbial growth, inactivation, cross-contamination, or partitioning [32], except for within-lot testing, which did not conform to any of the aforementioned processes.

Table 1. Sequence of stages, microbial processes represented, data sources, assumptions, and corresponding functions coded in R for the construction of the exposure assessment model of Listeria monocytogenes (LM) in RTE smoked and gravad fish.

Module	Stage	Microbial Process	Assumptions	Sources	Function in R
Primary processing	Generation of contaminated lots of pre-fillet (pre-processed) fish	None	LM prevalence in fish units is modelled from data found in incoming fish sampled at primary processing. Since LM numbers in incoming fish were generally low (<10 CFU/g), LM concentration in fish was calculated from the expected proportion of non-zeroes under a Poisson distribution.	Autio et al. [33], Cruz et al. [34], Dass [29], Di Ciccio et al. [35], Markkula et al. [36], Medrala et al. [37], Miettinen et al. [38], Rorvik et al. [39], Vogel et al. [40], Jarvis [41]	Lot2LotGen()
	Storage	Growth	LM in pre-filleted fish is assumed to follow the kinetics of a cocktail of CICC 21632 (serotype 1/2a), CICC 21633 (serotype 1/2a), CICC 21635 (serotype 4b), and CICC 21639 (serotype 1/2a) LM inoculated in raw salmon flesh. Lag phase is considered.	Jia et al. [42]	sfRawFishStorage(), served by sfGrowthLDP()
	Filleting	Cross-contamination	LM is transferred to the fillets during the slicing process of the raw whole fish using a compartmental model defined by two variability distributions: “a”, the transfer rate between slicer blade (or filleting knife) and product, and “e”, the transfer rate from the original contamination to the slicing system. Distribution parameters were obtained from published data.	Hoelzer et al. [43], Aarnisalo et al. [44]	sfSlicer()
	Holding-off time	Growth	The same predictive microbiology model as in Storage, yet the LM growth and the exhaustion of the lag phase were followed up in this stage.	Jia et al. [42]	sfRawFishStorage(), served by sfGrowthLDP()
Secondary processing	Brining or salting (smoked fish)	Cross-contamination	Fish fillets can be salted either through brine injection or by dry salting. Internal contamination by brining may occur during the injection of saline solution due to brine containers or the brine itself serving as reservoirs for LM, at a given probability. For dry-salted fillets, it is assumed that external contamination can occur during the procedure of smearing salt on fish fillets if surfaces are contaminated with LM, at a given probability. A published distribution for LM transfer rate is assumed.	Gudbjornsdottir et al. [45], Gudmundsdottir et al. [46] Hoelzer et al. [43]	sfBrineORsaltCC(), served by sfBriningCC() and sfSmearingCC()
	Smoking and maturation (smoked fish)	Inactivation	Salting, drying, and smoking produce a slight reduction in LM, which is different if fish fillets were brine-injected or dry-salted. The LM log10 reduction in brine-injected fish fillets was assumed to be that of the combined results of inoculation experiments in smoked salmon, first salted through brine injection, and then submitted to cycles of cold-smoking between 6 and 8 h until a total maturation time of 18–24 h. The LM log10 reduction in dry-salted fish was assumed to be that of the combined results of inoculation experiments on the surface of dry-salted salmon, determined before and after a total maturation time between 18 and 24 h.	Eklund et al. [47], Porsby et al. [48] Eklund et al. [47], Porsby et al. [48], Neunlist et al. [49]	sfSmoking()
	Smearing (gravad fish)	Cross-contamination	During smearing of fish fillets with salt, sugar, and spices, external contamination can occur if surfaces are contaminated with LM, at a given probability. A published distribution for LM transfer rate is assumed.	Hoelzer et al. [43]	sfSmearingCC()
	Maceration (gravad fish)	Inactivation	The maceration of fish smeared with gravlax curing agents is assumed to reduce the populations of LM, according to the results of a study where inoculated raw salmon was smeared with salt, brown sugar, black pepper, and dill and left to macerate at 4.5 °C for 72 h.	Lopes et al. [50]	sfMaceration()
	Slicing	Cross-contamination	The same compartmental model as in Filleting, but used to produce slices from smoked or gravad fish fillets.	Hoelzer et al. [43], Aarnisalo et al. [44]	sfSlicer()
	Packaging	Mixing	No cross-contamination is assumed during packaging. Consecutive slices of recently-sliced RTE smoked or gravad fish are gathered into packages of end-product.	-	sfPackaging()
	Within-lot testing	None	At a given probability, any lot of RTE smoked or gravad fish can be subjected to sampling and testing, according to a two-class or three-class microbiological sampling plan.	-	sfTesting()
Distribution	Cold chain	Growth	LM in smoked or gravad fish, as affected by populations of lactic acid bacteria (LAB), are assumed to grow during the various cold chain logistics stages, including transportation to retail, display at retail, and transportation to home. Specific growth rates for LM and LAB are estimated from secondary models, using validated kinetic parameters. LM numbers follow an extended Jameson-effect competition model, which uses an interaction Gamma parameter.	Mejlholm and Dalgaard [51,52,53,54,55], Mejlholm et al. [56] Gimenez et al. [57], Moller et al. [58]	sfColdChain(), served by sfMejlholmDalgaard(), sfMejlholmDalgaardLAB(), sfGrowthJameson()
Consumer’s handling	Home storage	Growth	The same as in Cold chain, following up the growth.	Mejlholm and Dalgaard [51,52,53,54,55], Mejlholm et al. [56], Gimenez et al. [57], Moller et al. [58]	sfColdChain(), served by sfMejlholmDalgaard(), sfMejlholmDalgaardLAB(), sfGrowthJameson()
	Portioning	Partitioning	The consumer is assumed to take a number of RTE fish slices from the pack. LM cells present in a contaminated pack are assumed to be moderately clustered within the package.	Nauta [59]	sfPortioning()

presents a synthesis of the modules, the sequence of stages and processes they consist of, the assumption and data sources employed, and the corresponding functions programmed in the R software version 4.4.2.

2.1.1. Contaminated Lots of Pre-Fillet Fish

The prevalence of L. monocytogenes in pre-processed fish was modeled using published data, assuming that the variability between different lots of fish follows a Beta distribution. The model was expressed as

$$\begin{gathered} s_j~Binomial\left(n_j,p_j\right) \\ {p}_j~Beta\left(\alpha ,\beta \right) \end{gathered}$$

This distribution is defined by parameters α and β, derived from the survey data listed in Table 2. Each sampling result, from a total of 12 fish lots, is considered a part of a binomial distribution, representing the unobserved prevalence in each lot.

The model employs Bayesian methods with non-informative priors for α and β, set to follow Gamma distributions. After 20,000 iterations in a Markov chain Monte Carlo (MCMC) simulation, the average values for α and β were calculated, with the mean prevalence of L. monocytogenes in any given fish lot modeled as Beta (0.8741, 5.880). This represents an average prevalence rate of approximately 14.86%, with a 95% confidence range from 0.037 to 55.41%.

Since, to the best of the authors’ knowledge, the numbers of L. monocytogenes in gutted and cleaned fish are very low, mostly below the limit of quantification (10 CFU/g) [7,60], the Poisson assumption was employed to approximate the concentration of L. monocytogenes per gram of raw fish from the unobservable prevalence of the lot j (p_j). Such a procedure calls for two assumptions: (1) that L. monocytogenes cells are randomly distributed in fish (i.e., are not clustered), and (2) that the analytical weight was the same (25 g) in all detection assays carried out by the sources shown in Table 2. A further assumption if that a clean, degutted fish unit weighs 3800 g.

Table 2. L. monocytogenes prevalence in raw fish sampled at processing facilities.

Country	Product	Sample Size, n	Positive Enrichment, s	Prevalence (%)	Source *
Finland	Raw rainbow trout	35	0	0.00	Autio et al. [33]
Brazil	Raw salmon	255	105	41.2	Cruz et al. [34]
Ireland	Raw salmon	60	17	28.3	Dass [29]
Italy	Raw salmon	21	5	23.8	Di Ciccio et al. [35]
Finland	Raw fish	45	2	4.40	Markkula et al. [36]
	Fish before processing	212	9	4.20
Poland	Incoming salmon	46	2	4.34	Medrala et al. [37]
	Incoming seatrout	26	4	15.4
Finland	Raw fish	18	2	11.1	Miettinen et al. [38]
Norway	Salmon pre filleting	24	4	16.6	Rorvik et al. [39]
Denmark	Raw fish	12	0	0.00	Vogel et al. [40]
	Raw fish	18	0	0.00

(*) Data extracted from the Pathogens-in-Foods Database [61], except for Cruz et al. [34].

Table 2. L. monocytogenes prevalence in raw fish sampled at processing facilities.

Country	Product	Sample Size, n	Positive Enrichment, s	Prevalence (%)	Source *
Finland	Raw rainbow trout	35	0	0.00	Autio et al. [33]
Brazil	Raw salmon	255	105	41.2	Cruz et al. [34]
Ireland	Raw salmon	60	17	28.3	Dass [29]
Italy	Raw salmon	21	5	23.8	Di Ciccio et al. [35]
Finland	Raw fish	45	2	4.40	Markkula et al. [36]
	Fish before processing	212	9	4.20
Poland	Incoming salmon	46	2	4.34	Medrala et al. [37]
	Incoming seatrout	26	4	15.4
Finland	Raw fish	18	2	11.1	Miettinen et al. [38]
Norway	Salmon pre filleting	24	4	16.6	Rorvik et al. [39]
Denmark	Raw fish	12	0	0.00	Vogel et al. [40]
	Raw fish	18	0	0.00

(*) Data extracted from the Pathogens-in-Foods Database [61], except for Cruz et al. [34].

A function, Lot2LotGen(), was built to generate a contamination matrix, N, of dimensions r × c, whose number of rows, r, represents the number of lots and can therefore be understood as number of iterations that will correspond to between-lot variability; the number of columns, c, represents the number of pre-filleting fish units of weight Unit_size (3800 g) to be produced in the lot, considered as fixed and equal for all lots (cf. Appendix A.1).

2.1.2. Storage of Pre-Fillet Fish

The primary processing module begins with storing gutted, clean fish prior to filleting. Even under very cold storage conditions, L. monocytogenes can still grow. Jia et al. [42] have provided models for this growth based on their studies, where they inoculated salmon with various strains of L. monocytogenes and stored them at temperatures ranging from 4 to 35 °C. They used a Lotka–Volterra-based equation to determine how temperature affects the growth rate and the initial delay before growth starts, considering the competitive effects of the fish’s natural microbiota.

The QRA model applies these findings to predict the growth rate and the initial delay period of the pathogen during storage, using a straightforward log-linear model to estimate the pathogen’s concentration at the end of the storage period. Jia et al. [42] also provided the maximum population density of L. monocytogenes in raw fish (9.20 log₁₀ CFU/g). The model assumes a brief holding period for the fish before filleting, with storage temperature and duration modeled to vary according to Pert distributions. Specifically, temperature and storage time for each lot are sampled from Pert (−2, 0, 4) and Pert (0.5, 2.0, 6.0), respectively.

Additionally, an auxiliary function, sfGrowthLDP(), calculates the number of L. monocytogenes in a unit of raw fish after storage, based on constant temperature and time settings using validated models for maximum growth rate and delay duration [42,62]. This function aids in multiple storage assessments for raw fish, as detailed in Appendix A.2. The function sfRawFishStorage() then uses these calculations to stochastically simulate the growth of L. monocytogenes during cold storage, pulling data from a contamination matrix and the storage variability parameters listed in Appendix A.3.

2.1.3. Filleting of Raw Fish

As early as 1995, Eklund et al. [47] and Rorvik et al. [39] observed that L. monocytogenes could be transferred from the exterior of fish to cut surfaces of fillets or sides, and concluded that filleting is a critical stage, since filleting tables, knives, and gloves of personnel could further spread the contamination. The representation of cross-contamination during fish slicing was regarded as relevant, because L. monocytogenes present on tables and cutting surfaces can adhere strongly after a short period of time, which entails the possibility that filleted fish become contaminated during the first stages of processing.

Aarnisalo et al. [44] carried out a study to investigate the transfer of L. monocytogenes from an inoculated slicing blade (slicer) to slices of gravad salmon, and from inoculated salmon fillet to the slicing machine and subsequently to slices of uninoculated fillets. A marked reduction in the counts from the slicing blade (5.9–9.0 log₁₀ CFU/blade) to the fillets (1.6 log₁₀ CFU/g) was observed after 39 slices; nonetheless, the first slices contained higher counts. Hoelzer et al. [43] utilized this bacterial transfer dataset, along with others produced from slicing meat and RTE meat products, to populate a compartment model consisting of four elements (slicer, chub, slice, and a bacterial loss bin). From this deterministic model, they produced various estimates of coefficients for slicing transfer (a) and for transfer from original contamination to the system (e) and derived thereof variability distributions regarding the transfer coefficients a and e. Additionally, Hoelzer et al. [43] proposed a probabilistic model of microbial transfer during a generic slicing process. This model and its parameters are employed in the present QRA to represent the transfer of L. monocytogenes cells during the filleting operation. A further assumption is that two fillets of weight 1300 g each can be obtained from one clean fish unit.

A function, sfSlicer(), was written to stochastically simulate the transfer of bacteria during this step. The function is fed by the outputs of the function sfRawFishStorage(), the weight of the fillet (w_Fillet), the load of L. monocytogenes cells on the slicer (knife or blade) (Init_Slicer), the parameters of the logistic distribution regarding the transfer coefficient a (location_a, scale_a), and the parameters of the normal distribution of the log₁₀ coefficient e (μ_loge, σ_loge). The outputs of the function sfSlicer() are as follows: the contamination matrix of L. monocytogenes numbers in fish fillets (N_Fillet), the previous expenditure of L. monocytogenes lag phase corresponding to those fillets (WorkDone_s), the (unchanged) probability vector of contaminated lots after filleting Prob_UnitPos, and the mean prevalence of contaminated lots after filleting (P_Fillet) (cf. Appendix A.4).

2.1.4. Holding-Off Time of Fish Fillets

The growth of L. monocytogenes in fish fillets during the holding-off stage is simulated. The objective of introducing a storage time after filleting is to allow for the transportation of fish fillets from the primary to the secondary processing facility, or, alternatively, for a short period of time before commencing processing (salting in the case of smoked fish or smearing with curing agents in the case of gravad fish). The latter case is represented in the present QRA model, assuming that the holding-off temperature (Temp_hold, °C) and time (time_hold, h) follow Pert distributions, Pert (−2, 0, 4) and Pert (1, 2.0, 6.0), respectively. The maximum population density (MPD) of L. monocytogenes in raw fish fillets is assumed to be the same as in raw clean fish (9.20 log₁₀ CFU/g).

The function sfRawFishStorage() and its auxiliary function sfGrowthLPD(), explained in detail in Appendix A.2 and Appendix A.3, are reused in the holding-off stage to stochastically estimate the growth of L. monocytogenes in raw fish fillets. In a lot, the fish fillets are considered to be exposed to the same Temp_hold and time_hold, which are sampled from the aforementioned Pert distributions. In addition to these parameters, the inputs of the function sfRawFishStorage() are the contamination matrix, N_Fillet, the microbial work done after the first storage (WorkDone_s, a matrix), the weight of the fillet (w_Fillet, a scalar), and the maximum population density of L. monocytogenes in the fish fillet (MPD, a scalar), whereas the outputs are the contamination matrix after L. monocytogenes growth (N_Hold) and the total microbial work done (WorkDone_h, a matrix).

$$\left\{N_{Hold ij}, {WorkDone}_{h ij}\right\}=sfGrowthLPD\left(N_{Fillet ij} , {time}_{hold i}, {Temp}_{hold i}, w_{Fillet}, MPD, {WorkDone}_s\right) j=1, 2\dots ,{ c}_f$$

sfRawFishStorage() returns unaffected the mean prevalence of contaminated lots of fish fillets (P_Hold, a scalar) and the probability that the sampled lot is contaminated (Prob_UnitPos, a vector).

2.1.5. Brining or Salting of Fish Fillets (Relevant to Smoked Fish)

In the processing of smoked fish, fish fillets can be salted in two ways: by dry-salting or by brining (i.e., injection of a saturated NaCl solution). Given that L. monocytogenes is able to survive in the salted fish due to its halotolerance, no microbial reduction or inactivation process is represented at this stage. Instead, opportunities for external or internal contamination during salting are contemplated. In the case of brining, recirculating brine is a source for contamination since L. monocytogenes can survive in NaCl solutions of up to 10% [63]. Gudmundsdottir et al. [46] and Gudbjornsdottir et al. [45] found L. monocytogenes at frequencies of 21.4% (3/14) and 8.7% (2/23), respectively, in injection brines, which supported the fact that brine containers and the brine itself may serve as reservoirs for L. monocytogenes. Based on these data, the QRA model assumed that, on a lot basis, the probability that the brine solution is contaminated (Pcc_Brine) is 13.5% (5/37). When this event occurs, an internal contamination of the fish fillet is produced through the brine injected. Nonetheless, no data were found on the likely L. monocytogenes levels in contaminated brine solution, and therefore the numbers were assumed.

With regards to dry-salting, it is considered that cross-contamination can take place through tables or other surfaces during the smearing of fish fillets with salt/sugar/spices, at a certain probability, Pcc_Smear. The probability Pcc_Smear is assumed to be 3.9% [46]. Every fillet is subjected to the same Pcc_Smear probability, and if the contamination event takes place, cells are partially transferred according to a transfer coefficient. The normal distribution parameters of the log₁₀ transfer coefficient of L. monocytogenes (TR_Smear) were taken from Hoelzer et al. [43] to represent the transfer from board to meat. Nonetheless, no data were found for the likely levels of L. monocytogenes on environmental elements in contact with fish fillets (N_Surface) while dry-salting or smearing with other ingredients (gravlax curing).

Two auxiliary functions were built, sfBriningCC() and sfSmearingCC(), to model the cross-contamination during brining and dry-salting, respectively. The function sfBriningCC() stochastically simulates the potential internal contamination of fish fillets during brining by the injection of salt solution [64] (cf. Appendix A.5). The function sfSmearingCC() stochastically simulates the potential external contamination of fish fillets during dry salting when producing smoked fish, or during smearing with gravlax curing agents when producing gravad or any macerated fish (cf. Appendix A.6). Those two auxiliary functions could be used on their own, if we were to assume that all the lots of fish fillets were processed either through brine injection or through dry salting. The main function sfBrineORsaltCC() was conceived to allow for the inclusion of both types of salting in the QRA model. Thus, some of the lots of fish fillets will be subjected to brining (at the probability p_Brine) and others to dry-salting (at the probability 1-p_Brine) (cf. Appendix A.7).

2.1.6. Smoking and Maturation of Salted Fish Fillets (Relevant to Smoked Fish)

The combination of hurdles that occur in the processing of smoked fish (salting, drying, and smoking) affect the homeostasis of L. monocytogenes; as they struggle to maintain their energy balance, they become metabolically exhausted during the subsequent step of maturation, causing the populations to drop [65]. Although the temperature during smoking is too low to eliminate L. monocytogenes, the phenolic compounds from smoke at a concentration of 20 ppm can inhibit L. monocytogenes [66].

In the literature, few studies investigating the overall effects of the individual processing steps for the production of smoked salmon have been undertaken for both types of salting (brine injection or dry-salting).

Table 3. Reduction in L. monocytogenes concentration in salted salmon fillets due to smoking and maturation, taken from challenge tests.

Type of Salting	Source	Conditions of Smoking and Maturation	Before Treatment (log10 CFU/g)	After Treatment (log10 CFU/g)	Reduction (log10 CFU/g)
Brine injection	Porsby et al. [48]	Cold smoking, 24 °C in cycles of 6 h	3.00 ± 0.3	2.50 ± 1.1	0.50
			3.30 ± 0.4	1.80 ± 0.9	1.50
			2.90 ± 0.2	1.00 ± 0.0	1.90
			3.30 ± 0.1	1.80 ± 0.6	1.50
			3.00 ± 0.1	1.10 ± 0.3	1.90
	Elkund et al. [47]	Cold smoking, 17–21 °C × 18 h	2.36	1.41	0.95
			3.34	2.63	0.71
		Cold smoking, 22–30 °C × 18 h	2.11	2.34	−0.23
			3.28	3.38	−0.10
			4.57	4.49	0.08
Dry-salting	Porsby et al. [48]	Cold smoking, 24 °C in cycles of 6 h	3.30 ± 0.2	1.30 ± 0.7	1.80
			3.10 ± 0.2	1.40 ± 0.7	1.70
	Neunlist et al. [49])	Liquid smoking and maturing, 4 °C × 24 h	5.70	4.70	1.00
	Elkund et al. [47]	Cold smoking, 17–21 °C × 18 h	2.04	0.83	1.21
			2.56	1.11	1.45
		Cold smoking, 22–30 °C ×18 h	0.11	−0.39	0.50
			1.04	0.54	0.50
			2.42	1.84	0.58

compiles the available information on the combined effect of cold-smoking and maturation on L. monocytogenes.

In each of those challenge studies, the conditions of smoking and the treatment prior to smoking were different. In the work of Neunlist et al. [49], smoking dry-salted salmon reduced L. monocytogenes counts by 0.5 log₁₀ CFU/g; however, during subsequent maturation, the effect of salt and smoke continued, and the mean number of L. monocytogenes was further reduced by another 0.5 log₁₀ CFU/g. Thus, the combination of the two steps significantly reduced the concentration of L. monocytogenes by 1.0 log₁₀ CFU/g.

In the paper by Porsby et al. [48], the effect of smoking and maturation was carried out in salmon fillets that were salted either by brine injection or by dry-salting and in fresh unsalted salmon fillets as the control. From different challenge studies, they determined that smoking and maturation could reduce L. monocytogenes in up to 1.9 log₁₀ after 24 h in brined fish, although they could not find statistical differences with salmon fillets that were dry-salted. Eklund et al. [47] inoculated L. monocytogenes in two different manners: on the surface, to simulate external contamination, and internally, to simulate contamination through brine injection; cold-smoking was the carried out at two distinct ranges of temperature: a lower temperature (17–21 °C) and a higher temperature (22–30 °C). The results of Elkund et al. [47] suggested that smoking and maturation may be more effective in reducing L. monocytogenes populations in dry-salted salmon fillets than in brine injected ones. In the experiment where cold-smoking was conducted at the higher temperatures of 22–30 °C on salmon samples that were inoculated (internally) via brine injection, smoking and maturation did not appear to have any effect on L. monocytogenes. These results agreed with Niedziela et al. [67], who earlier pointed out that dry-salting is more effective in controlling L. monocytogenes than brine injection. Dry-salting pulls water out of the flesh due to osmotic pressure. On the contrary, a brine-injected fillet will keep the water distributed in the flesh, and the surface water on the fish may even be pulled towards the higher salt concentration within the fish. As the surface is the part of the fish that tends to have higher concentrations of L. monocytogenes, the difference of losing surface water and keeping surface water will cause a different microbial concentration change in the fish.

Thus, the present QRA model estimates the log₁₀ reduction factors due to smoking and maturation for brine-injected fish fillets and dry-salted fish fillets by calculating the mean and sample standard deviations of the values compiled in Table 3. The mean (μ_rBrine) and standard deviation (σ_RBrine) of the normally distributed log₁₀ reduction factor for brined fish fillets (R_Brine) are 0.871 and 0.807 log₁₀, respectively, while the mean (μ_rDrysalt) and standard deviation (σ_RDrysalt) of the normally distributed log₁₀ reduction factor for dry-salted fish fillets (R_Drysalt) are 1.093 and 0.532 log₁₀, respectively.

A function, sfSmoking(), was written to simulate the microbial process of inactivation of L. monocytogenes in the salted fish fillets subject to smoking and 18–24 h maturation (cf. Appendix A.8). Its inputs are the outputs of the function sfBrineORsaltCC() and the normal distribution parameters μ_RBrine, σ_RBrine, μ_RDrysalt, and σ_RDrysalt that describe the microbial log₁₀ reduction in L. monocytogenes for brined or dry-salted fish fillets. These parameters are pre-defined, as described above.

2.1.7. Smearing of Fish Fillets with Ingredients (Relevant to Gravad Fish)

Different procedures have been reported for the elaboration of gravad fish. According to Wiernasz [68], for gravlax production in France, salmon fillets are cured with a mix of salt, sugar, pepper, and dill during a period of 14 h at 6 °C, whereas Peiris et al. [63] explain that, in Sweden, salmon or rainbow trout are rubbed with a mixture of sugar, salt, and pepper and are covered with dill and stored in plastic bags at low temperature for 48 h. A longer procedure has been described by Cruz et al. [34]; salmon fillets are hand-rubbed with a commercial mixture of NaCl, sodium nitrate, and sodium nitrite and stored at 4 °C for 24 h in high-density polypropylene boxes. After the excess of salt is washed out with chlorinated water and the fillets are drained, a mixture of sugar, NaCl, ground white pepper, and dried dill is hand-rubbed into the flesh side of the fillets, and they are then stored for 24 h at 4 °C in the boxes. Fillets are then layered on stainless steel supports, sprayed with sweet wine, and ripened for 48 h at 4 °C.

In a processing line for gravlax, Cruz et al. [34] found that, amongst the food contact surfaces, the pathogen was present in 40% of the samples from salting boxes, ripening trolleys after 48 h in the cold room, and weighting trays, whereas a lower frequency of positive samples was found in the salting table (30%). Therefore, as for dry-salting in the production of smoked fish, during smearing with salt/sugar/spices in the production of gravad fish it is assumed that bacterial transfer can occur from tables or other surfaces, at a probability Pcc_Smear (3.9% obtained from Gudmundsdottir et al. [46]). As with the smearing of fish with salt (Section 2.1.5), for the smearing of fish fillets with gravlax curing agents the same parameter values were employed: ${\mu }_{TRsmear}=-0.29$ and ${\sigma }_{TRSmear}=0.31$ for the normal distribution truncated on ]∞, 0] of the log₁₀ transfer coefficient of L. monocytogenes (TR_Smear) [43], and the load of L. monocytogenes on environmental elements in contact with a fish fillet (N_Surface = 10 CFU).

The function sfSmearingCC() is used to stochastically simulate the potential external contamination of fish fillets during smearing with salt, sugar, and condiments. The function is described in detail in Appendix A.6. In the context of the processing of gravad fish, sfSmearingCC() is fed by the outputs of the holding-off storage function (N_Hold, Prob_UnitPos, and P_Hold) and Pcc_Smear (0.039), $N_{surface}$ (100 CFU), μ_TRSmear (−0.29), and σ_TRSmear, (0.31). The function provides as outputs the contamination matrix N_Smear, the vector Prob_{UnitPos Smear}, and the scalar P_Smear.

2.1.8. Maceration or Curing of Fish Fillets (Relevant to Gravad Fish)

Lopes et al. [50] performed a challenge test on traditional gravlax salmon to determine the effect of processing on L. monocytogenes counts. Salmon fillets, contaminated at a level of 5.5 log₁₀ CFU/g, were premoistened with lemon juice and covered with a mixture of salt, brown sugar, black pepper, and dill and then cured at 4.5 °C. After 72 h of curing, the population of L. monocytogenes was reduced by 0.9 log₁₀ CFU/g. In the work of Neunlist et al. [49], the salting step had a weaker effect on L. monocytogenes culturability of 0.5 log₁₀ CFU/g. From this information, a log₁₀ reduction factor for gravlax maceration of fish fillets (R_Gravad) was derived, assuming that it followed a normal distribution with mean μ_RGravad = 0.70 and standard deviation σ_RGravad = 0.283 log₁₀ CFU/g.

A function, sfMaceration(), was built to stochastically represent the reduction of L. monocytogenes concentration in fish fillets smeared with gravlax curing agents during cold maceration or curing. The inputs of this function are the contamination matrix N_Smear and the normal distribution parameters μ_RGravad and σ_RGravad, defining the microbial log₁₀ reduction in L. monocytogenes in macerating fish fillets smeared with curing ingredients.

2.1.9. Slicing of Processed Fish Fillets

In the literature, a few environmental microbiological surveys have evidenced that slicing machines in RTE seafood processing facilities can be a source of L. monocytogenes spread. For instance, Di Ciccio et al. [35] repeatedly isolated L. monocytogenes serotypes 1/2a and 1/2b from slicer belts, distribution trays, slicing machines, and slicing covers for three years in a smoked-salmon production facility: out of 95 environmental samples tested, slicing machines (37%) and working tables (43%) had the highest frequencies of detection. In the USA, in a processing plant of catfish fillets, Chen et al. [69] determined that in 15% (7/45) of the sampling times, skinning, slicing, or blending equipment were contaminated with L. monocytogenes. In Ireland, Dass [29] isolated MLVA types L. monocytogenes serotypes c and b in the slicing machines over a one-year survey (2 positives out of 36). Thus, it was deemed necessary to represent cross-contamination during the slicing of processed fish fillets in the present QRA model. The function sfSlicer(), developed from the compartmental model of Hoelzer et al. [43] and used to simulate cross-contamination during fish filleting, is also employed at the slicing stage. In this case, the unit to be sliced is the processed fish fillet of weight w_Fillet and the sliced unit is the smoked fish slice or the gravad fish slice, of weight w_Slice.

The function sfSlicer(), already used during the filleting process and described in Appendix A.4, takes the outputs of the function sfSmoking() in the case of the QRA for smoked fish, or the outputs of the function sfMaturation() in the case of the QRA for gravad fish. In any case, the weight of a fish fillet (w_Fillet) is 1300 g and the weight of a slice (w_Slice) is 32.5 g. The parameters of the logistic distribution about the transfer coefficient a (location_a = 0.07; scale_a = 0.03), and the parameters of the normal distribution of the log₁₀ coefficient e (μ_loge = −2.12; σ_loge = 0.85) remain the same as those used in the stage of filleting [43], whereas the load of L. monocytogenes cells on the slicer (Init_Slicer) is set to 0.

2.1.10. Packaging of Processed Fish Slices

There is no assumption of cross-contamination during the packaging of processes fish slices. A function, sfPackaging(), was written. It consists of a simple grouping of consecutive slices to produce a pack of Slices_pack slices. It adds up the microbial load N_Slice, corresponding to the group of slices, and the weight of the slices (pack). Slices_pack is assumed to be 8 slices per pack; thus, the grouping produces a pack of weight Unit_SizePack, which equals Slices_pack × w_Slice (8 × 32.5 = 260 g). The inputs of the function sfPackaging() are N_Slice and Slices_pack, and the output is the contamination matrix N_Pack. The number of packs of end-product (c_p) produced in a lot will then be c_s/Slices_pack. Thus, the output N_Pack is a matrix of dimensions r lots by c_p packs. The mean prevalence of contaminated packs, P_Pack, returns the value of P_Smoked for smoked fish packs, or the value of P_Smear for gravad fish packs.

2.1.11. Within-Lot Testing

The QRA model enables the testing of L. monocytogenes in food unit samples taken from a lot, according to a two-class or a three-class mixed sampling plan. In the two-class plan, n samples are randomly extracted and analyzed per lot. For each sample, a sub-sample of g grams is used in the enrichment essay, and the lot is rejected if more than c samples are positive in detection. In a three-class mixed sampling plan, samples are also enumerated, and the lot is rejected if more than c units are positive in detection or if at least one unit is found to have a concentration greater than a predefined concentration M [70]. It is assumed that the enumeration assay is conducted only on positive samples in detection, by direct plating of $g_{TestedEnum}$ g taken from the same sample.

A function, sfTesting(), was built to consider this selection step. Contaminated lots detected after testing are not removed from the matrix, and so the input matrix output N_Pack is returned unchanged. The function fvTesting() only updates the probability for each lot to be put on the market, using the Bayes’ theorem (cf. Appendix A.10).

2.1.12. Cold Chain

The characteristics of smoked fish and gravad fish—namely, intrinsic factors (pH, aw, preserving compounds, lactic acid protective cultures) and extrinsic factors (temperature, atmosphere in the package) do not preclude L. monocytogenes from growing [26]. If the pathogen is present on processed fish fillets, salting will not be sufficient to inhibit its growth. The slow development of L. monocytogenes relies upon cold temperatures and vacuum packaging, which reduces the pathogen’s growth due to anaerobic conditions and the inhibitory effect of competitive LAB. The cold chain step in the present QRA models encompasses the transportation from end processing to retail, the time at retail, and the transportation from retail to home.

In the cold chain segment of this study, the growth of L. monocytogenes and lactic acid bacteria (LAB), which can inhibit the pathogen, is modeled under various conditions. This module encompasses several key steps: Predictive models are utilized to determine the growth rates of both the pathogen and LAB in seafood, taking into account internal factors like pH and external factors such as temperature (refer to Appendix A.11 and Appendix A.12 for model details). A comprehensive growth model that includes initial growth delays is employed to estimate the concentrations of the pathogen and LAB at the conclusion of the cold chain (see Appendix A.13). The characteristics of the seafood that influence microbial growth rates are evaluated through a specific function (see Appendix A.14). The principal cold chain function integrates this data to simulate the growth of both microbes as the seafood progresses from production to retail (refer to Appendix A.15).

Table 4. Parameters used for modelling the simultaneous growth of L. monocytogenes and lactic acid bacteria (LAB) in RTE seafood during cold chain distribution and home storage.

Type of Parameter	Parameter	Definition (Unit)	Value or Distribution	Source
Relative to kinetic parameters of LM in RTE seafood	${\mu }_{LM ref}$	Optimum growth rate of LM (h−1)	0.419	Mejlholm and Dalgaard [51]
	MPDLM	Maximum population density of LM (log10 CFU/g)	MPDLM min = 6.60 MPDLM mode = 7.36 MPDLM max = 8.20 MPDLM ~ Pert (MPDLM min, MPDLM mode, MPDLM max)	Pérez-Rodríguez et al. [24]
	h0 LM	Parameter regarding the initial physiological state of LM (-)	μh0 = 2.8 σh0 = 4.6 h0 LM ~ Normal (μh0, σh0), h0 > 0	Couvert et al. [71]: q0 then obtained by (1/(exp(h0)−1))
Relative to kinetic parameters of LAB in RTE seafood	${\mu }_{LAB ref}$	Optimum growth rate of LAB (h−1)	0.583	Mejlholm and Dalgaard [54]
	MPDLAB	Maximum population density of LAB (log10 CFU/g)	MPDLAB min = 8.0 MPDLAB mode = 8.5 MPDLAB max = 9.0 MPDLAB ~ Pert (MPDLAB min, MPDLAB mode, MPDLABM max)	MPDLAB min from Mejlholm and Dalgaard [54]
	q0 LAB	Parameter regarding the initial physiological state of LAB (-)	ln q0 LABmin = −12 ln q0 LABmode = 2.73 ln q0 LABmax = 1.26 q0 LAB~exp{ Pert (ln q0 LABmin, ln q0 LABmode, ln q0 LABmax)}	Couvert et al. [71]
Relative to smoked fish characteristics	${\overline{C}}_{0 LAB SF}$	Between-lot mean concentration of LAB in smoked fish after packaging (log10 CFU/g)	${\overline{\overline{C}}}_{0 LAB\min SF}$ = −1.00 ${\overline{C}}_{0 LAB\mathrm{mode}SF}$ = 0.28 ${\overline{C}}_{0 LAB\max SF }$ = 1.60 ${\overline{C}}_{0 LAB SF}$ ~ Pert (${\overline{\overline{\overline{C}}}}_{0 LAB\min SF}$, ${\overline{\overline{C}}}_{0 LAB\mathrm{mode}SF}, {\overline{C}}_{0 LAB\max SF })$	Wiernasz et al. [72]
	pHSF	pH of smoked fish (-)	pHminSF = 5.8 pHmodeSF = 6.1 pHmaxSF = 6.5 pHSF~Pert (pHminSF, pHmodeSF, pHmaxSF)	pHminSF from Mejlholm and Dalgaard [51] pHmodeSF from Porsby et al. [48] pHmaxSF from Hwang and Sheen [73]
	NaClSF	NaCl content in smoked fish (% wb)	NaClminSF = 1.5 NaClmodeSF = 3.4 NaClmaxSF = 5.3 NaClSF~Pert (NaClminSF, NaClmodeSF, NaClmaxSF)	NaClminSF from FAO-WHO [26] NaClmodeSF from Porsby et al. [48], Mejlholm and Dalgaard [51], FAO-WHO [26], and Orozco [74] NaClmaxSF from Mejlholm and Dalgaard [51]
	PheSF	Phenol compound in smoked fish (ppm)	PheminSF = 5 PhemodeSF = 10 PhemaxSF = 22 PheSF~Pert (PheminSF, PhemodeSF, PhemaxSF)	PheminSF from Leblanc et al. [72] PhemodeSF from Hwang and Sheen [73], Porsby et al. [48], Mejlholm and Dalgaard [51], FAO/WHO [26], Leblanc et al. [75], and Eklund et al. [47] PhemaxSF from Porsby et al. [48]
	CO2 equi SF	CO2 concentration at equilibrium in the packaging of smoked fish (fraction)	CO2equi SF min = 0.25 CO2equi SF mode = 0.25 CO2equi SF max = 0.30 CO2equi SF~Pert (CO2equi SF min, CO2equi SF mode, CO2equi SF max)	Mejlholm and Dalgaard [51]
	Others: NitSF, LAtot GF, AAtot SF, BAtot SF, CAtot SF, DAtot SF, SAtot SF	Nitrites concentration (ppm) and lactic acid, acetic acid, benzoic acid, citric acid, diacetate, lactic acid, and sorbic acid concentrations in water phase (ppm)	Allow for minimum, mode, and maximum for each compound to be sampled from Pert distribution. Values of zero set to all.
Relative to gravad fish characteristics	${\overline{C}}_{0 LAB GF}$	Between-lot mean concentration of LAB in gravad fish after packaging (log10 CFU/g)	${\overline{\overline{C}}}_{0 LAB\min GF}$ = −1.00 ${\overline{C}}_{0 LAB\mathrm{mode}GF}$= 0.28 ${\overline{C}}_{0 LAB\max GF }$= 1.60 ${\overline{C}}_{0 LAB GF}$ ~ Pert (${\overline{\overline{\overline{C}}}}_{0 LAB\min GF}$, ${\overline{\overline{C}}}_{0 LAB\mathrm{mode}GF}, {\overline{C}}_{0 LAB\max GF })$	Wiernacz et al. [72]
	pHGF	pH of gravad fish (-)	pHminGF = 6.1 pHmodeGF = 6.2 pHmaxGF = 6.3 pHSF~Pert (pHminSF, pHmodeSF, pHmaxSF)	Mejlholm and Dalgaard [51] and Orozco [74]
	NaClGF	NaCl content in gravad fish (% wb)	NaClminGF = 3.0 NaClmodeGF = 3.2 NaClmaxGF = 3.4 NaClGF~Pert (NaClminGF, NaClmodeGF, NaClmaxGF)	Mejlholm and Dalgaard [51] and Aarnisalo et al. [44]
	PheGF	Phenol compound in gravad fish (ppm)	PheminSF = 0 PhemodeSF = 0 PhemaxSF = 5 PheGF~Pert (PheminGF, PhemodeGF, PhemaxGF)	Mejlholm and Dalgaard [51]
	CO2 equi GF	CO2 concentration at equilibrium in the packaging of gravad fish (fraction)	CO2equi GF min = 0.25 CO2equi GF mode = 0.25 CO2equi GF max = 0.30 CO2equi GF~Pert (CO2equi SF min, CO2equi SF mode, CO2equi SF max)	Mejlholm and Dalgaard [51]
	Others: NitGF, LAtot GF, AAtot GF, BAtot GF, CAtot GF, DAtot GF, SAtot GF	Nitrite concentration and lactic acid, acetic acid, benzoic acid, citric acid, diacetate, lactic acid, and sorbic acid concentrations in water phase (ppm)	Allow for minimum, mode and maximum for each compound to be sampled from Pert distribution. Values of zero set to all.	-
Relative to cold chain distribution	timeCC	Time elapsed between end of production and arrival of the product at home (h)	timeCC min = 12 timeCC mode = 144 timeCC max = 720 timeCC~Pert (timeCC min, timeCC mode, timeCC max)	FDA-FSIS [23]
	TempCC	Average temperature between end of production and arrival of the product at home (°C)	TempCC min = 0.28 TempCC mode = 4.60 TempCC max = 7.00 TempCC~Pert (TempCC min, TempCC mode, TempCC max)	From Normal (4.6, 2.2) °C in Pouillot et al. [20]
	CorTimeTempCC	Correlation between time and temperature during cold chain	−0.16	Pouillot et al. [19]
Relative to home storage	TimeHome	Storage time at home (h)	TimeHome min = 0.73 TimeHome mode = 70 TimeHome max = 840 for smoked fish and 528 for gravad fish TimeHome ~ Pert (timeHome min, timeHome mode, timeHome max)	Minimum and mode values from Weibull (shape = 1.14, scale = 18.39) days in Endrikat et al. [76] Maximum is best guess: 35 days for smoked fish and 22 days for gravad fish
	TempHome	Storage temperature at home (°C)	TempHome min = 1.12 TempHome mode = 7.0 TempHome max = 12.9 TempCC~Pert (TempHome min, TempHome mode, TempHome max)	From Normal (7, 3) °C in Pouillot et al. [20]
	CorTimeTempHome	Correlation between time and temperature at home storage	−0.12	Pouillot et al. [19]

provides a list of the parameters used in these models, detailing their significance and data sources.

Briefly (see Appendix A for details), the function sfMejlholmDalgaard() deterministically computes the growth rate of L. monocytogenes in RTE seafood at given intrinsic and extrinsic characteristics, according to the secondary model based on the Gamma concept for lightly preserved seafood developed and validated by Mejlholm and Dalgaard [51,52,53,54,55,56] using the. Similarly, the function sfMejlholmDalgaardLAB() deterministically estimates the growth rate of LAB in RTE seafood at given environmental characteristics, according to the secondary model developed in [54]. The function sfGrowthJameson() estimates the numbers of L. monocytogenes and LAB (CFU) in RTE seafood at given environmental characteristics after a certain time period. The function is based on the Baranyi–Roberts-based Jameson-effect competition model put forward by Giménez and Dalgaard [57], supplemented with the Gamma (γ) interaction parameter later proposed by Moller et al. [58]. By default, γ is set to 1. Eventually, assuming Jameson-effect microbial competition and allowing for the presence of a lag phase, the function sfColdChain() stochastically estimates the concentration of L. monocytogenes and LAB populations in the RTE seafood product at the end of the cold chain distribution (i.e., arrival at consumer’s home).

2.1.13. Home Storage

During home storage, L. monocytogenes and LAB continue to grow. Endrikat et al. [76] represented the home storage time in days by a Weibull distribution with shape parameter 1.14 and scale parameter 18.39. This information was translated into a Pert distribution for the minimum value Time_{Home min} (0.73 h as the 2.5% percentile of the Weibull distribution) and the mode Time_{Home mode} (70 h as the mode of the Weibull distribution). The maximum home storage time, Time_{Home max}, consists of a best guess of 35 days (840 h). Likewise, a Pert distribution regarding the home storage temperature (Temp_Home) was approximated from the normal distribution used in Pouillot et al. [20], truncating at the 2.5% and 97.5% percentiles to obtain the minimum (Temp_{Home min} = 1.12) and the maximum (Temp_{Home max} = 12.9) parameters, respectively. The mean of the normal distribution was set as the mode (Temp_{Home mode} = 7.0). In addition, a low correlation between home storage time and temperature was quantified as rank correlation by Pouillot et al. [19], which will be used in the present QRA model (CorTimeTemp_Home = −0.12).

The simultaneous growth of L. monocytogenes and LAB are simulated using the function ColdChain() and its auxiliary functions (cf. previous section), as performed for the distribution module of cold chain, including interactions [77]. Values of home storage time (time_{Home ij}) and temperature (Temp_{Home ij}) are sampled for every RTE seafood pack (i.e., different consumers), targeting the rank correlation value of CorTimeTemp_Home [78].

$$\begin{gathered} {time}_{Home ij} ~ Pert \left({time}_{Home min},{time}_{Home mode},{time}_{Home max}\right) i=1, 2, \dots , r;j=1, 2, \dots , c_p \\ {Temp}_{Home ij} ~ Pert \left({Temp}_{Home min},{Temp}_{Home mode},{Temp}_{Home max}\right) i=1, 2, \dots , r;j=1, 2, \dots , c_p \end{gathered}$$

The growth rates of L. monocytogenes (μ_{LM ij}) and LAB (μ_{LAB ij}) in RTE seafood for every pack ij stored at the temperature Temp_{Home ij} are then computed by the functions sfMejlholmDalgaard() and sfMejlholmDalgaardLAB(), respectively, using the product’s environmental characteristics already generated in the cold chain stage by the function sfCharacteristics().

Next, the function sfGrowthJameson() is executed to produce the numbers of L. monocytogenes and LAB in the RTE seafood units at the point of consumption (N_{Home LM}, N_{Home LAB} in CFU).

$$\begin{gathered} \left\{N_{Home LM ij}, N_{Home LAB ij}, {\ln q}_{Home LM ij},{\ln q}_{Home LAB ij} \right\}=sfGrowthJameson \\ \left(N_{cc LM ij} , {N_{cc LAB ij}, time}_{Home ij}, {\mathrm{e}\mathrm{x}\mathrm{p}(q}_{cc LM ij}), {\mathrm{e}\mathrm{x}\mathrm{p}(q}_{cc LAB ij}), {\mu }_{LM ij}, {\mu }_{LAB ij}, {MPD}_{LM i},{MPD}_{LAB i},\gamma ,{Unit}_{SizePack} \right) \\ i=1, 2, \dots , r; j=1, 2\dots ,{ c}_p \end{gathered}$$

2.1.14. Portioning Before Consumption

During serving at home, it is assumed that the consumer removes a serving size Serv_size (g) from the pack of RTE seafood of net weight Unit_SizePack g. Then, the number of L. monocytogenes cells in this small unit can be considered to be a sample from a Beta-binomial distribution. Furthermore, a moderate clustering of L. monocytogenes cells in the RTE seafood contained in the pack is assumed (dispersion, b = 1; [59]). The dispersion parameter and the number of portions that can be obtained from a pack are assumed to be independent of the microbial numbers. A function, sfPortioning(), was developed to simulate the microbial process of partitioning (cf. Appendix A.16).

2.2. Hazard and Risk Characterisation

Several dose-response relationships based on the exponential model are available for L. monocytogenes [16,21,26,79,80]. The dose-response model chosen to estimate the risk of listeriosis per serving of RTE seafood is that of Pouillot et al. [79] for the elderly population (>65 years old) with unknown underlying conditions. According to this model, the probability r that an ingested L. monocytogenes cell causes an invasive listeriosis follows a log ₁₀ normal distribution, with mean −12.83 and standard deviation 1.62. The function DRQuick() from the doseresponsemodels R package [79] was employed to estimate the marginal probabilities of invasive listeriosis in the elderly population, RiskServing_ij, from the input matrix of doses, N_{Portion ij}. The mean risk for every lot i (RiskLot_i) was calculated as a risk averaged across servings, j, and weighed by the lot-specific probability, Prob_{UnitPos Tested i}:

$${RiskLot}_i={\displaystyle \frac{\sum _{j=1}^{c_p}{RiskServing}_{ij}\times {Prob}_{UnitPos Tested i}}{c_p}}$$

2.3. QRA Model’s Ouputs

The model’s outputs were summarized at three stages: end of processing, point of consumption, and risk characterization. The descriptors at the end of processing were computed from the prevalence and the contamination matrix outputs of the function sfTesting() and included the following: (1) descriptive statistics (mean, median, and 95% confidence interval) of the mean concentration of L. monocytogenes (CFU/g) in the fraction of contaminated lots; (2) the prevalence of contaminated packs; and (3) the probability that a contaminated pack contains more than 10 CFU L. monocytogenes per g RTE seafood.

The model’s descriptors at the point of consumption were estimated from the outputs of the function sfColdChain(), applied to depict home storage, and encompassed the following: (1) descriptive statistics (mean, median, and 95% confidence interval) of the concentration of L. monocytogenes in any serving; (2) the prevalence of contaminated servings; (3) the probability that a contaminated serving contains more than 10 CFU L. monocytogenes per g RTE seafood; and (4) the probability that a contaminated serving contains more than 100 CFU L. monocytogenes per g RTE seafood.

The descriptors for risk characterization include the summary statistics, mean, median, and 2.5, 97.5, and 99.5 percentiles of the lot-level mean risk per serving RiskLot.

2.4. QRA Model’s Functionality: Reference and What-If Scenarios

To illustrate the utility of the QRA model, reference scenarios were separately run for the three types of RTE seafood studied: smoked brine-injected fish, smoked dry-salted fish, and gravad fish. In addition, five what-if scenarios were run for each type of RTE seafood, which assessed the following: (1) reduction of the between-lot mean prevalence of L. monocytogenes; (2) absence of initial contamination load on filleting knives; (3) no transfer of contamination during salting (smoked fish) or smearing with ingredients (gravad fish); (4) application of protective LAB cultures; and (5) reduction of storage temperature at home.

(a)

Reference, constituted by the baseline scenarios of the three RTE seafood products, with parameter values supported as much as possible by current data and, in their absence, by reasonable assumptions.

(b)

Reduction of L. monocytogenes prevalence in a lot of incoming fish, assessed by setting the parameter $\widehat{\alpha }$ of the Beta distribution regarding prevalence to half its original value (0.437).

(c)

Reduction in cross-contamination during filleting, assessed by assuming that filleting knives are cleaned/disinfected after filleting every fish, and therefore setting the parameter Init_Slicer of the function sfSlicer() to zero.

(d)

Absence of contamination during salting or smearing of fish fillets, represented by a zero probability of contamination during brine injection (Pcc_Brine = 0%) for smoked brined fish and zero probability of contamination during smearing with salt or curing agents (Pcc_Smear = 0%) for both smoked dry-salted fish and gravad fish.

(e)

Application of protective cultures when processing fish fillets, represented by an increase in the mean lot concentration of LAB in RTE seafood after packaging (${\overline{C}}_{0 LAB}$) by 5 log₁₀ CFU/g. Therefore, the minimum (${\overline{C}}_{0 LAB min}$), mode (${\overline{C}}_{0 LAB mode}$), and maximum (${\overline{C}}_{0 LAB max}$) parameters defining the Pert distribution regarding ${\overline{C}}_{0 LAB}$ were set to 4.00, 5.28, and 6.60 log₁₀ CFU/g, respectively, for the RTE products.

(f)

Lower home storage temperature, represented by a decrease of 1.5 °C in the mode and maximum storage temperature at home (Temp_{Home mode} = 5.5 and Temp_{Home max} = 11.4).

The reference and model scenarios for the three RTE seafood products were run by setting an initial contamination matrix size of r = 5000 lots and c =100 fish units in the function Lot2LotGen().

2.5. QRA Model’s Implementation

All the functions described in Section 2.1 were programmed in R (R Core Team [81]) and compiled in the package qraLm, which can be installed from the Github repository: https://github.com/WorldHealthOrganization/qraLm, accessed on 10 September 2024. The reference manual can be found at: https://WorldHealthOrganization.github.io/qraLm/reference/, accessed on 10 September 2024.

3. Results and Discussion

3.1. Reference Scenario and Comparison with Other QRA Models

The outcomes of the references scenarios suggested that, despite the high prevalence of lots contaminated with L. monocytogenes at the end of processing—38.7% for smoked brined fish, 34.4% for smoked dry-salted fish, and 52.2% for gravad fish—the mean concentrations in the fraction of contaminated lots would be very low (0.0028 CFU/g for smoked brined fish, 0.0021 CFU/g for smoked dry-salted fish, and 0.0029 CFU/g for gravad fish;

Table 5. Simulation outcomes of the exposure assessment model of L. monocytogenes in RTE seafood products at the end of processing for the reference and selected what-if scenarios. Simulations ran for 5000 lots consisting of 100 fish units each, where packs of final product weigh 260 g. Prevalence estimates are shown as proportions (scale 0–1).

Scenario	Mean Counts (CFU/g) in Contaminated Lots (Mean, Median, [95% CI])	Prevalence of Contaminated Lots	Prevalence of Contaminated Packs	P (N > 10 CFU/g in a Contaminated Pack) *
Smoked brined fish
Reference	0.0028; 0.0017 [1.18 × 10−4–0.0130]	0.3870	0.0814	0
Lower prevalence of contaminated lots	0.0023; 0.0014 [9.49 × 10−5–0.0097]	0.2343	0.0497	0
No initial load on filleting knives	0.0023; 0.0012 [7.69 × 10−5–0.0117]	0.3298	0.0652	0
No contamination during brining	0.0028; 0.0017 [1.20 × 10−4–0.0129]	0.3767	0.0792	0
Addition of LAB culture	0.0028; 0.0017 [1.18 × 10−4–0.0130]	0.3870	0.0814	0
Lower home storage temperature	0.0028; 0.0017 [1.18 × 10−4–0.0130]	0.3870	0.0814	0
Smoked dry-salted fish
Reference	0.0021; 0.0013 [1.28 × 10−4–0.0095]	0.3443	0.0649	0
Lower prevalence of contaminated lots	0.0016; 0.0009 [7.18 × 10−5–0.0072]	0.1611	0.0311	0
No initial load on filleting knives	0.0018; 0.0010 [8.71 × 10−5–0.0085]	0.2927	0.0512	0
No contamination during smearing	0.0012; 0.0005 [3.58 × 10−5–0.0063]	0.3554	0.0406	0
Addition of LAB culture	0.0021; 0.0013 [1.28 × 10−4–0.0095]	0.3443	0.0649	0
Lower home storage temperature	0.0021; 0.0013 [1.28 × 10−4–0.0095]	0.3443	0.0649	0
Gravad fish
Reference	0.0029; 0.0020 [4.25 × 10−4–0.0102]	0.5215	0.1114	0
Lower prevalence of contaminated lots	0.0022; 0.0016 [2.66 × 10−4–0.0080]	0.3124	0.0542	0
No initial load on filleting knives	0.0023; 0.0016 [2.41 × 10−4–0.0091]	0.4521	0.0886	0
No contamination during smearing	0.0026/0.0018 [3.97 × 10−4–0.0100]	0.5031	0.1137	0
Addition of LAB culture	0.0029; 0.0020 [4.25 × 10−4–0.0102]	0.5215	0.1114	0
Lower home storage temperature	0.0029; 0.0020 [4.25 × 10−4–0.0102]	0.5215	0.1114	0

(*) In all scenarios, P (N > 100 CFU/g in a contaminated pack) = 0.

). The prevalences of contaminated packs leaving the processing facilities were estimated at 8.14%, 6.49%, and 11.14%, respectively, although virtually no pack would contain numbers higher than 10 CFU/g (Table 5). Higher levels of contamination in cold-smoked and salt-cured salmon produced in Finland were predicted by a Markov-chain-based model developed by Pasonen et al. [30]. They estimated that ~10% of the contaminated packs exceeded the 100 CFU/g limit, whereas the prevalence of contaminated packs was 22% (95% CI: 20–25%). Although the assumptions of such a QRA differed in many instances from our model, it is plausible that their higher exposure estimates primarily obey the greater L. monocytogenes initial prevalence levels from the Finnish survey data used as inputs in their QRA model (in the range of 15.7–31.8% in the period between 2004 and 2010). By contrast, in our QRA model the mean of the between-lot L. monocytogenes prevalence was 14.85%, as modelled by a Beta (0.8741, 5.880) distribution, built upon survey data from multiple countries. Nonetheless, despite these differences, our model did coincide with that of Pasonen et al. [30] in the prediction that L. monocytogenes contaminated packs tend to have low bacterial concentrations, but on a few rare occasions the concentrations at the point of consumption would be in the order of hundreds of CFU/g (

Table 6. Simulation outcomes of the exposure assessment model of L. monocytogenes in RTE seafood products at the point of consumption for the reference and selected what-if scenarios. Simulations ran for 5000 lots consisting of 100 fish units each, where packs of final product weigh 260 g and a serving is a slice of product (32.5 g). Prevalence estimates are shown as proportions (scale 0–1).

Scenario	Counts (CFU/g) in Any Serving (Mean, Median, [95% CI])	Prevalence of Contaminated Servings	P (N > 10 CFU/g in a Contaminated Serving)	P (N > 100 CFU/g in a Contaminated Serving)
Smoked brined fish
Reference	103.6; 0.00 [0–0.2261]	0.0459	0.0122	0.0040
Lower prevalence of contaminated lots	103.4; 0.00 [0–0.0605]	0.0267	0.0110	0.0037
No initial load on filleting knives	76.39; 0.00 [0–0.1195]	0.0453	0.0111	0.0035
No contamination during brining	100.9; 0.00 [0–0.2139]	0.0444	0.0120	0.0039
Addition of LAB culture	0.0548; 0.00 [0–0.0886]	0.0416	0.0021	0.0002
Lower home storage temperature	6.4923; 0.00 [0–0.0994]	0.0413	0.0064	0.0016
Smoked dry-salted fish
Reference	125.9; 0.00 [0–0.1197]	0.0352	0.0100	0.0032
Lower prevalence of contaminated lots	58.58; 0.00 [0–0.0206]	0.0165	0.0082	0.0027
No initial load on filleting knives	104.0; 0.00 [0–0.0642]	0.0274	0.0092	0.0029
No contamination during smearing	74.26; 0.00 [0–0.0763]	0.0210	0.0060	0.0019
Addition of LAB culture	0.0340; 0.00 [0–0.0508]	0.0320	0.0015	8.1 x 10−5
Lower home storage temperature	3.8733; 0.00 [0–0.0567]	0.0318	0.0052	0.0012
Gravad fish
Reference	162.6; 0.00 [0–3.3787]	0.0735	0.0260	0.0104
Lower prevalence of contaminated lots	85.84; 0.00 [0–0.4191]	0.0353	0.0213	0.0085
No initial load on filleting knives	124.9; 0.00 [0–1.8050]	0.0579	0.0237	0.0094
No contamination during smearing	156.5; 0.00 [0–3.3617]	0.0744	0.0255	0.0101
Addition of LAB culture	0.2100; 0.00 [0–0.4503]	0.0660	0.0057	0.0005
Lower home storage temperature	12.486; 0.00 [0–1.0267]	0.0675	0.0157	0.0052

).

According to the present QRA model, at the end of processing a production lot of gravad fish would be more likely to be contaminated with L. monocytogenes than a production lot of smoked fish, and, in turn, if the smoked fish was salted by brine-injection the probability of lot contamination would be higher than if dry-salted. The prevalence of L. monocytogenes predicted by our QRA model for smoked (6.40–8.14%) and gravad fish packs (11.15%) are fairly in concordance with the joint results of EFSA’s EU-wide baseline survey and monitoring data, which encountered overall prevalence values of 8.42% (219/2602) in cold-smoked fish and 11.11% (30/270) in gravad fish, as compiled by Pérez-Rodríguez et al. [24].

Outcomes from other epidemiological surveys are also comparable to our simulation findings for gravad fish; namely, the incidence of 15% for cold stored gravlax from a Brazilian salmon processing plant (Cruz et al. [34]) and the incidence of 8% in gravlax fish from Icelandic markets [82]. In terms of enumeration estimates at the point of consumption (Table 6), the simulations predicted mean concentrations of L. monocytogenes in any serving (contaminated or not) of 103.6 CFU/g for smoked brined fish and 125.9 CFU/g for smoked dry-salted fish, and a higher mean concentration, at 162.6 CFU/g, for gravad fish, although it is worth mentioning that these high mean estimates are a result of a few highly contaminated servings, as implied by their 95% confidence intervals ([0–0.2261 CFU/g], [0–0.1197 CFU/g], and [0–3.3787 CFU/g], respectively). In the same order of magnitude of these intervals fell the median values of L. monocytogenes concentration found for cold-smoked fish (2.48 CFU/g) and for gravad fish (3.34 CFU/g) predicted by Pérez-Rodríguez et al. [24].

In their short-scope QRA model, the consumption of gravad fish was linked to greater exposure than smoked fish, in agreement with our model. Nonetheless, our model’s estimate for the level of exposure from a contaminated serving of smoked fish (log₁₀ (103.6/0.0459) = 3.35 log₁₀ CFU/g, worked out from Table 5) was considerably higher than the estimates from the QRA models of L. monocytogenes in contaminated servings of French cold-smoked salmon (mean: 1.38 log₁₀ CFU/g; Pouillot et al. [19]) and smoked salmon consumed in Navarra (mean: 2.25 log₁₀ CFU/g; Garrido et al. [27]).

On the other hand, our mean concentrations in any serving of smoked fish (103.6 CFU/g and 125.9 CFU/g) are close to the output of a QRA model of L. monocytogenes in Irish cold-smoked salmon, which predicted a mean concentration in any serving of 71.6 CFU/g (as determined by the product of the mean prevalence of contaminated servings of 18% and the mean concentration in contaminated servings of 10^2.6 CFU/g) (Dass [29]).

The probability that a serving is contaminated with L. monocytogenes followed a decreasing order: 7.35% for gravad fish, 4.59% for smoked brined fish, and 3.52% for smoked dry-salted fish (Table 6). The same rank order was observed for the probabilities that a contaminated serving contains more than 10 CFU/g and 100 CFU/g. These were in the ranges 1.00–2.60% and 0.32–1.04%, respectively. At least three-fold higher probabilities of L. monocytogenes in RTE fish servings were found by the EFSA’s generic short-scope QRA model [16], which estimated that proportions of 8.01% cold-smoked fish servings and 4.66% gravad fish servings would exceed a concentration of 100 CFU/g. These considerable deviations may stem from the following assumptions: (1) initial prevalences of L. monocytogenes (at end of processing) that were higher (17.4% for cold-smoked fish and 12.2% for gravad fish) than the throughputs of our model (6.49–8.14% for smoked fish and 11.14% for gravad fish); (2) initial concentrations of L. monocytogenes in the contaminated fraction (at end of processing) that were higher (0.867 log₁₀ CFU/g for cold-smoked fish and 1.011 log₁₀ CFU/g for gravad fish) than the throughputs of our model (0.0021–0.0028 CFU/g for smoked fish and 0.0029 CFU/g for gravad fish); and (3) serving sizes (49–66 g for smoked fish and 129–154 g for gravad fish in the elderly population) that were higher than the serving size of one slice (32.5 g) assumed in our model.

The present QRA model revealed a high between-lot variability in the risk of listeriosis associated with the consumption of a slice of RTE fish product (

Table 7. Statistics of the mean risk of invasive listeriosis per lot of RTE seafood products in the elderly population for the reference and selected what-if scenarios. The logarithm base 10 of the mean risk reduction attained by each scenario in comparison to the reference one is shown (log₁₀ RR).

Scenario	Mean	Median	2.5 pct	97.5 pct	99.5 pct	log10 RR
Smoked brined fish
Reference	9.751 × 10−8	6.572 × 10−11	3.693 × 10−15	3.064 × 10−7	3.836 × 10−6	-
Lower prevalence of contaminated lots	8.778 × 10−8	1.478 × 10−11	3.911 × 10−16	1.422 × 10−7	2.881 × 10−6	0.05
No initial load on filleting knives	6.920 × 10−8	2.431 × 10−11	7.494 × 10−16	1.233 × 10−7	2.272 × 10−6	0.15
No contamination during brining	9.272 × 10−8	6.164 × 10−11	6.477 × 10−15	2.566 × 10−7	3.287 × 10−6	0.02
Addition of LAB culture	2.718 × 10−10	5.399 × 10−12	2.409 × 10−15	1.918 × 10−9	8.972 × 10−9	2.55
Lower home temperature	1.112 × 10−8	1.769 × 10−11	2.842× 10−15	3.082 × 10−8	2.113 × 10−7	0.94
Smoked dry-salted fish
Reference	9.634 × 10−8	5.352 × 10−11	1.183 × 10−14	1.703 × 10−7	4.087 × 10−6	-
Lower prevalence of contaminated lots	5.113 × 10−8	9.071 × 10−12	1.868 × 10−15	5.483 × 10−8	1.513 × 10−6	0.28
No initial load on filleting knives	7.428 × 10−8	2.415 × 10−11	4.059 × 10−15	8.223 × 10−8	2.386 × 10−6	0.11
No contamination during smearing	5.984 × 10−8	6.953 × 10−11	1.835 × 10−13	9.772 × 10−8	2.608 × 10−6	0.20
Addition of LAB culture	1.693 × 10−10	4.462 × 10−12	6.363 × 10−15	1.502 × 10−9	5.888 × 10−9	2.76
Lower home temperature	6.899 × 10−9	1.607 × 10−11	9.780 × 10−15	1.991 × 10−8	1.817 × 10−7	1.14
Gravad fish
Reference	2.086 × 10−7	1.376 × 10−9	4.863 × 10−13	1.402 × 10−6	9.436 × 10−6	-
Lower prevalence of contaminated lots	1.133 × 10−7	1.900 × 10−10	3.531 × 10−14	5.372 × 10−7	3.815 × 10−6	0.27
No initial load on filleting knives	1.623 × 10−7	6.810 × 10−10	1.344 × 10−13	9.013 × 10−7	7.841 × 10−6	0.10
No contamination during smearing	2.037 × 10−7	1.347 × 10−9	6.725 × 10−13	1.368 × 10−6	1.036 × 10−5	0.01
Addition of LAB culture	1.019 × 10−9	2.940 × 10−11	7.731 × 10−14	7.457 × 10−9	2.977 × 10−8	2.31
Lower home temperature	2.761 × 10−8	2.343 × 10−10	1.328 × 10−13	1.500 × 10−7	9.657 × 10−7	0.88

). The heterogeneity in the mean risk from lot to lot is represented in Figure 2, Figure 3 and Figure 4 for smoked brine-injected fish, smoked dry-salted fish, and gravad fish, respectively. The mean lot risks of listeriosis per serving were estimated at 9.751 × 10⁻⁸ (median 6.572 × 10⁻¹¹) for smoked brined fish, 9.634 × 10⁻⁸ (median 5.352 × 10⁻¹¹) for smoked dry-salted fish, and 2.086 × 10⁻⁷ (1.376 × 10⁻⁹) for gravad fish (Table 7). On a serving basis of one slice, the mean risk linked to gravad fish was 0.33 or 0.34 log₁₀ higher than those of smoked brined or dry-salted fish. Similarly, the EFSA’s generic QRA model [16] calculated a 1.12 log₁₀ higher risk of listeriosis for sliced reduced-oxygen packaged gravad fish in comparison to sliced reduced-oxygen packaged cold-smoked fish in the elderly population, using the same dose-response model (Pouillot et al. [80]). Nevertheless, the median risk values were considerably higher in the EFSA model (4.37 × 10⁻⁹ for cold-smoked fish and 5.86 × 10⁻⁸ for gravad fish), which is likely to have arisen from the higher values of the model inputs discussed above.

In comparison to our QRA simulation, other models attained very different mean risk estimates for cold-smoked salmon, namely, the model of Pouillot et al. [20] (1.3 × 10⁻⁶ in the French elderly population) and the model of Garrido et al. [27] (4.17 × 10⁻¹⁰ in the normal and immunocompromised population of Navarra, Spain). Dass [29], in his listeriosis QRA model from Irish cold-smoked salmon, estimated a median risk in the high-risk population (6.165 × 10⁻¹¹), which was in closer agreement with our median risk estimates for smoked fish.

For comparison with the EFSA’s model outputs of number of annual illnesses [16], the mean risk estimates of our model were used to compute the mean number of invasive listeriosis in the 28 EU MS per year, based on the same estimated number of annual servings in the elderly population (1.99 × 10¹⁰ servings of smoked fish and 8.35 × 10⁹ servings of gravad fish). According to these values, our QRA model would predict a mean of 1940 annual cases of listeriosis linked to the consumption of smoked fish and 1741 annual cases linked to gravad fish. Using the same consumption data, EFSA [16] estimated a considerably lower number of listeriosis cases, at 201 and 230 cases in the elderly EU population linked to cold-smoked fish and gravad fish, respectively.

It is worth highlighting that the listeriosis dose-response model chosen has a high impact on the final risk estimate, and hence on the number of cases predicted. For instance, if the dose-response model of FAO-WHO [26] for increased susceptible population had been used, the present QRA model would have produced lower mean risk per serving estimates of 3.566 × 10⁻⁹ (median 4.511 × 10⁻¹³) for the smoked brined fish and 5.599 × 10⁻⁹ (median 9.776 × 10⁻¹²) for the gravad fish. These would have in turn predicted 72 and 48 annual listeriosis cases in the EU, 28 linked to these two products, respectively, assuming a consumption of 2.03 × 10¹⁰ servings of smoked fish and 8.53 × 10⁹ servings of gravad fish in the increased susceptible population (elderly plus pregnant population, taken from Pérez-Rodríguez et al. [24]).

3.2. What-If Scenarios

Among the what-if scenarios evaluated, reducing the prevalence of L. monocytogenes in fish entering the processing facilities would produce a decrease in the proportion of contaminated lots at the end of processing in about 40–53% for the three RTE seafood products, followed by the use of disinfected filleting knives (13–15%; Table 5). The absence of bacterial transfer during brine injection or smearing with ingredients would generate a lower effect on the prevalence of contaminated lots (decreasing it in 2.5–3.5% in smoked brined fish and gravad fish) because the frequency of contamination during salting assumed in the reference scenario is already low. Likewise, the what-if scenario that would reduce the prevalence of contaminated packs at the end of processing the most would be the reception of incoming fish with a lower prevalence of L. monocytogenes; this would reduce the proportion of outgoing contaminated packs from 8.14% to 4.97% in smoked brined fish, from 6.49% to 3.11% in smoked dry-salted fish, and from 11.14% to 5.42% in gravad fish (Table 5). The effectiveness of the what-if scenarios in decreasing the L. monocytogenes concentration in contaminated lots is, however, less evident in the mean and median estimates, given the high variability in L. monocytogenes contamination from lot to lot; however, the lowering effect can be appreciated in the 97.5th percentile. A scenario of absence of contamination during brine injection or smearing with ingredients would not practically lower the mean concentration of L. monocytogenes in contaminated lots at the end of processing, as it would lower the prevalence of contaminated lots and contaminated packs. As deduced by the 97.5th percentile in the concentration of contaminated lots (0.2261 CFU/g in smoked brined fish, 0.1197 CFU/g in smoked dry-salted fish, and 3.3787 CFU/g in gravad fish in the reference scenario), halving the prevalence of incoming fish into processing lines would be far more effective (to 0.0605 CFU/g, 0.0206 CFU/g, and 0.4191 CFU/g, respectively) than keeping filleting knives free of L. monocytogenes at all times (to 0.1195 CFU/g, 0.0642 CFU/g, and 1.8050 CFU/g, respectively; Table 5).

The effectiveness of the use of protective LAB cultures and the lower storage temperature at home can be compared with the other scenarios through the exposure assessment simulation outputs at the point of consumption (Table 6). Whereas reducing the contamination prevalence of incoming fish was the scenario that produced the highest reduction (52–54%) in the prevalence of contaminated servings (from 4.59% to 2.67% in smoked brined fish, from 3.52% to 1.65% in smoked dry-salted fish, and from 7.35% to 3.53% in gravad fish), in terms of L. monocytogenes numbers, the greatest control was achieved by the use of protective cultures, which led to the lowest L. monocytogenes concentration in any serving, and, as a consequence, the lowest probabilities of finding counts greater than 10 CFU/g or 100 CFU/g in a contaminated serving. However, the efficacy of the use of LAB cultures was found to depend on the RTE seafood product: for the smoked brined/dry-salted fish, the application of protective cultures produced a 2000/3000-fold decrease in the mean L. monocytogenes concentration in any serving and an 83/85% decrease in the probability that a contaminated serving contains more than 10 CFU/g (Table 6). In gravad fish, the efficacy of using protective culture was lower, causing a 780-fold decrease in the mean L. monocytogenes concentration in any serving and a 78% decrease in the probability that a contaminated serving contains more than 10 CFU/g. Nonetheless, considering the 100 CFU/g limit, there was no real difference in LAB culture effectiveness between the types of RTE product, since in all products the application of protective cultures led to a ~96% reduction in the probability of finding L. monocytogenes counts greater than 100 CFU/g in a contaminated serving.

After the use of protective cultures, the next “most effective scenario” to control L. monocytogenes up to the point of consumption was the maintenance of the RTE seafood product to lower storage temperatures at home. Reductions in the mean concentration in any serving and its 97.5th percentile were in the order of 13-to-32-fold and 2.1-to-3.3-fold, respectively. From L. monocytogenes mean concentrations at the point of consumption of 103.6 CFU/g (97.5th pct 0.2261 CFU/g), 125.9 CFU/g (97.5th pct 0.1197 CFU/g), and 162.6 CFU/g (97.5th pct 3.378 CFU/g) in the smoked brined fish, smoked dry-salted fish, and gravad fish, respectively, the sole proper storage practice taken by the consumers would reduce the concentrations to 6.4923 CFU/g (97.5th pct 0.0994 CFU/g), 3.8733 CFU/g (97.5th pct 0.0567 CFU/g), and 12.486 CFU/g (97.5th pct 1.0267 CFU/g) in the smoked brined fish, smoked dry-salted fish, and gravad fish, respectively. The frequency of contaminated servings above the 100 CFU L. monocytogenes per gram of serving would also be reduced by 50-60% in the three RTE products. In terms of prevalence of contaminated servings, however, the level of reduction attained by keeping colder home temperatures would be nearly the same as that of applying protective cultures (4.13% versus 4.16% for smoked brined fish, 3.18% versus 3.20% for smoked dry-salted fish, and 6.75% versus 6.60% for gravad fish; Table 6).

An inverse result on the relative importance of these two scenarios (colder home temperatures and added protective LAB cultures) in the concentration of L. monocytogenes in the servings of smoked salmon was purported by the listeriosis QRA model of Pouillot et al. [19], where they found that the mean storage temperature at the consumer phase (p = 10⁻²⁰) was a stronger determinant of exposure than the initial background microbiota (LAB) concentration (p = 0.002). Other QRA models whose outcomes helped to reinforce the relevance of improving consumer’s awareness of the correct storage conditions of RTE seafood and avoid temperature abuse were the ones developed by Dass [29], Garrido et al. [27], Pérez-Rodríguez et al. [24], and Pasonen et al. [30].

In our QRA model, the two scenarios relative to decreasing cross-contamination during secondary processing—namely, no initial contamination load on filleting knives and no bacterial transfer during brine injection or smearing with ingredients—were effective but, among the five what-if scenarios evaluated, yielded the least extent of reduction in L. monocytogenes prevalence and concentrations in RTE seafood servings (Table 6). In both smoked brined fish and gravad fish, maintaining filleting knives free of L. monocytogenes was more effective than the absence of contamination during brining/smearing in decreasing the mean concentration in any serving (23–25% reduction against 3.4–3.7%), the 97.5th percentile in the mean concentration in any serving (47% reduction against 0.5–1.2%), the proportion of contaminated servings with more than 10 CFU/g L. monocytogenes (9.0% reduction versus 1.6–1.9%), and the proportion of contaminated servings with more than 100 CFU/g L. monocytogenes (7.1–9.6% reduction versus 2.5–2.8%). The opposite outcome was encountered for the smoked dry-salted fish, where avoiding bacterial transfer during dry-salting would be more effective than keeping filleting knives disinfected, producing a reduction by 41% in the mean concentration in any serving versus 17%, a reduction by 44% in the prevalence of contaminated servings versus 22%, and a reduction by 40% in the proportion of servings with L. monocytogenes concentrations greater than 10 or 100 CFU/g against 8.0% or 9.3%, respectively.

In the three RTE seafood products, it was evident—within the constraints of the model assumptions—that the cross-contamination that could occur during processing (filleting and salting or smearing with curing agents) would be a less important determinant of the consumer’s exposure to L. monocytogenes when compared to the maintenance of proper cold storage at home or to the pathogen’s occurrence in fish entering processing.

The impact of the what-if scenarios on the between-lot mean risk per serving of RTE seafood product can be visualized in the density curves of Figure 2, Figure 3 and Figure 4, which also show the shifts in the 2.5th and the 97.5th percentiles. Among the scenarios evaluated, the application of protective LAB cultures (which increases the initial LAB counts by 5 log₁₀ CFU/g) would be the most effective strategy to hinder the growth of L. monocytogenes, thereby reaching a mean risk reduction of 2.55 log₁₀ for smoked brined fish, 2.76 log₁₀ for smoked dry-salted fish, and 2.31 log₁₀ for gravad fish. This strategy also reduced, to the greatest extent, the median (by >1.0 log₁₀) and the highest (97.5 and 99.5th by >2.2 log₁₀) percentiles of the mean lot risk posed by the three RTE products (Table 7). Decreasing the home storage temperature by 1.5 °C (mode and maximum) turned out to be the second most important risk control strategy, causing a risk reduction level in the range of 0.52–0.76 log₁₀ for the 50th percentile, 0.93–1.00 log₁₀ for the 97.5th percentile, and 0.98–1.25 log₁₀ for the 99.5th percentile among the three RTE products. In terms of mean risk reduction, keeping the RTE products at lower temperatures would bring about a mean risk reduction of 0.94 log₁₀ in smoked brined fish, 1.14 log₁₀ in smoked dry-salted fish, and 0.88 log₁₀ in gravad fish. A very similar prediction was computed in a QRA model in Irish cold-smoked salmon (Dass [29]), where, by fixing the storage temperature from 3–10 °C to 4 °C, the mean risk was decreased by 1.10 log₁₀ in the increased susceptibility population. In comparison to our findings, lower mean risk reductions have been reported by QRA models in analogous scenarios, such as decreasing the distribution temperature profiles by 1–2 °C caused a risk reduction of 0.60 log₁₀ (Pérez-Rodríguez et al. [24]), decreasing home storage temperature from 7 °C to 4 °C caused a median risk reduction of 0.52 log₁₀ (Pasonen et al. [30]), and all domestic temperatures with a mean temperature of 5.4 °C led to a risk reduction of 0.46 log₁₀ (Garrido et al. [27]).

In smoked dry-salted fish and gravad fish, decreasing the mean lot prevalence by half was the what-if scenario that followed in effectiveness, as attested by the mean risk reductions of 0.28 and 0.27 log₁₀, respectively. The risk lot medians linked to these two RTE products were also reduced by 0.77 and 0.86 log₁₀, respectively, whereas the high percentiles were reduced by 0.40- 0.50 log₁₀. This would imply that, if contamination was assumed to be carried over between lots (within the same processing facility), the mean lot risk would in turn be increased.

Nonetheless, in smoked brine-injected fish, the risk reduction attained by decreasing the prevalence of contaminated lots was low (0.05 log₁₀) and comparable to the effect of avoiding contamination during brine injection (0.02 log₁₀ risk reduction), although the former provided a slightly higher risk reduction, as also attested by the median (1.478 × 10⁻¹¹ versus 6.164 × 10⁻¹¹), the 97.5th percentile (1.422 × 10⁻⁷ versus 2.566 × 10⁻⁷), and the 99.5th percentile (2.881 × 10⁻⁶ versus 3.287 × 10⁻⁶). The relatively lower importance of the scenario of reduced initial prevalence, found in the present QRA, is in agreement with the findings of other authors, who determined that the strategies that reduced growth (such as the use of protective cultures or the reduced storage temperature tested in our simulations) present a greater impact on the final risk than strategies that reduce the occurrence [19,22,83]. Furthermore, measures aimed at reducing the initial prevalence of L. monocytogenes in fish are not well documented.

The strategy of keeping filleting knives free of L. monocytogenes at all times (i.e., the very ideal scenario of having no initial load on knives before filleting every fish) rendered a mean risk reduction that was comparable across the three RTE products evaluated—0.15 log₁₀ in smoked brined fish, 0.11 log₁₀ in smoked dry-salted fish, and 0.10 log₁₀ in gravad fish—with fairly important reductions in the median risk (0.30–0.43 log₁₀) and high percentiles (0.10–0.40 log₁₀). In relation to the absence of bacterial transfer during fillet smearing with salt or with curing agents, the effectiveness of this scenario to diminish the final risk of listeriosis depended on the type of product, which in gravad fish provided a very negligible mean risk reduction of 0.01 log₁₀, while in dry-salted fish the mean risk reduction was higher at 0.20 log₁₀ (very close to the effect of lowering the prevalence of contaminated lots). This difference is explained by the fact that, although the parameters assumed for smearing were the same in both products, gravad fish has intrinsic characteristics that are more favorable for L. monocytogenes growth. Thus, in gravad fish, the growth of L. monocytogenes after the product leaves the factory has a contribution to final risk that nearly overcomes the risk reduction effect of the absence of contamination during fillet smearing.

Taking together the results of the five scenarios across the three RTE seafood products, it became clear that the consumer’s conventional measure of maintaining the product at the recommended cold temperatures is very effective in decreasing the risk of listeriosis (overall mean risk reduction of ~1.00 log₁₀). Nevertheless, since this measure is at the same time the hardest to implement (and impossible to control), strategies that are placed at the processing level must be considered, namely reducing the prevalence of L. monocytogenes in incoming un-filleted fish (overall mean risk reduction of ~0.20 log₁₀) and/or slowing down the pathogen’s growth in the product. The latter can be achieved by the use of protective cultures (overall mean risk reduction of ~2.56 log₁₀) or even the modification of the physicochemical properties of the product (lower water activity, addition of organic acids or other growth inhibitors [84]). Avoiding cross-contamination is an effective control measure; however, the model has evidenced that, once L. monocytogenes has entered the processing lot, minimizing the chances for cross-contamination during filleting, salting, or smearing with curing agents would result in a comparatively small risk reduction, which is lower for gravad fish (0.01–0.10 log₁₀) compared to smoked fish (0.02–0.15 log₁₀).

4. Conclusions and Perspectives

We have developed a robust four-module quantitative risk assessment (QRA) model of L. monocytogenes in ready-to-eat (RTE) fish products. This model, composed of 18 functions and a total of 128 parameters, is freely available and designed with flexibility in mind. It provides, for the first time, the ability to check for the efficacy of batch testing on risk. Our findings highlight the critical importance of strict hygiene practices during processing, the benefits of using protective LAB cultures, and the necessity of proper storage conditions by consumers to effectively mitigate the risk of listeriosis. The model’s inherent adaptability ensures that food safety authorities can tailor interventions to meet specific needs, enhancing food safety measures for various RTE seafood products.

As new data become available, the parameters can be updated, and the functions, programmed in R software, can be reassembled to represent different seafood products. For example, to adapt the model for exposure assessment of sashimi, the smoking function (sfSmoking()) can be omitted, while for ceviche, it would be appropriate to remove the smoking function and add the maceration function (sfMaceration()), adjusting the parameters to suit these specific raw fish dishes.

This QRA model is intended to assist food safety authorities worldwide in obtaining risk estimates and evaluating risk management options tailored to their specific RTE seafood chains and product characteristics. Moreover, the model facilitates the evaluation of a variety of intervention strategies (such as the use of bacteriocinogenic cultures, organic acids, extended smoking periods) by allowing adjustments to interaction parameters in the Jameson model or modifying growth rates for L. monocytogenes and lactic acid bacteria (LAB). This QRA model not only provides a framework for assessing the risk of listeriosis from the consumption of smoked and gravad fish but also offers the potential to apply its methodology to a broader range of seafood products, thereby supporting global efforts in improving food safety.