Food Microbiology xxx (2014) 1e11
Contents lists available at ScienceDirect
Food Microbiology journal homepage: www.elsevier.com/locate/fm
Comparison of individual-based modeling and population approaches for prediction of foodborne pathogens growth Jean-Christophe Augustin a, *, Rachel Ferrier a, b, Bernard Hezard b, Adrienne Lintz b, Valérie Stahl b a b
Université Paris-Est, Ecole Nationale Vétérinaire d’Alfort, Maisons-Alfort F-94704, France Aérial, Institut technique agro-industriel, Parc d’Innovation, 250 rue Laurent Fries, Illkirch F-67412, France
a r t i c l e i n f o
a b s t r a c t
Article history: Available online xxx
Individual-based modeling (IBM) approach combined with the microenvironment modeling of vacuumpacked cold-smoked salmon was more effective to describe the variability of the growth of a few Listeria monocytogenes cells contaminating irradiated salmon slices than the traditional population models. The IBM approach was particularly relevant to predict the absence of growth in 25% (5 among 20) of artificially contaminated cold-smoked salmon samples stored at 8 C. These results confirmed similar observations obtained with smear soft cheese (Ferrier et al., 2013). These two different food models were used to compare the IBM/microscale and population/macroscale modeling approaches in more global exposure and risk assessment frameworks taking into account the variability and/or the uncertainty of the factors influencing the growth of L. monocytogenes. We observed that the traditional population models significantly overestimate exposure and risk estimates in comparison to IBM approach when contamination of foods occurs with a low number of cells ( > > > dN 1 > > > ¼ $mmax $ð1 expðN Nmax ÞÞ < dt 1 þ expð Q Þ with Q0 ¼ ln expðKÞ 1 > dQ > > > > > dt ¼ mmax > :
observed that exposure and risk estimates were still highly dependent on the modeling approach and that population models
where K, the initial physiological state characteristic, is equal to: K ¼ mmax $lag
Please cite this article in press as: Augustin, J.-C., et al., Comparison of individual-based modeling and population approaches for prediction of foodborne pathogens growth, Food Microbiology (2014), http://dx.doi.org/10.1016/j.fm.2014.04.006
10
J.-C. Augustin et al. / Food Microbiology xxx (2014) 1e11
The effect of storage temperature and food characteristics on
mmax and lag was described by the following multiplicative function with interactions derived from cardinal and square root models:
gðT; pH; aw Þ ¼ CM2 ðTÞCM1 ðpHÞSR1 ðaw ÞxðT; pH; aw Þ
where c ¼ 7.6, Tinf ¼ 3.6 C, Tsup ¼ 17.3 C, pHinf ¼ 4.34, pHsup ¼ 5.93, aw,inf ¼ 0.917, and aw,sup ¼ 0.988. Individual-cell lag times, lagi, were derived from individual physiological states k, following an extreme value type II with parameters a and b (Guillier and Augustin, 2006):
with
CMn ðXÞ ¼
; X Xmin
0
8 >
$ Xopt Xmin $ X Xopt Xopt Xmax $ ðn 1Þ$Xopt þ Xmin nX : Xopt Xmin
; Xmin < X < Xmax ; X Xmax
0
and SRn X ¼
(
0 X Xmin Xopt Xmin
n
; Xmin < X Xopt
8
< expðT=cÞ exp Tinf c . with p T ¼ > : exp Tsup c exp Tinf c
; T Tinf ; Tinf < T < Tsup ; ; T Tsup
0 ;pH pHinf 8 > < expðpHÞexp pHinf ;pHinf < pH < pHsup ; p pH ¼ > : exp pHsup exp pHinf 1 8 > > > 0 > > > > < aw a w;inf and p aw ¼ a aw;inf > w;sup > > > > > > 1 :
lnðE½kÞ ¼ 0:0103$lnðKÞ5 þ 0:0065$lnðKÞ4 0:039$lnðKÞ3 þ 0:0586$lnðKÞ2 þ 1:1941$lnðKÞ þ 0:1549 where E[k] and S[k] are the expected value and the standard deviation of k, respectively. K is the population initial physiological state characteristic. References
where Xmin, Xopt and Xmax are the minimal, optimal and maximal temperature, pH and water activity for growth. In the IBM approach, the following equations were used to describe the impact of growth conditions on the single-cell growth probability, p, of L. monocytogenes (Augustin and CzarneckaKwasiborski, 2012):
1
(10)
with S½k ¼ e1:004$lnðE½kÞ0:447 and
uðX Þ Q i and u X 2$ jsi 1 u Xj
Xopt X Xopt Xmin
1:1642 S½k $S½k and b ¼ 0:3658 0:3658
;pH pHsup
; aw aw;inf ; aw;inf < aw < aw;sup ; ; aw aw;sup
Anonymous, 2003. Quantitative Assessment of the Relative Risk to Public Health From Foodborne Listeria monocytogenes Among Selected Categories of Readyto-eat Foods. U.S. Food and Drug Administration/US. Department of Agriculture-Food Safety and Inspection Service. Avaible at: http://www.fsis. usda.gov/Science/Risk_Assessment/index.asp. Anonymous, 2004a. Risk Assessment of Listeria monocytogenes in Ready-to-eat Foods. In: Microbial Risk Assessment Serie No 5. World Health Organisation/ Food and Agriculture Organization of the United Nations, Geneva, Switzerland. Anonymous, 2004b. ISO 21807:2004 Microbiology of Food and Animal Feeding Stuffs. Determination of Water Activity. International Organization for Standardization, Geneva, Switzerland. Anonymous, 2005. Commission regulation (EC) No 2073/2005 of 15 november 2005 on microbiological criteria for foodstuffs. Off. J. Eur. Union L338 (1). Anonymous, 2008. Exposure Assessment of Microbiological Hazards in Food. In: Microbiological Risk Assessment Series No 7. World Health Organisation/Food and Agriculture Organization of the United Nations, Geneva, Switzerland. Anonymous, 2009. NF V 04-035 Lait et produits laitiers e Détermination du pH. Association Française de Normalisation, Saint-Denis, France. Augustin, J.-C., Bergis, H., Midelet-Bourdin, G., Cornu, M., Couvert, O., Denis, C., Huchet, V., Lemonnier, S., Pinon, A., Vialette, M., Zuliani, V., Stahl, V., 2011. Design of challenge testing experiments to assess the variability of Listeria monocytogenes growth in foods. Food Microbiol. 28, 746e754. Augustin, J.-C., Czarnecka-Kwasiborski, A., 2012. Single-Cell growth probability of Listeria monocytogenes at suboptimal temperature, pH, and water Activity. Front. Microbiol. 3, 157 http://dx.doi.org/10.3389/fmicb.2012.00157. Augustin, J.-C., Zuliani, V., Cornu, M., Guillier, L., 2005. Growth rate and growth probability of Listeria monocytogenes in dairy, meat and seafood products in suboptimal conditions. J. Appl. Microbiol. 99, 1019e1042. Bates, D.M., 2010. lme4: Mixed-Effects Modeling with R. Available at:https://r-forge. r-project.org/projects/lme4/2010. Baranyi, J., 1998. Comparison of stochastic and deterministic concepts of bacterial lag. J. Theor. Biol. 192, 403e408. Baranyi, J., Roberts, T.A., 1994. A dynamic approach to predicting bacterial growth in food. Int. J. Food Microbiol. 23, 277e294. Cabezas, L., Marcos, A., Esteban, M.A., Fernández-Salguero, J., Alcalá, M., 1988. Improved equation for cryoscopic estimation of water activity in cheese. Food Chem. 30, 59e66. Couvert, O., Pinon, A., Bergis, H., Bourdichon, F., Carlin, F., Cornu, M., Denis, C., Gnanou Besse, N., Guillier, L., Jamet, E., Mettler, E., Stahl, V., Thuault, D., Zuliani, V., Augustin, J.-C., 2010. Validation of a stochastic modelling approach for Listeria monocytogenes growth in refrigerated foods. Int. J. Food Microbiol. 144, 236e242.
Please cite this article in press as: Augustin, J.-C., et al., Comparison of individual-based modeling and population approaches for prediction of foodborne pathogens growth, Food Microbiology (2014), http://dx.doi.org/10.1016/j.fm.2014.04.006
J.-C. Augustin et al. / Food Microbiology xxx (2014) 1e11 Elfwing, A., LeMarc, Y., Baranyi, J., Ballagi, A., 2004. Observing growth and division of large numbers of individual bacteria by image analysis. Appl. Environ. Microbiol. 70, 675e678. European Food Safety Authority, EFSA, 2013. Analysis of the baseline survey on the prevalence of Listeria monocytogenes in certain ready-to-eat (RTE) foods in the EU, 2010-2011 part A: Listeria monocytogenes prevalence estimates. EFSA J. 11, 3241, 75. Ferrier, R., Hezard, B., Lintz, A., Stahl, V., Augustin, J.-C., 2013. Combining individualbased modeling and food microenvironment descriptions to predict the growth of Listeria monocytogenes on smear soft cheese. Appl. Environ. Microbiol. 79, 5870e5881. Guillier, L., Augustin, J.-C., 2006. Modelling the individual cell lag time distributions of Listeria monocytogenes as a function of the physiological state and the growth conditions. Int. J. Food Microbiol. 111, 241e251. Guillier, L., Pardon, P., Augustin, J.-C., 2005. Influence of stress on individual lag time distributions of Listeria monocytogenes. Appl. Environ. Microbiol. 71, 2940e 2948. Grimm, V., 1999. Ten years of individual-based modelling in ecology: what have we learned and what could we learn in the future? Ecol. Model. 115, 129e148. Koutsoumanis, K., 2008. A study on the variability in the growth limits of individual cells and its effect on the behavior of microbial populations. Int. J. Food Microbiol. 128, 116e121. Koutsoumanis, K., Angelidis, A.S., 2007. Probabilistic modeling approach for evaluating the compliance of ready-to-eat foods with new European Union safety criteria for Listeria monocytogenes. Appl. Environ. Microbiol. 73, 4996e5004. Koutsoumanis, K., Lianou, A., 2013. Stochasticity in colonial growth dynamics of individual bacterial cells. Appl. Environ. Microbiol. 79, 2294e2301. Koutsoumanis, K.P., Sofos, J.N., 2005. Effect of inoculum size on the combined temperature, pH and aw limits for growth of Listeria monocytogenes. Int. J. Food Microbiol. 104, 83e91. Kreft, J.U., Picioreanu, C., Wimpenny, J.W., van Loosdrecht, M., 2001. Individualbased modelling of biofilms. Microbiology 147, 2897e2912.
11
Le Marc, Y., Huchet, V., Bourgeois, C.M., Guyonnet, J.P., Mafart, P., Thuault, D., 2002. Modelling the growth kinetics of Listeria as a function of temperature, pH and organic acid concentration. Int. J. Food Microbiol. 73, 219e237. Mataragas, M., Zwietering, M.H., Skandamis, P.N., Drosinos, E.H., 2010. Quantitative microbiological risk assessment as a tool to obtain useful information for risk managers - specific application to Listeria monocytogenes and ready-to-eat meat products. Int. J. Food Microbiol. 141, S170eS179. Métris, A., George, S.M., Mackey, B.M., Baranyi, J., 2008. Modeling the variability of single-cell lag times for Listeria innocua populations after sublethal and lethal heat treatments. Appl. Environ. Microbiol. 74, 6949e6955. Nauta, M.J., 2000. Separation of uncertainty and variability in quantitative microbial risk assessment models. Int. J. Food Microbiol. 57, 9e18. Pin, C., Baranyi, J., 2006. Kinetics of single cells: observation and modeling of a stochastic process. Appl. Environ. Microbiol. 72, 2163e2169. Pouillot, R., Goulet, V., Delignette-Muller, M.L., Mahé, A., Cornu, M., 2009. Quantitative risk assessment of Listeria monocytogenes in French cold-smoked salmon: II. Risk characterization. Risk Anal. 29, 806e819. Pouillot, R., Miconnet, N., Afchain, A.L., Delignette-Muller, M.L., Beaufort, A., Rosso, L., Denis, J.-B., Cornu, M., 2007. Quantitative risk assessment of Listeria monocytogenes in French cold-smoked salmon: I. Quantitative exposure assessment. Risk Anal. 27, 683e700. Ross, T., Rasmussen, S., Fazil, A., Paoli, G., Sumner, J., 2009a. Quantitative risk assessment of Listeria monocytogenes in ready-to-eat meats in Australia. Int. J. Food Microbiol. 131, 128e137. Ross, T., Rasmussen, S., Sumner, J., 2009b. Using a quantitative risk assessment to mitigate risk of Listeria monocytogenes in ready-to-eat meats in Australia. Food Control 20, 1058e1062. Tenenhaus-Aziza, F., Daudin, J.-J., Maffre, A., Sanaa, M., 2014. Risk-based approach for microbiological food safety management in the dairy industry: the case of Listeria monocytogenes in soft cheese made from pasteurized milk. Risk Anal. 34, 56e74.
Please cite this article in press as: Augustin, J.-C., et al., Comparison of individual-based modeling and population approaches for prediction of foodborne pathogens growth, Food Microbiology (2014), http://dx.doi.org/10.1016/j.fm.2014.04.006
Contents lists available at ScienceDirect
Food Microbiology journal homepage: www.elsevier.com/locate/fm
Comparison of individual-based modeling and population approaches for prediction of foodborne pathogens growth Jean-Christophe Augustin a, *, Rachel Ferrier a, b, Bernard Hezard b, Adrienne Lintz b, Valérie Stahl b a b
Université Paris-Est, Ecole Nationale Vétérinaire d’Alfort, Maisons-Alfort F-94704, France Aérial, Institut technique agro-industriel, Parc d’Innovation, 250 rue Laurent Fries, Illkirch F-67412, France
a r t i c l e i n f o
a b s t r a c t
Article history: Available online xxx
Individual-based modeling (IBM) approach combined with the microenvironment modeling of vacuumpacked cold-smoked salmon was more effective to describe the variability of the growth of a few Listeria monocytogenes cells contaminating irradiated salmon slices than the traditional population models. The IBM approach was particularly relevant to predict the absence of growth in 25% (5 among 20) of artificially contaminated cold-smoked salmon samples stored at 8 C. These results confirmed similar observations obtained with smear soft cheese (Ferrier et al., 2013). These two different food models were used to compare the IBM/microscale and population/macroscale modeling approaches in more global exposure and risk assessment frameworks taking into account the variability and/or the uncertainty of the factors influencing the growth of L. monocytogenes. We observed that the traditional population models significantly overestimate exposure and risk estimates in comparison to IBM approach when contamination of foods occurs with a low number of cells ( > > > dN 1 > > > ¼ $mmax $ð1 expðN Nmax ÞÞ < dt 1 þ expð Q Þ with Q0 ¼ ln expðKÞ 1 > dQ > > > > > dt ¼ mmax > :
observed that exposure and risk estimates were still highly dependent on the modeling approach and that population models
where K, the initial physiological state characteristic, is equal to: K ¼ mmax $lag
Please cite this article in press as: Augustin, J.-C., et al., Comparison of individual-based modeling and population approaches for prediction of foodborne pathogens growth, Food Microbiology (2014), http://dx.doi.org/10.1016/j.fm.2014.04.006
10
J.-C. Augustin et al. / Food Microbiology xxx (2014) 1e11
The effect of storage temperature and food characteristics on
mmax and lag was described by the following multiplicative function with interactions derived from cardinal and square root models:
gðT; pH; aw Þ ¼ CM2 ðTÞCM1 ðpHÞSR1 ðaw ÞxðT; pH; aw Þ
where c ¼ 7.6, Tinf ¼ 3.6 C, Tsup ¼ 17.3 C, pHinf ¼ 4.34, pHsup ¼ 5.93, aw,inf ¼ 0.917, and aw,sup ¼ 0.988. Individual-cell lag times, lagi, were derived from individual physiological states k, following an extreme value type II with parameters a and b (Guillier and Augustin, 2006):
with
CMn ðXÞ ¼
; X Xmin
0
8 >
$ Xopt Xmin $ X Xopt Xopt Xmax $ ðn 1Þ$Xopt þ Xmin nX : Xopt Xmin
; Xmin < X < Xmax ; X Xmax
0
and SRn X ¼
(
0 X Xmin Xopt Xmin
n
; Xmin < X Xopt
8
< expðT=cÞ exp Tinf c . with p T ¼ > : exp Tsup c exp Tinf c
; T Tinf ; Tinf < T < Tsup ; ; T Tsup
0 ;pH pHinf 8 > < expðpHÞexp pHinf ;pHinf < pH < pHsup ; p pH ¼ > : exp pHsup exp pHinf 1 8 > > > 0 > > > > < aw a w;inf and p aw ¼ a aw;inf > w;sup > > > > > > 1 :
lnðE½kÞ ¼ 0:0103$lnðKÞ5 þ 0:0065$lnðKÞ4 0:039$lnðKÞ3 þ 0:0586$lnðKÞ2 þ 1:1941$lnðKÞ þ 0:1549 where E[k] and S[k] are the expected value and the standard deviation of k, respectively. K is the population initial physiological state characteristic. References
where Xmin, Xopt and Xmax are the minimal, optimal and maximal temperature, pH and water activity for growth. In the IBM approach, the following equations were used to describe the impact of growth conditions on the single-cell growth probability, p, of L. monocytogenes (Augustin and CzarneckaKwasiborski, 2012):
1
(10)
with S½k ¼ e1:004$lnðE½kÞ0:447 and
uðX Þ Q i and u X 2$ jsi 1 u Xj
Xopt X Xopt Xmin
1:1642 S½k $S½k and b ¼ 0:3658 0:3658
;pH pHsup
; aw aw;inf ; aw;inf < aw < aw;sup ; ; aw aw;sup
Anonymous, 2003. Quantitative Assessment of the Relative Risk to Public Health From Foodborne Listeria monocytogenes Among Selected Categories of Readyto-eat Foods. U.S. Food and Drug Administration/US. Department of Agriculture-Food Safety and Inspection Service. Avaible at: http://www.fsis. usda.gov/Science/Risk_Assessment/index.asp. Anonymous, 2004a. Risk Assessment of Listeria monocytogenes in Ready-to-eat Foods. In: Microbial Risk Assessment Serie No 5. World Health Organisation/ Food and Agriculture Organization of the United Nations, Geneva, Switzerland. Anonymous, 2004b. ISO 21807:2004 Microbiology of Food and Animal Feeding Stuffs. Determination of Water Activity. International Organization for Standardization, Geneva, Switzerland. Anonymous, 2005. Commission regulation (EC) No 2073/2005 of 15 november 2005 on microbiological criteria for foodstuffs. Off. J. Eur. Union L338 (1). Anonymous, 2008. Exposure Assessment of Microbiological Hazards in Food. In: Microbiological Risk Assessment Series No 7. World Health Organisation/Food and Agriculture Organization of the United Nations, Geneva, Switzerland. Anonymous, 2009. NF V 04-035 Lait et produits laitiers e Détermination du pH. Association Française de Normalisation, Saint-Denis, France. Augustin, J.-C., Bergis, H., Midelet-Bourdin, G., Cornu, M., Couvert, O., Denis, C., Huchet, V., Lemonnier, S., Pinon, A., Vialette, M., Zuliani, V., Stahl, V., 2011. Design of challenge testing experiments to assess the variability of Listeria monocytogenes growth in foods. Food Microbiol. 28, 746e754. Augustin, J.-C., Czarnecka-Kwasiborski, A., 2012. Single-Cell growth probability of Listeria monocytogenes at suboptimal temperature, pH, and water Activity. Front. Microbiol. 3, 157 http://dx.doi.org/10.3389/fmicb.2012.00157. Augustin, J.-C., Zuliani, V., Cornu, M., Guillier, L., 2005. Growth rate and growth probability of Listeria monocytogenes in dairy, meat and seafood products in suboptimal conditions. J. Appl. Microbiol. 99, 1019e1042. Bates, D.M., 2010. lme4: Mixed-Effects Modeling with R. Available at:https://r-forge. r-project.org/projects/lme4/2010. Baranyi, J., 1998. Comparison of stochastic and deterministic concepts of bacterial lag. J. Theor. Biol. 192, 403e408. Baranyi, J., Roberts, T.A., 1994. A dynamic approach to predicting bacterial growth in food. Int. J. Food Microbiol. 23, 277e294. Cabezas, L., Marcos, A., Esteban, M.A., Fernández-Salguero, J., Alcalá, M., 1988. Improved equation for cryoscopic estimation of water activity in cheese. Food Chem. 30, 59e66. Couvert, O., Pinon, A., Bergis, H., Bourdichon, F., Carlin, F., Cornu, M., Denis, C., Gnanou Besse, N., Guillier, L., Jamet, E., Mettler, E., Stahl, V., Thuault, D., Zuliani, V., Augustin, J.-C., 2010. Validation of a stochastic modelling approach for Listeria monocytogenes growth in refrigerated foods. Int. J. Food Microbiol. 144, 236e242.
Please cite this article in press as: Augustin, J.-C., et al., Comparison of individual-based modeling and population approaches for prediction of foodborne pathogens growth, Food Microbiology (2014), http://dx.doi.org/10.1016/j.fm.2014.04.006
J.-C. Augustin et al. / Food Microbiology xxx (2014) 1e11 Elfwing, A., LeMarc, Y., Baranyi, J., Ballagi, A., 2004. Observing growth and division of large numbers of individual bacteria by image analysis. Appl. Environ. Microbiol. 70, 675e678. European Food Safety Authority, EFSA, 2013. Analysis of the baseline survey on the prevalence of Listeria monocytogenes in certain ready-to-eat (RTE) foods in the EU, 2010-2011 part A: Listeria monocytogenes prevalence estimates. EFSA J. 11, 3241, 75. Ferrier, R., Hezard, B., Lintz, A., Stahl, V., Augustin, J.-C., 2013. Combining individualbased modeling and food microenvironment descriptions to predict the growth of Listeria monocytogenes on smear soft cheese. Appl. Environ. Microbiol. 79, 5870e5881. Guillier, L., Augustin, J.-C., 2006. Modelling the individual cell lag time distributions of Listeria monocytogenes as a function of the physiological state and the growth conditions. Int. J. Food Microbiol. 111, 241e251. Guillier, L., Pardon, P., Augustin, J.-C., 2005. Influence of stress on individual lag time distributions of Listeria monocytogenes. Appl. Environ. Microbiol. 71, 2940e 2948. Grimm, V., 1999. Ten years of individual-based modelling in ecology: what have we learned and what could we learn in the future? Ecol. Model. 115, 129e148. Koutsoumanis, K., 2008. A study on the variability in the growth limits of individual cells and its effect on the behavior of microbial populations. Int. J. Food Microbiol. 128, 116e121. Koutsoumanis, K., Angelidis, A.S., 2007. Probabilistic modeling approach for evaluating the compliance of ready-to-eat foods with new European Union safety criteria for Listeria monocytogenes. Appl. Environ. Microbiol. 73, 4996e5004. Koutsoumanis, K., Lianou, A., 2013. Stochasticity in colonial growth dynamics of individual bacterial cells. Appl. Environ. Microbiol. 79, 2294e2301. Koutsoumanis, K.P., Sofos, J.N., 2005. Effect of inoculum size on the combined temperature, pH and aw limits for growth of Listeria monocytogenes. Int. J. Food Microbiol. 104, 83e91. Kreft, J.U., Picioreanu, C., Wimpenny, J.W., van Loosdrecht, M., 2001. Individualbased modelling of biofilms. Microbiology 147, 2897e2912.
11
Le Marc, Y., Huchet, V., Bourgeois, C.M., Guyonnet, J.P., Mafart, P., Thuault, D., 2002. Modelling the growth kinetics of Listeria as a function of temperature, pH and organic acid concentration. Int. J. Food Microbiol. 73, 219e237. Mataragas, M., Zwietering, M.H., Skandamis, P.N., Drosinos, E.H., 2010. Quantitative microbiological risk assessment as a tool to obtain useful information for risk managers - specific application to Listeria monocytogenes and ready-to-eat meat products. Int. J. Food Microbiol. 141, S170eS179. Métris, A., George, S.M., Mackey, B.M., Baranyi, J., 2008. Modeling the variability of single-cell lag times for Listeria innocua populations after sublethal and lethal heat treatments. Appl. Environ. Microbiol. 74, 6949e6955. Nauta, M.J., 2000. Separation of uncertainty and variability in quantitative microbial risk assessment models. Int. J. Food Microbiol. 57, 9e18. Pin, C., Baranyi, J., 2006. Kinetics of single cells: observation and modeling of a stochastic process. Appl. Environ. Microbiol. 72, 2163e2169. Pouillot, R., Goulet, V., Delignette-Muller, M.L., Mahé, A., Cornu, M., 2009. Quantitative risk assessment of Listeria monocytogenes in French cold-smoked salmon: II. Risk characterization. Risk Anal. 29, 806e819. Pouillot, R., Miconnet, N., Afchain, A.L., Delignette-Muller, M.L., Beaufort, A., Rosso, L., Denis, J.-B., Cornu, M., 2007. Quantitative risk assessment of Listeria monocytogenes in French cold-smoked salmon: I. Quantitative exposure assessment. Risk Anal. 27, 683e700. Ross, T., Rasmussen, S., Fazil, A., Paoli, G., Sumner, J., 2009a. Quantitative risk assessment of Listeria monocytogenes in ready-to-eat meats in Australia. Int. J. Food Microbiol. 131, 128e137. Ross, T., Rasmussen, S., Sumner, J., 2009b. Using a quantitative risk assessment to mitigate risk of Listeria monocytogenes in ready-to-eat meats in Australia. Food Control 20, 1058e1062. Tenenhaus-Aziza, F., Daudin, J.-J., Maffre, A., Sanaa, M., 2014. Risk-based approach for microbiological food safety management in the dairy industry: the case of Listeria monocytogenes in soft cheese made from pasteurized milk. Risk Anal. 34, 56e74.
Please cite this article in press as: Augustin, J.-C., et al., Comparison of individual-based modeling and population approaches for prediction of foodborne pathogens growth, Food Microbiology (2014), http://dx.doi.org/10.1016/j.fm.2014.04.006
