The Annals of Applied Statistics

Bayesian modeling of bacterial growth for multiple populations

A. Paula Palacios, J. Miguel Marín, Emiliano J. Quinto, and Michael P. Wiper

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Bacterial growth models are commonly used for the prediction of microbial safety and the shelf life of perishable foods. Growth is affected by several environmental factors such as temperature, acidity level and salt concentration. In this study, we develop two models to describe bacterial growth for multiple populations under both equal and different environmental conditions. First, a semi-parametric model based on the Gompertz equation is proposed. Assuming that the parameters of the Gompertz equation may vary in relation to the running conditions under which the experiment is performed, we use feedforward neural networks to model the influence of these environmental factors on the growth parameters. Second, we propose a more general model which does not assume any underlying parametric form for the growth function. Thus, we consider a neural network as a primary growth model which includes the influencing environmental factors as inputs to the network. One of the main disadvantages of neural networks models is that they are often very difficult to tune, which complicates fitting procedures. Here, we show that a simple Bayesian approach to fitting these models can be implemented via the software package WinBugs. Our approach is illustrated using real experimental Listeria monocytogenes growth data.

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Ann. Appl. Stat., Volume 8, Number 3 (2014), 1516-1537.

First available in Project Euclid: 23 October 2014

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Bacterial population modeling growth functions neural networks Bayesian inference


Palacios, A. Paula; Marín, J. Miguel; Quinto, Emiliano J.; Wiper, Michael P. Bayesian modeling of bacterial growth for multiple populations. Ann. Appl. Stat. 8 (2014), no. 3, 1516--1537. doi:10.1214/14-AOAS720.

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  • Andrieu, C., de Freitas, N. and Doucet, A. (2001). Robust full Bayesian learning for radial basis networks. Neural Comput. 13 2359–2407.
  • Augustin, J. C. and Carlier, V. (2000). Modelling the growth rate of Listeria monocytogenes with a multiplicative type model including interactions between environmental factors. Int. J. Food Microbiol. 56 53–70.
  • Baranyi, J., Ross, T., McMeekin, T. A. and Roberts, T. A. (1996). Effects of parameterization on the performance of empirical models used in predictive microbiology. Food Microbiology 13 83–91.
  • Brooks, S. P. and Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Statist. 7 434–455.
  • CAC (1996). Principles and guidelines for the application of microbiological risk assessment. Technical report, Codex Alimentarius Commission, CX/FH 96/10 FAO, Rome, Italy.
  • CEC (2002). Regulation (EC) 178/2002 of 28 January 2002 on laying down the general principles and requirements of food law, establishing the European Food Safety Authority and laying down procedures in matters of food safety. OJ L 31. Technical report, Commission of the European Communities.
  • CEC (2005). Regulation (EC) 2073/2005 of 15 November 2005 on microbiological criteria for foodstuffs. OJ L 338. Technical report, Commission of the European Communities.
  • Celeux, G., Forbes, F., Robert, C. P. and Titterington, D. M. (2006). Deviance information criteria for missing data models. Bayesian Anal. 1 651–673 (electronic).
  • Davey, K. R. (1989). A predictive model for combined temperature and water activity on microbial growth during the growth phase. J. Appl. Microbiol. 67 483–488.
  • Davidson, P. M. (2001). Chemical preservatives and natural antimicrobial compounds. In Food Microbiology: Fundamentals and Frontiers. ASM Press, Washington, DC.
  • Donnet, S., Foulley, J.-L. and Samson, A. (2010). Bayesian analysis of growth curves using mixed models defined by stochastic differential equations. Biometrics 66 733–741.
  • Eduardo, A. J. S., Quinto, E. J., Castro, M. J. and Mora, M. T. (2011). Analyzing the time-to-detection–generation time relationship of Escherichia coli. CyTA-Journal of Food 9 271–277.
  • FAO/WHO (1995). Uncertainty and variability in the risk assessment process. In Application of Risk Analysis to Food Standards Issues. Agriculture and Consumer Protection Dept., Geneva. Available at
  • Fennema, O. R. and Tannenbaum, S. R. (1996). Introduction to food chemistry. In Food Chemistry, 3th ed. (O. R. Fennema, ed.) 1–16. Dekker, New York.
  • Fine, T. L. (1999). Feedforward Neural Network Methodology. Springer, New York.
  • García-Gimeno, R. M., Hervás-Martínez, C. et al. (2002). Improving artificial neural networks with a pruning methodology and genetic algorithms for their application in microbial growth prediction in food. Int. J. Food Microbiol. 72 19–30.
  • Geeraerd, A. H., Herremans, C. H., Cenens, C. and Impe, J. F. V. (1998). Application of artificial neural networks as a non-linear modular modeling technique to describe bacterial growth in chilled food products. Int. J. Food Microbiol. 44 49–68.
  • Gelfand, A. E. (1996). Model determination using sampling-based methods. In Markov Chain Monte Carlo in Practice (W. Gilks, S. Richardson and D. Spiegelhalter, eds.) 145–161. Chapman & Hall, London.
  • Gelfand, A. E. and Ghosh, S. K. (1998). Model choice: A minimum posterior predictive loss approach. Biometrika 85 1–11.
  • Gilks, W. R., Richardson, S. and Spiegelhalter, D. J. (1996). Markov Chain Monte Carlo in Practice. Chapman & Hall, London.
  • Gompertz, B. (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philos. Trans. R. Soc. Lond. 115 513–583.
  • Hajmeer, M. N., Basheer, I. A. and Najjar, Y. M. (1997). Computational neural networks for predictive microbiology. II. Application to microbial growth. Int. J. Food Microbiol. 34 51–66.
  • ICMSF (1980). Microbial Ecology of Foods. Factors Affecting the Life and Death of Microorganisms 1. Academic Press, London.
  • IOM, Institute of Medicine of National Academies (2010). Preservation and physical property roles of sodium in foods. In Strategies to Reduce Sodium Intake in the United States. Committee on Strategies to Reduce Sodium Intake. National Academies Press, Washington, DC. Available at
  • Lavine, M. and West, M. (1992). A Bayesian method for classification and discrimination. Canad. J. Statist. 20 451–461.
  • Lee, H. K. H. (2004). Bayesian Nonparametrics via Neural Networks. SIAM, Philadelphia, PA.
  • Leistner, L. (2000). Basic aspects of food preservation by hurdle technology. Int. J. Food Microbiol. 55 181–186.
  • MacKay, D. J. C. (1995). Bayesian methods for neural networks: Theory and applications. Technical report, Cavendish Laboratory, Cambridge Univ., Cambridge.
  • McClure, P. J., Baranyi, J., Boogard, E., Kelly, T. M. and Roberts, T. A. (1993). A predictive model for the combined effect of pH, sodium chloride and storage temperature on the growth of Brochothrix thermosphacta. Int. J. Food Microbiol. 19 161–178.
  • McKellar, R. C. and Lu, X. (2004). Modeling Microbial Responses in Food. CRC Press, Boca Raton, FL.
  • McMeekin, T. A., Chandler, R. E., Doe, P. E., Garland, C. D., Olley, J., Putro, S. and Ratkowsky, D. A. (1987). Model for combined effect of temperature and salt concentration/water activity on the growth rate of Staphylococcus xylosus. J. Appl. Microbiol. 62 543–550.
  • Miles, D. W., Ross, T., Olley, J. and McMeekin, T. A. (1997). Development and evaluation of a predictive model for the effect of temperature and water activity on the growth rate of Vibrio parahaemolyticus. Int. J. Food Microbiol. 38 133–142.
  • Montville, T. J. and Matthews, K. R. (2001). Principles which influence microbial growth, survival, and death of foods. In Food Microbiology: Fundamentals and Frontiers. ASM Press, Washington, DC.
  • Montville, T. J. and Matthews, K. R. (2005). Food Microbiology: An Introduction, 1st ed. ASM Press, Washington, DC.
  • Müller, P. and Insua, D. R. (1998). Issues in Bayesian analysis of neural network models. Neural Comput. 10 749–770.
  • NACMCF (1997). Principles and guidelines for the application of microbiological risk assessment. Technical report, USA Dept. Agriculture, Food Safety and Inspection Service, Washington, DC.
  • Neal, R. M. (1996). Bayesian Learning for Neural Networks. Springer, Berlin.
  • Palacios, A. P., Marín, J. M., Quinto, E. J. and Wiper, M. P. (2014a). Supplement to “Bayesian modeling of bacterial growth for multiple populations.” DOI:10.1214/14-AOAS720SUPPA.
  • Palacios, A. P., Marín, J. M., Quinto, E. J. and Wiper, M. P. (2014b). Supplement to “Bayesian modeling of bacterial growth for multiple populations.” DOI:10.1214/14-AOAS720SUPPB.
  • Potter, N. N. and Hotchkiss, J. H. (1998). Food Science. Springer, Berlin.
  • Pouillot, R., Albert, I., Cornu, M. and Denis, J. B. (2003). Estimation of uncertainty and variability in bacterial growth using Bayesian inference. Application to Listeria monocytogenes. Int. J. Food Microbiol. 81 87–104.
  • Ratkowsky, D. A., Olley, J., McMeekin, T. A. and Ball, A. (1982). Relationship between temperature and growth rate of bacterial cultures. J. Bacteriol. 149 1–5.
  • Robert, C. P. and Mengersen, K. L. (1999). Reparameterisation issues in mixture modelling and their bearing on MCMC algorithms. Comput. Statist. Data Anal. 29 325–344.
  • Roeder, K. and Wasserman, L. (1997). Practical Bayesian density estimation using mixtures of normals. J. Amer. Statist. Assoc. 92 894–902.
  • Ross, T. and Dalgaard, P. (2004). Secondary models. In Modeling Microbial Responses in Food. CRC Press, Boca Raton, FL.
  • Ross, T. and McMeekin, T. A. (1994). Predictive microbiology. Int. J. Food Microbiol. 23 241–264.
  • Rosso, L., Lobry, J. R., Bajard, S. and Flandrois, J. P. (1995). Convenient model to describe the combined effects of temperature and pH on microbial growth. Appl. Environ. Microbiol. 61 610–616.
  • Shelef, L. A. and Seiter, J. (2005). Indirect and miscellaneous antimicrobials. In Antimocrobials in Food, 3rd ed. 573–598. Taylor and Francis, Boca Raton, FL.
  • Skinner, G. U. Y. E., Larkin, J. W. and Rhodehamel, E. J. (1994). Mathematical modeling of microbial growth: A review. J. Food. Saf. 14 175–217.
  • Spiegelhalter, D. J., Thomas, A. and Best, N. G. (1999). WinBUGS version 1.2 user manual. MRC Biostatistics Unit.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 583–639.
  • Stern, H. S. (1996). Neural networks in applied statistics. Technometrics 38 205–220.
  • Valero, A., Hervas, C., Garcia-Gimeno, R. M. and Zurera, G. (2007). Searching for new mathematical growth model approaches for Listeria monocytogenes. J. Food Sci. 72 M016–M025.
  • Vehtari, A. and Lampinen, J. (2003). Expected utility estimation via cross-validation. Bayesian Stat. 7 701–710.
  • Watanabe, S. (2010). Equations of states in singular statistical estimation. Neural Netw. 23 20–34.
  • Wijtzes, T., De Wit, J. C. et al. (1995). Modelling bacterial growth of Lactobacillus curvatus as a function of acidity and temperature. Appl. Environ. Microbiol. 61 2533–2539.
  • Wijtzes, T., Rombouts, F. M., Kant-Muermans, M. L. T., Van’t Riet, K. and Zwietering, M. H. (2001). Development and validation of a combined temperature, water activity, pH model for bacterial growth rate of Lactobacillus curvatus. Int. J. Food Microbiol. 63 57–64.
  • Zwietering, M. H., Jongenburger, I., Rombouts, F. M. and Van’t Riet, K. (1990). Modeling of the bacterial growth curve. Appl. Environ. Microbiol. 56 1875–1881.

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