Journal of Nonlinear Science

, Volume 28, Issue 1, pp 371–393 | Cite as

The Stochastic Quasi-chemical Model for Bacterial Growth: Variational Bayesian Parameter Update

  • Panagiotis Tsilifis
  • William J. Browning
  • Thomas E. Wood
  • Paul K. Newton
  • Roger G. Ghanem


We develop Bayesian methodologies for constructing and estimating a stochastic quasi-chemical model (QCM) for bacterial growth. The deterministic QCM, described as a nonlinear system of ODEs, is treated as a dynamical system with random parameters, and a variational approach is used to approximate their probability distributions and explore the propagation of uncertainty through the model. The approach consists of approximating the parameters’ posterior distribution by a probability measure chosen from a parametric family, through minimization of their Kullback–Leibler divergence.


Bayes rule Kullback–Leibler divergence Evidence lower bound Quasi-chemical model Gradient-based optimization 



The authors gratefully acknowledge support from US Army Research Office Contract W911NF-14-C-0151.


  1. Banks, H., Bihari, K.: Modelling and estimating uncertainty in parameter estimation. Inverse Prob. 17, 95 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. Banks, H., Browning, W., Catenacci, J., Wood, T.: Analysis of a Quasi-chemical Kinetic Food Chemistry Model. Center for Research in Scientific Computation Technical Report CRSC-TR16-05. NC State University, Raleigh, NC (2016)Google Scholar
  3. Baranyi, J., Roberts, T.: A dynamic approach to predicting bacterial growth in food. Int. J. Food Microbiol. 23, 277–294 (1994)CrossRefGoogle Scholar
  4. Baranyi, J., Roberts, T., McClure, P.: A non-autonomous differential equation to model bacterial growth. Food Microbiol. 10, 43–59 (1993)CrossRefMATHGoogle Scholar
  5. Bickel, P., Doksum, K.: Mathematical Statistics: Basic Ideas and Selected Topics, vol. 2. CRC Press, Boca Raton (2015)MATHGoogle Scholar
  6. Bishop, C.: Pattern Recognition and Machine Learning, Information Science and Statistics. Springer, New York (2006)MATHGoogle Scholar
  7. Browning, W.J.: Near real-time quantification of stochastic model parameters. Tech. rep., prepared by Applied Mathematics Inc., Small Business Technology Transfer, Phase II Final Report, Army STTR Topic A13A-009 (28 September 2016)Google Scholar
  8. Buchanan, R.: Predictive microbiology: Mathematical modeling of microbial growth in foods. In: ACS Symposium Series-American Chemical Society, (1992)Google Scholar
  9. Buchanan, R., Whiting, R., Damert, W.: When is simple good enough: a comparison of the Gompertz. Baranyi, and three-phase linear models for fitting bacterial growth curves. Food Microbiol. 14, 313–326 (1997)CrossRefGoogle Scholar
  10. Byrd, R., Lu, P., Nocedal, J., Zhu, C.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16, 1190–1208 (1995)MathSciNetCrossRefMATHGoogle Scholar
  11. Chaloner, K., Verdinelli, I.: Bayesian experimental design: a review. Stat. Sci. 10(3), 273–304 (1995)MathSciNetCrossRefMATHGoogle Scholar
  12. Chaspari, T., Tsiartas, A., Tsilifis, P., Narayanan, S.: Markov chain monte carlo inference of parametric dictionaries for sparse bayesian approximations. IEEE Trans. Signal Process. 64, 3077–3092 (2016)MathSciNetCrossRefGoogle Scholar
  13. Chen, P., Zabaras, N., Bilionis, I.: Uncertainty propagation using infinite mixture of gaussian processes and variational bayesian inference. J. Comput. Phys. 284, 291–333 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. Doona, C., Feeherry, F., Ross, E.: A quasi-chemical model for the growth and death of microorganisms in foods by non-thermal and high-pressure processing. Int. J. Food Microbiol. 100, 21–32 (2005)CrossRefGoogle Scholar
  15. Doona, C., Feeherry, F., Ross, E., Kustin, K.: Inactivation kinetics of listeria monocytogenes by highpressure processing: pressure and temperature variation. J. Food Sci. 77, M458–M465 (2012)CrossRefGoogle Scholar
  16. Gershman, S., Hoffman, M., Blei, D.: Nonparametric variational inference. In: International Conference on Machine Learning (2012)Google Scholar
  17. Goldbeter, A.: Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  18. Gompertz, B.: 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 (1825)CrossRefGoogle Scholar
  19. Haario, H., Saksman, E., Tamminen, J.: An adaptive Metropolis algorithm. Bernoulli 7, 223–242 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. Hastings, W.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)MathSciNetCrossRefMATHGoogle Scholar
  21. Huber, M., Bailey, T., Durrant-Whyte, H., Hanebeck, U.: On entropy approximation for Gaussian mixture random vectors, In: IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems, MFI 2008, (pp. 181–188). IEEE, (2008)Google Scholar
  22. Kullback, S., Leibler, R.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)MathSciNetCrossRefMATHGoogle Scholar
  23. McMeekin, T., Brown, J., Krist, K., Miles, D., Neumeyer, K., Nichols, D., Olley, J., Presser, K., Ratkowsky, D., Ross, T., Salter, M.: Quantitative microbiology: a basis for food safety. Emerg. Infect. Dis. 3, 541 (1997)CrossRefGoogle Scholar
  24. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1091 (1953)CrossRefGoogle Scholar
  25. Pinski, F., Simpson, G., Stuart, A., Weber, H.: Algorithms for Kullback–Leibler approximation of probability measures in infinite dimensions. SIAM J. Sci. Comput. 37, A2733–A2757 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. Ricker, W.: Growth rates and models. Fish Physiol. 8, 677–743 (1979)CrossRefGoogle Scholar
  27. Robert, C., Casella, G.: Monte Carlo Statistical Methods. Springer, Berlin (2013)MATHGoogle Scholar
  28. Roberts, G., Rosenthal, J.: Optimal scaling of discrete approximations to Langevin diffusions. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 60, 255–268 (1998)MathSciNetCrossRefMATHGoogle Scholar
  29. Ross, E., Taub, I., Doona, C., Feeherry, F., Kustin, K.: The mathematical properties of the quasi-chemical model for microorganism growth-death kinetics in foods. Int. J. Food Microbiol. 99, 157–171 (2005)CrossRefGoogle Scholar
  30. Schnute, J.: A versatile growth model with statistically stable parameters. Can. J. Fish. Aquat. Sci. 38, 1128–1140 (1981)CrossRefGoogle Scholar
  31. Silverman, B.: Density estimation for statistics and data analysis, vol. 26. CRC Press, Boca Raton (1986)CrossRefMATHGoogle Scholar
  32. Stuart, A.: Inverse problems: a bayesian perspective. Acta Numer. 19, 451–559 (2010)MathSciNetCrossRefMATHGoogle Scholar
  33. Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, Philadelphia (2005)CrossRefMATHGoogle Scholar
  34. Taub, I., Ross, E., Feeherry, F.: Model for predicting the growth and death of pathogenic organisms. In: Van Impe, J.F.M., Gernaerts, K. (eds.) Proceedings of the Third International Conference on Predictive Modeling in Foods (2000)Google Scholar
  35. Taub, I., Feeherry, F., Ross, E., Kustin, K., Doona, C.: A quasi-chemical kinetics model for the growth and death of Staphylococcus aureus in intermediate moisture bread. J. Food Sci. 68, 2530–2537 (2003)CrossRefGoogle Scholar
  36. Tsilifis, P., Bilionis, I., Katsounaros, I., Zabaras, N.: Computationally efficient variational approximations for bayesian inverse problems. J. Verif. Valid. Uncertain. Quantif. 1, 031004 (2016)CrossRefGoogle Scholar
  37. Tsilifis, P., Ghanem, R., Hajali, P.: Efficient bayesian experimentation using an expected information gain lower bound. SIAM/ASA J. Uncertain. Quantif. 5, 30–62 (2017)MathSciNetCrossRefMATHGoogle Scholar
  38. Vrettas, M., Cornford, D., Opper, M.: Estimating parameters in stochastic systems: a variational bayesian approach. Phys. D 240, 1877–1900 (2011)CrossRefMATHGoogle Scholar
  39. Whiting, R.: Modeling bacterial survival in unfavorable environments. J. Ind. Microbiol. 12, 240–246 (1993)CrossRefGoogle Scholar
  40. Whiting, R., Sackitey, S., Calderone, S., Morely, K., Phillips, J.: Model for the survival of Staphylococcus aureus in nongrowth environments. Int. J. Food Microbiol. 31, 231–243 (1996)CrossRefGoogle Scholar
  41. Ye, J., Rey, D., Kadakia, N., Eldridge, M., Morone, U., Rozdeba, P., Abarbanel, H., Quinn, J.: Systematic variational method for statistical nonlinear state and parameter estimation. Phys. Rev. E 92, 052901 (2015)MathSciNetCrossRefGoogle Scholar
  42. Zwietering, M., Jongenburger, I., Rombouts, F., Van’t Riet, K.: Modeling of the bacterial growth curve. Appl. Environ. Microbiol. 56, 1875–1881 (1990)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Panagiotis Tsilifis
    • 1
  • William J. Browning
    • 2
  • Thomas E. Wood
    • 2
  • Paul K. Newton
    • 3
  • Roger G. Ghanem
    • 1
  1. 1.Department of Civil EngineeringUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Applied Mathematics Inc.Gales FerryUSA
  3. 3.Department of Aerospace and Mechanical Engineering and MathematicsUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations