Improvement of random coefficient differential models of growth of anaerobic photosynthetic bacteria by combining Bayesian inference and gPC

Abstract

The time evolution of microorganisms, such as bacteria, is of great interest in biology. In the article by D. Stanescu et al. [Electronic Transactions on Numerical Analysis, 34, 44–58 (2009)], a logistic model was proposed to model the growth of anaerobic photosynthetic bacteria. In the laboratory experiment, actual data for two species of bacteria were considered: Rhodobacter capsulatus and Chlorobium vibrioforme. In this paper, we suggest a new nonlinear model by assuming that the population growth rate is not proportional to the size of the bacteria population, but to the number of interactions between the microorganisms, and by taking into account the beginning of the death phase in the kinetic curve. Stanescu et al. evaluated the effect of randomness into the model coefficients by using generalized polynomial chaos (gPC) expansions, by setting arbitrary distributions without taking into account the likelihood of the data. By contrast, we utilize a Bayesian inverse approach for parameter estimation to obtain reliable posterior distributions for the random input coefficients in both the logistic and our new model. Since our new model does not possess an explicit solution, we use gPC expansions to construct the Bayesian model and to accelerate the Markov chain Monte Carlo algorithm for the Bayesian inference.