Uncertainty Quantification of Random Microbial Growth in a Competitive Environment via Probability Density Functions

The Baranyi–Roberts model describes the dynamics of the volumetric densities of two interacting cell populations. We randomize this model by considering that the initial conditions are random variables whose distributions are determined by using sample data and the principle of maximum entropy. Subsequenly, we obtain the Liouville–Gibbs partial differential equation for the probability density function of the two-dimensional solution stochastic process. Because the exact solution of this equation is unaffordable, we use a finite volume scheme to numerically approximate the aforementioned probability density function. From this key information, we design an optimization procedure in order to determine the best growth rates of the Baranyi–Roberts model, so that the expectation of the numerical solution is as close as possible to the sample data. The results evidence good fitting that allows for performing reliable predictions.