The mathematical properties of the quasi-chemical model for microorganism growth-death kinetics in foods

Knowledge of the mathematical properties of the quasi-chemical model [Taub, Feeherry, Ross, Kustin, Doona, 2003. A quasi-chemical kinetics model for the growth and death of Staphylococcus aureus in intermediate moisture bread. J. Food Sci. 68 (8), 2530-2537], which is used to characterize and predict microbial growth-death kinetics in foods, is important for its applications in predictive microbiology. The model consists of a system of four ordinary differential equations (ODEs), which govern the temporal dependence of the bacterial life cycle (the lag, exponential growth, stationary, and death phases, respectively). The ODE system derives from a hypothetical four-step reaction scheme that postulates the activity of a critical intermediate as an antagonist to growth (perhaps through a quorum sensing biomechanism). The general behavior of the solutions to the ODEs is illustrated by several examples. In instances when explicit mathematical solutions to these ODEs are not obtainable, mathematical approximations are used to find solutions that are helpful in evaluating growth in the early stages and again near the end of the process. Useful solutions for the ODE system are also obtained in the case where the rate of antagonist formation is small. The examples and the approximate solutions provide guidance in the parameter estimation that must be done when fitting the model to data. The general behavior of the solutions is illustrated by examples, and the MATLAB programs with worked examples are included in the appendices for use by predictive microbiologists for data collected independently.