Primary models measure the behavior of the bacteria over time to a set of conditions. They include bacterial growth models [76,77], bacterial death models [78], growth rate models [42], thermal inactivation models [79], and others.

The most common to fit the microbial growth data seem to be the sigmoidal functions. The sigmoidal functions are composed of four distinct phases as is the case of the microbial growth curve. Two models proposed by Gibson et al. (1997) [76], the modified logistic model and the modified Gompertz, are broadly used.

In this purpose, primary models use the curve-fitting tool of Matlab 7.0 (Math Works, Natick, MA, USA) with which 95% confidence limit (CL) for growth parameters is usually applied. Hence, it is believed that a considerable part of microbial population under the same environmental conditions present similar growth potential. However, when modelling growth curves are obtained by different methodologies (p.e. colony forming units counting or optical density), the fitted parameters show differentiations, as the rate of increase of the optical absorbance does not utter as the maximum specific growth rate and the detection time is not equal to the lag time; furthermore, the initial inoculum is much higher than the detection threshold. Recently, new techniques were developed on processing microscopic procedures issued from monitoring bacterial colony growth. The microscopic images are collected and related to bacterial growth [80].

From image processing, information regarding, e.g., morphology, the colony radius and colony area is gathered and related to bacterial growth. 

Nevertheless, the Baranyi model proposed in the 1990s [59] has been extensively investigated and used for modeling purposes of the microbial growth. Moreover, by the use of the curve-fitting tool programs DMFit, an Excel add-in, and MicroFit, a stand-alone fitting program, which is allocated by the Institute of Food Research in the U.K. (http://www.ifr.bbsrc.ac.uk/Safety/DMFit/default.html), its use has become widely known.

As far as the model of Buchanan is concerned, which is a three-phase linear model (lag phase; exponential growth phase; and stationary phase), its use seems to be limited. The model was used to fit experimental data for E. coli O157:H7.

Albeit, the above models are fitting results in case of homogenous populations. McKellar proposes a model in case that growth is expressed as a function of two distinct cell populations.

Lastly, the Gamma concept model assumes that the effects of controlling variables can be broadening and that the cardinal parameters of temperature, pH, and water activity are not dependent on the other variables [54].

The secondary models control factors of primary models changing the kinetic parameters (p.e. modeling of lag phase and growth rate with respect to one or more environmental or physicochemical factors [61,54]. In other terms, we can say that secondary models characterize all those biotic and abiotic parameters able to modify the microbial kinetics, such as temperature, water activity, pH, and other factors [81].

Finally, the tertiary models are applications of one or more secondary models for providing predictions by including algorithms to calculate shifting conditions. These models are computer tools to consolidate the primary and secondary models used broadly in the food industry and research [82].

This publication can be found here:https://www.mdpi.com/2304-8158/8/12/654