Modelling Bacterial Dynamics in Food Products: Role of Environmental Noise and Interspecific Competition

In this paper we review some results obtained within the context of the predictive microbiology, which is a specific field of the population dynamics. In particular we discuss three models, which exploit tools of statistical mechanics, for bacterial dynamics in food of animal origin. In the first model, the random fluctuating behaviour, experimentally measured, of the temperature is considered. In the second model stochastic differential equations are introduced to take into account the influence of physical and chemical variables, such as temperature, pH and activity water, subject to deterministic and random variations. The third model, which is an extended version of the second one, neglects the environmental fluctuations, and concentrates on the role of the interspecific bacterial interactions. The comparison between expected results and observed data indicates that the presence of noise sources and interspecific bacterial interactions improves the predictive features of the models analyzed.


Introduction
Predictive microbiology exploits mathematical models to describe bacterial dynamics in different products of food industry. The models take into account the role played by environmental variables, whose variations can affect, sometimes dramatically, the quality and safety of the food products. Predictive models belong to three different types: primary, secondary and tertiary [1]. The first class of models allows to obtain the time evolution of microbial populations. The models belonging to the second type give information on the relationship between parameters which appear in primary models, and physical and chemical variables such as T (temperature), pH (hydrogen ion concentration), and aw (activity water). The third class of models puts together the primary and secondary ones, letting the evolution of physical and chemical variables be considered, when analysis and prediction of the concentration of spoiling or pathogen bacteria of the food are performed [2].
A well-known method for the theoretical analysis of microbial growth exploits generalized Lotka-Volterra (LV) equations [3,4], which allow to describe the dynamics of two competing bacterial populations in different food products. A prototype model structure for mixed microbial populations in food products was proposed by Dens et al. [5]. A similar approach indicated that experimental data for Escherichia coli O157:H7 in ground beef could be well reproduced by an interspecific competition model for two bacterial populations. In the same work the effects of random fluctuations were considered using growth rates whose values are obtained from uniform random distributions [6]. An extensive review on predictive microbiology showed that in general a stochastic approach provides predictions which exclude the worst-case scenario [7]. In particular, stochastic terms were introduced to reproduce and predict bacterial dynamics, exploiting an approach based on primary and secondary growth models [8]. Moreover other authors presented a stochastic model which interprets the bacterial growth as the average evolution of many cells: measured values of the growth rate for many different Copyright © 2013 SciRes. JMP D. VALENTI ET AL. 1060 cells allow to describe the theoretical growth rate used in the model as a stochastic variable with a corresponding probability distribution [9][10][11]. Finally, a stochastic ecological model, based on the Verhulst logistic differential equation, was devised [12].
The previous models however do not include explicitly stochastic terms in the equations of motion of the systems analyzed. In other words, the models used in predictive microbiology are not usually based on stochastic differential equations.
Aim of this paper is to analyze how predictions for bacterial dynamics are affected by the three following features: 1) use of differential equations (dynamical approach); 2) presence of interactions among bacterial populations; 3) introduction of stochastic terms, i.e. noise sources, which mimic the random fluctuations of environmental variables. In the following we present a general approach to model the bacterial dynamics in food products, taking into account three different situations of microbial growth in real food systems.

The Model
The theoretical approach is based on generalized Lotka-Volterra (LV) equations, in which the bacterial growth rates depend on environmental variables, such as temperature, pH, and activity water, whose randomly fluctuating behaviour can be modeled by inserting terms of additive white Gaussian noise.

Bacterial Growth in Fish Products
In this section we consider an interspecific competition model to describe the dynamics, during the refrigerated storage, of two bacterial populations, i.e. Aeromonas hydrophila and the aerobic mesophilic bacteria (AMB), located on gilthead seabream (Sparus aurata) surfaces. Aeromonas hydrophila is a foodborne disease agent bacterium, present in water and many food products of animal origin, such as seafood, shellfish, milk, meat-based products and in general raw foods [13][14][15][16][17][18], while AMB represents the count of total microflora normally distributed on fish surfaces. The dynamics of the two populations can be described by the following primary model based on generalized LV equations max / max d d 1 for the dynamics of the two populations. In Equations Ah and AMB Q represent the physiological state of the two bacterial populations. Moreover, the behaviour of the growth rates is given by the following secondary model where 2 and 2 indicate the concentrations of oxygen and carbon dioxide, respectively. The values of the parameters in Equations (5) and (6) are The whole dynamics of the system is described by a tertiary model, which combines the previous primary model for the time evolution of the microbial populations [19] with the secondary model [20,21] connecting the growth rates of A. hydrophila and AMB with physical and chemical variables.
Equations (5), (6) are solved by numerical integration, setting pH 7.0  , 2 CO % 1.0  , and letting the temperature vary. The growth rate curves are shown in Figure 1. The temperature values used in the model were obtained experimentally and are also shown in Figure 1 (gray line).

O% 20 
In the figure it is possible to observe that the theoretical results both for A. hydrophila (dashed black line) and AMB (full black line) are in a very good agreement with the corresponding experimental data (black squares for A. hydrophila, black circles for AMB). Specifically, the theoretical values are within the experimental errors (vertical bars) for both populations. Conversely, previous results showed a much worse agreement between experimental and theoretical growth curves, when no interaction terms between the two bacterial populations are included in the model [21]. This confirms the presence of interspecific interaction and the critical role played by the randomly fluctuating temperature in the dynamics of A. hydrophila and AMB.

Bacterial Growth in Meat Products: Two Interacting Populations
In this section we introduce a model for the dynamics of two competing bacterial populations, Listeria monocytogenes and lactic acid bacteria (LAB), present in a meat product, i.e. a traditional Sicilian salami (Salame S. Angelo PGI (Protected Geographical Indication)) very important from the point of view of the Italian food industry. Specifically, L. monocytogenes is a microbial agent of foodborne disease, while LAB constitute the normal bacterial flora of the substrate. The primary model is based on the following system of generalized Lotka-Volterra equations [5,6] max / max d d pH and max , respectivley. Temperature, pH, and activity water are described as stochastic processes. In particular, their dynamics is given by two different contributions: 1) a linearly decreasing deterministic behaviour within a time interval of 168 h, according to the procedure followed in the production process (a fermentation period of 7 days); 2) terms of additive white Gaussian noise, which account for the presence of random fluctuations due to environmental perturbations. By this way the following system of three stochastic differential equations is obtained [22] , are statistically independent Gaussian white noises with the following properties shown in Figure 3. Here, the histograms indicate that the best agreement between the theoretical distributions (white bars) and experimental one (black bars) is observed when the bacterial dynamics is obtained for values of the noise intensities different from zero (stochastic dynamics), and in particular for , Here we note that the theoretical curves of L. monocytogenes (dashed black line) and LAB (full black line) fit very well the corresponding experimental data (black squares for L. monocytogenes, black circles for LAB). This indicates that the interaction, present in the model, between the two bacterial populations reproduces a feature of the real biological system [24][25][26]. In particular, we note that the condition ), determines conditions for the coexistence of the two populations, according to empirical data [23,27,28]

Bacterial Growth in Meat Products: Three Interacting Populations
In the previous section we applied an interspecific competition model, based on Lotka-Volterra equations, to describe the time behaviour of L. monocytogenes and LAB during the fermentation step of S. Angelo salami. The aim of this section is to extend this approach, taking into account a third bacterial population, that is Enterobacteria, whose role is critical for the safety and quality of several meat products, since this bacterial family contains many foodborne human disease agents as well as spoiling bacteria for salami. In particular, we analyze the effects due to the interaction among the three populations, in view of reproducing the biological competition and better simulating the real bacterial growth. The competition among different bacterial populations can be explained recalling that some species determine substrate modifications, which can favour or inhibit the growth of other populations. These effects can be modeled by interaction terms, each one reproducing the influence of a specific population on the dynamics of another bacterial group. To stress the importance of the interspecific competition in modeling bacterial dynamics, we consider temperature, pH and aw as deterministic variables, subject to a decreasing time behaviour, without any random . As a second step, we analyzed the role of the random fluctuations on the dynamics of the system. For this purpose, we solved again Equations (7)  fluctuations. Specifically, the model is obtained starting from Equations (7)  and max LAB  are the same as those used in the previous section (see Equations (11), (12)). The expression for max Ent  , here not given, is very similar to that for (see Equation (6) in conditions and strong LAB interaction. Finally we note that our study could play a key role in view of incorporating stochastic microbial predictive models into a risk assessment process, contributing to improve the precision of the expected concentrations of a foodborne disease agent. This aspect agrees to the new European approach to food safety assessment and management.