^{1}

^{*}

^{2}

^{*}

^{2}

^{2}

^{1}

^{*}

^{2}

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.

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 [

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. [

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 theoretical approach is based on generalized LotkaVolterra (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.

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 [13-18], 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

for the dynamics of the two populations. In Equations (1)-(4) and are, respectively, the population concentrations of A. hydrophila and AMB at time t, and are the maximum specific growth rates of the two bacterial populations, and are the theoretical maximum population concentrations under monospecific growth conditions, and are the interspecific competition parameters of AMB on A. hydrophila and vice-versa, respectively; and represent the physiological state of the two bacterial populations. Moreover, the behaviour of the growth rates is given by the following secondary model

where and indicate the concentrations of oxygen and carbon dioxide, respectively. The values of the parameters in Equations (5) and (6) are, , , , , , and, ,.

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 [

Equations (5), (6) are solved by numerical integration, setting, , and letting the temperature vary. The growth rate curves are shown in

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 [

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 LotkaVolterra equations [5,6]

Here, and are the population concentrations of L. monocytogenes and LAB, respectively; and represent the maximum specific growth rates of the two populations, and and are the theoretical maximum population concentrations. The coefficients and are the interspecific competition parameters of LAB on L. monocytogenes and vice-versa. and represent the physiological state of the two populations.

To solve Equations (7)-(10) it is necessary to set how and vary. This can be done by introducing for the maximum growth rates the following secondary model

obtained by a phenomenological approach (see Ref. [

where, with, are statistically independent Gaussian white noises with the following properties

and are the noise intensities.

As a first step, Equations (7)-(15) have been solved numerically within the Ito scheme, performing 1000 realizations and obtaining the mean growth curves in absence of noise (, ,). The initial concentrations of the two populations, however, have been set randomly. Specifically, in each realization the initial values of and have been extracted from two Gaussian distributions, whose mean values and standard deviations were equal to those of the distributions experimentally observed [

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)-(15) both in deterministic regime and for three different values of the noise intensities, and, obtaining the theoretical probability distributions of L. monocytogenes concentration at the end of the fermentation period (168 hours).

Predicted results, together with observed data, are shown in

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

fluctuations. Specifically, the model is obtained starting from Equations (7)-(12) and introducing two additional equations, one for (Enterobacteria concentration), the other for (physiological state of Enterobacteria). By this way, we obtain the primary model for the three populations [

where, and, referring to Enterobacteria, are bacterial concentration, maximum specific growth rate and theoretical maximum population concentration, respectively. The coefficients and are the interspecific competition parameters of Enterobacteria on L. monocytogenes and vice-versa. Analogously and are the interspecific competition parameters of Enterobacteria on LAB and vice-versa. Finally, represents the physiological state of Enterobacteria. All other variables and parameters are the same as defined is the previous section.

The secondary models for and are the same as those used in the previous section (see Equations (11), (12)). The expression for, here not given, is very similar to that for (see Equation (6) in Section 2.1) and was devised by using 107 experimental growth curves (see Ref. [

Predictive microbiology is an interesting tool which allows to describe microbial evolution in food products as a function of environmental conditions, especially when models take into account bacterial interactions and random fluctuations of chemical and physical variables. In this perspective, the results presented here can be useful to better understand the role of the microbial interaction and environmental noise. Our findings, obtained in three different systems, indicate that interspecific bacterial interaction and environmental random fluctuations are essential for a more precise and reliable prediction of the bacterial dynamics. In particular we note that the time evolution of the microbial concentration both in fish and meat products shows the same characteristics: a slow increase is present during the first part (lag-time) of the dynamics; afterwards a rapid increase (log-phase) of the bacterial concentration takes place until the curve reaches a saturation value (stationary phase), which corresponds to the maximum bacterial concentration measured. In the case of L. monocytogenes behavior in meat products, the log and stationary phases are not evident since the

bacterial growth is affected by adverse environmental 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.

Authors acknowledge the financial support by Ministero dell’Istruzione, dell’Università e della Ricerca of Italian Government.