## 3 Results and Discussion

Table 1: Water activity (\(a_{w}\)), proximate composition, and bacterial growth/survival in nine different type of cakes. Sigmoidal growth models are excellent to predict bacterial growth at constant temperatures; however, they have some limitations, e.g., isothermal models are not suitable for varying temperatures. To predict bacterial growth under fluctuating environmental conditions, a dynamic model integrating the primary and the secondary models was developed [20]. Accurate predictions using dynamic models provide valuable industrial data to evaluate their current modules during processing, distribution, and storage, and risk assessment to improve their performance in evaluating the risk of bacterial growth in food products. To describe the kinetic behavior of _Salmonella_ spp. and _S. aureus_ in cake at varying temperatures, a dynamic model was developed, using calculated kinetic parameters at a constant temperature (Figure 2). In Korea, refrigerated products, including cakes, should be stored and distributed at 0-10 \({}^{\circ}\)C [24]. The average temperature in summer in Korea over the last 5 years was 24.5 \({}^{\circ}\)C [25]. Therefore, we set the variable temperature range from 10 to 25 \({}^{\circ}\)C. During this temperature profile, _S. aureus_ cell populations in cake increased from 3.7 to 7.5 log CFU/g for 6 d; however, _Salmonella_ spp. did not grow (Figure 2). RMSE \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Bacteria** & **Storage Temperature (\({}^{\circ}\)C)** & _LPD \({}^{\text{1}}\) (h)_ & \begin{tabular}{c} \(\mathbf{\mu_{max}}\)2 \\ **(Log CFU/g/h)** \\ \end{tabular} & \begin{tabular}{c} \(\mathbf{N_{0}}\)3 \\ **(Log CFU/g)** \\ \end{tabular} & \
\begin{tabular}{c} \(\mathbf{N_{max}}\)4 \\ **(Log CFU/g)** \\ \end{tabular} &
\begin{tabular}{c} \(\mathbf{R^{2}}\) \\ **(Log CFU/g)** \\ \end{tabular} \\ \hline \multirow{3}{*}{_Salmonella_ spp.} & 20 & 1.6 \(\pm\) 0.0 \({}^{\text{A}}\) & 0.10 \(\pm\) 0.00 \({}^{\text{B}}\) & 1.9 \(\pm\) 0.1 & 7.0 \(\pm\) 0.0 & 0.990–0.993 \\  & 25 & 4.0 \(\pm\) 1.9 \({}^{\text{A}}\) & 0.19 \(\pm\) 0.00 \({}^{\text{B}}\) & 3.3 \(\pm\) 0.5 & 7.5 \(\pm\) 0.0 & 0.992–0.994 \\  & 30 & 5.2 \(\pm\) 3.8 \({}^{\text{A}}\) & 0.35 \(\pm\) 0.05 \({}^{\text{A}}\) & 3.8 \(\pm\) 0.0 & 7.1 \(\pm\) 0.3 & 0.955–0.972 \\ \hline \multirow{3}{*}{_S. aureus_} & 15 & 30.1 \(\pm\) 8.5 \({}^{\text{a}}\) & 0.05 \(\pm\) 0.02 \({}^{\text{b}}\) & 3.7 \(\pm\) 0.0 & 7.2 \(\pm\) 0.1 & 0.971–0.996 \\  & 25 & 6.3 \(\pm\) 0.2 \({}^{\text{b}}\) & 0.29 \(\pm\) 0.07 \({}^{\text{a}}\) & 4.0 \(\pm\) 0.3 & 8.1 \(\pm\) 0.4 & 0.985–0.989 \\ \cline{1-1}  & 30 & 4.2 \(\pm\) 2.5 \({}^{\text{b}}\) & 0.33 \(\pm\) 0.02 \({}^{\text{a}}\) & 3.7 \(\pm\) 0.7 & 7.6 \(\pm\) 0.3 & 0.942–0.965 \\ \hline \hline \end{tabular}  
* \({}^{\text{1}}\) Lag phase duration; 2 the maximum specific growth rate; 3 initial cell concentration; 4 maximum cell concentration; 4 the same column with different superscript letters are significantly different (_p_ \(<\) 0.05); \({}^{\text{A-B}}\) means within the same column with different superscript letters are significantly different (_p_ \(<\) 0.05).  
\end{table} 
Table 2: Kinetic parameters calculated by the Baranyi model for _Salmonella_ spp. and _Staphylococcus aureus_ growth in the cake. Figure 1: Secondary predictive model for the maximum specific growth rate (\(\mu_{\text{max}}\)) and lag phase duration (_LPD_) of _Salmonella_ spp. (**A,B**) and _Staphylococcus aureus_ (**C,D**) in the cake as a function of temperature. values of the developed dynamic model were 0.22 and 0.19 for _Salmonella_ spp. and _S. aureus_, respectively, indicating that the developed dynamic model had a good performance. Therefore, the developed dynamic model can be used to predict _Salmonella_ spp. and _S. aureus_ growth in preventing foodborne outbreaks.