## 2 Materials and Methods
### Estimation of the L. monocytogenes Inactivation Parameters  
The Weibull-Mafart and Bigelow primary inactivation models, available in the user-friendly software Bioinactivation4 (version 0.1.0, [[https://foodlab-upct.shinyapps.io/bioinactivation4/](https://foodlab-upct.shinyapps.io/bioinactivation4/)]([https://foodlab-upct.shinyapps.io/bioinactivation4/](https://foodlab-upct.shinyapps.io/bioinactivation4/)), accessed on 1 April 2025) [50], were fitted to the experimental data obtained (log CFU/mL) over time (s) for each treatment (PEO, FIV and control without OEO) and temperature condition (52, 54, 57.5 and 63 \({}^{\circ}\)C). The Mafart model describes the time required to reduce a microbial population by one Log\({}_{10}\) unit at a constant temperature, represented by the \(\delta\)-value. This model is often combined with the Weibull model to interpret non-linear regression [51; 52]. In contrast, the Bigelow model represents microbial inactivation at a constant temperature but follows a first-order, linear relationship [53; 54]. To assess the goodness-of-fit of the primary models, several statistical indices were calculated to select the best model to fit the obtained results from the thermal inactivation of _L. monocytogenes_ (RMSE, AIC, B\({}_{f}\), and A\({}_{f}\), as described below). The Bigelow primary model is described by Equation (1) [53]. \[Log_{10}N(t)=Log_{10}N_{0}-\frac{1}{D_{T}}t \tag{1}\]  
where the D-value at a constant temperature T is identified as D\({}_{T}\); N(t) is the population at time t (CFU/mL); and N\({}_{0}\) is the initial population (CFU/mL). The Weibull-Mafart model is described by Equation (2) [52]:  
\[Log_{10}N(t)=log_{10}\ N_{0}-(\frac{t}{\delta_{T}})^{p} \tag{2}\]  
where the time to reduce 90% of the microbial population at temperature T is represented by \(\delta_{\text{T}}\) (scale parameter), and \(p\) is the shape parameter (dimensionless). When \(p>1\), the curve is convex; when \(p<1\), the curve is concave; and when \(p=1\), the curve follows a linear trend [55]. For the secondary models, both one-step and two-step approaches were employed to estimate the z-values, which describe the temperature dependence of the inactivation process. These secondary models were fitted to the complete experimental dataset, incorporating all temperature and treatment conditions. The time-temperature data were adjusted using 57.5 \({}^{\circ}\)C as the reference temperature (T\({}_{\text{R}}\)), as recommended for similar studies on _L. monocytogenes_[7; 43; 56; 57]. The z-values were calculated by fitting the secondary models to the entire dataset, ensuring that temperature variation was adequately accounted for in the inactivation curves. For the Mafart model, Equation (3) describes the secondary behavior and evaluates the sensitivity of microorganisms at a certain temperature [58]:  
\[Log_{10}\delta_{T}=log_{10}\delta_{R}+\frac{T_{R}-T}{z} \tag{3}\]  
where \(\delta_{\text{R}}\) is the \(\delta\)-value at the T\({}_{\text{R}}\). In evaluating the goodness-of-fit of each model, the Root Mean Square Error (RMSE, Equation (4)) was calculated. Values lower than 0.50 indicate an acceptable fit between the model and the experimental data [59; 60]. \[\text{RMSE}=\sqrt{\sum_{i=1}^{n}\frac{\left(X_{\text{obs}}-X_{\text{fit}} \right)^{2}}{n-s}} \tag{4}\]  
where Xobs and Xfit represent the observed and fitted values, respectively; n is the total number of observations; and s is the number of model parameters. In the two-step inactivation model, the Residual Standard Error (RSE) is used to measure how well the residuals fit in the regression model, accounting for the degrees of freedom (d/). It is calculated using Equation (5) [59], which is closely related to RMSE in the one-step but is not identical.