## 2 Materials and Methods
### Challenge Testing  
GSE-MIC, GE-MIC, pure GSE, pure GE, and a control in TSBYE not containing any extract were inoculated with 200 \(\upmu\)L of a stationary phase culture of _L. monocytogenes_ (to a concentration of approximately 10\({}^{7}\) CFU/mL), and incubated at 30 \({}^{\circ}\)C while shaking at 180 rpm. Samples were taken at different time intervals, decimally diluted and plated on a nonselective and a selective medium with the drop technique. TSAYE and Listeria selective agar--Oxford formulation (Oxford, VWR Chemicals, Leuven, Belgium) were used as a nonselective and selective medium, respectively. The plates were incubated at 30 \({}^{\circ}\)C for 24 h, after which the colonies were counted. The experimental data were plotted as a function of time. To model the growth and inactivation of _L. monocytogenes_, the primary models of [PERSON] and [PERSON] [25] and [PERSON] et al. [26] were fitted to the experimental data, respectively.  
The growth model of [PERSON] and [PERSON] [25] is described by Equations (1)-(3).  
\[\frac{\mathrm{d}N(\mathrm{t})}{\mathrm{d}t}\ =\ \left(\frac{Q(t)}{1+Q(t)} \right)\mu_{max}\left(1-\frac{N(t)}{N_{max}}\right)N(t) \tag{1}\]\[\frac{\text{d}Q(t)}{\text{d}t}\ =\ \mu_{max}Q(t) \tag{2}\]  
\[\lambda\cdot\mu_{max}\ =\ \ln\left(1+\frac{1}{Q(0)}\right) \tag{3}\]  
With \(N(t)\) [CFU/mL], the cell density at time \(t\); \(N_{max}\) [CFU/mL], the maximum cell density; \(\mu_{max}\) [1/h], the maximum specific growth rate; \(Q(t)\) [\(-\)], representing the physiological state of the cells; \(Q(0)\) [\(-\)], a measure of the initial physiological state of the cells; and \(\lambda\) [h], the lag phase duration. Since experiments were conducted at static environmental conditions, the model was simplified to  
\[\ln(N(t))\ =\ \ln(N(0))+\mu_{max}\cdot A(t)-\ln\left(1+\frac{\text{e}^{\mu_{ max}\cdot A(t)}-1}{\text{e}^{(N_{max}-N(0))}}\right) \tag{4}\]  
\[A(t)\ =\ t+\frac{1}{\mu_{max}}\ln\left(\frac{\text{e}^{-\mu_{max}\cdot t}+Q(0)}{ 1+Q(0)}\right) \tag{5}\]  
The inactivation model of [PERSON] et al. [26] is described by Equations (6)-(8).  
\[\frac{\text{d}N(t)}{\text{d}t}\ =\ -\left(\frac{1}{1+C_{c}(t)}\right)k_{max} \left(1-\frac{N_{res}}{N(t)}\right)N(t) \tag{6}\]  
\[\frac{\text{d}C_{C}(t)}{\text{d}t}\ =\ -k_{max}C_{C}(t) \tag{7}\]  
\[C_{C}(0)\ =\ \text{e}^{k_{max}S_{L}}-1 \tag{8}\]  
With \(N(t)\) [CFU/mL], the cell density at time \(t\); \(N_{res}\) [CFU/mL], the residual cell density (or stress resistant subpopulation); \(k_{max}\) [1/h], the maximum specific inactivation rate; \(C_{C}(t)\) [\(-\)], representing a critical intracellular or extracellular component for cell survival; \(Cc(0)\) [\(-\)], the initial concentration of this critical component; and \(S_{L}\) [h], the shoulder length. Since experiments were conducted at static environmental conditions, the model was simplified to  
\[N(t)\ =\ (N_{0}-N_{res})\cdot e^{-k_{max}t}\left(\frac{e^{-k_{\text{max}}S_{L}}}{ 1+(e^{-k_{\text{max}}S_{L}}-1)\cdot e^{-k_{\text{max}}t}}\right)+N_{res} \tag{9}\]  
Parameters of the models were estimated via the minimization of the sum of squared errors, using the lsqnonlin routine of the Optimization Toolbox of MATLAB version R2018b (The Mathworks, Inc., Natick, MA, USA). For fitting purposes the results were logarithmically transformed. For the growth model of [PERSON] and [PERSON] [25], results were expressed as \(\ln\) (CFU/mL), because the model involves the natural exponential function to describe the growth phase. For the inactivation model of [PERSON] et al. [26], results were expressed as \(\log\) (CFU/mL), because the model, as all inactivation models, requires the reduction of microorganisms to be expressed in \(\log\) units. In order to avoid apparent SI caused by small fitting differences, the initial cell density \(N_{0}\) was set to be equal to the average of the cell densities at time zero of the individual replicates on the selective and nonselective medium for each condition, i.e., approximately 7 \(\log\) (CFU/mL) or 16 \(\ln\) (CFU/mL).