Cardinal Model Research Articles

Limit models in strictly stable abstract elementary classes

AbstractIn this paper, we examine the locality condition for non‐splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove the following. Suppose that is an abstract elementary class satisfying the joint embedding and amalgamation properties with no maximal model of cardinality , stability in , , continuity for (i.e., if and is a limit model witnessed by for some limit ordinal and there exists so that does not ‐split over for all , then does not ‐split over ). Then for and limit ordinals both with cofinality , if satisfies symmetry for (or just ‐symmetry), then, for any and that are and ‐limit models over , respectively, we have that and are isomorphic over . Note that no tameness is assumed.

Categoricity and multidimensional diagrams

We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high enough cardinality will have a single model in any high enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation.

Characterisation of the surface growth of Mucor circinelloides in cheese agar media using predictive mathematical models

The main objective of this work was to characterise the mycelial growth of Mucor circinelloides, one of the fungal contaminants that appear frequently in the artisan cheese production environment. The study uses primary Baranyi and Huang models to compare their parameters and predict M. circinelloides on cheese-based medium (CBA) under diverse environmental conditions (temperature range from 6 to 37 °C and 0 and 1 % NaCl concentration). However, the Baranyi model consistently estimated longer lag phases and higher surface growth rates (sgr) than the Huang model; both models showed adequate best-fit performance (exactly with the mean coefficient of determination R2 = (0.993 ± 0.020 × 10−1). The groups of primary growth parameters were analysed against temperature using the cardinal model (CM) with the following main outputs. The optimal surface growth rates (sgropt) on CBA were 6.8 and 6.5 mm/d calculated with the Baranyi and Huang models, respectively. They were reduced by approximately 46 % on the surface of the agar medium when 1 % NaCl was added. Topt was estimated in a very narrow range of 32.1–32.5 °C from both primary sgr data sets (0 % and 1 % NaCl). Similarly, Tmax values of 37.2 °C and 37.3 °C were estimated for the Baranyi and Huang models, respectively; however, they decreased at 2 °C in CBA with 1 % NaCl (Tmax = 35.1 °C). The application of CM for sgr provided an estimation of the parameter Tmin with negative values that are considered only as a theoretical output. The results provide insight into the modelling and prediction of fungi growth as a function of time and salt concentration, including the times to detect visible mycelial growth of Mucor circinelloides. The mere quantification of this phenomenon can be useful for practice. Adjusting the frequency of the cheese surface washing step with a salt solution at the early stage of ripening properly can prevent the growth of not only fast fungal growers.

Detection of risk areas in dairy powder processes: The development of thermophilic spore forming bacteria taking into account their growth limits

Anoxybacillus flavithermus, Geobacillus stearothermophilus and Bacillus licheniformis are the main contaminants found in dairy powders. These spore-forming thermophilic bacteria, rarely detected in raw milk, persist, and grow during the milk powder manufacturing process. Moreover, in the form of spores, these species resist and concentrate in the powders during the processes. The aim of this study was to determine the stages of the dairy powder manufacturing processes that are favorable to the growth of such contaminants. A total of 5 strains were selected for each species as a natural contaminant of dairy pipelines in order to determine the minimum and maximum growth enabling values for temperature, pH, and aw and their optimum growth rates in milk. These growth limits were combined with the environmental conditions of temperature, pH and aw encountered at each step of the manufacture of whole milk, skim milk and milk protein concentrate powders to estimate growth capacities using cardinal models and the Gamma concept. These simulations were used to theoretically calculate the population sizes reached for the different strains studied at each stage in between two successive cleaning in place procedures. This approach highlights the stages at which risk occurs for the development of spore-forming thermophilic bacterial species. During the first stages of production, i.e. pre-treatment, pasteurization, standardization and pre-heating before concentration, physico-chemical conditions encountered are suitable for the development and growth of A. flavithermus, G. stearothermophilus and B. licheniformis. During the pre-heating stage and during the first effects in the evaporators, the temperature conditions appear to be the most favorable for the growth of G. stearothermophilus. The temperatures in the evaporator during the last evaporator effects are favorable for the growth of B. licheniformis. In the evaporation stage, low water activity severely limits the development of A. flavithermus.

AMAnD: an automated metagenome anomaly detection methodology utilizing DeepSVDD neural networks.

The composition of metagenomic communities within the human body often reflects localized medical conditions such as upper respiratory diseases and gastrointestinal diseases. Fast and accurate computational tools to flag anomalous metagenomic samples from typical samples are desirable to understand different phenotypes, especially in contexts where repeated, long-duration temporal sampling is done. Here, we present Automated Metagenome Anomaly Detection (AMAnD), which utilizes two types of Deep Support Vector Data Description (DeepSVDD) models; one trained on taxonomic feature space output by the Pan-Genomics for Infectious Agents (PanGIA) taxonomy classifier and one trained on kmer frequency counts. AMAnD's semi-supervised one-class approach makes no assumptions about what an anomaly may look like, allowing the flagging of potentially novel anomaly types. Three diverse datasets are profiled. The first dataset is hosted on the National Center for Biotechnology Information's (NCBI) Sequence Read Archive (SRA) and contains nasopharyngeal swabs from healthy and COVID-19-positive patients. The second dataset is also hosted on SRA and contains gut microbiome samples from normal controls and from patients with slow transit constipation (STC). AMAnD can learn a typical healthy nasopharyngeal or gut microbiome profile and reliably flag the anomalous COVID+ or STC samples in both feature spaces. The final dataset is a synthetic metagenome created by the Critical Assessment of Metagenome Annotation Simulator (CAMISIM). A control dataset of 50 well-characterized organisms was submitted to CAMISIM to generate 100 synthetic control class samples. The experimental conditions included 12 different spiked-in contaminants that are taxonomically similar to organisms present in the laboratory blank sample ranging from one strain tree branch taxonomic distance away to one family tree branch taxonomic distance away. This experiment was repeated in triplicate at three different coverage levels to probe the dependence on sample coverage. AMAnD was again able to flag the contaminant inserts as anomalous. AMAnD's assumption-free flagging of metagenomic anomalies, the real-time model training update potential of the deep learning approach, and the strong performance even with lightweight models of low sample cardinality would make AMAnD well-suited to a wide array of applied metagenomics biosurveillance use-cases, from environmental to clinical utility.

Cardinal models to describe the effect of temperature and pH on the growth of Anoxybacillus flavithermus & Bacillus licheniformis

Anoxybacillus flavithermus and Bacillus licheniformis are among the predominant spore-formers of heat-processed foods. To our knowledge, no systematic analysis of growth kinetic data of A. flavithermus or B. licheniformis is currently available. In the present study, the growth kinetics of A. flavithermus and B. licheniformis in broth at various temperature and pH conditions were studied. Cardinal models were used to model the effect of the above-mentioned factors on the growth rates. The estimated values for the cardinal parameters Tmin,Topt,Tmax,pHmin and pH1/2 for A. flavithermus were 28.70 ± 0.26, 61.23 ± 0.16 and 71.52 ± 0.32 °C, 5.52 ± 0.01 and 5.73 ± 0.01, respectively, while for B. licheniformis they were 11.68 ± 0.03, 48.05 ± 0.15, 57.14 ± 0.01 °C, 4.71 ± 0.01 and 5.670 ± 0.08, respectively. The growth behaviour of these spoilers was also investigated in a pea beverage at 62 and 49 °C, respectively, to adjust the models to this product. The adjusted models were further validated at static and dynamic conditions and demonstrated good performance with 85.7 and 97.4% of predicted populations for A. flavithermus and B. licheniformis, respectively, being within the −10%–10% relative error (RE) zone. The developed models can be useful tools in assessing the potential of spoilage of heat-processed foods including plant-based milk alternatives.

The Löwenheim-Skolem theorem for Gödel logic

We prove the following Löwenheim-Skolem theorems for first-order Gödel logic:(1)For the Gödel logic G[0,1], a sentence ϕ has models of every infinite cardinality if and only if it has a model of cardinality ℶω(=sup⁡{ℵ0,2ℵ0,…}).(2)For an arbitrary Gödel logic GT, a sentence ϕ has models of every infinite cardinality if and only if it has a model of cardinality ℶω1. Moreover, (1) becomes false if ℶω is replaced by a smaller cardinality, and (2) becomes false if ℶω1 is replaced by a smaller cardinality.

High-Performance Row Pattern Recognition Using Joins

The SQL standard introduced MATCH_RECOGNIZE in 2016 for row pattern recognition. Since then, MATCH_RECOGNIZE has been supported by several leading relation systems, they implemented this function using Non-Deterministic Finite Automaton (NFA). While NFA is suitable for pattern recognition in streaming scenarios, the current uses of NFA by the relational systems for historical data analysis scenarios overlook important optimization opportunities. We propose a new approach to use Join to speed up row pattern recognition in historical analysis scenarios for relational systems. Implemented as a logical plan rewrite rule, the new approach first filters the input relation to MATCH_RECOGNIZE using Joins constructed based on a subset of symbols taken from the PATTERN expression, then run the NFA-based MATCH_RECOGNIZE on the filtered rows, reducing the net cost. The rule also includes a specialized cardinality model for the Joins and a cost model for the NFA-based MATCH_RECOGNIZE operator for choosing an appropriate symbol set. The rewrite rule is applicable when the query pattern's definition is self-contained and either the input table has no duplicates or there is a window condition. Applying the rewrite rule to a query benchmark with 1,800 queries spanning over 6 patterns and 3 pattern definitions, we observed median speedups of 5.4X on Trino (v373 with ORC files on Hive), 57.5X on SQL Server (2019) using column store and 41.6X on row store.

Re-parameterization of the asymmetric model for fungal spore germination

The asymmetric model (Dantigny et al., 2011, A new model for germination of fungi, International Journal of Food Microbiology, 146, 176–181) is capable to fit both symmetric and asymmetric curves of fungal spore germination. However, the shape parameter in the asymmetric model must be derived from a regression of the model with experimental data of fungal spore germination, thus the asymmetric model can fit spore germination under tested conditions only. In this paper, to solve this limitation, the previously proposed shape parameter-based asymmetric model has been replaced by using the pure biological parameter-based asymmetric model as: P=Pmax·1–11+tτ4τ·μPmax, where P is the percentage of germinated spores at time t, Pmax is the maximum percentage of germinated spores, τ is the germination time (the time that P = 0.5Pmax), and μ is the slope of fungal spore germination at t = τ, representing the rate of fungal spore germination. The modified asymmetric model was validated against the germination data of six fungal species collected from the literature. The results showed that the modified asymmetric model described the spore germination of all fungi studied well. Moreover, the τ and μ in the modified asymmetric model at various environmental conditions were estimated properly via the Cardinal Model with Inflection. This solved the limitation of the previous asymmetric model. So, it can be concluded that the modified asymmetric model is an improvement over the previous asymmetric model for predicting fungal spore germination.

Finding and Recognizing Popular Coalition Structures

An important aspect of multi-agent systems concerns the formation of coalitions that are stable or optimal in some well-defined way. The notion of popularity has recently received a lot of attention in this context. A partition is popular if there is no other partition in which more agents are better off than worse off. In this paper, we study popularity, strong popularity, and mixed popularity (which is particularly attractive because existence is guaranteed by the Minimax Theorem) in a variety of coalition formation settings. Extending previous work on marriage games, we show that mixed popular partitions in roommate games can be found efficiently via linear programming and a separation oracle. This approach is quite universal, leading to efficient algorithms for verifying whether a given partition is popular and for finding strongly popular partitions (resolving an open problem). By contrast, we prove that both problems become computationally intractable when moving from coalitions of size 2 to coalitions of size 3, even when preferences are strict and globally ranked. Moreover, we show that finding popular, strongly popular, and mixed popular partitions in symmetric additively separable hedonic games and symmetric fractional hedonic games is NP-hard. Together, these results indicate strong boundaries to the tractability of popularity in both ordinal and cardinal models of hedonic games.

Raw vs. frozen pork “gyros”: Predicting simultaneous growth of pathogenic and spoilage microorganisms under commercially occurring roasting scenarios

The objectives of the present study were: (i) to record time-temperature data in the geometrical center (“coldest spots”) of different sizes and states (raw vs. frozen) of pork “gyros” during realistic cooking scenarios, (ii) to model the growth of Salmonella spp., Escherichia coli O157:H7, Staphylococcus aureus, Clostridium sporogenes (as a surrogate of Clostridiumbotulinum), and lactic acid bacteria (LAB) (dominant indigenous microbiota) as a function of temperature in the range likely occurring during roasting, using existing and newly fitted cardinal growth models and (iii) to evaluate the performance of these models (incorporating or not the Jameson effect) in simulating the growth of Salmonella spp. or C. sporogenes at different starting levels, in parallel to that of dominant microbiota, under commercial roasting scenarios of raw and frozen gyros. In frozen gyros, the residence of “coldest spots” within the dangerous temperature zone (4.4–60οC) was either non-existing (60 and 180 kg) or limited (30 kg) compared to raw gyros during the 8.5–11 h roasting. Regarding microbiological data, growth rate was increased from 10 °C to 40 °C for Salmonella spp. and E. coli O157: H7 or from 12 °C to 45 °C for C. sporogenes, while limited growth (Salmonella spp. and E. coli O157:H7), or inactivation (S. aureus) of pathogens was recorded during storage at 45 °C. Growth of S. aureus was observed only during storage at 40 °C. Primary modeling was performed by using the Baranyi model, while secondary cardinal models were adequately applied to describe growth rate as a function of temperature, according to the low RMSE values (0.017–0.050). Under the worst-case scenario (2-days roasting), Salmonella spp. growth was observed not only in raw gyros (ca. 2.5 log increase) but also in frozen (ca. 2.0 log increase). On the contrary, C. sporogenes growth potential was significantly lower, i.e., 0.5–1.0 log increase, suggesting a non-considerable risk of C. botulinum toxin production. A good agreement between simulated and observed growth of Salmonella spp. or C. sporogenes against LAB was indicated by the ASZ approach (Salmonella spp.: 83%; C. sporogenes: 84%; LAB: 81%). Overall, the developed secondary models may adequately predict the growth of Salmonella spp. or C. sporogenes against LAB under the examined contamination levels and roasting scenarios; however further testing of models applicability is prompted in relevant products i.e., gyros made of chicken, kebab, or döner.

Adaptation of Cardinal Points Symmetry Landmarks Distribution Model to in-vivo B-Mode Ultrasound Images of Transverse Cross-section of Carotid Arteries

The geometry of the imaged transverse cross-section of carotid arteries in in-vivo B-mode ultrasound images are most times irregular, unsymmetrical, full of speckles and usually non-uniform. We had earlier developed a technique of cardinal point symmetry landmark distribution model (CPS-LDM) to completely characterize the Region of Interest (ROI) of the geometric shape of thick-walled simulated B-mode ultrasound images of carotid artery imaged in the transverse plane, but this was based on the symmetric property of the image. In this paper, this developed technique was applied to completely characterize the region of interest of the geometric shape of in-vivo B-mode ultrasound images of non-uniform carotid artery imaged in the transverse plane. In order to adapt the CPS-LD Model to the in-vivo carotid artery images, the single VS-VS vertical symmetry line common to the four ROIs of the symmetric image is replaced with each ROI having its own VS-VS vertical symmetry line. This adjustment enables the in-vivo carotid artery images possess symmetric properties, hence, ensuring that all mathematical operations of the CPS-LD Model are conveniently applied to them. This adaptability was observed to work well in segmenting the in-vivo carotid artery images. This paper shows the adaptive ability of the developed CPS-LD Model to successfully annotate and segment in-vivo B-mode ultrasound images of carotid arteries in the transverse cross-sectional plane either they are symmetrical or unsymmetrical.

Isomorphic limit ultrapowers for infinitary logic

The logic \({\mathbb{L}}_\theta ^1\) was introduced in [She12]; it is the maximal logic below \({{\mathbb{L}}_{\theta, \theta}}\) in which a well ordering is not definable. We investigate it for θ a compact cardinal. We prove that it satisfies several parallels of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are \({\mathbb{L}}_\theta ^1\)-equivalent iff for some ω-sequence of θ-complete ultrafilters, the iterated ultrapowers by it of those two models are isomorphic.Also for strong limit λ>θ of cofinality \({\aleph _0}\), every complete \({\mathbb{L}}_\theta ^1\)-theory has a so-called special model of cardinality λ, a parallel of saturated. For first order theory T and singular strong limit cardinal λ, T has a so-called special model of cardinality λ. Using “special” in our context is justified by: it is unique (fixing T and λ), all reducts of a special model are special too, so we have another proof of interpolation in this case.

Growth Modeling Kinetics of Aspergillus flavus in Dried Jujube at Different Temperatures

Dried jujube is a characteristic fruit of Xinjiang. Aspergillus is one of the main pathogens that causes mold on dried jujube, and A. flavus is a toxin-producing species, the aflatoxin produced by A. flavus is extremely toxic and carcinogenic. In this study, the growth kinetic models of A. flavus isolated from red jujube at different temperatures and times were fitted to Huang model and linear equation respectively, the Cardinal model was used to describe the growth rate and lag time of A. flavus on dried jujube agar. It turned out that 30–35 °C was the optimal temperature for growing A. flavus, so dried jujube should avoid storing in this temperature range. The kinetic model established in this study will help to understand the growth characteristics of A. flavus, and lay a foundation for evaluating the quality of stored dried jujube and predictions of shelf life, which are conducive to optimizing storage methods for dried jujube. It can be judged according to the value of Af and Bf, the Huang model had a better fitting effect than the Baranyi model, The two models all had the highest growth rate at 35 °C, and A. flavus grew more vigorously and the lag period shortened as the temperature was increased. The secondary Cardinal model had a good fitting effect on the growth rate and lag time, and the secondary Ratkowsky model had a good fitting effect on the growth rate. This study may have theoretical and application value to strengthen the safety of jujube storage in the future.

SkinnerDB: Regret-bounded Query Evaluation via Reinforcement Learning

SkinnerDB uses reinforcement learning for reliable join ordering, exploiting an adaptive processing engine with specialized join algorithms and data structures. It maintains no data statistics and uses no cost or cardinality models. Also, it uses no training workloads nor does it try to link the current query to seemingly similar queries in the past. Instead, it uses reinforcement learning to learn optimal join orders from scratch during the execution of the current query. To that purpose, it divides the execution of a query into many small time slices. Different join orders are tried in different time slices. SkinnerDB merges result tuples generated according to different join orders until a complete query result is obtained. By measuring execution progress per time slice, it identifies promising join orders as execution proceeds. Along with SkinnerDB, we introduce a new quality criterion for query execution strategies. We upper-bound expected execution cost regret, i.e., the expected amount of execution cost wasted due to sub-optimal join order choices. SkinnerDB features multiple execution strategies that are optimized for that criterion. Some of them can be executed on top of existing database systems. For maximal performance, we introduce a customized execution engine, facilitating fast join order switching via specialized multi-way join algorithms and tuple representations. We experimentally compare SkinnerDB’s performance against various baselines, including MonetDB, Postgres, and adaptive processing methods. We consider various benchmarks, including the join order benchmark, TPC-H, and JCC-H, as well as benchmark variants with user-defined functions. Overall, the overheads of reliable join ordering are negligible compared to the performance impact of the occasional, catastrophic join order choice.

Development and validation of a predictive model for the effect of temperature, pH and water activity on the growth kinetics of Bacillus coagulans in non-refrigerated ready-to-eat food products

A cardinal model (CM) for the effects of temperature (range: 32–59 °C), pH (range: 5.0–8.5) and water activity (aw) (range: 0.980–0.995) on Bacillus coagulans DSM 1 growth rate was developed in brain heart infusion broth (BHI), using the Bioscreen C method and further validated in selected food products. The estimated values for the cardinal parameters Tmin, Topt, Tmax, pHmin, pHopt, pHmax, awmin and awopt were 23.77 ± 0.19 °C, 52.89 ± 0.01 °C, 59.37 ± 0.07 °C, 4.70 ± 0.02, 6.43 ± 0.02, 8.56 ± 0.01, 0.969 ± 0.0007 and 0.998 ± 0.0011, respectively. The growth behaviour of B. coagulans was studied in five commercial non-refrigerated ready-to-eat food products under static conditions at 53 °C in order to estimate the optimum specific growth rate for each tested food product. The developed models were validated in the five selected food products under four different dynamic temperature profiles by comparing predicted and observed growth behaviour of B. coagulans. The validation results indicated a good performance of the model for all tested products with the overall Bias factor (Bf) and Accuracy factor (Af) estimated at 1.00 and 1.12, respectively. The developed model can be considered an effective tool in predicting B. coagulans growth and spoilage risks of non-refrigerated ready-to-eat food products during distribution and storage.

Effect of temperature on mycelial growth of Alternaria alternata and Colletotrichum gloeosporioides isolated from papaya fruit

The objective of this study was to evaluate the influence of temperature on mycelial appearance time (tv ) for Alternaria alternata and Colletotrichum gloeosporioides isolated from papaya rots by predictive mathematical models and to evaluate their performance in predicting fungal growth. The Baranyi and Roberts models were used to estimate the radial growth rates (µ). The cardinal model with inflexion was applied to model µ as a function of the temperature for A. alternate. A polynomial function was used to model tv for A. alternata, whereas the Arrhenius-Davey model for C. gloeosporioides. Simulations of tv were conducted with data obtained on the papaya fruit. The resulted models seem satisfactory to describe tv . The model provides insight into the dynamics of postharvest fungal disease and forms the basis of a predictive model to be used by transporters to determine how long a fruit can be conserved before the infection starts. Highlights The main postharvest rots of papaya are C. gloeosporioides and A. alternata. The time of mycelium appearance may be useful to determine the shell life. Correlations exist between fungal growth in vitro and in the real matrix. The results may help us to improve the supply chain.

Intraspecific variability in cardinal growth temperatures and water activities within a large diversity of Penicillium roqueforti strains

Different strains of a given fungal species may display heterogeneous growth behavior in response to environmental factors. In predictive mycology, the consideration of such variability during data collection could improve the robustness of predictive models. Among food-borne fungi, Penicillium roqueforti is a major food spoiler species which is also used as a ripening culture for blue cheese manufacturing. In the present study, we investigated the intraspecific variability of cardinal temperatures and water activities (aw), namely, minimal (Tmin and awmin), optimal (Topt and awopt) and maximal (Tmax) temperatures and/or aw estimated with the cardinal model for radial growth, of 29 Penicillium roqueforti strains belonging to 3 genetically distinct populations. The mean values of cardinal temperatures and aw for radial growth varied significantly across the tested strains, except for Tmax which was constant. In addition, the relationship between the intraspecific variability of the biological response to temperature and aw and putative genetic populations (based on microsatellite markers) within the selected P. roqueforti strains was investigated. Even though no clear relationship was identified between growth parameters and ecological characteristics, PCA confirmed that certain strains had marginal growth response to temperature or aw. Overall, the present data support the idea that a better knowledge of the response to abiotic factors such as temperature and aw at an intraspecific level would be useful to model fungal growth in predictive mycology approaches.

Quantitative Characterization of Geotrichum candidum Growth in Milk

The study of microbial growth in relation to food environments provides essential knowledge for food quality control. With respect to its significance in the dairy industry, the growth of Geotrichum candidum isolate J in milk without and with 1% NaCl was investigated under isothermal conditions ranging from 6 to 37 °C. The mechanistic model by Baranyi and Roberts was used to fit the fungal counts over time and to estimate the growth parameters of the isolate. The effect of temperature on the growth of G. candidum in milk was modelled with the cardinal models, and the cardinal temperatures were calculated as Tmin = −3.8–0.0 °C, Topt = 28.0–34.6 °C, and Tmax = 35.2–37.2 °C. The growth of G. candidum J was slightly faster in milk with 1% NaCl and in temperature regions under 21 °C. However, in a temperature range that was close to the optimum, its growth was slightly inhibited by the lowered water activity level. The present study provides useful cultivation data for understanding the behaviour of G. candidum in milk and can serve as an effective tool for assessing the risk of fungal spoilage, predicting the shelf life of dairy products, or assessing the optimal conditions for its growth in relation to the operational parameters in dairy practices.

Development and validation of an extended predictive model for the effect of pH and water activity on the growth kinetics of Geobacillus stearothermophilus in plant-based milk alternatives

The cardinal model for the effect of temperature on Geobacillus stearothermophilus ATCC 7953 growth developed by Kakagianni, Gougouli, & Koutsoumanis, 2016 was expanded for the effect of pH and water activity (aw). The effect of pH (range: 5.7–8.5) and aw (range: 0.985–0.999) on G. stearothermophilus growth rate was studied in tryptone soy broth (TSB) using the Bioscreen C method and further modelled using a Cardinal Model (CM). The estimated values for the cardinal parameters pHmin,pHopt,pHmax,awmin, and awopt were 5.65 ± 0.14, 6.74 ± 0.03, 8.71 ± 0.03, 0.984 ± 0.007 and 0.998 ± 0.001, respectively. The growth behaviour of G. stearothermophilus was investigated in 7 commercial non-refrigerated plant-based milk alternatives under static conditions (62 °C) and the estimated maximum specific growth rates were used to determine the optimum growth rate for each product. The developed model was validated against observed growth of G. stearothermophilus in the 7 products during storage at non-isothermal conditions (testing 4 different temperature profiles). The validation results showed a good performance of the model with overall Bias factor (Bf) = 1.06 and Accuracy factor (Af) = 1.12. The developed model can be used as an effective tool by the food industry in predicting spoilage of plant-based milk alternatives during distribution and storage at retail and domestic levels.