Mathematical Models to Describe the Kinetic Behavior of Staphylococcus aureus in Jerky

Jimyeong Ha1,2, Jeeyeon Lee1,2, Soomin Lee1,2, Sejeong Kim1,2, Yukyung Choi1, Hyemin Oh1, Yujin Kim1, Yewon Lee1, Yeongeun Seo1, Yohan Yoon1,2,*
Author Information & Copyright
1Department of Food and Nutrition, Sookmyung Women’s University, Seoul 04310, Korea
2Risk Analysis Research Center, Sookmyung Women’s University, Seoul 04310, Korea
*Corresponding author : Yohan Yoon, Department of Food and Nutrition, Sookmyung Women’s University, Seoul 04310, Korea Tel: +82-2-2077-7585 Fax: +82-2-710-9479 E-mail:

© Korean Society for Food Science of Animal Resources. This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received: Mar 28, 2019 ; Revised: Apr 04, 2019 ; Accepted: Apr 04, 2019

Published Online: Jun 30, 2019


The objective of this study was to develop mathematical models for describing the kinetic behavior of Staphylococcus aureus (S. aureus) in seasoned beef jerky. Seasoned beef jerky was cut into 10-g pieces. Next, 0.1 mL of S. aureus ATCC13565 was inoculated into the samples to obtain 3 Log CFU/g, and the samples were stored aerobically at 10°C, 20°C, 25°C, 30°C, and 35°C for 600 h. S. aureus cell counts were enumerated on Baird Parker agar during storage. To develop a primary model, the Weibull model was fitted to the cell count data to calculate Delta (required time for the first decimal reduction) and ρ (shape of curves). For secondary modeling, a polynomial model was fitted to the Delta values as a function of storage temperature. To evaluate the accuracy of the model prediction, the root mean square error (RMSE) was calculated by comparing the predicted data with the observed data. The surviving S. aureus cell counts were decreased at all storage temperatures. The Delta values were longer at 10°C, 20°C, and 25°C than at 30°C and 35°C. The secondary model well-described the temperature effect on Delta with an R2 value of 0.920. In validation analysis, RMSE values of 0.325 suggested that the model performance was appropriate. S. aureus in beef jerky survives for a long period at low storage temperatures and that the model developed in this study is useful for describing the kinetic behavior of S. aureus in seasoned beef jerky.

Keywords: jerky; mathematical model; Staphylococcus aureus; Weibull model


Jerky is a nutritional snack with a high protein content and light weight, and thus it is consumed by many people (Holley, 1985). It is also easy to store because of its long shelf-life and low Aw (Calicioglu et al., 2003). However, outbreaks of foodborne illness have occurred in many countries (Eidson et al., 2000; Keene et al., 1997). These outbreaks may be caused by cross contamination during jerky processing, molding, packaging, and cutting. Also, most of jerkies are made in small companies, and these companies have difficulties for food hygiene management. Thus, foodborne pathogen growth need to be simulated in jerky for exposure assessment.

Staphylococcus aureus can produce enterotoxin, leading to foodborne intoxication (Le Loir et al., 2003). Generally, the symptoms of foodborne illness include abdominal cramps, vomiting, and diarrhea (Jones et al., 2002). S. aureus can grow under various conditions, such as a wide range of temperatures, pH, and low AW (Bergdoll, 1989; Schmitt et al., 1990) and most S. aureus isolates from food exhibit antimicrobial resistance (Can et al., 2017). The pathogen is commonly found on human skin (Otto, 2008), and may be cross-contaminated from human hands to jerky. Thus, there is high possibility for jerky contamination by S. aureus.

Predictive models are useful for estimating microbial growth or death in food using mathematical models (Zwietering et al., 1996). The purpose of a predictive model is to secure food safety in advance by identifying risk factors (Yoon, 2010). A primary model describes changes in bacterial cell counts over storage time to calculate kinetic parameters such as growth rate and lag phase duration (Ha et al., 2016). A secondary model describes the effects of environmental factors such as pH, AW, and temperature on kinetic parameters (Buchanan, 1993; Ha et al., 2016).

Therefore, the objective of this study was to develop mathematical models for describing the kinetic behavior of S. aureus in beef jerky.

Materials and Methods

Preparation of inocula

S. aureus ATCC13565 was cultured in 10 mL of tryptic soy broth (TSB; BD Biosciences, Franklin Lakes, NJ, USA) at 37°C for 24 h. For subculture, 0.1 mL of the culture was transferred into 10 mL fresh TSB at 37°C for 24 h. The sample was centrifuged at 1,912×g and 4°C for 15 min and washed twice with phosphate-buffered saline (PBS: pH 7.4; 0.2 g of KH2PO4, 1.5 g of Na2HPO4·7H2O, 8.0 g of NaCl, and 0.2 g of KCl in 1 L of distilled water). The supernatants were discarded, and the cell pellets were resuspended in PBS. Cell suspensions were diluted with PBS to 3–4 Log CFU/mL for inoculation.

Development of predictive model

Seasoned beef jerky was purchased from an online shop in Korea. Ten-gram portions of the samples were placed into sterile filter bags (3M, St. Paul, MN, USA), and 0.1-mL aliquots of S. aureus were dotted on several places of the beef jerky surface for inoculation to obtain 3 Log CFU/g in the sample bags. The samples were rubbed 20 times and the sample bags were sealed, followed by aerobic storage at 10°C (600 h), 20°C (600 h), 25°C (480 h), 30°C (192 h), and 35°C (96 h). These time intervals were determined according to the time that S. aureus cell counts were below detection limit. Beef jerky samples were analyzed at different time intervals. Thirty milliliters of 0.1% buffered peptone water (BPW; BD Biosciences) were added to each sample and homogenized with a BagMixer (Interscience, St. Nom, France) for 90 s. The homogenates were serially diluted with BPW. The 0.1 mL of the diluents were plated onto Baird-Parker agar (MB Cell, Los Angeles, CA, USA) for S. aureus, and the plates were incubated at 37°C for 48 h. Typical colonies on the plates were counted, and the Weibull model was fitted to the S. aureus cell count data (Van Boekel, 2002).

Log ( N ) = Log ( N 0 ) ( time/ δ ) ρ

where N0 is the initial cell count, ρ is the shape of the curve, and δ is the time required for the first decimal reduction. The polynomial model (δ=N0+a×T+b×T2) was used to evaluate the effect of storage temperature on δ.


S. aureus cell count data were obtained at 15°C and 23°C in additional experiments to evaluate the model performance. These observed data were compared to predicted data, which were calculated from the predictive model. The differences between the observed and predicted data were quantified by calculating the root mean square error (RMSE) (Baranyi et al., 1996);

RMSE = 1 / n × ( observed data - predicticed data ) 2

where n represents the number of data points.

Statistical analysis

The experimental data were analyzed with the general linear model procedure of SAS® version 9.3 (SAS Institute, Inc., Cary, NC, USA). The mean comparisons were performed by a pairwise t-test at α=0.05.

Results and Discussion

Because various types of jerky are available made from different meat types and marinades, the behavior of S. aureus may differ among jerkies. Thus, a predictive model should be developed for each jerky type to describe the behavior of S. aureus. However, this effort requires a long time and is costly. Developing a model with the jerky type, allowing the highest S. aureus growth, may be appropriate for the most severe case, which would save time and expense. To determine a model for developing a predictive model, we examined the pH and water activities of 75 samples of 15 original jerky products (Table 1) and 50 samples of 10 seasoned jerky products (Table 2). The pH values were highly similar among the samples (6.13–6.17), but the water activities were higher in the seasoned jerky (0.810±0.045) than in the original jerky (0.656±0.134) (Tables 1 and 2). The 10-seasoned jerky products contained sodium nitrite, potassium sorbate, and sodium sorbate. The growth of most bacteria is inhibited when AW is reduced. Particularly, the growth of S. aureus is inhibited when AW is less than 0.850 in an aerobic environment (Jay 1992). Holley (1985) indicated that the AW of jerky was 0.620 when stored at refrigeration temperature for 26 days, but there was no significant difference in S. aureus cell counts compared to the cell counts on day 0. This result suggests that S. aureus can survive in beef jerky even if AW is less than 0.850. Additionally, Lee et al. (2016) indicated that S. aureus did not grow under vacuum conditions. Hence, we developed predictive models using the seasoned beef jerky products as a model product under aerobic conditions to predict the most severe case of S. aureus growth.

Table 1. General information of original jerky samples purchased from online shops
Sample Meat (%) Aw pH
A Beef (85.82) 0.739 6.17±0.05
B Beef (87.00) 0.811 5.97±0.07
C Beef (85.23) 0.747 6.12±0.07
D Chicken (88.00) 0.620 6.17±0.04
E Beef (85.06) 0.771 6.09±0.08
F Beef (85.76) 0.770 5.71±0.04
G Beef (85.76) 0.792 5.92±0.05
H Beef (91.59) 0.520 6.19±0.03
I Chicken (90.12) 0.833 6.52±0.04
J Beef (88.33) 0.481 6.13±0.03
K Beef (86.13) 0.479 6.22±0.09
L Pork (94.00) 0.545 6.39±0.03
M Beef (88.47) 0.703 6.06±0.04
N Beef (85.27) 0.504 6.57±0.07
O Chicken (87.00) 0.527 6.30±0.04
Average 0.656 6.17±0.22
Download Excel Table
Table 2. General information of seasoned jerky samples purchased from online shops
Sample Meat (%) Aw pH
a Beef (93.51) 0.833 6.12±0.11
b Beef (86.20) 0.834 5.97±0.09
c Beef (85.07) 0.813 5.95±0.05
d Beef (85.14) 0.792 5.90±0.09
e Beef (86.76) 0.808 5.99±0.02
f Chicken (87.00) 0.718 6.33±0.03
g Beef (85.07) 0.804 6.46±0.03
h Beef (85.22) 0.767 6.09±0.02
i Beef (88.15) 0.858 6.16±0.04
j Beef (90.12) 0.874 6.36±0.06
Average 0.810 6.13±0.19
Download Excel Table

S. aureus-inoculated seasoned beef jerky samples were stored in aerobic packaging at 10°C, 20°C, 25°C, 30°C, and 35°C. The cell counts were gradually decreased at 10°C and 20°C, but a tail effect was observed at 10°C and S. aureus cell counts survived through the end of storage at 20°C (Fig. 1). However, the S. aureus cell counts greatly decreased as the temperature was increased to 25°C, 30°C, and 35°C (Fig. 1). The cell counts decreased to below the detection limit (0.48 Log CFU/g) after 432, 144, and 120 h at 20°C, 25°C, and 30°C, respectively (Fig. 1).

Fig. 1. Cell counts of Staphylococcus aureus in jerky during aerobic storage at 10°C, 20°C, 25°C, 30°C, and 35°C. Symbol, observed cell counts; line, fitted line with the Weibull model (van Boekel, 2002).
Download Original Figure

To describe the kinetic behavior of S. aureus in beef jerky, primary models were developed and R2 values ranged from 0.868 to 0.967, indicating that the developed primary models were appropriate. These primary models showed that the δ values generally decreased as temperature increased (Table 3). This result agrees with those of Moon et al. (2017) who showed that S. aureus in dried julienned squid survived longer at 10°C than at 35°C. These results suggest that if S. aureus is contaminated in beef jerky stored at low temperature, the pathogen can survive for a long time and cause food safety issues. Because S. aureus cell counts decreased as shown in Fig. 1, the ρ values were less than 1, indicating that all curves were concave (Coroller et al., 2006). To evaluate the effect of ρ on temperature, a secondary model was developed and R2 was 0.920 (Fig. 2), indicating that the developed model was appropriate. The equation was δ=(−4.4271)+(13.9841×T)+(−0.3605×T2) (Fig. 2). The secondary model showed that the δ values were generally influenced by temperature. The RMSE value was calculated to evaluate model performance. A value close to zero indicates that the predicted values are the same as the observed values (Kim et al., 2017). In this study, the value was 0.326, indicating that the developed models were appropriate for describing the kinetic behavior of S. aureus in beef jerky.

Fig. 2. δ values from the primary model and the fitted line by a secondary model describing the effect of temperature on δ for Staphylococcus aureus in jerky.
Download Original Figure
Table 3. δ and ρ calculated by the Weibull model for Staphylococcus aureus survival in jerky during aerobic storage at 10°C, 20°C, 25°C, 30°C, and 35°C
Kinetic parameters Temperature (°C)
10 20 25 30 35
δ 99.335±2.072B 128.950±15.910A 126.450±8.556A 83.910±6.986B 45.670±5.586C
ρ 0.432±0.037C 0.671±0.046B 0.611±0.021B 0.916±0.020A 0.666±0.074B
R2 0.869 0.931 0.954 0.868 0.967

δ, required time for the first decimal reduction; ρ, shape of curve.

A–C Means within the same row with different superscript letters are significantly different (p<0.05).

Download Excel Table

In conclusion, the developed predictive models are useful in describing the kinetic behavior of S. aureus in beef jerky. Additionally, because the model beef jerky was selected according to the most optimum growth conditions for S. aureus, the developed models can be applied to other jerkies. Although beef jerky has a low AW, if S. aureus is contaminated in the beef jerky, the cells can survive for a long period at low temperature and cause food safety issues. Therefore, beef jerky should not be considered as a microbiologically safe food and thus, cross-contamination should be controlled during processing.


Conflict of Interest

The authors declare no potential conflict of interest.


This research was supported by a grant (16162MFDS584) from Ministry of Food and Drug Safety, Korea in 2016.


Author Contributions

Conceptualization: Ha J, Yoon Y. Data curation: Lee S, Kim S. Formal analysis: Lee J. Methodology: Choi Y, Oh H. Validation: Kim Y, Lee Y, Seo Y. Investigation: Yoon Y. Writing - original draft: Ha J, Yoon Y. Writing - review & editing: Ha J, Lee J, Lee S, Kim S, Choi Y, Oh H, Kim Y, Lee Y, Seo Y, Yoon Y.


Ethics Approval

This article does not require IRB/IACUC approval because there are no human and animal participants.



Baranyi J, Ross T, McMeekin TA, Roberts TA. 1996; Effects of parameterization on the performance of empirical models used in ‘predictive microbiology’. Food Microbiol. 13:83-91


Bergdoll MS. 1989; Staphylococcus aureus. In Foodborne bacterial pathogens. In: Doyle MP, editor.(ed.)Marcel Dekker. New York, NY, USA: p. 463-523.


Buchanan RL. 1993; Predictive food microbiology. Trends Food Sci Technol. 4:6-11


Calicioglu M, Sofos JN, Kendall PA. 2003; Influence of marinades on survival during storage of acid-adapted and nonadapted Listeria monocytogenes inoculated post-drying on beef jerky. Int J Food Microbiol. 86:283-292


Can HY, Elmali M, Karagoz A. 2017; Molecular typing and antimicrobial susceptibility of Staphylococcus aureus strains isolated from raw milk, cheese, minced meat, and chicken meat samples. Korean J Food Sci Anim Resour. 37:175-180


Coroller L, Leguerinel I, Mettler E, Savy N, Mafart P. 2006; General model, based on two mixed Weibull distributions of bacterial resistance, for describing various shapes of inactivation curves. Appl Environ Microbiol. 72:6493-6502


Eidson M, Sewell CM, Graves G, Olson R. 2000; Beef jerky gastroenteritis outbreaks. J Environ Health. 62:9-13.


Ha J, Gwak E, Oh MH, Park B, Lee J, Kim S, Lee H, Lee S, Yoon Y, Choi KH. 2016; Kinetic behavior of Salmonella on low NaNO2 sausages during aerobic and vacuum storage. Korean J Food Sci Anim Resour. 36:262-266


Holley RA. 1985; Beef jerky: Fate of Staphylococcus aureus in marinated and corned beef during jerky manufacture and 2.5°C storage. J Food Prot. 48:107-111


Jay JM. 1992 Modern food microbiology. 4th edChapman & Hall. New York, NY, USA:


Jones TF, Kellum ME, Porter SS, Bell M, Schaffner W. 2002; An outbreak of community-acquired foodborne illness caused by methicillin-resistant Staphylococcus aureus. Emerg Infect Dis. 8:82-84


Keene WE, Sazie E, Kok J, Rice DH, Hancock DD, Balan VK, Zhao T, Doyle MP. 1997; An outbreak of Escherichia coli O157:H7 infections traced to jerky made from deer meat. JAMA. 277:1229-1231


Kim S, Jeong J, Lee H, Lee J, Lee S, Ha J, Choi Y, Yoon Y, Choi KH. 2017; Kinetic behavior of Campylobacter jejuni in beef tartare at cold temperatures and transcriptomes related to its survival. J Food Prot. 80:2127-2131


Le Loir Y, Baron F, Gautier M. 2003; Staphylococcus aureus and food poisoning. Genet Mol Res. 2:63-76.


Lee J, Gwak E, Ha J, Kim S, Lee S, Lee H, Oh MH, Park BY, Oh NS, Choi KH, Yoon Y. 2016; Mathematical model for predicting the growth probability of Staphylococcus aureus in combinations of NaCl and NaNO2 under aerobic or evacuated storage conditions. Korean J Food Sci Anim Resour. 36:752-759


Moon HJ, Min KJ, Park NY, Park HJ, Yoon KS. 2017; Survival of Staphylococcus aureus in dried fish products as a function of temperature. Food Sci Biotechnol. 26:823-828


Otto M. 2008; Staphylococcal biofilms. Curr Top Microbiol Immunol. 322:207-228


Schmitt M, Schuler-Schmid U, Scmidt-Lorenz W. 1990; Temperature limits of growth, TNase and enterotoxin production of Staphylococcus aureus strains isolated from foods. Int J Food Microbiol. 11:1-19


van Boekel MA. 2002; On the use of the Weibull model to describe thermal inactivation of microbial vegetative cells. Int J Food Microbiol. 74:139-159


Yoon Y. 2010; Principal theory and application of predictive microbiology. Food Sci Ind. 43:70-74.


Zwietering MH, De Wit JC, Notermans S. 1996; Application of predictive microbiology to estimate the number of Bacillus cereus in pasteurised milk at the point of consumption. Int J Food Microbiol. 30:55-70