Probability of Escherichia coli contamination spread in ground beef production

Petko M. Kitanov; Allan R. Willms

doi:10.3934/mbe.2018045

Article Contents

2018, Volume 15, Issue 4: 1011-1032. Doi: 10.3934/mbe.2018045

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Probability of Escherichia coli contamination spread in ground beef production

Petko M. Kitanov^1, and
Allan R. Willms^2, ,

1.
School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623, USA
2.
Department of Mathematics & Statistics, University of Guelph, Guelph ON N1G 2W1, Canada

* Corresponding author: Allan R. Willms
* Corresponding author: Allan R. Willms

Received: September 7, 2017

Accepted: October 27, 2017

Published online: August 2018

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Abstract

Human illness due to contamination of food by pathogenic strains of Escherichia coli is a serious public health concern and can cause significant economic losses in the food industry. Recent outbreaks of such illness sourced from ground beef production motivates the work in this paper. Most ground beef is produced in large facilities where many carcasses are butchered and various pieces of them are ground together in sequential batches. Assuming that the source of contamination is a single carcass and that downstream from the production facility ground beef from a particular batch has been identified as contaminated by E. coli, the probability that previous and subsequent batches are also contaminated is modelled. This model may help the beef industry to identify the likelihood of contamination in other batches and potentially save money by not needing to cook or recall unaffected batches of ground beef.

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Mathematics Subject Classification: Primary: 92B05; Secondary: 92F05, 62P10.

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Figure 1. Schematic diagram showing spread of meat in ground beef production. Each carcass is spread in a like manner within a region of a raw source. The centres of regions from sequential carcasses are shifted forward in the raw source. The model can account for carcasses being spread across raw sources (dashed line). Material from the raw sources is sequentially input to consecutive batches of ground beef

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Figure 2. Spread of carcasses in a raw source. (a): Example base probability density function for pieces from a carcass. The density is symmetric, and piece-wise linear. In this example there are $K = 2$ linear segments both to the left and right of zero, and there are discontinuities at $\pm N_2$. The parameters $K$, $N$, and $H$ must be chosen so that the area under the curve equals one. (b): Distributions of carcasses in a raw source. The centres, $\mu_c$, of the distributions of consecutive carcasses are evenly spaced along the raw source. The individual distributions overlap (more than depicted in the figure). For carcasses in the middle of the source, the probability density function $G_{sc}$ is just a shifted version of the base base probability $F_s$, as illustrated by $G_{s5}$. At the ends, distributions that extend beyond the boundary are reflected back, as indicated by the arrow and dashed line at the bottom left, and the reflected portion is added to the distribution already there yielding the dotted line distribution $G_{s1}$.

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Figure 3. Mass input from raw source $s$ to batches. Batch $b$ receives a mass of $m_{sb}$ from source $s$. This mass is located in the interval $B_{sb} = (M_{sb}, M_{sb}+m_{sb}]$, where $M_{sb}$ is the total mass used from this source in batches prior to $b$

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Figure 4. Probability (%) that a particular carcass in a hot source is hot given that batch $h$ is the hot batch, Equation (10), for the synthetic data set. The four separate curves in each plot correspond to hot batch $h = 5, 8, 10$, and $12$, left to right, respectively

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Figure 5. Probability (%) that carcass number 100 in Source Ⅵ is present in other batches given that it is present in Batch 2, Equation (12), for the synthetic data set

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Figure 6. Probability of contamination for each batch given that a fixed batch is contaminated (hot). The five separate curves correspond to Batches 2, 5, 8, 11, and 14 being the hot batch

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Figure 7. Probability of contamination for each batch given that a fixed batch is contaminated (hot). The fifteen separate curves correspond to Batches 2, 5, 8, $\dotsc$, 41 and 44 being the hot batch

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Table 1. List of Symbols

$S$	Total number of raw sources.
$B$	Total number of ground beef batches.
$h$	The contaminated (hot) batch.
$C_s$	Number of carcasses in raw source $s$.
$p_s$	Number of pieces supplied by each carcass in raw source $s$.
$a_s$	Average size of pieces from each carcass in raw source $s$.
$M_s$	Total mass in raw source $s$.
$x$	Mass location in raw source.
$\mu_c$	Mid point of piece distribution for carcass $c$ (in source $s$).
$F_s$	Base probability density function for piece distribution in source $s$.
$G_{sc}(x)$	Probability density function for piece distribution for carcass $c$ in source $s$.
$Q_{sc}(R)$	Probability that a piece from carcass $c$ is located in region $R$ in source $s$.
$K$	Half the number of piece-wise linear segments of $F_s$.
$N_i$	Boundaries of piece-wise linear segments of $F_s$, measured in number of carcasses from centre, $\mu_c$, of distribution.
$H^\pm_i$	Values of $F_s$ at boundary $N_i$, approaching from the left, $-$, or the right, $+$.
$m_{sb}$	Mass from source $s$ input to batch $b$.
$M_{sb}$	Mass from source $s$ input to batches $1$ through $b-1$.
$B_{sb}$	Interval of mass locations in source $s$ input to batch $b$.
$A_{sc}(B_{sb})$	Probability that carcass $c$ is absent from the set $B_{sb}$, that is, carcass $c$ contributes no pieces to batch $b$ through source $s$.
$f_s$	Fraction of fat in raw source $s$.
$g_s$	Relative susceptibility to contamination factor for source $s$.
$V_{s_1 s_2}$	Fraction of carcasses present in both raw sources $s_1$ and $s_2$.

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Table 2. Model parameters for the raw sources in the synthetic data set

Source	$g_s$	$f_s$	$p_s$	$a_s$ (kg)	$N_{1s}$	$C_s$	$M_s$ (kg)
Ⅰ(frozen lean)	0.2	0.05	25	0.5	15	160	2000
Ⅱ (frozen lean)	0.2	0.09	25	0.5	15	160	2000
Ⅲ (frozen lean)	0.2	0.07	25	0.5	15	160	2000
Ⅳ(fresh lean)	0.8	0.10	20	0.25	20	500	2500
Ⅴ (fresh lean)	0.8	0.08	20	0.25	20	600	3000
Ⅵ (fresh fat)	1.0	0.40	40	0.2	30	250	2000
Ⅶ (fresh fat)	1.0	0.45	40	0.2	30	250	2000

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Table 3. Source input mass, $m_{sb}$, (kg) and total fat percentage for the synthetic data set

	Source
	frozen lean			fresh lean		fresh fat
Batch	Ⅰ	Ⅱ	Ⅲ	Ⅳ	Ⅴ	Ⅵ	Ⅶ	fat %
1	312			136		552		25
2	384			52		564		25
3	114			404		260	222	25
4	262			239		231	268	25
5	201	205		89		293	212	25
6	320	180		292		100	108	15
7	407	105		284			204	15
8		390		456			154	15
9		300		205	325		170	15
10		209		211	543		37	10
11		293		132	536		39	10
12		318	94		540		48	10
13			479		454		67	10
14			701		226		73	10

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Table 4. Probability (%) that sources are hot, given hot batch $h$, for the synthetic data set. ${\rm Prob}(s_1\text{ is hot }| \;h)$ is computed from Equation (9) and the data from Tables 2 and 3. Blank entries indicate zero probability due to no mass input

	Source
hot	frozen lean			fresh lean		fresh fat
Batch	Ⅰ	Ⅱ	Ⅲ	Ⅳ	Ⅴ	Ⅵ	Ⅶ
1	1.3			4.6		94.0
2	1.6			1.8		96.6
3	0.5			13.6		43.8	42.1
4	1.1			8.2		39.4	51.4
5	0.9	1.6		3.2		52.0	42.3
6	2.7	2.7		19.7		33.8	41.1
7	3.4	1.6		18.9			76.2
8		6.2		32.3			61.4
9		4.5		13.8	17.5		64.2
10		5.2		23.4	48.2		23.1
11		7.8		15.6	50.7		25.9
12		9.1	2.1		54.7		34.2
13			10.2		44.1		45.7
14			17.2		25.3		57.5

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Table 5. Probability of contamination for each batch in percent. Each column corresponds to a different hot batch

	hot batch
Batch	1	2	3	4	5	6	7	8	9	10	11	12	13	14
1	100	46	3	0	0	0	0	1	0	1	0	0	0	0
2	60	100	33	12	0	1	0	1	0	1	1	0	0	0
3	4	46	100	63	32	1	1	1	1	1	1	1	1	0
4	1	20	75	100	70	40	16	1	1	1	1	1	1	0
5	1	1	32	63	100	78	41	16	1	1	1	1	0	0
6	1	1	1	29	61	100	63	28	10	1	0	0	0	0
7	1	1	1	10	23	44	100	58	31	7	5	4	1	0
8	1	1	1	1	10	20	61	100	56	14	13	15	15	11
9	1	1	1	1	1	8	35	58	100	48	24	26	29	29
10	1	1	1	1	1	1	16	29	62	100	53	28	31	32
11	1	1	1	1	1	1	11	25	43	47	100	50	35	36
12	1	1	1	1	1	1	7	21	39	20	41	100	56	42
13	1	1	1	1	1	1	2	17	34	18	22	47	100	67
14	0	0	0	0	0	0	0	10	27	14	18	27	57	100

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Table 6. Number of profiles (on the diagonal) and profile matches between batches

Batch	1	2	3	4	5	6	7	8	9	10	11	12	13	15	16	20	21	22	23	27	28	29	30	33	40	41	42	43	44	45
1	87	7	15	12	4	7	2	2	1	1	1	0	1	1	1	0	0	1	1	0	0	3	0	0	0	0	1	0	0	0
2		56	10	2	3	2	1	1	0	0	0	0	1	1	0	0	1	1	0	0	0	1	0	0	0	0	1	1	0	0
3			64	6	0	2	4	2	0	0	0	0	0	0	1	0	0	0	0	0	0	1	1	0	0	0	1	1	0	0
4				61	4	8	3	4	0	2	2	1	0	0	0	2	0	0	0	0	0	2	2	0	0	0	0	0	0	0
5					57	14	1	4	2	1	4	3	0	0	1	1	1	0	0	0	0	1	0	0	0	0	0	0	0	0
6						62	5	6	3	3	1	2	0	0	0	0	0	0	0	0	0	6	2	0	0	1	0	1	0	0
7							66	12	6	4	3	3	0	0	0	0	4	1	0	0	0	0	0	0	0	1	0	1	0	0
8								50	3	2	3	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	2	0	0
9									56	9	6	5	1	1	0	0	1	0	0	0	0	0	0	0	0	0	1	1	1	0
10										50	4	4	0	0	0	1	1	0	0	0	0	0	0	0	0	1	0	0	0	1
11											58	9	2	0	2	1	1	0	0	0	0	0	1	0	0	0	0	0	0	0
12												61	3	1	3	0	0	0	0	0	1	0	2	0	0	0	2	2	0	0
13													67	3	1	0	0	1	0	0	2	2	2	0	1	0	0	0	0	0
15														63	11	0	3	2	2	1	0	4	1	0	2	0	0	0	0	2
16															57	0	6	3	2	0	1	3	3	0	0	0	0	0	0	0
20																49	7	7	0	1	1	1	0	0	0	0	0	0	0	2
21																	87	11	4	0	2	3	0	5	0	0	1	0	0	0
22																		45	3	1	2	1	0	0	0	0	1	0	0	0
23																			56	2	1	1	1	0	4	0	0	0	0	0
27																				52	3	4	2	2	4	0	0	0	0	1
28																					51	3	3	0	0	0	0	0	0	1
29																						63	10	5	1	1	0	0	0	0
30																							71	0	0	1	0	0	0	0
33																								62	0	1	1	0	0	1
40																									53	5	4	4	8	3
41																										50	7	3	6	8
42																											78	27	13	5
43																												81	15	11
44																													65	14
45																														62

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Table 7. Fit carcass distribution parameters for the combined raw sources. The overlap fractions $V_{24}$ and $V_{34}$ are equal to $V_{23}$

$p$	$a$ (kg)	$N_1$	$N_2$	$H^\pm_0$	$H^\pm_1$	$V_{23}$	$V_{56}$	$V_{78}$
8	0.045	27	6383	$4.351\times 10^{-2}$	$ 1.177\times 10^{-5}$	0.08	0	0.20

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Table 8. Estimated source input mass, $m_{sb}$, (kg) for real data set

	Source
	frozen lean				fresh lean		fresh fat
Batch	Ⅰ	Ⅱ	Ⅲ	Ⅳ	Ⅴ	Ⅵ	Ⅶ	Ⅷ
1-10	355	355			264		255
11-13		355	355		264		255
14-17		355	355		132	132	255
18-20		355	355			264	255
21-22		355	355			264		255
23-40			709			264		255
41-45				709		264		255

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Table 9. Estimated and derived model parameters for the raw sources

Source	$g_s$	$f_s$	$C_s$	$M_s$ (kg)
Ⅰ (frozen lean)	0.2	0.05	1,950	3,545
Ⅱ (frozen lean)	0.2	0.05	4,290	7,800
Ⅲ (frozen lean)	0.2	0.05	9,360	17,018
Ⅳ (frozen lean)	0.2	0.05	1,950	3,545
Ⅴ(fresh lean)	0.8	0.10	2,175	3,955
Ⅵ (fresh lean)	0.8	0.10	4,350	7,909
Ⅶ (fresh fat)	1.0	0.24	2,800	5,091
Ⅷ (fresh fat)	1.0	0.24	3,500	6,364

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Table 10. Probability of contamination for Batches 1-24 in percent for hot batches (columns) 1-20. Batches 25-45 (not shown) all had probability of contamination of 2 percent for hot Batches 1-20

	hot batch
Batch	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
1	100	15	12	10	8	6	5	3	1	1	0	0	0	0	0	0	0	0	0	0
2	16	100	14	10	8	7	5	3	2	1	0	0	0	0	0	0	0	0	0	0
3	12	14	100	12	8	7	5	4	3	2	1	0	0	0	0	0	0	0	0	0
4	10	10	12	100	10	7	6	5	4	3	2	1	0	0	0	0	0	0	0	0
5	9	8	8	11	100	10	7	6	5	4	3	2	1	0	0	0	0	0	0	0
6	7	7	7	7	10	100	10	7	6	5	4	3	2	1	0	0	0	0	0	0
7	5	5	5	6	7	10	100	10	7	6	5	4	3	2	1	1	0	0	0	0
8	3	3	4	5	6	7	10	100	10	7	5	5	4	3	2	1	1	0	0	0
9	2	2	3	4	5	6	7	10	100	10	6	5	5	3	3	2	1	0	0	0
10	1	1	2	3	4	5	6	7	10	100	9	6	6	4	4	3	2	1	0	0
11	0	0	1	2	3	4	5	5	6	9	100	10	7	6	5	4	3	2	1	1
12	0	0	0	1	2	3	4	5	5	6	10	100	10	6	6	5	4	2	2	1
13	0	0	0	0	1	2	3	4	5	6	7	10	100	9	6	6	5	3	3	2
14	0	0	0	0	0	1	2	3	3	4	6	6	9	100	10	7	6	5	5	5
15	0	0	0	0	0	0	1	2	3	4	5	6	7	10	100	10	7	7	6	6
16	0	0	0	0	0	0	1	1	2	3	4	5	6	7	10	100	11	8	8	7
17	0	0	0	0	0	0	0	1	1	2	3	4	5	6	7	11	100	11	9	9
18	0	0	0	0	0	0	0	0	0	1	2	2	3	5	7	8	11	100	13	11
19	0	0	0	0	0	0	0	0	0	0	1	2	2	5	6	7	9	13	100	14
20	0	0	0	0	0	0	0	0	0	0	0	1	2	5	6	7	8	11	14	100
21	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	3	4	4	5
22	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	3	4	4
23	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3
24	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	3	3

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Table 11. Probability of contamination for Batches 1-4 and 16-45 in percent for hot batches (columns) 21-45. Batches 5-15 (not shown) all had probability of contamination of 2 percent for hot batches 21-45

	hot batch
Batch	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45
1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2
2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2
3	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
4	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
$\vdots$
16	3	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
17	3	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
18	4	3	3	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	1	1	2	2	2	2	2
19	4	4	3	2	2	2	2	2	2	2	2	1	1	1	1	1	1	1	1	1	2	2	2	2	2
20	5	4	3	3	2	2	2	2	2	2	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2
21	100	14	10	9	7	5	4	3	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
22	14	100	12	9	7	6	4	3	2	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
23	10	12	100	12	8	6	5	4	3	2	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
24	9	9	12	100	11	7	5	4	4	3	2	1	0	0	0	0	0	0	0	0	0	0	0	0	0
25	7	7	8	11	100	10	7	5	4	4	3	2	1	0	0	0	0	0	0	0	0	0	0	0	0
26	6	6	6	7	10	100	10	7	5	4	4	3	2	1	0	0	0	0	0	0	0	0	0	0	0
27	4	4	5	6	7	10	100	10	7	5	4	4	3	2	1	0	0	0	0	0	0	0	0	0	0
28	3	3	4	4	5	7	10	100	10	7	5	4	4	3	2	1	0	0	0	0	0	0	0	0	0
29	1	2	3	4	4	5	7	10	100	10	7	5	4	4	3	2	1	0	0	0	0	0	0	0	0
30	1	1	2	3	4	4	5	7	10	100	10	7	5	4	4	3	2	1	0	0	0	0	0	0	0
31	0	0	1	2	3	4	4	5	7	10	100	10	7	5	4	4	3	2	1	0	0	0	0	0	0
32	0	0	0	1	2	3	4	4	5	7	10	100	10	7	5	4	4	3	2	1	1	0	0	0	0
33	0	0	0	0	1	2	3	4	4	5	7	10	100	10	7	5	4	4	3	2	1	1	0	0	0
34	0	0	0	0	0	1	2	3	4	4	5	7	10	100	10	7	5	4	4	3	2	1	1	0	0
35	0	0	0	0	0	0	1	2	3	4	4	5	7	10	100	10	7	5	4	4	3	2	1	1	0
36	0	0	0	0	0	0	0	1	2	3	4	4	5	7	10	100	10	7	5	4	4	3	2	1	1
37	0	0	0	0	0	0	0	0	1	2	3	4	4	5	7	10	100	10	7	5	5	4	3	2	2
38	0	0	0	0	0	0	0	0	0	1	2	3	4	4	5	7	10	100	10	7	5	5	4	3	3
39	0	0	0	0	0	0	0	0	0	0	1	2	3	4	4	5	7	10	100	11	6	6	5	5	5
40	0	0	0	0	0	0	0	0	0	0	0	1	2	3	4	4	5	7	11	100	9	6	6	6	7
41	0	0	0	0	0	0	0	0	0	0	0	1	1	2	3	4	5	5	6	9	100	12	9	8	8
42	0	0	0	0	0	0	0	0	0	0	0	0	1	1	2	3	4	5	5	6	12	100	13	10	10
43	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	2	3	4	5	6	9	12	100	14	12
44	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	2	3	5	6	8	10	14	100	16
45	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	2	3	5	6	8	9	12	16	100

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References

	M. Aslam , G. G. Greer , F. M. Nattress , C. O. Gill and L. M. McMullen , Genotypic analysis of Escherichia coli recovered from product and equipment at a beef-packing plant, Journal of Applied Microbiology, 97 (2004) , 78-86. doi: 10.1111/j.1365-2672.2004.02277.x.
	M. Aslam , F. M. Nattress , G. G. Greer , C. Yost , C. O. Gill and L. McMullen , Origin of contamination and genetic diversity of Eschericia coli in beef cattle, Applied and Environmental Microbiology, 69 (2003) , 2794-2799.
	R. G. Bell , Distribution and sources of microbial contamination on beef carcasses, Journal of Applied Microbiology, 82 (1997) , 292-300. doi: 10.1046/j.1365-2672.1997.00356.x.
	M. H. Cassin , A. M. Lammerding , E. C. D. Todd , W. Ross and R. S. McColl , Quantitative risk assessment for Escherichia coli O157: H7 in ground beef hamburgers, International Journal of Food Microbiology, 41 (1998) , 21-44. doi: 10.1016/S0168-1605(98)00028-2.
	D. A. Drew , G. A. Koch , H. Vellante , R. Talati and O. Sanchez , Analyses of mechanisms for force generation during cell septation in Escherichia coli, Bulletin of Mathematical Biology, 71 (2009) , 980-1005. doi: 10.1007/s11538-008-9390-6.
	G. Duffy, F. Butler, E. Cummins, S. O'Brien, P. Nally, E. Carney, M. Henchion, D. Mahone and C. Cowan, E. coli O157: H7 in Beef burgers Produced in the Republic of Ireland: A Quantitative Microbial Risk Assessment, Technical report, Ashtown Food Research Centre, Teagasc, Dublin 15, Ireland, 2006.
	G. Duffy , E. Cummins , P. Nally , S. O'Brien and F. Butler , A review of quantitative microbial risk assessment in the management of Escherichia coli O157:H7 on beef, Meat Science, 74 (2006) , 76-88. doi: 10.1016/j.meatsci.2006.04.011.
	E. Ebel , W. Schlosser , J. Kause , K. Orloski , T. Roberts , C. Narrod , S. Malcolm , M. Coleman and M. Powell , Draft risk assessment of the public health impact of Escherichia coli O157:H7 in ground beef, Journal of Food Protection, 67 (2004) , 1991-1999. doi: 10.4315/0362-028X-67.9.1991.
	P. M. Kitanov and A. R. Willms, Estimating Escherichia coli contamination spread in ground beef production using a discrete probability model, in Mathematical and Computational Approaches in Advancing Modern Science and Engineering (eds. J. Bélair, I. F. I, H. Kunze, R. Makarov, R. Melnik and R. Spiteri), Springer, AMMCS-CAIMS 2015, Waterloo, Canada, 2016,245-254. doi: 10.1007/978-3-319-30379-6_23.
	A. Rabner , E. Martinez , R. Pedhazur , T. Elad , S. Belkin and Y. Shacham , Mathematical modeling of a bioluminescent E. Coli based biosensor, Nonlinear Analysis: Modelling and Control, 14 (2009) , 505-529.
	E. Salazar-Cavazos and M. Santillán , Optimal performance of the tryptophan operon of E. coli: A stochastic, dynamical, mathematical-modeling approach, Bulletin of Mathematical Biology, 76 (2014) , 314-334. doi: 10.1007/s11538-013-9920-8.
	E. Scallan , R. M. Hoekstra , F. J. Angulo , R. V. Tauxe , M. A. Widdowson , S. L. Roy , J. L. Jones and P. M. Griffin , Foodborne illness acquired in the United States -Major pathogens, Emerging Infectious Diseases, 17 (2011) , 7-15. doi: 10.3201/eid1701.P11101.
	M. Signorini and H. Tarabla , Quantitative risk assessment for verocytotoxigenic Escherichia coli in ground beef hamburgers in Argentina, International Journal of Food Microbiology, 132 (2009) , 153-161. doi: 10.1016/j.ijfoodmicro.2009.04.022.
	B. A. Smith , A. Fazil and A. M. Lammerding , A risk assessment model for Escherichia coli O157:H7 in ground beef and beef cuts in canada: Evaluating the effects of interventions, Food Control, 29 (2013) , 364-381. doi: 10.1016/j.foodcont.2012.03.003.
	J. Turner , R. G. Bowers , M. Begon , S. E. Robinson and N. P. French , A semi-stochastic model of the transmission of Escherichia coli O157 in a typical UK dairy herd: Dynamics, sensitivity analysis and intervention prevention strategies, Journal of Theoretical Biology, 241 (2006) , 806-822. doi: 10.1016/j.jtbi.2006.01.020.
	J. Turner , R. G. Bowers , D. Clancy , M. C. Behnke and R. M. Christley , A network model of E.coli O157 transmission within a typical UK dairy herd: The effect of heterogeneity and clustering on the prevalence of infection, Journal of Theoretical Biology, 254 (2008) , 45-54. doi: 10.1016/j.jtbi.2008.05.007.
	J. Tuttle , T. Gomez , M. P. Doyle , J. G. Wells , T. Zhao , R. V. Tauxe and P. M. Griffin , Lessons from a large outbreak of Escherichia coli O157:H7 infections: Insights into the infectious dose and method of widespread contamination of hamburger patties, Epidemiology and Infection, 122 (1999) , 185-192. doi: 10.1017/S0950268898001976.
	X. Wang , R. Gautam , P. J. Pinedo , L. J. S. Allen and R. Ivanek , A stochastic model for transmission, extinction and outbreak of Escherichia coli O157:H7 in cattle as affected by ambient temperature and cleaning practices, Journal of Mathematical Biology, 69 (2014) , 501-532. doi: 10.1007/s00285-013-0707-1.
	X. Yang , M. Badoni , M. K. Youssef and C. O. Gill , Enhanced control of microbiological contamination of product at a large beef packing plant, Journal of Food Protection, 75 (2012) , 144-149. doi: 10.4315/0362-028X.JFP-11-291.
	X. S. Zhang , M. E. Chase-Topping , I. J. McKendrick , N. J. Savill and M. E. J. Woolhouse , Spread of Escherichia coli O157:H7 infection among Scottish cattle farms: Stochastic models and model selection, Epidemics, 2 (2010) , 11-20.