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August  2018, 15(4): 1011-1032. doi: 10.3934/mbe.2018045

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

Received  September 07, 2017 Accepted  October 27, 2017 Published  March 2018

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.

Keywords: Escherichia coli, ground beef, contamination probability, probability model, beef recall.
Mathematics Subject Classification: Primary: 92B05; Secondary: 92F05, 62P10.
Citation: Petko M. Kitanov, Allan R. Willms. Probability of Escherichia coli contamination spread in ground beef production. Mathematical Biosciences & Engineering, 2018, 15 (4) : 1011-1032. doi: 10.3934/mbe.2018045
References:
[1]

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.

[2]

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. 

[3]

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.

[4]

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.

[5]

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.

[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.

[7]

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.

[8]

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.

[9]

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.

[10]

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. 

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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.

[16]

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.

[17]

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.

[18]

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.

[19]

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.

[20]

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. 

show all references

References:
[1]

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.

[2]

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. 

[3]

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.

[4]

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.

[5]

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.

[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.

[7]

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.

[8]

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.

[9]

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.

[10]

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. 

[11]

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.

[12]

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.

[13]

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.

[14]

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.

[15]

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.

[16]

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.

[17]

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.

[18]

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.

[19]

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.

[20]

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. 

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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$S$ 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 $s_2$.
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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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
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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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
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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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
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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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
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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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
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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$p$ $a$ (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
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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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
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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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
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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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
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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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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