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Stability analysis on an economic epidemiological model with vaccination
Modeling environmental transmission of MAP infection in dairy cows
1. | Department of Mathematics, University of Peradeniya, Peradeniya, KY 20400, Sri Lanka |
2. | Department of Forestry, Wildlife and Fisheries, University of Tennessee, Knoxville, TN 37996, USA |
3. | Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA |
Johne's disease is caused by Mycobacterium avium subspecies paratuberculosis(MAP). It is a chronic, progressive, and inflammatory disease which has a long incubation period. One main problem with the disease is the reduction of milk production in infected dairy cows. In our study we develop a system of ordinary differential equations to describe the dynamics of MAP infection in a dairy farm. This model includes the progression of the disease and the age structure of the cows. To investigate the effect of persistence of this bacteria on the farm on transmission in our model, we include environmental compartments, representing the pathogen input in an explicit way. The effect of indirect transmission from the bacteria in the environment and the culling of high-shedding adults can be seen in the numerical simulations. Since culling usually only happens once a year, we include a novel feature in the simulations with a discrete action of removing high-shedding adults once a year. We conclude that with culling of high shedders even at a high rate, the infection will persist in the modeled farm setting.
References:
[1] |
D. J. Begg and R. J. Whittington,
Experimental animal infection models for Johne's disease, an infectious enteropathy caused by Mycobacterium avium subsp. paratuberculosis, The Veterinary Journal, 176 (2008), 129-145.
doi: 10.1016/j.tvjl.2007.02.022. |
[2] |
R. Breban,
Role of environmental persistence in pathogen transmission: A mathematical modeling approach, Journal of Mathematical Biology, 66 (2013), 535-546.
doi: 10.1007/s00285-012-0520-2. |
[3] |
K. L. Cook, J. S. Britt and C. H. Bolster,
Survival of Mycobacterium avium subsp. paratuberculosis in biofilms on livestock watering trough materials, Veterinary Microbiology, 141 (2010), 103-109.
doi: 10.1016/j.vetmic.2009.08.013. |
[4] |
O. Diekmann, H. Heesterbeek and T. Britton,
Mathematical Tools for Understanding Infectious Disease Dynanics Princeton University Press, 2013. |
[5] |
O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts,
The construction of next-generation matrices for compartmental epidemic models, Journal of the Royal Society Interface, 7 (2010), 873-885.
doi: 10.1098/rsif.2009.0386. |
[6] |
E. Doré, J. Paré, G. Côté, S. Buczinski, O. Labrecque, J. P. Roy and G. Fecteau,
Risk factors associated with transmission of Mycobacterium avium subsp. paratuberculosis to calves within dairy herd: A systematic review, Journal of Veterinary Internal Medicine, 26 (2012), 32-45.
doi: 10.1111/j.1939-1676.2011.00854.x. |
[7] |
P. van den Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[8] |
P. van den Driessche and J. Watmough,
Further notes on the basic reproduction number, Mathematical Epidemiology, 1945 (2008), 159-178.
doi: 10.1007/978-3-540-78911-6_6. |
[9] |
L. A. Espejo, S. Godden, W. L. Hartmann and S. J. Wells,
Reduction in incidence of Johne's disease associated with implementation of a disease control program in Minnesota demonstration herds, Journal of Dairy Science, 95 (2012), 4141-4152.
doi: 10.3168/jds.2011-4550. |
[10] |
A. B. Garcia and L. Shalloo,
Invited review: The economic impact and control of paratuberculosis in cattle, Journal of Dairy Science, 98 (2015), 5019-5039.
doi: 10.3168/jds.2014-9241. |
[11] |
I. A. Gardner, S. S. Nielsen, R. J. Whittington, M. T. Collins, D. Bakker, B. Harris, S. Sreevatsan, J. E. Lombard, R. Sweeney, D. R. Smith, J. Gavalchin and S. Eda,
Consensus-based reporting standards for diagnostic test accuracy studies for paratuberculosis in ruminants, Preventive Veterinary Medicine, 101 (2011), 18-34.
doi: 10.1016/j.prevetmed.2011.04.002. |
[12] |
G. F. Gerlach,
Paratuberculosis: the pathogen and routes of infection, Dtsch Tierarztl Wochenschr, 109 (2002), 504-506.
|
[13] |
R. W. Humphry, A. W. Stott, C. Adams and G. J. Gunn,
A model of the relationship between the epidemiology of Johne's disease and the environment in suckler-beef herds, The Veterinary Journal, 172 (2006), 432-445.
doi: 10.1016/j.tvjl.2005.07.017. |
[14] |
Z. Lu, R. M. Mitchell, R. L. Smith, J. S. Van Kessel, P. P. Chapagain, Y. H. Schukken and Y. T. Gröhn,
The importance of culling in Johne's disease control, Journal of Theoretical Biology, 254 (2008), 135-146.
doi: 10.1016/j.jtbi.2008.05.008. |
[15] |
C. Marcé, P. Ezanno, M. F. Weber, H. Seegers, D. U. Pfeiffer and C. Fourichon,
Invited review: Modeling within-herd transmission of Mycobacterium avium subspecies paratuberculosis in dairy cattle: A review, Journal of Dairy Science, 93 (2010), 4455-4470.
doi: 10.3168/jds.2010-3139. |
[16] |
C. Marcé, P. Ezanno, H. Seegers, D. U. Pfeiffer and C. Fourichon,
Predicting fadeout versus persistence of paratuberculosis in a dairy cattle herd for management and control purposes: a modelling study, Preventive Veterinary Medicine, 42 (2011), p36.
doi: 10.1186/1297-9716-42-36. |
[17] |
C. Marcé, P. Ezanno, H. Seegers, D. U. Pfeiffer and C. Fourichon,
Within-herd contact structure and transmission of Mycobacterium avium subspecies paratuberculosis in a persistently infected dairy cattle herd, Preventive Veterinary Medicine, 100 (2011), 116-125.
doi: 10.1016/j.prevetmed.2011.02.004. |
[18] |
T. Massaro, S. Lenhart, M. Spence, C. Drakes, G. Yang, F. Agusto, R. Johnson, B. Whitlock, A. Wadhwa and S. Eda,
Modeling for cost analysis of Johne's disease control based on EVELISA testing, Journal of Biological Systems, 21 (2013), 1340010.
doi: 10.1142/S021833901340010X. |
[19] |
R. M. Mitchell, G. F. Medley, M. T. Collins and Y. H. Schukken,
A meta-analysis of the effect of dose and age at exposure on shedding of Mycobacterium avium subsp. paratuberculosis (MAP) in experimentally infected calves and cows, Epidemiology and Infection, 140 (2012), 231-246.
doi: 10.1017/S0950268811000689. |
[20] |
R. M. Mitchell, Y. Schukken, A. Koets, M. Weber, D. Bakker, J. Stabel, R. H. Whitlock and Y. Louzoun,
Differences in intermittent and continuous fecal shedding patterns between natural and experimental Mycobacterium avium subsp. paratuberculosis infections in cattle, Veterinary Research, 46 (2015), p66.
doi: 10.1186/s13567-015-0188-x. |
[21] |
R. A. Mortier, H. W. Barkema, T. A. Wilson, T. T. Sajobi, R. Wolf and J. De Buck,
Dose-dependent interferon-gamma release in dairy calves experimentally infected with Mycobacterium avium subsp. paratuberculosis, Veterinary Immunology and Immunopathology, 161 (2014), 205-210.
doi: 10.1016/j.vetimm.2014.08.007. |
[22] |
S. L. Ott, S. J. Wells and B. A. Wagner,
Herd-level economic losses associated with Johne's disease on US dairy operations, Preventive Veterinary Medicine, 40 (1999), 179-192.
doi: 10.1016/S0167-5877(99)00037-9. |
[23] |
E. A. Raizman, J. Fetrow, S. J. Wells, S. M. Godden, M. J. Oakes and G. Vazquez,
The association between Mycobacterium avium subsp. paratuberculosis fecal shedding or clinical \textrm{Johne's} disease and lactation performance on two Minnesota, USA dairy farms, Preventive veterinary medicine, 78 (2007), 179-195.
doi: 10.1016/j.prevetmed.2006.10.006. |
[24] |
J. Robins, S. Bogen, A. Francis, A. Westhoek, A. Kanarek, S. Lenhart and S. Eda,
Agent-based model for Johne's disease dynamics in a dairy herd, Veterinary Research, 46 (2015), p68.
doi: 10.1186/s13567-015-0195-y. |
[25] |
H. J. W. van Roermund, D. Bakker, P. T. J. Willemsen and M. C. M. de Jong,
Horizontal transmission of Mycobacterium avium subsp. paratuberculosis in cattle in an experimental setting: Calves can transmit the infection to other calves, Veterinary Microbiology, 122 (2007), 270-279.
doi: 10.1016/j.vetmic.2007.01.016. |
[26] |
A. M. Scanu, T. J. Bull, S. Cannas, J. D. Sanderson, L. A. Sechi, G. Dettori, S. Zanetti and J. H. Taylor,
Mycobacterium avium subspecies paratuberculosis infection in cases of irritable bowel syndrome and comparison with Crohn's disease and Johne's disease: Common neural and immune pathogenicities, Journal of Clinical Microbiology, 45 (2007), 3883-3890.
doi: 10.1128/JCM.01371-07. |
[27] |
M. C. Scott, J. P. Bannantine, Y. Kaneko, A. J. Branscum, R. H. Whitlock, Y. Mori, C. A. Speer and S. Eda,
Absorbed EVELISA: A diagnostic test with improved specificity for Johne's disease in cattle, Foodborne Pathogens and Disease, 7 (2010), 1291-1296.
doi: 10.1089/fpd.2010.0541. |
[28] |
S. Singh and K. Gopinath,
Mycobacterium avium subspecies paratuberculosis and Crohn's regional ileitis: How strong is association?, Journal of Laboratory Physicians, 3 (2011), 69-74.
doi: 10.4103/0974-2727.86836. |
[29] |
R. L. Smith, Y. T. Gröhn, A. K. Pradhan, R. H. Whitlock, J. S. Van Kessel, J. M. Smith, D. R. Wolfgang and Y. H. Schukken,
The effects of progressing and nonprogressing Mycobacterium avium subsp. paratuberculosis infection on milk production in dairy cows, Journal of Dairy Science, 99 (2016), 1383-1390.
doi: 10.3168/jds.2015-9822. |
[30] |
J. H. Taylor,
Review Mycobacterium avium subspecies paratuberculosis, Crohn's disease and the doomsday scenario, Gut Pathogens, 1 (2009), p15.
doi: 10.1186/1757-4749-1-15. |
[31] |
R. H. Whitlock, R. W. Sweeney, T. L. Fyock and J. Smith, MAP supershedders: Another factor in the control of Johne's disease, In Proceedings of the 8th International Colloquium on Paratuberculosis}(2005). |
[32] |
R. J. Whittington, I. B. Marsh and L. A. Reddacliff,
Survival of Mycobacterium avium subsp. paratuberculosis in dam water and sediment, Applied and Environmental Microbiology, 71 (2005), 5304-5308.
doi: 10.1128/AEM.71.9.5304-5308.2005. |
[33] |
R. J. Whittington and P. A. Windsor,
In utero infection of cattle with Mycobacterium avium subsp. paratuberculosis: A critical review and meta-analysis, The Veterinary Journal, 179 (2009), 60-69.
doi: 10.1016/j.tvjl.2007.08.023. |
[34] |
M. Bani-Yaghoub, R. Gautam, Z. Shuai, P. van den Driessche and R. Ivanek,
Reproduction numbers for infections with free-living pathogens growing in the environment, Journal of Biological Dynamics, 6 (2012), 923-940.
doi: 10.1080/17513758.2012.693206. |
[35] |
USDA. Johne's Disease on U. S. Dairies, 1991-2007, Fort Collins, CO, USA, NAHMS USDA-APHIS-VS-CEAH |
[36] |
Cow in and out game
http://fergusonfoundation.org/lessons/cow_in_out/cowmoreinfo.shtml, Alice Ferguson Foundation, 2012. |
show all references
References:
[1] |
D. J. Begg and R. J. Whittington,
Experimental animal infection models for Johne's disease, an infectious enteropathy caused by Mycobacterium avium subsp. paratuberculosis, The Veterinary Journal, 176 (2008), 129-145.
doi: 10.1016/j.tvjl.2007.02.022. |
[2] |
R. Breban,
Role of environmental persistence in pathogen transmission: A mathematical modeling approach, Journal of Mathematical Biology, 66 (2013), 535-546.
doi: 10.1007/s00285-012-0520-2. |
[3] |
K. L. Cook, J. S. Britt and C. H. Bolster,
Survival of Mycobacterium avium subsp. paratuberculosis in biofilms on livestock watering trough materials, Veterinary Microbiology, 141 (2010), 103-109.
doi: 10.1016/j.vetmic.2009.08.013. |
[4] |
O. Diekmann, H. Heesterbeek and T. Britton,
Mathematical Tools for Understanding Infectious Disease Dynanics Princeton University Press, 2013. |
[5] |
O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts,
The construction of next-generation matrices for compartmental epidemic models, Journal of the Royal Society Interface, 7 (2010), 873-885.
doi: 10.1098/rsif.2009.0386. |
[6] |
E. Doré, J. Paré, G. Côté, S. Buczinski, O. Labrecque, J. P. Roy and G. Fecteau,
Risk factors associated with transmission of Mycobacterium avium subsp. paratuberculosis to calves within dairy herd: A systematic review, Journal of Veterinary Internal Medicine, 26 (2012), 32-45.
doi: 10.1111/j.1939-1676.2011.00854.x. |
[7] |
P. van den Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[8] |
P. van den Driessche and J. Watmough,
Further notes on the basic reproduction number, Mathematical Epidemiology, 1945 (2008), 159-178.
doi: 10.1007/978-3-540-78911-6_6. |
[9] |
L. A. Espejo, S. Godden, W. L. Hartmann and S. J. Wells,
Reduction in incidence of Johne's disease associated with implementation of a disease control program in Minnesota demonstration herds, Journal of Dairy Science, 95 (2012), 4141-4152.
doi: 10.3168/jds.2011-4550. |
[10] |
A. B. Garcia and L. Shalloo,
Invited review: The economic impact and control of paratuberculosis in cattle, Journal of Dairy Science, 98 (2015), 5019-5039.
doi: 10.3168/jds.2014-9241. |
[11] |
I. A. Gardner, S. S. Nielsen, R. J. Whittington, M. T. Collins, D. Bakker, B. Harris, S. Sreevatsan, J. E. Lombard, R. Sweeney, D. R. Smith, J. Gavalchin and S. Eda,
Consensus-based reporting standards for diagnostic test accuracy studies for paratuberculosis in ruminants, Preventive Veterinary Medicine, 101 (2011), 18-34.
doi: 10.1016/j.prevetmed.2011.04.002. |
[12] |
G. F. Gerlach,
Paratuberculosis: the pathogen and routes of infection, Dtsch Tierarztl Wochenschr, 109 (2002), 504-506.
|
[13] |
R. W. Humphry, A. W. Stott, C. Adams and G. J. Gunn,
A model of the relationship between the epidemiology of Johne's disease and the environment in suckler-beef herds, The Veterinary Journal, 172 (2006), 432-445.
doi: 10.1016/j.tvjl.2005.07.017. |
[14] |
Z. Lu, R. M. Mitchell, R. L. Smith, J. S. Van Kessel, P. P. Chapagain, Y. H. Schukken and Y. T. Gröhn,
The importance of culling in Johne's disease control, Journal of Theoretical Biology, 254 (2008), 135-146.
doi: 10.1016/j.jtbi.2008.05.008. |
[15] |
C. Marcé, P. Ezanno, M. F. Weber, H. Seegers, D. U. Pfeiffer and C. Fourichon,
Invited review: Modeling within-herd transmission of Mycobacterium avium subspecies paratuberculosis in dairy cattle: A review, Journal of Dairy Science, 93 (2010), 4455-4470.
doi: 10.3168/jds.2010-3139. |
[16] |
C. Marcé, P. Ezanno, H. Seegers, D. U. Pfeiffer and C. Fourichon,
Predicting fadeout versus persistence of paratuberculosis in a dairy cattle herd for management and control purposes: a modelling study, Preventive Veterinary Medicine, 42 (2011), p36.
doi: 10.1186/1297-9716-42-36. |
[17] |
C. Marcé, P. Ezanno, H. Seegers, D. U. Pfeiffer and C. Fourichon,
Within-herd contact structure and transmission of Mycobacterium avium subspecies paratuberculosis in a persistently infected dairy cattle herd, Preventive Veterinary Medicine, 100 (2011), 116-125.
doi: 10.1016/j.prevetmed.2011.02.004. |
[18] |
T. Massaro, S. Lenhart, M. Spence, C. Drakes, G. Yang, F. Agusto, R. Johnson, B. Whitlock, A. Wadhwa and S. Eda,
Modeling for cost analysis of Johne's disease control based on EVELISA testing, Journal of Biological Systems, 21 (2013), 1340010.
doi: 10.1142/S021833901340010X. |
[19] |
R. M. Mitchell, G. F. Medley, M. T. Collins and Y. H. Schukken,
A meta-analysis of the effect of dose and age at exposure on shedding of Mycobacterium avium subsp. paratuberculosis (MAP) in experimentally infected calves and cows, Epidemiology and Infection, 140 (2012), 231-246.
doi: 10.1017/S0950268811000689. |
[20] |
R. M. Mitchell, Y. Schukken, A. Koets, M. Weber, D. Bakker, J. Stabel, R. H. Whitlock and Y. Louzoun,
Differences in intermittent and continuous fecal shedding patterns between natural and experimental Mycobacterium avium subsp. paratuberculosis infections in cattle, Veterinary Research, 46 (2015), p66.
doi: 10.1186/s13567-015-0188-x. |
[21] |
R. A. Mortier, H. W. Barkema, T. A. Wilson, T. T. Sajobi, R. Wolf and J. De Buck,
Dose-dependent interferon-gamma release in dairy calves experimentally infected with Mycobacterium avium subsp. paratuberculosis, Veterinary Immunology and Immunopathology, 161 (2014), 205-210.
doi: 10.1016/j.vetimm.2014.08.007. |
[22] |
S. L. Ott, S. J. Wells and B. A. Wagner,
Herd-level economic losses associated with Johne's disease on US dairy operations, Preventive Veterinary Medicine, 40 (1999), 179-192.
doi: 10.1016/S0167-5877(99)00037-9. |
[23] |
E. A. Raizman, J. Fetrow, S. J. Wells, S. M. Godden, M. J. Oakes and G. Vazquez,
The association between Mycobacterium avium subsp. paratuberculosis fecal shedding or clinical \textrm{Johne's} disease and lactation performance on two Minnesota, USA dairy farms, Preventive veterinary medicine, 78 (2007), 179-195.
doi: 10.1016/j.prevetmed.2006.10.006. |
[24] |
J. Robins, S. Bogen, A. Francis, A. Westhoek, A. Kanarek, S. Lenhart and S. Eda,
Agent-based model for Johne's disease dynamics in a dairy herd, Veterinary Research, 46 (2015), p68.
doi: 10.1186/s13567-015-0195-y. |
[25] |
H. J. W. van Roermund, D. Bakker, P. T. J. Willemsen and M. C. M. de Jong,
Horizontal transmission of Mycobacterium avium subsp. paratuberculosis in cattle in an experimental setting: Calves can transmit the infection to other calves, Veterinary Microbiology, 122 (2007), 270-279.
doi: 10.1016/j.vetmic.2007.01.016. |
[26] |
A. M. Scanu, T. J. Bull, S. Cannas, J. D. Sanderson, L. A. Sechi, G. Dettori, S. Zanetti and J. H. Taylor,
Mycobacterium avium subspecies paratuberculosis infection in cases of irritable bowel syndrome and comparison with Crohn's disease and Johne's disease: Common neural and immune pathogenicities, Journal of Clinical Microbiology, 45 (2007), 3883-3890.
doi: 10.1128/JCM.01371-07. |
[27] |
M. C. Scott, J. P. Bannantine, Y. Kaneko, A. J. Branscum, R. H. Whitlock, Y. Mori, C. A. Speer and S. Eda,
Absorbed EVELISA: A diagnostic test with improved specificity for Johne's disease in cattle, Foodborne Pathogens and Disease, 7 (2010), 1291-1296.
doi: 10.1089/fpd.2010.0541. |
[28] |
S. Singh and K. Gopinath,
Mycobacterium avium subspecies paratuberculosis and Crohn's regional ileitis: How strong is association?, Journal of Laboratory Physicians, 3 (2011), 69-74.
doi: 10.4103/0974-2727.86836. |
[29] |
R. L. Smith, Y. T. Gröhn, A. K. Pradhan, R. H. Whitlock, J. S. Van Kessel, J. M. Smith, D. R. Wolfgang and Y. H. Schukken,
The effects of progressing and nonprogressing Mycobacterium avium subsp. paratuberculosis infection on milk production in dairy cows, Journal of Dairy Science, 99 (2016), 1383-1390.
doi: 10.3168/jds.2015-9822. |
[30] |
J. H. Taylor,
Review Mycobacterium avium subspecies paratuberculosis, Crohn's disease and the doomsday scenario, Gut Pathogens, 1 (2009), p15.
doi: 10.1186/1757-4749-1-15. |
[31] |
R. H. Whitlock, R. W. Sweeney, T. L. Fyock and J. Smith, MAP supershedders: Another factor in the control of Johne's disease, In Proceedings of the 8th International Colloquium on Paratuberculosis}(2005). |
[32] |
R. J. Whittington, I. B. Marsh and L. A. Reddacliff,
Survival of Mycobacterium avium subsp. paratuberculosis in dam water and sediment, Applied and Environmental Microbiology, 71 (2005), 5304-5308.
doi: 10.1128/AEM.71.9.5304-5308.2005. |
[33] |
R. J. Whittington and P. A. Windsor,
In utero infection of cattle with Mycobacterium avium subsp. paratuberculosis: A critical review and meta-analysis, The Veterinary Journal, 179 (2009), 60-69.
doi: 10.1016/j.tvjl.2007.08.023. |
[34] |
M. Bani-Yaghoub, R. Gautam, Z. Shuai, P. van den Driessche and R. Ivanek,
Reproduction numbers for infections with free-living pathogens growing in the environment, Journal of Biological Dynamics, 6 (2012), 923-940.
doi: 10.1080/17513758.2012.693206. |
[35] |
USDA. Johne's Disease on U. S. Dairies, 1991-2007, Fort Collins, CO, USA, NAHMS USDA-APHIS-VS-CEAH |
[36] |
Cow in and out game
http://fergusonfoundation.org/lessons/cow_in_out/cowmoreinfo.shtml, Alice Ferguson Foundation, 2012. |






Variable | Defining the variable | Initial value |
Sc | Number of susceptible calves | 130 |
Sh | Number of susceptible heifers | 520 |
Sa | Number of susceptible adults | 650 |
Ec | Number of exposed calves | 70 |
Eh | Number of exposed heifers | 248 |
Ea | Number of exposed adults | 250 |
Lh | Number of low-shedding heifers | 32 |
La | Number of low-shedding adults | 80 |
Ha | Number of high-shedding adults | 20 |
B1 | Amount of bacteria (MAP) in the environment 1(Scaled in 108) | 0.2 |
B2 | Amount of bacteria (MAP) in the environment 2(Scaled in 108) | 590 |
Variable | Defining the variable | Initial value |
Sc | Number of susceptible calves | 130 |
Sh | Number of susceptible heifers | 520 |
Sa | Number of susceptible adults | 650 |
Ec | Number of exposed calves | 70 |
Eh | Number of exposed heifers | 248 |
Ea | Number of exposed adults | 250 |
Lh | Number of low-shedding heifers | 32 |
La | Number of low-shedding adults | 80 |
Ha | Number of high-shedding adults | 20 |
B1 | Amount of bacteria (MAP) in the environment 1(Scaled in 108) | 0.2 |
B2 | Amount of bacteria (MAP) in the environment 2(Scaled in 108) | 590 |
Parameter | Defining the parameter | Parameter value |
b | Birth rate of calves from susceptible and exposed adults | 0.00127 |
bLa | Birth rate of calves from low-shedding adults | 0.00127 |
bHa | Birth rate of calves from high-shedding adults | 0.00127 |
µSc | Death rate of susceptible calves | 0.00028 |
µEc | Death rate of exposed calves | 0.00028 |
µSh | Death rate of susceptible heifers | 0.000063 |
µEh | Death rate of exposed heifers | 0.000063 |
µLh | Death rate of low-shedding heifers | 0.000063 |
µSa | Death rate of susceptible adults | 0.0012 |
µEa | Death rate of exposed adults | 0.0012 |
µLa | Death rate of low-shedding adults | 0.0012 |
µHa | Death rate of high-shedding adults | 0.0012 |
µB1 | Decay rate of bacteria in the heifer environment | 0.0027 |
µB1 | Decay rate of bacteria in the adult environment | 0.0027 |
δ | Culling rate of high-shedding adults | 0.9 |
νL | Probability of getting infected through vertical transmission from low-shedding adults | 0 |
νH | Probability of getting infected through vertical transmission from high-shedding adults | 0.22 |
a1 | Transfer rate from calves to heifers due to age progression | 0.0168 |
a2 | Transfer rate from heifers to adults due to age progression | 0.00151 |
d1 | Transfer rate from exposed heifers to low-shedding heifers | 0.0014 |
d2 | Transfer rate from exposed adults to low-shedding adults | 0.0014 |
d3 | Transfer rate from low-shedding adults to high-shedding adults | 0.00078 |
β1 | Transmission rate for susceptible calves due to the colostrum and milk from low-shedding adults | 0.000021 |
β2 | Transmission rate for susceptible calves due to the colostrum and milk from high-shedding adults | 0.000028 |
γ1 | Transmission rate for susceptible heifers due to direct contact with low-shedding heifers | 0.0000024 |
γ2 | Transmission rate for susceptible adults due to direct contact with low-shedding adults | 0.0000012 |
γ3 | Transmission rate for susceptible adults due to direct contact with high-shedding adults | 0.0000018 |
p | Probability of newborn susceptible calves getting infected by MAP in the adult environment | 0.3 |
r1 | Probability of susceptible heifers getting infected by MAP in the heifer environment | 0.06 |
r2 | Probability of susceptible adults getting infected by MAP in the adult environment | 0.06 |
λ1 | Rate at which the bacteria is added to the heifer environment from the low-shedding heifers | 0.007 |
λ2 | Rate at which the bacteria is added to the adult environment from the low-shedding adults | 0.007 |
λ3 | Rate at which the bacteria is added to the adult environment from the high-shedding adults | 29.5 |
Parameter | Defining the parameter | Parameter value |
b | Birth rate of calves from susceptible and exposed adults | 0.00127 |
bLa | Birth rate of calves from low-shedding adults | 0.00127 |
bHa | Birth rate of calves from high-shedding adults | 0.00127 |
µSc | Death rate of susceptible calves | 0.00028 |
µEc | Death rate of exposed calves | 0.00028 |
µSh | Death rate of susceptible heifers | 0.000063 |
µEh | Death rate of exposed heifers | 0.000063 |
µLh | Death rate of low-shedding heifers | 0.000063 |
µSa | Death rate of susceptible adults | 0.0012 |
µEa | Death rate of exposed adults | 0.0012 |
µLa | Death rate of low-shedding adults | 0.0012 |
µHa | Death rate of high-shedding adults | 0.0012 |
µB1 | Decay rate of bacteria in the heifer environment | 0.0027 |
µB1 | Decay rate of bacteria in the adult environment | 0.0027 |
δ | Culling rate of high-shedding adults | 0.9 |
νL | Probability of getting infected through vertical transmission from low-shedding adults | 0 |
νH | Probability of getting infected through vertical transmission from high-shedding adults | 0.22 |
a1 | Transfer rate from calves to heifers due to age progression | 0.0168 |
a2 | Transfer rate from heifers to adults due to age progression | 0.00151 |
d1 | Transfer rate from exposed heifers to low-shedding heifers | 0.0014 |
d2 | Transfer rate from exposed adults to low-shedding adults | 0.0014 |
d3 | Transfer rate from low-shedding adults to high-shedding adults | 0.00078 |
β1 | Transmission rate for susceptible calves due to the colostrum and milk from low-shedding adults | 0.000021 |
β2 | Transmission rate for susceptible calves due to the colostrum and milk from high-shedding adults | 0.000028 |
γ1 | Transmission rate for susceptible heifers due to direct contact with low-shedding heifers | 0.0000024 |
γ2 | Transmission rate for susceptible adults due to direct contact with low-shedding adults | 0.0000012 |
γ3 | Transmission rate for susceptible adults due to direct contact with high-shedding adults | 0.0000018 |
p | Probability of newborn susceptible calves getting infected by MAP in the adult environment | 0.3 |
r1 | Probability of susceptible heifers getting infected by MAP in the heifer environment | 0.06 |
r2 | Probability of susceptible adults getting infected by MAP in the adult environment | 0.06 |
λ1 | Rate at which the bacteria is added to the heifer environment from the low-shedding heifers | 0.007 |
λ2 | Rate at which the bacteria is added to the adult environment from the low-shedding adults | 0.007 |
λ3 | Rate at which the bacteria is added to the adult environment from the high-shedding adults | 29.5 |
Susceptible | Exposed | Low-shedding | High-shedding | |
Calves | 65% | 35% | 0% | 0% |
Heifers | 65% | 31% | 4% | 0% |
Adults | 65% | 25% | 8% | 2% |
Susceptible | Exposed | Low-shedding | High-shedding | |
Calves | 65% | 35% | 0% | 0% |
Heifers | 65% | 31% | 4% | 0% |
Adults | 65% | 25% | 8% | 2% |
Compartment | Without culling | With annual testing & culling |
Sc | 26 | 42 |
Ec | 53 | 32 |
Sh | 196 | 530 |
Eh | 349 | 244 |
Lh | 301 | 208 |
Sa | 5 | 90 |
Ea | 321 | 388 |
La | 456 | 424 |
Ha | 284 | 10 |
Compartment | Without culling | With annual testing & culling |
Sc | 26 | 42 |
Ec | 53 | 32 |
Sh | 196 | 530 |
Eh | 349 | 244 |
Lh | 301 | 208 |
Sa | 5 | 90 |
Ea | 321 | 388 |
La | 456 | 424 |
Ha | 284 | 10 |
Compartment | Equilibrium values after 25 years without culling | Final values with annual testing & culling |
Sc | 25 | 40 |
Ec | 54 | 33 |
Sh | 184 | 498 |
Eh | 349 | 252 |
Lh | 311 | 230 |
Sa | 4 | 67 |
Ea | 308 | 370 |
La | 455 | 439 |
Ha | 296 | 11 |
B1 | 807×108 | 610×108 |
B2 | 3234523×108 | 786526×108 |
Compartment | Equilibrium values after 25 years without culling | Final values with annual testing & culling |
Sc | 25 | 40 |
Ec | 54 | 33 |
Sh | 184 | 498 |
Eh | 349 | 252 |
Lh | 311 | 230 |
Sa | 4 | 67 |
Ea | 308 | 370 |
La | 455 | 439 |
Ha | 296 | 11 |
B1 | 807×108 | 610×108 |
B2 | 3234523×108 | 786526×108 |
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