Modeling environmental transmission of MAP infection in dairy cows

Kokum R. De Silva; Shigetoshi Eda; Suzanne Lenhart

doi:10.3934/mbe.2017052

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August 2017, 14(4): 1001-1017. doi: 10.3934/mbe.2017052

Modeling environmental transmission of MAP infection in dairy cows

Kokum R. De Silva ^1, , Shigetoshi Eda ^2, and Suzanne Lenhart ^3,,

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

* Corresponding author: Suzanne Lenhart

Received March 20, 2016 Accepted January 31, 2017 Published February 2017

Fund Project: This work was partially supported by the National Institute for Mathematical Biological Synthesis, sponsored by the National Science Foundation Award NSF DBI-1300426.

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.

Keywords: Johne's disease for dairy cattle, environmental transmission, system of differential equations, discrete testing, basic reproduction number.

Mathematics Subject Classification: Primary: 92B05; Secondary: 34B15.

Citation: Kokum R. De Silva, Shigetoshi Eda, Suzanne Lenhart. Modeling environmental transmission of MAP infection in dairy cows. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1001-1017. doi: 10.3934/mbe.2017052

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.

Figure 1. Flow diagram of the transitions in our model (Sc, Sh, Sa -Susceptible calves, heifers, adults, Ec, Eh, Ea -Exposed calves, heifers, adults, Lh, La -Low shedding heifers, adults, Ha -High shedding adults, B₁ -Bacteria in the heifer environment, B₂ -Bacteria in the adult environment)

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Figure 2. Environmental transmission coefficient $f(B)$ with $K_1 = 1000$ and $K_2 = 100$

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Figure 3. Dynamics of the animals in each compartment with no testing or culling and with annual testing and culling

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Figure 4. Dynamics of the total animals in each disease class with no testing or culling and with annual testing and culling

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Figure 5. Number of exposed cows from the bacteria in the environment 1 and 2 when $p = 0.3 , r_1 = 0.06$, and $ r_2 = 0.06$ with no testing or culling and with annual testing and culling

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Figure 6. Dynamics of the bacteria in the two environments with no testing or culling and with annual testing and culling

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Figure 7. Number of exposed cows due to different infection routes without testing or culling and with annual testing and culling

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Table 1. Initial number of animals in each compartment

Variable	Defining the variable	Initial value
S_c	Number of susceptible calves	130
S_h	Number of susceptible heifers	520
S_a	Number of susceptible adults	650
E_c	Number of exposed calves	70
E_h	Number of exposed heifers	248
E_a	Number of exposed adults	250
L_h	Number of low-shedding heifers	32
L_a	Number of low-shedding adults	80
H_a	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

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Variable	Defining the variable	Initial value
S_c	Number of susceptible calves	130
S_h	Number of susceptible heifers	520
S_a	Number of susceptible adults	650
E_c	Number of exposed calves	70
E_h	Number of exposed heifers	248
E_a	Number of exposed adults	250
L_h	Number of low-shedding heifers	32
L_a	Number of low-shedding adults	80
H_a	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

Table 2. Parameters and their values

Parameter	Defining the parameter	Parameter value
b	Birth rate of calves from susceptible and exposed adults	0.00127
b_La	Birth rate of calves from low-shedding adults	0.00127
b_Ha	Birth rate of calves from high-shedding adults	0.00127
µ_{S_c}	Death rate of susceptible calves	0.00028
µ_{E_c}	Death rate of exposed calves	0.00028
µ_{S_h}	Death rate of susceptible heifers	0.000063
µ_{E_h}	Death rate of exposed heifers	0.000063
µ_{L_h}	Death rate of low-shedding heifers	0.000063
µ_{S_a}	Death rate of susceptible adults	0.0012
µ_{E_a}	Death rate of exposed adults	0.0012
µ_{L_a}	Death rate of low-shedding adults	0.0012
µ_{H_a}	Death rate of high-shedding adults	0.0012
µ_B₁	Decay rate of bacteria in the heifer environment	0.0027
µ_B₁	Decay rate of bacteria in the adult environment	0.0027
δ	Culling rate of high-shedding adults	0.9
ν_L	Probability of getting infected through vertical transmission from low-shedding adults	0
ν_H	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

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Parameter	Defining the parameter	Parameter value
b	Birth rate of calves from susceptible and exposed adults	0.00127
b_La	Birth rate of calves from low-shedding adults	0.00127
b_Ha	Birth rate of calves from high-shedding adults	0.00127
µ_{S_c}	Death rate of susceptible calves	0.00028
µ_{E_c}	Death rate of exposed calves	0.00028
µ_{S_h}	Death rate of susceptible heifers	0.000063
µ_{E_h}	Death rate of exposed heifers	0.000063
µ_{L_h}	Death rate of low-shedding heifers	0.000063
µ_{S_a}	Death rate of susceptible adults	0.0012
µ_{E_a}	Death rate of exposed adults	0.0012
µ_{L_a}	Death rate of low-shedding adults	0.0012
µ_{H_a}	Death rate of high-shedding adults	0.0012
µ_B₁	Decay rate of bacteria in the heifer environment	0.0027
µ_B₁	Decay rate of bacteria in the adult environment	0.0027
δ	Culling rate of high-shedding adults	0.9
ν_L	Probability of getting infected through vertical transmission from low-shedding adults	0
ν_H	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

Table 3. Initial prevalence of the disease in each age class

	Susceptible	Exposed	Low-shedding	High-shedding
Calves	65%	35%	0%	0%
Heifers	65%	31%	4%	0%
Adults	65%	25%	8%	2%

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	Susceptible	Exposed	Low-shedding	High-shedding
Calves	65%	35%	0%	0%
Heifers	65%	31%	4%	0%
Adults	65%	25%	8%	2%

Table 4. Comparison of the number of animals in each compartment at the end of 10 years without culling and with annual culling

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

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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

Table 5. Equilibrium values for the number of animals in each compartment at the end of 25 years without culling and the final values for the number of animals in each compartment at the end of 10 years with these equilibrium values as the initial values and annual culling

Compartment	Equilibrium values after 25 years without culling	Final values with annual testing & culling
S_c	25	40
E_c	54	33
S_h	184	498
E_h	349	252
L_h	311	230
S_a	4	67
E_a	308	370
L_a	455	439
H_a	296	11
B₁	807×10⁸	610×10⁸
B₂	3234523×10⁸	786526×10⁸

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Compartment	Equilibrium values after 25 years without culling	Final values with annual testing & culling
S_c	25	40
E_c	54	33
S_h	184	498
E_h	349	252
L_h	311	230
S_a	4	67
E_a	308	370
L_a	455	439
H_a	296	11
B₁	807×10⁸	610×10⁸
B₂	3234523×10⁸	786526×10⁸

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2018 Impact Factor: 1.313

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2018 Impact Factor: 1.313

American Institute of Mathematical Sciences

Modeling environmental transmission of MAP infection in dairy cows

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