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

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

* 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.

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. CookJ. 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. DiekmannJ. 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. DoréJ. ParéG. CôtéS. BuczinskiO. LabrecqueJ. 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. EspejoS. GoddenW. 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. GardnerS. S. NielsenR. J. WhittingtonM. T. CollinsD. BakkerB. HarrisS. SreevatsanJ. E. LombardR. SweeneyD. R. SmithJ. 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. HumphryA. W. StottC. 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. LuR. M. MitchellR. L. SmithJ. S. Van KesselP. P. ChapagainY. 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. MarcéP. EzannoM. F. WeberH. SeegersD. 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. MarcéP. EzannoH. SeegersD. 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. MarcéP. EzannoH. SeegersD. 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. MassaroS. LenhartM. SpenceC. DrakesG. YangF. AgustoR. JohnsonB. WhitlockA. 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. MitchellG. F. MedleyM. 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. MitchellY. SchukkenA. KoetsM. WeberD. BakkerJ. StabelR. 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. MortierH. W. BarkemaT. A. WilsonT. T. SajobiR. 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. OttS. 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. RaizmanJ. FetrowS. J. WellsS. M. GoddenM. 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. RobinsS. BogenA. FrancisA. WesthoekA. KanarekS. 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 RoermundD. BakkerP. 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. ScanuT. J. BullS. CannasJ. D. SandersonL. A. SechiG. DettoriS. 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. ScottJ. P. BannantineY. KanekoA. J. BranscumR. H. WhitlockY. MoriC. 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. SmithY. T. GröhnA. K. PradhanR. H. WhitlockJ. S. Van KesselJ. M. SmithD. 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. WhittingtonI. 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-YaghoubR. GautamZ. ShuaiP. 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. CookJ. 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. DiekmannJ. 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. DoréJ. ParéG. CôtéS. BuczinskiO. LabrecqueJ. 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. EspejoS. GoddenW. 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. GardnerS. S. NielsenR. J. WhittingtonM. T. CollinsD. BakkerB. HarrisS. SreevatsanJ. E. LombardR. SweeneyD. R. SmithJ. 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. HumphryA. W. StottC. 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. LuR. M. MitchellR. L. SmithJ. S. Van KesselP. P. ChapagainY. 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. MarcéP. EzannoM. F. WeberH. SeegersD. 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. MarcéP. EzannoH. SeegersD. 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. MarcéP. EzannoH. SeegersD. 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. MassaroS. LenhartM. SpenceC. DrakesG. YangF. AgustoR. JohnsonB. WhitlockA. 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. MitchellG. F. MedleyM. 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. MitchellY. SchukkenA. KoetsM. WeberD. BakkerJ. StabelR. 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. MortierH. W. BarkemaT. A. WilsonT. T. SajobiR. 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. OttS. 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. RaizmanJ. FetrowS. J. WellsS. M. GoddenM. 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. RobinsS. BogenA. FrancisA. WesthoekA. KanarekS. 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 RoermundD. BakkerP. 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. ScanuT. J. BullS. CannasJ. D. SandersonL. A. SechiG. DettoriS. 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. ScottJ. P. BannantineY. KanekoA. J. BranscumR. H. WhitlockY. MoriC. 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. SmithY. T. GröhnA. K. PradhanR. H. WhitlockJ. S. Van KesselJ. M. SmithD. 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. WhittingtonI. 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-YaghoubR. GautamZ. ShuaiP. 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)
Figure 2.  Environmental transmission coefficient $f(B)$ with $K_1 = 1000$ and $K_2 = 100$
Figure 3.  Dynamics of the animals in each compartment with no testing or culling and with annual testing and culling
Figure 4.  Dynamics of the total animals in each disease class with no testing or culling and with annual testing and culling
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
Figure 6.  Dynamics of the bacteria in the two environments with no testing or culling and with annual testing and culling
Figure 7.  Number of exposed cows due to different infection routes without testing or culling and with annual testing and culling
Table 1.  Initial number of animals in each compartment
VariableDefining the variableInitial value
ScNumber of susceptible calves130
ShNumber of susceptible heifers520
SaNumber of susceptible adults650
EcNumber of exposed calves70
EhNumber of exposed heifers248
EaNumber of exposed adults250
LhNumber of low-shedding heifers32
LaNumber of low-shedding adults80
HaNumber of high-shedding adults20
B1Amount of bacteria (MAP) in the environment 1(Scaled in 108)0.2
B2Amount of bacteria (MAP) in the environment 2(Scaled in 108)590
VariableDefining the variableInitial value
ScNumber of susceptible calves130
ShNumber of susceptible heifers520
SaNumber of susceptible adults650
EcNumber of exposed calves70
EhNumber of exposed heifers248
EaNumber of exposed adults250
LhNumber of low-shedding heifers32
LaNumber of low-shedding adults80
HaNumber of high-shedding adults20
B1Amount of bacteria (MAP) in the environment 1(Scaled in 108)0.2
B2Amount of bacteria (MAP) in the environment 2(Scaled in 108)590
Table 2.  Parameters and their values
ParameterDefining the parameterParameter value
bBirth rate of calves from susceptible and exposed adults0.00127
bLaBirth rate of calves from low-shedding adults0.00127
bHaBirth rate of calves from high-shedding adults0.00127
µScDeath rate of susceptible calves0.00028
µEcDeath rate of exposed calves0.00028
µShDeath rate of susceptible heifers0.000063
µEhDeath rate of exposed heifers0.000063
µLhDeath rate of low-shedding heifers0.000063
µSaDeath rate of susceptible adults0.0012
µEaDeath rate of exposed adults0.0012
µLaDeath rate of low-shedding adults0.0012
µHaDeath rate of high-shedding adults0.0012
µB1Decay rate of bacteria in the heifer environment0.0027
µB1Decay rate of bacteria in the adult environment0.0027
δCulling rate of high-shedding adults0.9
νLProbability of getting infected through vertical transmission from low-shedding adults0
νHProbability of getting infected through vertical transmission from high-shedding adults0.22
a1Transfer rate from calves to heifers due to age progression0.0168
a2Transfer rate from heifers to adults due to age progression0.00151
d1Transfer rate from exposed heifers to low-shedding heifers0.0014
d2Transfer rate from exposed adults to low-shedding adults0.0014
d3Transfer rate from low-shedding adults to high-shedding adults0.00078
β1Transmission rate for susceptible calves due to the colostrum and milk from low-shedding adults0.000021
β2Transmission rate for susceptible calves due to the colostrum and milk from high-shedding adults0.000028
γ1Transmission rate for susceptible heifers due to direct contact with low-shedding heifers0.0000024
γ2Transmission rate for susceptible adults due to direct contact with low-shedding adults0.0000012
γ3Transmission rate for susceptible adults due to direct contact with high-shedding adults0.0000018
pProbability of newborn susceptible calves getting infected by MAP in the adult environment0.3
r1Probability of susceptible heifers getting infected by MAP in the heifer environment0.06
r2Probability of susceptible adults getting infected by MAP in the adult environment0.06
λ1Rate at which the bacteria is added to the heifer environment from the low-shedding heifers0.007
λ2Rate at which the bacteria is added to the adult environment from the low-shedding adults0.007
λ3Rate at which the bacteria is added to the adult environment from the high-shedding adults29.5
ParameterDefining the parameterParameter value
bBirth rate of calves from susceptible and exposed adults0.00127
bLaBirth rate of calves from low-shedding adults0.00127
bHaBirth rate of calves from high-shedding adults0.00127
µScDeath rate of susceptible calves0.00028
µEcDeath rate of exposed calves0.00028
µShDeath rate of susceptible heifers0.000063
µEhDeath rate of exposed heifers0.000063
µLhDeath rate of low-shedding heifers0.000063
µSaDeath rate of susceptible adults0.0012
µEaDeath rate of exposed adults0.0012
µLaDeath rate of low-shedding adults0.0012
µHaDeath rate of high-shedding adults0.0012
µB1Decay rate of bacteria in the heifer environment0.0027
µB1Decay rate of bacteria in the adult environment0.0027
δCulling rate of high-shedding adults0.9
νLProbability of getting infected through vertical transmission from low-shedding adults0
νHProbability of getting infected through vertical transmission from high-shedding adults0.22
a1Transfer rate from calves to heifers due to age progression0.0168
a2Transfer rate from heifers to adults due to age progression0.00151
d1Transfer rate from exposed heifers to low-shedding heifers0.0014
d2Transfer rate from exposed adults to low-shedding adults0.0014
d3Transfer rate from low-shedding adults to high-shedding adults0.00078
β1Transmission rate for susceptible calves due to the colostrum and milk from low-shedding adults0.000021
β2Transmission rate for susceptible calves due to the colostrum and milk from high-shedding adults0.000028
γ1Transmission rate for susceptible heifers due to direct contact with low-shedding heifers0.0000024
γ2Transmission rate for susceptible adults due to direct contact with low-shedding adults0.0000012
γ3Transmission rate for susceptible adults due to direct contact with high-shedding adults0.0000018
pProbability of newborn susceptible calves getting infected by MAP in the adult environment0.3
r1Probability of susceptible heifers getting infected by MAP in the heifer environment0.06
r2Probability of susceptible adults getting infected by MAP in the adult environment0.06
λ1Rate at which the bacteria is added to the heifer environment from the low-shedding heifers0.007
λ2Rate at which the bacteria is added to the adult environment from the low-shedding adults0.007
λ3Rate at which the bacteria is added to the adult environment from the high-shedding adults29.5
Table 3.  Initial prevalence of the disease in each age class
SusceptibleExposedLow-sheddingHigh-shedding
Calves65%35%0%0%
Heifers65%31%4%0%
Adults65%25%8%2%
SusceptibleExposedLow-sheddingHigh-shedding
Calves65%35%0%0%
Heifers65%31%4%0%
Adults65%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
CompartmentWithout cullingWith annual testing & culling
Sc2642
Ec5332
Sh196530
Eh349244
Lh301208
Sa590
Ea321388
La456424
Ha28410
CompartmentWithout cullingWith annual testing & culling
Sc2642
Ec5332
Sh196530
Eh349244
Lh301208
Sa590
Ea321388
La456424
Ha28410
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
CompartmentEquilibrium values after 25 years without cullingFinal values with annual testing & culling
Sc2540
Ec5433
Sh184498
Eh349252
Lh311230
Sa467
Ea308370
La455439
Ha29611
B1807×108610×108
B23234523×108786526×108
CompartmentEquilibrium values after 25 years without cullingFinal values with annual testing & culling
Sc2540
Ec5433
Sh184498
Eh349252
Lh311230
Sa467
Ea308370
La455439
Ha29611
B1807×108610×108
B23234523×108786526×108
[1]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37

[2]

Olga Vasilyeva, Tamer Oraby, Frithjof Lutscher. Aggregation and environmental transmission in chronic wasting disease. Mathematical Biosciences & Engineering, 2015, 12 (1) : 209-231. doi: 10.3934/mbe.2015.12.209

[3]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595

[4]

Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455

[5]

Mahin Salmani, P. van den Driessche. A model for disease transmission in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 185-202. doi: 10.3934/dcdsb.2006.6.185

[6]

Burcu Adivar, Ebru Selin Selen. Compartmental disease transmission models for smallpox. Conference Publications, 2011, 2011 (Special) : 13-21. doi: 10.3934/proc.2011.2011.13

[7]

Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239

[8]

Dina Kalinichenko, Volker Reitmann, Sergey Skopinov. Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion. Conference Publications, 2013, 2013 (special) : 407-414. doi: 10.3934/proc.2013.2013.407

[9]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159

[10]

Jing-Jing Xiang, Juan Wang, Li-Ming Cai. Global stability of the dengue disease transmission models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2217-2232. doi: 10.3934/dcdsb.2015.20.2217

[11]

Lars Grüne, Peter E. Kloeden, Stefan Siegmund, Fabian R. Wirth. Lyapunov's second method for nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 375-403. doi: 10.3934/dcds.2007.18.375

[12]

Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565

[13]

Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457

[14]

Ketty A. De Rezende, Mariana G. Villapouca. Discrete conley index theory for zero dimensional basic sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056

[15]

Yayun Zheng, Xu Sun. Governing equations for Probability densities of stochastic differential equations with discrete time delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3615-3628. doi: 10.3934/dcdsb.2017182

[16]

Ariel Cintrón-Arias, Carlos Castillo-Chávez, Luís M. A. Bettencourt, Alun L. Lloyd, H. T. Banks. The estimation of the effective reproductive number from disease outbreak data. Mathematical Biosciences & Engineering, 2009, 6 (2) : 261-282. doi: 10.3934/mbe.2009.6.261

[17]

Ionel S. Ciuperca, Matthieu Dumont, Abdelkader Lakmeche, Pauline Mazzocco, Laurent Pujo-Menjouet, Human Rezaei, Léon M. Tine. Alzheimer's disease and prion: An in vitro mathematical model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-36. doi: 10.3934/dcdsb.2019057

[18]

Hongbin Guo, Michael Yi Li. Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2413-2430. doi: 10.3934/dcdsb.2012.17.2413

[19]

W.R. Derrick, P. van den Driessche. Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 299-309. doi: 10.3934/dcdsb.2003.3.299

[20]

Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51

2017 Impact Factor: 1.23

Metrics

  • PDF downloads (11)
  • HTML views (6)
  • Cited by (0)

[Back to Top]