# American Institute of Mathematical Sciences

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:

show all references

##### References:
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)
Environmental transmission coefficient $f(B)$ with $K_1 = 1000$ and $K_2 = 100$
Dynamics of the animals in each compartment with no testing or culling and with annual testing and culling
Dynamics of the total animals in each disease class with no testing or culling and with annual testing and culling
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
Dynamics of the bacteria in the two environments with no testing or culling and with annual testing and culling
Number of exposed cows due to different infection routes without testing or culling and with annual testing and culling
Initial number of animals in each compartment
 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
Parameters and their values
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%
 Susceptible Exposed Low-shedding High-shedding Calves 65% 35% 0% 0% Heifers 65% 31% 4% 0% Adults 65% 25% 8% 2%
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
 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
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 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
 [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] Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [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

2018 Impact Factor: 1.313