Modeling environmental transmission of MAP infection in dairy cows

Kokum R. De Silva; Shigetoshi Eda; Suzanne Lenhart

doi:10.3934/mbe.2017052

Article Contents

2017, Volume 14, Issue 4: 1001-1017. Doi: 10.3934/mbe.2017052

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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
* Corresponding author: Suzanne Lenhart

Received: March 19, 2016

Accepted: January 30, 2017

Published: May 2017

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

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Abstract

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.

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

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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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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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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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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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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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