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Modeling environmental transmission of MAP infection in dairy cows

  • * Corresponding author: Suzanne Lenhart

    * Corresponding author: Suzanne Lenhart
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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  • 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.

    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)

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

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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

    CompartmentWithout cullingWith annual testing & culling
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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

    CompartmentEquilibrium values after 25 years without cullingFinal values with annual testing & culling
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