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Development of a mathematical model |
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We utilized the longitudinal study dataset to define an
appropriate deterministic compartmental model, incorporating age- and seasonal
effects.
As the disease almost exclusively occurred in the
lactating and dry cow groups, the young stock was excluded from the model. For
the definition of disease states, observations of clinical BDD status were
combined with serologic results. As BDD is defined as a clinical disease, we
assumed that presence of lesions constituted infection, I, regardless of
serologic status. The seasonal nature of BDD incidence has been discussed
above. Investigating the serological distributions, a cut-off PP of 25 was
considered appropriate. It was shown that few clinical positives are
serologically negative; on the other hand, a large proportion of clinical
negatives were serologically positive. We therefore defined two additional
states: clinical negatives that were seronegative, which were considered
susceptible (S), and clinical negatives that were seropositive, which
were considered exposed (E). Further investigation showed that there was
a strong age effect, with cows <5 years of age having substantially fewer
exposed animals than cows >5 years of age. The lactating and dry cows were
similar, and were hence considered together. We formulated an SEIS model, where
the transitions from S to E and E to I were
bidirectional, and transitions from I to S were also possible. We
defined the two age groups to capture the age effect, and specified the housing
and grazing periods separately to account for the seasonality. An environmental
component, representing the reservoir of BDD-associated Treponema spp.,
was also defined.
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Fig. W7. Flow diagram of a compartmental SEIS model for BDD,
incorporating two age categories of cows (y: <5 years and o: >5 years)
and an environmental compartment of infectious Treponema spp. bacteria.
Solid lines show the movement of animals; dashed red lines show the movement of
BDD-associated Treponema spp. For definition of parameters, refer to
thesis text |
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From the longitudinal study data, we could determine the
number of cows in each state at a given time; we also knew the numbers of cows
that had transferred from another compartment since the previous time point,
including from S to E, which were defined as 'new' infections.
Using this information, we estimated the transmission parameter, ß, which
determines the rate at which S animals move to E, following the
methodology of Monti et al. (2006). Separate ßs were estimated for
younger and older cows, during the housing and grazing periods. We also
calculated the transmission coefficients governing the rates of cows between
the other compartments.
The model was formulated
using a series of coupled differential equations, and run using the parameter
estimates. The environmental compartment was iteratively parameterised, in the
absence of suitable information to assist with this. The model output was
generally consistent with our data and empirical experience, but there were
several discrepancies, which were possibly due to over- or underestimation of
transmission parameters, or a result of inappropriate model formulation.
Further exploration, analysis and model fitting is required. This model was
deterministic, and thus did not incorporate any stochastic effects;
incorporation of stochastic processes could prove to be more effective.
The model confirmed our findings from the observational studies and
the Bayesian model that use of serology in addition to clinical inspection for
BDD identifies a very high level of exposure to the infection in the cow
groups. Use of serology assumes a direct association with infection by
BDD-associated Treponema spp. This cannot be determined with certainty;
however, the results of our observational studies strongly support this
assumption. This indicates that the organisms causing BDD are ubiquitous in the
cows' environment, and that the 'force of infection' is high. Not all cows
exposed to the treponemes develop clinical BDD; we do not yet understand the
component causes required to constitute a 'sufficient cause'.
The model
presented in this thesis should be considered a preliminary effort at exploring
the use of SEIR-type models; further refinement and diagnostics are required.
The benefits of a mathematical model that effectively simulates the condition
are great: it can be used predictively to assess the effect of interventions,
and hence to inform putative control strategies. For instance, the effects of
improving environmental hygiene can be assessed (by increasing the removal of
treponemes from the environment), the likely effect of treatments can be
estimated, and the effectiveness of husbandry-related factors can be
explored. |
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©
Willem Daniel Vink 2006 |
