BOVINE TUBERCULOSIS TESTING IN MICHIGAN

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BOVINE TUBERCULOSIS TESTING IN MICHIGAN
David Blum Ioannis Giannakakis Abra Jeffers Ahren
Lacy

• MSE 220Probabilistic Analysis
• (Group Project)

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Background

• Tuberculosis is a widespread, potentially fatal
  disease. The bovine form of Tb, M. bovis, has a
  wide range of hosts including humans.
• A series of tests is performed by the state on a
  percentage of all Michigan herds per year,
  including
• Caudal Fold Test (CFT),
• Comparative Cervical Test (CCT), and
• Gamma interferon.

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Model formulation

• A decision tree is constructed which includes
• all the successive tests,
• the probabilities of being infected given the
  results of the tests,
• the decisions taken about whether or not to
  continue testing, and
• the total cost in each of the different cases.
• We treated each herd as existing in one of two
  following states
• high frequency of Tb infection,
• low frequency of Tb infection
• Tb transmission within herds with high frequency
  of Tb infection is sufficiently high that it is
  less costly to cull the herd than to eradicate Tb
  though ongoing testing and slaughtering of
  individual cattle

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Model formulation

• Before administering a test, one has a prior
  belief as to whether the herd has a low infection
  rate or a high infection rate
• We flip the tree, to determine ones posterior
  belief that the herd is highly infected given
  that there are k positive readings from the
  current test.
• Assumption Each cow tests independently of each
  other cow in the herd.
• The probability of seeing k positives (given that
  the herd is highly or low infected) can be
  modeled as a binomial distribution.

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Test Decision Space
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Assessing Posterior Belief of Infection

• For any single test
• Probability of seeing k positive results follows
  a binomial dist

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Assessing Posterior Belief of Infection

• The probability a cow tests positive on a test
  (given it has tested positive previously)
• where

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Baseline Parameters

• H Herd with high frequency of infection
• L Herd with low frequency of infection
• I the cow being tested is actually infected
  with Tb
• P(IH) 2.49
• P(IL) 0.01
• P(H) 0.22

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Baseline Parameters

• Test Sensitivity Specificity
• CFT 85 95
• CCT 75 98
• Gamma 85 93

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Analysis

• Plot P(HK) versus k at every model.
• The curve is S-shaped.
• As the number of positive observations within a
  herd increases, the herd infection belief
  increases more steeply
• A sharp S curve implies a good test,
• Strong distinction in posterior beliefs

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Findings

• Increasing the accuracy of any given test
  sharpens the corresponding S-curve and shifts
  left,
• Thus fewer positive observations are required to
  convince an observer that the herd is highly
  infected.
• Improving the accuracy of the first test (CF) is
  the most efficient way to improve the information
  available to an observer.
• The information conveyed through subsequent tests
  is less sensitive to a change in test accuracy
  than each preceding test.

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Findings

• P(IH) is an important factor affecting the shape
  of the P(Hk) curve.
• P(IL) has little to no effect on the shape of
  the posterior P(Hk) curve consistent with
  assumptions
• The prior belief P(H) does not affect the shape
  of the P(Hk) curve, though it does shift the
  curve horizontally
• Thus test results greatly outweigh the prior
  within a range of likely priors (0.001 to 5)

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Figures
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Figures
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Figures
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Figures
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Figures
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References

• Jeffers, K. J. (2008). Personal Communication.
  October 15, 2008.
• OReilly, L. M. and C.M. Daborn. (1995). The
  Epidemiology of Mycobacterium bovis Infections in
  Animals and Man A Review. Tubercule and Lung
  Disease. 76(S.1) 1 46.
• VanderKlok, M. S. (2008). Bovine Tuberculosis in
  Michigan Where We Are Today. Bovine TB
  Scientific Meeting, East Lansing, MI online.
  Available http//www.michigan.gov/documents/emerg
  ingdiseases/MDA_Update_Part_2_249465_7.pdf
  Accessed October 20, 2008.
• Judge, L. J (2005). Epidemiologic update for the
  Michigan bovine TB program, online. Available
  http//www.michigan.gov/documents/MDA_2005_BTB_Rep
  ort2_148142_7.pdf
• Radintz, T, and DiConstanzo, A (2008). Impancts
  of Bovine Tb Testing and Associated Costs on
  Cow-Calf Producer Profitability in 2008-2009.
  University of Minnesota Extension.