BOVINE TUBERCULOSIS TESTING IN
MICHIGAN
David Blum
Ioannis Giannakakis
Abra Jeffers
Ahren Lacy
MS&E 220:Probabilistic
Analysis
(Group Project)
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:
1.
2.
3.
Caudal Fold Test (CFT),
Comparative Cervical Test (CCT), and
Gamma interferon.
Model formulation
A decision tree is constructed which includes
1.
2.
3.
4.
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:
1.
2.
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
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 one’s 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.
Test Decision Space
Figure 1
Depopulate herd i
Kill herd
+Blood Test(2)
+Blood Test(1)
Autopsy
+ Tb
More Blood Test
Test more cows
-Blood Test(2)
Depopulate herd i
+Caudal Fold
Autopsy
+ Tb
Test more cows
Selected Cow j has Tb
-Blood Test(1)
-Caudal Fold
Depopulate herd i
Test Herd i
+Blood Test(2)
+Blood Test(1)
More Blood Test
-Blood Test(2)
+Caudal Fold
Autopsy
… doesn't j have Tb
-Blood Test(1)
-Caudal Fold
- Tb
Autopsy
- Tb
Assessing Posterior Belief of Infection
For any single test
P( H , k )
P(k | H )  P( H )
P( H | k ) 

P(k )
P ( k | L)  P ( L)  P ( k | H )  P ( H )
Probability of seeing k positive results follows a
binomial dist
P(k | H )  (
)
 P( | H )  1  P( | H ) 
k
N
k
( N k )
Assessing Posterior Belief of Infection
The probability a cow tests positive on a test (given it
has tested positive previously)
P(CF | H )  P(CF | I , H )  P( I | H )  P(CF | I ', H )  P( I ' | H )
P( CCT |  CF , H ) 
P(CCT , CF | I , H )  P( I | H )  P(  CCT ,  CF | I ', H )  P( I ' | H )
P( CF | I , H )  P( I | H )  P(  CF | I ', H )  P( I ' | H )
P( G |  CCT ,  CF , H ) 
P(G , CCT , CF | I , H )  P( I | H )  P( G ,  CCT ,  CF | I ', H )  P( I ' | H )
P( CCT ,  CF | I , H )  P( I | H )  P( CCT ,  CF | I ', H )  P( I ' | H )
where
P(TG , TCCT , TCF | I , H )  P(TG | I )  P(TCCT | I )  P(TCF | I )
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(I|H) =2.49%
P(I|L) =0.01%
P(H) =0.22%
Baseline Parameters
Test
CFT
CCT
Gamma
Sensitivity
85%
75%
85%
Specificity
95%
98%
93%
Analysis

Plot P(H|K) 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
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.
Findings

P(I|H) is an important factor affecting the shape of the P(H|k)
curve.

P(I|L) has little to no effect on the shape of the posterior P(H|k)
curve consistent with assumptions

The prior belief P(H) does not affect the shape of the P(H|k)
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%)
Figures
P os terior B elief Herd is Infec ted vs # P os itive C F O bs (rd 1)
P arametriz ed by C F S ens itivity, (C F S pec ific ity = 0.95)
1
0.9
0.8
0.65
P (H|k)
0.7
0.6
0.75
0.5
0.88
0.4
0.98
0.3
0.2
0.1
0
50
60
70
80
k
90
Figures
P os terior B elief Herd is Infec ted vs # P os itive C F O bs (rd 1)
P arameteriz ed by C F S pec ific ity, (C F S ens iivity = 0.85)
1
P (H|k)
0.9
0.8
0.65
0.7
0.6
0.75
0.88
0.5
0.4
0.98
0.3
0.2
0.1
0
0
100
200
300
k
400
500
600
Figures
P os terior B elief Herd is Infec ted vs # P os itive C F O bs (rd 1)
P arametriz ed by infec tion frequenc y of highly infec ted herd
1
0.9
P (H|k)
0.8
0.7
0.01
0.6
0.025
0.5
0.05
0.4
0.1
0.3
0.2
0.1
0
0
20
40
60
80
k
100
120
140
Figures
P os terior B elief Herd is Infec ted vs # P os itive C F O bs (rd 1)
P arameteriz ed by infec tion frequenc y of low infec ted herd
1.2
P (H|k)
1
0.00001
0.8
0.0001
0.6
0.0005
0.4
0.001
0.2
0
0
20
40
60
k
80
100
120
Figures
P os terior B elief Herd is Infec ted vs # P os itive C F O bs (rd 1)
P arameteriz ed by prior belief herd is infec ted
1
0.001
P (H|k)
0.8
0.0022
0.6
0.005
0.4
0.01
0.2
0.05
0
0
50
100
k
150
References
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



Jeffers, K. J. (2008). Personal Communication. October 15, 2008.
O’Reilly, 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/emergingdiseases/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_Report2_14814
2_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.