Table 14-5
The following data consists of a matrix of transition probabilities (P)of three potential diseases, and the initial incidence of each disease π(0).Assume that each state represents a disease (Disease 1, Disease 2, Disease 3, respectively)and the transition probabilities represent changes from one checkup to the next.
P =

π(0)= (.3, .3, .4)
-Using the data in Table 14-5, which disease will have the highest incidence when absorbing states are reached?
A)Disease 1
B)Disease 2
C)Disease 3
D)Disease 1 and Disease 3 are tied for the highest incidence

Question

Table 14-5
The following data consists of a matrix of transition probabilities (P)of three potential diseases, and the initial incidence of each disease π(0).Assume that each state represents a disease (Disease 1, Disease 2, Disease 3, respectively)and the transition probabilities represent changes from one checkup to the next.
P =  
 
π(0)= (.3, .3, .4)
-Using the data in Table 14-5, which disease will have the highest incidence when absorbing states are reached?
A)Disease 1 
B)Disease 2 
C)Disease 3 
D)Disease 1 and Disease 3 are tied for the highest incidence

Quizplus · Accepted Answer

To find the incidence of each disease when absorbing states are reached, we need to calculate the limiting probabilities using the formula:
lim n→∞ P^n = [I-Q]^-1 R
where Q is the submatrix of P consisting of non-absorbing states, and R is the matrix of probabilities of moving from non-absorbing states to absorbing states.

In this case, the absorbing states are Disease 1 and Disease 3 (states 1 and 3), so Q is:
[0.8 0.1;
 0.4 0.2]

And R is:
[0.9 0;
 0 0.6;
 0.1 0.4]

Therefore, 
lim n→∞ P^n = [(I-Q)^-1]R = 
[0.8182 0.1818]*[0.9 0;
               0 0.6;
               0.1 0.4] = 
[0.7455 0.2545]

This means that Disease 1 will have an incidence of 0.7455, Disease 2 will have an incidence of 0.0 (it is an absorbing state), and Disease 3 will have an incidence of 0.2545. Therefore, the answer is C, Disease 3 will have the highest incidence when absorbing states are reached.

Table 14-5 The Following Data Consists of a Matrix of Transition Probabilities