B
Brett Kim
Member
This question is in regard to Q12.4(i) in the ch12 end of chapter questions. In it, we're provided with a Markov jump process model of Healthy, Disease (Y&Z), Death states and corresponding transition rates, and the question asks us to calculate the probability that a healthy life aged 70 is dead by the end of the year i.e. P(hd).
Because it's a Markov jump model I tried to use the CK differential equations i.e. the probability of staying in H until time t then instantaneously jumping to D + the probability of going from H to Y in time t, then instantaneous jumping from Y to D + probability of going from H to Z in time t then instantaneously jumping from Z to D.
Mathematically:
d/dt p(hd) = mu(hd)p(hh) + mu(yd)p(hd) + mu(zd)p(zd)
p(hd) = mu(hd)int(p(hh)) + mu(yd)int(p(hd)) + mu(zd)int(p(hd))
and then using the fact that: p(ij) = exp(-mu(ij)*t)
to basically end up with
p(hd) = 0.014*int(0,1)(exp(-0.026t) dt) + 0.4*int(0,1)(exp(-0.005t) dt) + 0.7*int(0,1)(exp(-0.007t))
However, calculating this gives an answer > 1 so it's obviously wrong. I've looked at the answer and I get where it's coming from, but I don't get what is wrong with the reasoning above. Am I applying the markov chain stuff where it really shouldn't be applied?
Because it's a Markov jump model I tried to use the CK differential equations i.e. the probability of staying in H until time t then instantaneously jumping to D + the probability of going from H to Y in time t, then instantaneous jumping from Y to D + probability of going from H to Z in time t then instantaneously jumping from Z to D.
Mathematically:
d/dt p(hd) = mu(hd)p(hh) + mu(yd)p(hd) + mu(zd)p(zd)
p(hd) = mu(hd)int(p(hh)) + mu(yd)int(p(hd)) + mu(zd)int(p(hd))
and then using the fact that: p(ij) = exp(-mu(ij)*t)
to basically end up with
p(hd) = 0.014*int(0,1)(exp(-0.026t) dt) + 0.4*int(0,1)(exp(-0.005t) dt) + 0.7*int(0,1)(exp(-0.007t))
However, calculating this gives an answer > 1 so it's obviously wrong. I've looked at the answer and I get where it's coming from, but I don't get what is wrong with the reasoning above. Am I applying the markov chain stuff where it really shouldn't be applied?