Ch12 Q12.4

Discussion in 'CS2' started by Brett Kim, Sep 18, 2021.

  1. Brett Kim

    Brett Kim Keen 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?
     
  2. Dave Johnson

    Dave Johnson ActEd Tutor Staff Member

    Hi Brett

    The error you are making is writing p_ij(t) = exp{-mu_ij * t}

    (Sorry - I've altered your notation to make it clear this is a function of t.)

    You've used the formula for a holding probability (ie remaining in a state for a period) and it doesn't generalise when there are transitions. It would be OK to write p_HH(t) = exp{-mu_HH * t} for example, because return to state H is impossible, so this probability implies no jumps, but we can't write p_HD(t) = exp{-mu_HD * t}.

    The problem you have with this method more generally is that this system is sufficiently complex that we can't write p_HH(t), p_HY(t) and p_HZ(t) in terms of p_HD(t), which means we can't use our usual tools to integrate the expression.

    So I would focus on the model solution - it's the only way I can see that you will solve this.

    Hope that helps.

    Dave
     

Share This Page