Subject CT4, April 2015, Question 11

Hi Molly

When faced with this question it is often a good idea to think about the type of paths for the given scenario. In order for the person currently in H to be alive in three years' time, they must be either in H or the state related to having recovered from the condition (let's call it R).

I presume you're talking specifically about the probability PHR(3)? So we can think about the possible paths to end up in State R after 3 years. For this to happen, we must stay in H for some period of time, we then have to jump to R (we can't jump to any other state as otherwise we can't get back to R), we then have to stay in R for the remaining time period.

Putting these pieces together and integrating over the times of the jump we have:

int(0,3) PHHbar(w) * 0.4 * sigma * PRRbar(3-w) dw

In this case, we can also get to the same equation using either the forward or backward equations. For the backward equations, we have:

int(0,3) PHHbar(w) * (0.4 * sigma * PRR(3-w) + 0.6 * sigma * PDR(3-w) + mu * PDR(3-w)) dw

where D is the dead state and mu the transition rate from healthy to dead. We could have two different dead states D1 / D2 representing death from this cause or death from other causes but it doesn't make a difference to the final equation.

but PDR(t) = 0 for any t (or PD1R(t) = PD2R(t) = 0 for all t) and PRR(t) = PRRbar(t) so we get the same expression as above.

For the forward equation, we have:

int(0,3) (PHH(3-w) * 0.4 * sigma + PHD(3-w) * 0) * PRRbar(w) dw
= int(0,3) PHHbar(3-w) * 0.4 * sigma * PRRbar(w) dw
which is the same as the above (do the substitution u = 3-w for it to match exactly).

In general you won't necessarily get the same expression for the forward and backwards equations, it is a consequence of this specific jump process.

As a rule of thumb, I would ask myself whether the question appears to be about the first jump out of a state (so this may relate to a backwards equation) or the last jump into a state (so may relate to a forwards equation). For this particular jump process, to go from H to R, the first jump must be to R. Similarly, the last jump must be from H. This is why the equations match here.

However, it is also a good idea to get into the habit of just thinking about the possible paths and constructing an equation from scratch as we did initially. Not every probability question will fall into the categories of relating directly to a backwards or forwards equation.

Hope this helps!

Andy

 

Hi Molly

The question may make it clear what they want you to use by eg getting you to write down the relevant differential or integral equation first. If it doesn't then try thinking about whether you have the relevant information to be able to solve one or the other.

If the question asks for the probability of something happening that can't be written down in the usual notation then I would say it is likely wanting you to construct an integral from scratch, as I went through above. For example see CT4 April 2013 Question 8 and a similar question in the latest CS2 paper.

Hope this helps!

Andy