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 April 23rd, 2017, 12:54 PM #1 Senior Member   Joined: Feb 2015 From: london Posts: 121 Thanks: 0 Calculating expected time Lets say we have 3 states 0 - Infection Dormant 1 - Infection Active 2 - Individual Dead Transitions 0 -> 1, at rate a 1 -> 2, at rate b 1 -> 0, at rate g The question asks to prove that the expected time until an individual with a dormant infection will die is: $\displaystyle E[T] = \frac{a + b + g}{ab}$ where T denotes the length of time until death. Any ideas how to solve this, I thought I would be able to solve in the following way: $\displaystyle M_{02} = 1 + aM_{12}$ $\displaystyle M_{12} = 1 + bM_{22} + gM_{02}$ $\displaystyle M_{22} = 1$ $\displaystyle M_{02} = \frac{1 + a + ab}{1 - ab}$ However as you can see, my answer is not even close to what the question states the answer should be.
 April 23rd, 2017, 03:04 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,638 Thanks: 1474 are these rates Poisson?
 April 23rd, 2017, 11:16 PM #3 Senior Member   Joined: Feb 2015 From: london Posts: 121 Thanks: 0 The rates are instantaneous transition rates
 April 24th, 2017, 11:28 AM #4 Senior Member   Joined: Feb 2015 From: london Posts: 121 Thanks: 0 After a bit of googling, it looks like you can calculate expected time for a state transition of a continuous markov chain with the following method: $\displaystyle E[0->2] = \frac{1}{a} + E[1->2]$(1) $\displaystyle E[1->2] = \frac{1}{g+b} + \frac{b}{g+b} .0 + \frac{g}{g+b}E[0->2]$ (2) when you sub (2) into (1), $\displaystyle E[0->2] = \frac{a + b+ g}{ab}$ as required

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