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April 23rd, 2017, 11:54 AM  #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, 02:04 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,500 Thanks: 1372 
are these rates Poisson?

April 23rd, 2017, 10:16 PM  #3 
Senior Member Joined: Feb 2015 From: london Posts: 121 Thanks: 0 
The rates are instantaneous transition rates

April 24th, 2017, 10:28 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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