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April 23rd, 2017, 11:54 AM   #1
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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.
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April 23rd, 2017, 02:04 PM   #2
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are these rates Poisson?
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April 23rd, 2017, 10:16 PM   #3
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The rates are instantaneous transition rates
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April 24th, 2017, 10:28 AM   #4
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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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