How many days until recovery?

Question

A patient is either Ill or Recovering, until they become Healthy. Each day: - From Ill: stay Ill with probability 0.6, or move to Recovering with probability 0.4. - From Recovering: relapse to Ill with probability 0.2, or become Healthy with probability 0.8. Healthy ends the process. The patient starts Ill. On average, how many days until the patient is Healthy?

Accepted Answer

Let h(X) be the expected days to Healthy from state X; h(	ext{Healthy}) = 0. First-step analysis: one day passes, then add the expected time from the next state. h(	ext{Ill}) = 1 + 0.6\,h(	ext{Ill}) + 0.4\,h(	ext{Rec}), h(	ext{Rec}) = 1 + 0.2\,h(	ext{Ill}) + 0.8 \cdot 0. From the first equation, 0.4\,h(	ext{Ill}) = 1 + 0.4\,h(	ext{Rec}), so h(	ext{Ill}) = 2.5 + h(	ext{Rec}). Put this into the second: h(	ext{Rec}) = 1 + 0.2\,(2.5 + h(	ext{Rec})) = 1.5 + 0.2\,h(	ext{Rec}) \implies 0.8\,h(	ext{Rec}) = 1.5, so h(	ext{Rec}) = 	frac{15}{8}. Then h(	ext{Ill}) = 2.5 + 	frac{15}{8} =…

How many days until recovery?

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