Markov chains: long-run proportions (a) In a very simplistic model, suppose that

Question

Markov chains: long-run proportions (a) In a very simplistic model, suppose that at any given time period an individual can contract a particular disease with probability 1%. A sick person will recover during any particular time period with probability 10% (in which case they will be considered healthy at the beginning of the next time period). Assume that people do not develop resistance (or that the disease has sufcient variability), so that previous sickness does not inuence the chances of contracting the diseaseagain. Find the long-run proportion of the population who are sick. Clearlyidentifyyour answer!

Explanation / Answer

Ans:

Transition probability matrix(P):

ph=long run proportion of healthy people

ps=long run proportion of sick people

For long run proportion:

0.99ph+0.1ps=ph

0.01ph+0.9ps=ps

ph+ps=1

from first eqn:

0.1ps=0.01ph

ps=0.1ph

So,

ph+0.1ph=1

1.1ph=1

ph=1/1.1=0.91

ph=0.91

ps=1-ph=1-0.91=0.09

ps=0.09

Hence,long run proportion of people who are sick is 9%

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