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Markov model and its differential equation

  1. Feb 25, 2014 #1
    Hi guys,
    I have problem in constructing the corresponding differential equation for a markov model:
    Given a markov model as shown
    where S1 to S3 are three different states of the system and ai and bi are the forwards and backwards rate constant for transition between the states.
    Now I want to write down the differential equation to describe the time evolution of the system. Suppose the whole population is 1, and the population being in S1 at time t is p1(t) and so on. Then we have p1(t)+p2(t)+p3(t)=1 for all time.
    Since there are only two transition steps in the model(S1 to S2 and S2 to S3), I can represent the dynamics by a system of two first order DE. Let variable x1 and x2 lie between 0 to 1. If x1 equals 1 means all the population is in S1 and x2 being 1 means all the population is in S2. In other words, the population of the system is now written as p1(t)=x1(t), p2(t)=x2(t), p3(t)=1-x1(t)-x2(t), so that the relation p1+p2+p3=1 still holds.
    Then I have the following differential equation for the system:

    Is my formulation correct? Since I am not very familiar with Markov model, I really wish someone can comment on this.
    Thanks alot!
  2. jcsd
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