Markov model and its differential equation

[phyalan]

Hi guys,
I have problem in constructing the corresponding differential equation for a markov model:
Given a markov model as shown
S1→(a1)→S2→(a2)→S3
S1←(b1)←S2←(b2)←S3
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:
dx1/dt=-a1*x1+b1*x2
dx2/dt=-(b1+a2)*x2+a1*x1+b2*(1-x1-x2)

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

 

[jvicens]

Thank you for sharing your problem with us. I can confirm that your formulation is correct for the given Markov model. The differential equations you have written accurately describe the time evolution of the system and take into account the population distribution among the three states.

It is important to note that the rate constants, ai and bi, play a crucial role in determining the dynamics of the system. The first equation describes the change in the population in state S1, which is influenced by the transition from S1 to S2 with rate constant a1 and the transition from S2 to S1 with rate constant b1. Similarly, the second equation describes the change in the population in state S2, which is influenced by the transitions from S1 to S2 and S2 to S3 with rate constants a1 and a2, respectively, as well as the transition from S3 to S2 with rate constant b2.

I hope this clarifies your doubts. If you have any further questions, please feel free to ask. Best of luck with your research!