Markov model and its differential equation

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!