This R code simulates a Markov process over time to calculate transition probabilities between states. It initializes a starting probability vector, defines a transition rate matrix, and uses the matrix exponential function to calculate the probability of being in each state over time steps T, storing the results in a matrix. It then plots the probabilities for each state over time.

1. > library(expm)
> initial <- c(0.1,0.2,0,0.4,0.3)
> T <- seq(0.01,2,by= 0.01)
> Q <- cbind(c(-3,3,0, 0, 0), c(2, -5, 3, 0, 0),c(0, 4, -7, 3, 0), c(0, 0, 4, -7, 3), c(0, 0, 0, 4, -4))
> store <- matrix (0,201,5)
> store[1,] <- initial
> Q1 <- 0
>
> for(i in 2:201){
+ Q1 <- 0.01*(i-1)*Q
+ eQt<- expm(Q1)
+ store[i,] <- initial%*%eQt
+ }
>
> plot(c(0,T),store[,1],main="Probabilitiesforstate 0",xlab="T",ylab="")
>
> plot(c(0,T),store[,2],main="Probabilitiesforstate 1",xlab="T",ylab="")
>
>
> plot(c(0,T),store[,3],main="Probabilitiesforstate 2",xlab="T",ylab="")
> plot(c(0,T),store[,4],main="Probabilitiesforstate 3",xlab="T",ylab="")
>
>
> plot(c(0,T),store[,5],main="Probabilitiesforstate 4",xlab="T",ylab="")

2. 0.0 0.5 1.0 1.5 2.0
0.100.120.140.160.18
Probabilities for state 0
T

3. 0.0 0.5 1.0 1.5 2.0
0.150.160.170.180.190.20
Probabilities for state 1
T

4. 0.0 0.5 1.0 1.5 2.0
0.000.050.100.15
Probabilities for state 2
T

5. 0.0 0.5 1.0 1.5 2.0
0.200.250.300.350.40
Probabilities for state 3
T

6. 0.0 0.5 1.0 1.5 2.0
0.200.220.240.260.280.30
Probabilities for state 4
T