ADM Assignment 1

DTMC= new("markovchain",states=c("1","2","3","H"),
          transitionMatrix=matrix(data = c(0.6,0.3 ,0   ,0.1,
                                           0  ,0.75,0.15,0.1,
                                           0  ,0   ,0.9 ,0.1,
                                           0  ,0   ,0   ,1),
                                  byrow = T,nrow = 4), name="Stages")
initialState = c(1,0,0,0)
probFever=c(0.1,0.5,0.8,0)

#show(DTMC)

plot(DTMC,package="diagram")

timesteps <- 76
dis.df <- data.frame( "timestep" = numeric(),
 "S1" = numeric(), "S2" = numeric(), "S3" = numeric(),
 "H" = numeric(), "PF" = numeric(), stringsAsFactors=FALSE)

for (i in 0:timesteps) {
  dis.df[nrow(dis.df) + 1, 1:5] <- as.list(c(i,round(as.numeric(initialState * DTMC ^ i),3)))
  dis.df[i+1, 6] <- t(as.vector(dis.df[i + 1, 2:5],mode = "numeric")) %*% probFever
}
#dis.df

#plot(dis.df$timestep,dis.df$S1)
#points(dis.df$timestep,dis.df$S2, col="yellow")
#points(dis.df$timestep,dis.df$S3, col="red")
#points(dis.df$timestep,dis.df$H, col="green")
#points(dis.df$timestep,dis.df$PF, col="blue")

What is probability that the patient is still in stage 1 after 8 days?

dis.df[9,1:2] 

Prob = 1.7 %

What is the total probability to be sick after 8 days?

sum(dis.df[9,2:4])

Prob = 43.1 %

What is the probability of detecting fever on the next day?

dis.df[9,6]

Prob = 28.28 %

Does this model have limiting or stationary solutions? Why?

Stationary solution cause of absorbing state ‘H’

How long does the system need to reach a stationary solution?

dis.df[min(which(dis.df[,5]==1) ),1]

73 days

When will the patient be healed with a probability of 99%?

dis.df[min(which(dis.df[,5]==0.99) ),1]

On day 44