Markov Chain Question

A particular virus infecting a population of bats is such that it periodically mutates into one of Kpossible virus strains. At each time-step, the virus is equally likely to mutate from its current to any of the other K-1strains with probability a/(K-1), and will remain the same with probability (1-a). Mutations at every time step are assumed to be independent of any other prior mutations. Let X_ndenote the specific strain of the virus at time n, and note that {X_n: n >=0}is a discrete-time Markov chain on the state space {1, 2,..., K}.

• Find the stationary distribution for the chain for the general case with parameters K and a. (Hint: Look at the column sums of the one-step transition matrix.)
• Of the K strains of the virus, only the last two (K-1 and K) are capable of infecting humans. In stationarity, what is the probability that an interaction between a human and a bat infected with the virus leads to the human becoming infected? Assume that the probability that a human will be infected during a given interaction with a bat infected with one of the last two strains is p. You may use symbolically the stationary probabilities pi_1, pi_2, ..., pi_k