In this question we work with a continuous-time Markov chain where the rate of jump $i \rightarrow j$ may depend on $j$ but not on $i$. A virus can be in one of $s$ strains $1, \ldots, s$, and it mutates to strain $j$ with rate $r_{j} \geqslant 0$ from each strain $i \neq j$. (Mutations are caused by the chemical environment.) Set $R=r_{1}+\ldots+r_{s}$.

(a) Write down the Q-matrix (the generator) of the chain $\left(X_{t}\right)$ in terms of $r_{j}$ and $R$.

(b) If $R=0$, that is, $r_{1}=\ldots=r_{s}=0$, what are the communicating classes of the chain $\left(X_{t}\right)$ ?

(c) From now on assume that $R>0$. State and prove a necessary and sufficient condition, in terms of the numbers $r_{j}$, for the chain $\left(X_{t}\right)$ to have a single communicating class (which therefore should be closed).

(d) In general, what is the number of closed communicating classes in the chain $\left(X_{t}\right)$ ? Describe all open communicating classes of $\left(X_{t}\right)$.

(e) Find the equilibrium distribution of $\left(X_{t}\right)$. Is the chain $\left(X_{t}\right)$ reversible? Justify your answer.

(f) Write down the transition matrix $\widehat{P}=\left(\widehat{p}_{i j}\right)$ of the discrete-time jump chain for $\left(X_{t}\right)$ and identify its equilibrium distribution. Is the jump chain reversible? Justify your answer.