 Solution

Use the compartment counts $Z_n=(S_n,I_n,R_n)$ as the state of the SIRS Markov chain. Its state space is $\{(S,I,R)\in {\mathbb{Z}}_{\ge 0}^3:S+I+R=n\}$. The transition rates areEach susceptible has infection hazard $\lambda I/n$, so summing over the $S$ susceptibles gives the first rate. The memorylessness of the exponential distribution makes each of the $I$ current infectives have recovery hazard $\gamma$, irrespective of its infection time. Each removed individual has immunity-loss hazard $\nu$. These are all the possible off-diagonal transitions. The diagonal entry of the transition intensity matrix is minus their sum, and a transition with no eligible individual has rate zero. Thus the three total event rates are $\lambda SI/n$, $\gamma I$ and $\nu R$, and every event conserves population size.