 Solution

Write $a={\beta }^{\ast }=\beta N+\gamma$. If the first event is recovery, the epidemic has ended and a second event is impossible. If the first event is infection at time $s$, its density is $\beta N{e}^{-as}$, and the new state is $(N-1,2,0)$. The total next-event rate isThus, by the Markov property, the two-event probability in a stochastic SIR model by time one isEvaluate the first term as $\beta N(1-e^{-a})/a$. For the second, $b-a=a-2\beta$, soThereforeFor $N\ge 2$ and $\gamma >0$, the denominator ${\beta }^{\ast }-2\beta =\beta (N-2)+\gamma$ is positive. Event times are continuous, so inclusion or exclusion of the endpoint one makes no difference.