 Solution

At time zero the chain-binomial epidemic model has $B(0)\sim \mathrm{Bin}(N,\beta )$ and $C(0)\sim \mathrm{Bin}(1,1-e^{-\gamma })$, independently. The update gives $I(1)=1+B(0)-C(0)$. Since $B(0)\ge 0$ and $C(0)\le 1$, extinction at day one is equivalent to $B(0)=0$ and $C(0)=1$. Hence the one-day extinction probability isThis is the exact probability for the printed discrete-time approximation, which requires $\beta I(t)\delta t\in [0,1]$ wherever used. It should not be substituted for the distinct continuous-time extinction calculation.