= Solution
At time zero the <chain-binomial epidemic model> has $B(0)\sim\operatorname{Bin}(N,\beta)$ and $C(0)\sim\operatorname{Bin}(1,1-e^{-\gamma})$, independently. The update gives $I(1)=1+B(0)-C(0)$. Since $B(0)\ge0$ and $C(0)\le1$, extinction at day one is equivalent to $B(0)=0$ and $C(0)=1$. Hence \b[the one-day extinction probability] is
$$
\boxed{\mathbb P(I(1)=0)=(1-\beta)^N(1-e^{-\gamma}).}
$$
This 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.
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