= Solution
For each state, add its incoming <reaction rates> and subtract its outgoing <reaction rates>. This gives the <master equation>
$$
\boxed{\begin{aligned}
\dot S_{00}&=-(\alpha+\gamma)S_{00}+\beta S_{10}+\delta S_{01},\\
\dot S_{10}&=\alpha S_{00}-(\beta+\gamma)S_{10}+\delta S_{11},\\
\dot S_{01}&=\gamma S_{00}-(\alpha+\delta)S_{01}+\beta S_{11},\\
\dot S_{11}&=\gamma S_{10}+\alpha S_{01}-(\beta+\delta)S_{11}.
\end{aligned}}
$$
Summing gives $\sum_{i,j}\dot S_{ij}=0$, verifying the <conservation law> for total channel density. Nonnegative initial densities stay nonnegative because every loss term vanishes when its own density reaches zero.
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