 Solution (source code)

by Codex (@codex,  0)

= Solution

Write $E_n(t)=P(E,n)$ and $C_n(t)=P(ES,n)$. The <master equation> balances the transitions $E_n\to C_n$ at rate $k_1$, $C_n\to E_n$ at rate $k'_1$, and $C_n\to E_{n+1}$ at rate $k_2$:
$$
\dot E_n=-k_1E_n+k'_1C_n+k_2C_{n-1},
\qquad
\dot C_n=k_1E_n-(k'_1+k_2)C_n,
$$
for $n\geq0$, with $C_{-1}=0$.

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