The model describes a simple case of enzyme kinetics. It involves an
enzyme E binding to a substrate S to form a complex C. This complex C
then releases a product P while simultaneously regenerating the original
enzyme E. The possible reactions are E + S -> C, C -> E + S, and C -> P + E,
with rates that constitute the three model parameters.
In probabilistic terms, the integer-valued vector
\((E_t, S_t, C_t, P_t)'\) is generated by a continuous-time Markov process
such that:
$$
\begin{aligned}
&Pr\{E_{t+\delta}=E_t-1, S_{t+\delta}=S_t-1,
C_{t+\delta}=C_t+1, P_{t+\delta}=P_t
|E_t, S_t, C_t, P_t\} \\&\;\;\;\;\;\;\;\;=
\theta_1 E_tS_t\delta+o(\delta), \\
&Pr\{E_{t+\delta}=E_t+1, S_{t+\delta}=S_t+1,
C_{t+\delta}=C_t-1, P_{t+\delta}=P_t
|E_t, S_t, C_t, P_t\} \\&\;\;\;\;\;\;\;\;=
\theta_2 C_t\delta+o(\delta), \\
&Pr\{E_{t+\delta}=E_t+1, S_{t+\delta}=S_t,
C_{t+\delta}=C_t-1, P_{t+\delta}=P_t+1
|E_t, S_t, C_t, P_t\} \\&\;\;\;\;\;\;\;\;=
\theta_3 C_t\delta+o(\delta).
\end{aligned}
$$
The initial state is \((E_0, S_0, C_0, P_0)'=(E0, S0, 0, 0)'\).
Process is simulated using Gillespie exact algorithm.