Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 356 / 3 / e / Solution

The stochastic quasi-steady-state simulation evolves only $X$:

1. At the current integer $x\ge 1$, compute $q(x)={\alpha }_3{V}^2/({\alpha }_{4}x)$ and the averaged rates ${\lambda }_1(x)={\alpha }_{1}q(x{)}^2/V$ and ${\lambda }_2(x)={\alpha }_{2}x(x-1)/V$.
2. Draw a waiting time $\Delta t\sim \mathrm{Exp}({\lambda }_1+{\lambda }_2)$.
3. Set $x\leftarrow x+1$ with probability ${\lambda }_1/({\lambda }_1+{\lambda }_2)$; otherwise set $x\leftarrow x-1$.
4. Advance time by $\Delta t$ and repeat.

This is a Gillespie algorithm for the averaged slow master equation. It samples the fast conditional equilibrium analytically through its factorial moment and never simulates individual fast $Y$ births or deaths.