Error Bound for Piecewise Deterministic Processes Modeling Stochastic Reaction Systems

Biological processes involving the random interaction of $d$ species with integer particle numbers are often modeled by a Markov jump process on $\mathbb{N}_{0}^{d}$. Realizations of this process can, in principle, be generated with Gillespie's classical stochastic simulation algorithm, but for very reactive systems this method is usually inefficient. Hybrid models based on piecewise deterministic processes offer an attractive alternative which can decrease the simulation time considerably in applications where species with rather low particle numbers interact with very abundant species. We investigate the convergence of the hybrid model to the original one for a class of reaction systems with two distinct scales. Our main result is an error bound which states that, under suitable assumptions, the hybrid model approximates the marginal distribution of the discrete species and the conditional moments of the continuous species up to an error of $\mathcal{O}\!\left(M^{-1}\right)$, where $M$ is the scaling pa...