Error bound for piecewise deterministic processes modeling stochastic reaction systems. (English) Zbl 1266.60131
Summary: Biological processes involving the random interaction of \(d\) species with integer particle numbers are often modeled by a Markov jump process on \(\mathbb{N}_{0}^{d}\). A realization 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 \(\mathcal{O}(M^{-1})\), where \(M\) is the scaling parameter of the partial thermodynamic limit.
MSC:
60J22 | Computational methods in Markov chains |
60J27 | Continuous-time Markov processes on discrete state spaces |
65C20 | Probabilistic models, generic numerical methods in probability and statistics |
65C40 | Numerical analysis or methods applied to Markov chains |
92-08 | Computational methods for problems pertaining to biology |
92C42 | Systems biology, networks |
92D25 | Population dynamics (general) |