  A biochemical network can be simulated by a set of ordinary differential
equations (ODE) under well stirred reactor conditions, for large numbers of
molecules, and frequent reactions. This is no longer a robust representation
when some molecular species are in small numbers and reactions changing them
are infrequent. In this case, discrete stochastic events trigger changes of the
smooth deterministic dynamics of the biochemical network.
Piecewise-deterministic Markov processes (PDMP) are well adapted for describing
such situations. Although PDMP models are now well established in biology,
these models remain computationally challenging. Previously we have introduced
the push-forward method to compute how the probability measure is spread by the
deterministic ODE flow of PDMPs, through the use of analytic expressions of the
corresponding semigroup. In this paper we provide a more general simulation
algorithm that works also for non-integrable systems. The method can be used
for biochemical simulations with applications in fundamental biology,
biotechnology and biocomputing.This work is an extended version of the work
presented at the conference CMSB2019.
