  Delays are an important phenomenon arising in a wide variety of real world
systems. They occur in biological models because of diffusion effects or as
simplifying modeling elements. We propose here to consider delayed stochastic
reaction networks. The difficulty here lies in the fact that the state-space of
a delayed reaction network is infinite-dimensional, which makes their analysis
more involved. We demonstrate here that a particular class of stochastic
time-varying delays, namely those that follow a phase-type distribution, can be
exactly implemented in terms of a chemical reaction network. Hence, any
delay-free network can be augmented to incorporate those delays through the
addition of delay-species and delay-reactions. Hence, for this class of
stochastic delays, which can be used to approximate any delay distribution
arbitrarily accurately, the state-space remains finite-dimensional and,
therefore, standard tools developed for standard reaction network still apply.
In particular, we demonstrate that for unimolecular mass-action reaction
networks that the delayed stochastic reaction network is ergodic if and only if
the non-delayed network is ergodic as well. Bimolecular reactions are more
difficult to consider but an analogous result is also obtained. These results
tell us that delays that are phase-type distributed, regardless of their
distribution, are not harmful to the ergodicity property of reaction networks.
We also prove that the presence of those delays adds convolution terms in the
moment equation but does not change the value of the stationary means compared
to the delay-free case. Finally, the control of a certain class of delayed
stochastic reaction network using a delayed antithetic integral controller is
considered. It is proven that this controller achieves its goal provided that
the delay-free network satisfy the conditions of ergodicity and
output-controllability.
