Models for Multi-Specie Chemical Reactions Using Polynomial Stochastic Hybrid Systems

A procedure for constructing approximate stochastic models for chemical reactions is presented. This is done by representing the population of various species involved in a chemical reaction as the continuous state of a polynomial Stochastic Hybrid System (pSHS). An important property of pSHSs is that the dynamics of all the statistical moments of its continuous states, evolves according to a infinite-dimensional linear ordinary differential equation (ODE). Under appropriate conditions, this infinite-dimensional ODE can be accurately approximated by a finite-dimensional nonlinear ODE, the state of which typically contains the moments of interest. In this paper, for a very general class of chemical reactions, we provide existence and uniqueness conditions for these finite-dimensional nonlinear ODEs. Furthermore, explicit formulas to construct them are also provided. To illustrate the applicability of our results, we construct an approximate stochastic model for a decaying and dimerizing chemical reaction set. Moment estimates obtained from the finite-dimensional nonlinear ODE are compared with estimates obtained from a large number of Monte Carlo simulations.