Approximate analysis of biological systems by hybrid switching jump   diffusion
  In this paper we consider large state space continuous time Markov chains (MCs) arising in the field of systems biology. For density dependent families of MCs that represent the interaction of large groups of identical objects, Kurtz has proposed two kinds of approximations. One is based on ordinary differential equations, while the other uses a diffusion process. The computational cost of the deterministic approximation is significantly lower, but the diffusion approximation retains stochasticity and is able to reproduce relevant random features like variance, bimodality, and tail behavior. In a recent paper, for particular stochastic Petri net models, we proposed a jump diffusion approximation that aims at being applicable beyond the limits of Kurtz's diffusion approximation, namely when the process reaches the boundary with non-negligible probability. Other limitations of the diffusion approximation in its original form are that it can provide inaccurate results when the number of objects in some groups is often or constantly low and that it can be applied only to pure density dependent Markov chains. In order to overcome these drawbacks, in this paper we propose to apply the jump-diffusion approximation only to those components of the model that are in density dependent form and are associated with high population levels. The remaining components are treated as discrete quantities. The resulting process is a hybrid switching jump diffusion. We show that the stochastic differential equations that characterize this process can be derived automatically both from the description of the original Markov chains or starting from a higher level description language, like stochastic Petri nets. The proposed approach is illustrated on three models: one modeling the so called crazy clock reaction, one describing viral infection kinetics and the last considering transcription regulation. 
Alessio Angius, Gianfranco Balbo, Marco Beccuti, Enrico Bibbona,   Andras Horvath, Roberta Sirovich
cs.PF math.PR q-bio.QM