\begin{document}

\begin{frontmatter}


\dochead{}

\title{Approximate analysis of biological systems by hybrid switching jump diffusion}


\author[l1]{Alessio Angius}
\author[l1]{Gianfranco Balbo}
\author[l1]{Marco Beccuti}
\author[l2]{Enrico Bibbona}
\author[l1]{Andras Horvath}
\author[l2]{Roberta Sirovich}

\address[l1]{Universit\`a di Torino,
Dipartimento di Informatica\\
\{beccuti,angius,horvath,balbo\}@di.unito.it}

\address[l2]{Universit\`a di Torino,
Dipartimento di Matematica\\
\{roberta.sirovich,enrico.bibbona\}@unito.it}


\begin{abstract}
In this paper we consider large state space continuous time Markov chains
arising in the field of systems biology.  For a class of such models,
namely, for density dependent families of Markov chains that represent the
interaction of large groups of identical objects, Kurtz has proposed two
kinds of approximations.  One is based on ordinary differential equations
and provides a deterministic approximation, while the other uses a diffusion
process with which the resulting approximation is stochastic.  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 that cannot be captured by a single deterministic quantity.

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 in order to cover the case when
the process reaches the boundary with non-negligible probability. In this
paper we generalize the method so that it can be applied to any density
dependent Markov chains.  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, i.e., a diffusion with hybrid
state space and jumps where the discrete state changes can be seen as
switches that take the diffusion from a condition to another.  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.
\end{abstract}


\begin{keyword}
diffusion approximation, jump diffusion, stochastic differential equations with barriers.
\end{keyword}

\end{frontmatter}


\section{Introduction}\label{Sec:int}
\input{intro}
\section{Background}\label{Sec:mat}
\input{mater}
\section{Jump-diffusion approximations and the hybrid switching extention}\label{Sec:the}
\input{theor}

\section{Analytic calculation\label{Sec:analytic}}
\input{analytic}

\section{Simulation based results}\label{Sec:res}
\input{resul}


\section{Conclusions}\label{Sec:con}
\input{concl}


\bibliographystyle{elsarticle-num}
\bibliography{bibl}

\appendix
\input{dd_app}\label{App:1}
\input{algo}\label{Algo:1}
{"config": "arxiv", "file": "1406.1352/main.tex"}