Skip to main content
SpringerLink
Log in

A first step to implement Gillespie’s algorithm with rejection sampling

  • Published:
Statistical Methods & Applications Aims and scope Submit manuscript

Abstract

It is well known that firings of a well-stirred chemically reacting system can be described by a continuous-time Markov chain. The currently-used exact implementations of Gillespie’s algorithm simulate every reaction event individually and thus the computational cost is inevitably high. In this paper, we present an exact implementation of a continuous-time Markov chain with bounded intensity which can simulate the process at given time points. The implementation involves rejection sampling, with a trajectory either accepted or rejected based on just a few reaction events. A simulation study on the Schlögl model is presented and supplementary materials for this article are available online.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Arkin A, Ross J, McAdams H (1998) Stochastic kinetic analysis of developmental pathway bifurcation in phage \(\lambda \)-infected E. Coli cells. Genetics 149:1633–1648

    Google Scholar 

  • Beskos A, Papaspiliopoulos O, Roberts GO, Fearnhead P (2006) Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J R Stat Soc B 68(3):333–382

    Article  MATH  MathSciNet  Google Scholar 

  • Beskos A, Roberts GO (2005) Exact simulation of diffusions. Ann Appl Probab 15(4):2422–2444

    Article  MATH  MathSciNet  Google Scholar 

  • Cao Y, Gillespie D, Petzold L (2006) Efficient step size selection for the tau-leaping method. J Chem Phys 124:044109

    Article  Google Scholar 

  • Elliott RJ (1982) Stochastic calculus and applications. Springer, New York

    MATH  Google Scholar 

  • Elliott RJ, Chen Z, Duan Q (2009) Insurance claims modulated by a hidden Brownian marked point process. Insur Math Econ 45:163–172

    Article  MATH  MathSciNet  Google Scholar 

  • Higham DJ (2008) Modeling and simulating chemical reactions. SIAM Rev 50:347–368

    Article  MATH  MathSciNet  Google Scholar 

  • Gillespie D (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22:403–434

    Article  MathSciNet  Google Scholar 

  • Gonçalves F, Roberts GO (2013) Exact simulation problems for jump diffusions. Methodol Comput Appl Probab. doi:10.1007/s11009-013-9330-2

  • Pahle J (2009) Biochemical simulations: stochastic, approximate stochastic and hybrid approaches. Brief Bioinform 10(1):53–64

    Article  Google Scholar 

  • Pu Y, Watson LT, Cao Y (2011) Stiffness detection and reduction in discrete stochastic simulation of biochemical systems. J Chem Phys 134:054105

    Article  Google Scholar 

  • Ross SM (2007) Introdution to probability models, 9th edn. Elsevier, Singapore

    Google Scholar 

  • Sehl M, Zhou H, Sinsheimer JS, Lange KL (2011) Extinction models for cancer stem cell therapy. Math Biosci 234(2):132–146

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We are indebted to two anonymous referees and an associate editor. Their comments and suggestions were extremely useful in improving the article, and they rewrote parts of the paper.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shannxi, People’s Republic of China

    Qihong Duan

  2. Department of Mathematics, NorthWest University, Xi’an, 710069, Shannxi, People’s Republic of China

    Junrong Liu

Authors
  1. Qihong Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junrong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qihong Duan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duan, Q., Liu, J. A first step to implement Gillespie’s algorithm with rejection sampling. Stat Methods Appl 24, 85–95 (2015). https://doi.org/10.1007/s10260-014-0283-6

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10260-014-0283-6

Keywords

Search

Navigation