A comparison of computational efficiencies of stochastic algorithms in terms of two infection models

Previous Article
Fast two dimensional to three dimensional registration of fluoroscopy and CT-scans using Octrees on segmentation maps
MBE Home
This Issue
Next Article
Stochastic models for competing species with a shared pathogen

2012, 9(3): 487-526. doi: 10.3934/mbe.2012.9.487

A comparison of computational efficiencies of stochastic algorithms in terms of two infection models

H. Thomas Banks ^1, , Shuhua Hu ^1, , Michele Joyner ^2, , Anna Broido ^3, , Brandi Canter ^2, , Kaitlyn Gayvert ^4, and Kathryn Link ^5,

1.	Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, North Carolina State University, Raleigh, NC 27695-8212, United States, United States
2.	Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614-70663, United States, United States
3.	Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, United States
4.	Department of Mathematics, State University of New York at Geneseo, Geneseo, NY 14454, United States
5.	Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010-2899, United States

Received November 2011 Revised May 2012 Published July 2012

Abstract
Related Papers

In this paper, we investigate three particular algorithms: a stochastic simulation algorithm (SSA), and explicit and implicit tau-leaping algorithms. To compare these methods, we used them to analyze two infection models: a Vancomycin-resistant enterococcus (VRE) infection model at the population level, and a Human Immunodeficiency Virus (HIV) within host infection model. While the first has a low species count and few transitions, the second is more complex with a comparable number of species involved. The relative efficiency of each algorithm is determined based on computational time and degree of precision required. The numerical results suggest that all three algorithms have the similar computational efficiency for the simpler VRE model, and the SSA is the best choice due to its simplicity and accuracy. In addition, we have found that with the larger and more complex HIV model, implementation and modification of tau-Leaping methods are preferred.

Keywords: continuous time Markov chain models, Gillespie, tau-leaping, Dynamical models, stochastic simulation algorithms, bacterial and viral infection models..

Mathematics Subject Classification: 60J27, 60J22, 92D2.

Citation: H. Thomas Banks, Shuhua Hu, Michele Joyner, Anna Broido, Brandi Canter, Kaitlyn Gayvert, Kathryn Link. A comparison of computational efficiencies of stochastic algorithms in terms of two infection models. Mathematical Biosciences & Engineering, 2012, 9 (3) : 487-526. doi: 10.3934/mbe.2012.9.487

References:

[1]	B. M. Adams, H. T. Banks, M. Davidian, H. Kwon, H. T. Tran, S. N. Wynne and E. S. Rosenberg, HIV dynamics: Modeling, data analysis, and optimal treatment protocols, J. Computational and Applied Mathematics, 184 (2005), 10-49. doi: 10.1016/j.cam.2005.02.004.
[2]	B. M. Adams, H. T. Banks, M. Davidian and E. S. Rosenberg, Model fitting and prediction with HIV treatment interruption data, CRSC-TR05-40, NCSU, October, 2005, Bulletin of Mathematical Biology, 69 (2007), 563-584. doi: 10.1007/s11538-006-9140-6.
[3]	L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," Second edition, CRC Press, Boca Raton, FL, 2011.
[4]	P. Bai, H. T. Banks, S. Dediu, A. Y. Govan, M. Last, A. L. Lloyd, H. K. Nguyen, M. S. Olufsen, G. Rempala and B. D. Slenning, Stochastic and deterministic models for agricultural production networks, Mathematical Biosciences and Engineering, 4 (2007), 373-402.
[5]	H. T. Banks, M. Davidian, S. Hu, G. Kepler and E. S. Rosenberg, Modelling HIV immune response and validation with clinical data, Journal of Biological Dynamics, 2 (2008), 357-385.
[6]	D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bulletin of Mathematical Biology, 64 (2001), 29-64.
[7]	Y. Cao, D. T. Gillespie and L. R. Petzold, Avoiding negative populations in explicit Poisson tau-leaping, The Journal of Chemical Physics, 123 (2005), 054104.
[8]	Y. Cao, D. T. Gillespie and L. R. Petzold, Efficient step size selection for the tau-leaping simulation method, The Journal of Chemical Physics, 124 (2006), 044109.
[9]	Y. Cao, D. T. Gillespie and L. R. Petzold, Adaptive explicit-implicit tau-leaping method with automatic tau selection, The Journal of Chemical Physics, 126 (2007), 224101.
[10]	N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: Influence of pharmacokinetics and intracellular delay, J. of Theoretical Biology, 226 (2004), 95-109.
[11]	D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, The Journal of Computational Physics, 22 (1976), 403-434. doi: 10.1016/0021-9991(76)90041-3.
[12]	D. T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems, The Journal of Chemical Physics, 115 (2001), 1716-1733. doi: 10.1063/1.1378322.
[13]	D. T. Gillespie and L. R. Petzold, Improved leap-size selection for accelerated stochastic simulation, The Journal of Chemical Physics, 119 (2003), 8229-8234. doi: 10.1063/1.1613254.
[14]	D. T. Gillespie and L. R. Petzold, Stochastic simulation of chemical kinetics, Annual Review of Physical Chemistry, 58 (2007), 25-55.
[15]	G. M. Kepler, H. T. Banks, M. Davidian and E. S. Rosenberg, A model for HCMV infection in immunosuppressed patients, Mathematical and Computer Modelling, 49 (2009), 1653-1663. doi: 10.1016/j.mcm.2008.06.003.
[16]	T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Prob., 7 (1970), 49-58.
[17]	T. G. Kurtz, Limit theorems for sequences of pure jump Markov processes approximating ordinary differential processes, J. Appl. Prob., 8 (1971), 344-356.
[18]	H. H. McAdams and A. Arkin, Stochastic mechanisms in gene expression, PNAS, 94 (1997), 814-819. doi: 10.1073/pnas.94.3.814.
[19]	A. R. Ortiz, H. T. Banks, C. Castillo-Chavez, G. Chowell and X. Wang, A deterministic methodology for estimation of parameters in dynamic Markov chain models, Journal of Biological Systems, 19 (2011), 71-100. doi: 10.1142/S0218339011003798.
[20]	J. Pahle, Biochemical simulations: Stochastic, approximate stochastic and hybrid approaches, Brief Bioinform, 10 (2009), 53-64. doi: 10.1093/bib/bbn050.
[21]	A. S. Perelson, P. Essunger, Y. Z. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz and D. D. Ho, Decaycharacteristics of HIV-1-infected compartments during combination therapy, Nature, 387 (1997), 187-191.
[22]	J. E. Pearson, P. Krapivsky and A. S. Perelson, Stochastic theory of early viral infection: Continuous versus burst production of virions, PLoS Comput. Biol., 7 (2011), e1001058, 17 pp.
[23]	A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41 (1999), 3-44.
[24]	M. Rathinam, L. R. Petzold, Y. Cao and D. T. Gillespie, Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method, The Journal of Chemical Physics, 119 (2003), 12784-12794. doi: 10.1063/1.1627296.
[25]	E. Renshaw, "Modelling Biological Populations in Space and Time," Cambridge Studies in Mathematical Biology, 11, Cambridge Univ. Press, Cambridge, 1991.
[26]	D. J. Wilkinson, Stochastic modelling for quantitative description of heterogeneous biological systems, Nature Reviews Genetics, 10 (2009), 122-133. doi: 10.1038/nrg2509.

show all references

References:

[1]	B. M. Adams, H. T. Banks, M. Davidian, H. Kwon, H. T. Tran, S. N. Wynne and E. S. Rosenberg, HIV dynamics: Modeling, data analysis, and optimal treatment protocols, J. Computational and Applied Mathematics, 184 (2005), 10-49. doi: 10.1016/j.cam.2005.02.004.
[2]	B. M. Adams, H. T. Banks, M. Davidian and E. S. Rosenberg, Model fitting and prediction with HIV treatment interruption data, CRSC-TR05-40, NCSU, October, 2005, Bulletin of Mathematical Biology, 69 (2007), 563-584. doi: 10.1007/s11538-006-9140-6.
[3]	L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," Second edition, CRC Press, Boca Raton, FL, 2011.
[4]	P. Bai, H. T. Banks, S. Dediu, A. Y. Govan, M. Last, A. L. Lloyd, H. K. Nguyen, M. S. Olufsen, G. Rempala and B. D. Slenning, Stochastic and deterministic models for agricultural production networks, Mathematical Biosciences and Engineering, 4 (2007), 373-402.
[5]	H. T. Banks, M. Davidian, S. Hu, G. Kepler and E. S. Rosenberg, Modelling HIV immune response and validation with clinical data, Journal of Biological Dynamics, 2 (2008), 357-385.
[6]	D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bulletin of Mathematical Biology, 64 (2001), 29-64.
[7]	Y. Cao, D. T. Gillespie and L. R. Petzold, Avoiding negative populations in explicit Poisson tau-leaping, The Journal of Chemical Physics, 123 (2005), 054104.
[8]	Y. Cao, D. T. Gillespie and L. R. Petzold, Efficient step size selection for the tau-leaping simulation method, The Journal of Chemical Physics, 124 (2006), 044109.
[9]	Y. Cao, D. T. Gillespie and L. R. Petzold, Adaptive explicit-implicit tau-leaping method with automatic tau selection, The Journal of Chemical Physics, 126 (2007), 224101.
[10]	N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: Influence of pharmacokinetics and intracellular delay, J. of Theoretical Biology, 226 (2004), 95-109.
[11]	D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, The Journal of Computational Physics, 22 (1976), 403-434. doi: 10.1016/0021-9991(76)90041-3.
[12]	D. T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems, The Journal of Chemical Physics, 115 (2001), 1716-1733. doi: 10.1063/1.1378322.
[13]	D. T. Gillespie and L. R. Petzold, Improved leap-size selection for accelerated stochastic simulation, The Journal of Chemical Physics, 119 (2003), 8229-8234. doi: 10.1063/1.1613254.
[14]	D. T. Gillespie and L. R. Petzold, Stochastic simulation of chemical kinetics, Annual Review of Physical Chemistry, 58 (2007), 25-55.
[15]	G. M. Kepler, H. T. Banks, M. Davidian and E. S. Rosenberg, A model for HCMV infection in immunosuppressed patients, Mathematical and Computer Modelling, 49 (2009), 1653-1663. doi: 10.1016/j.mcm.2008.06.003.
[16]	T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Prob., 7 (1970), 49-58.
[17]	T. G. Kurtz, Limit theorems for sequences of pure jump Markov processes approximating ordinary differential processes, J. Appl. Prob., 8 (1971), 344-356.
[18]	H. H. McAdams and A. Arkin, Stochastic mechanisms in gene expression, PNAS, 94 (1997), 814-819. doi: 10.1073/pnas.94.3.814.
[19]	A. R. Ortiz, H. T. Banks, C. Castillo-Chavez, G. Chowell and X. Wang, A deterministic methodology for estimation of parameters in dynamic Markov chain models, Journal of Biological Systems, 19 (2011), 71-100. doi: 10.1142/S0218339011003798.
[20]	J. Pahle, Biochemical simulations: Stochastic, approximate stochastic and hybrid approaches, Brief Bioinform, 10 (2009), 53-64. doi: 10.1093/bib/bbn050.
[21]	A. S. Perelson, P. Essunger, Y. Z. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz and D. D. Ho, Decaycharacteristics of HIV-1-infected compartments during combination therapy, Nature, 387 (1997), 187-191.
[22]	J. E. Pearson, P. Krapivsky and A. S. Perelson, Stochastic theory of early viral infection: Continuous versus burst production of virions, PLoS Comput. Biol., 7 (2011), e1001058, 17 pp.
[23]	A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41 (1999), 3-44.
[24]	M. Rathinam, L. R. Petzold, Y. Cao and D. T. Gillespie, Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method, The Journal of Chemical Physics, 119 (2003), 12784-12794. doi: 10.1063/1.1627296.
[25]	E. Renshaw, "Modelling Biological Populations in Space and Time," Cambridge Studies in Mathematical Biology, 11, Cambridge Univ. Press, Cambridge, 1991.
[26]	D. J. Wilkinson, Stochastic modelling for quantitative description of heterogeneous biological systems, Nature Reviews Genetics, 10 (2009), 122-133. doi: 10.1038/nrg2509.

[1]	Sukhitha W. Vidurupola, Linda J. S. Allen. Basic stochastic models for viral infection within a host. Mathematical Biosciences & Engineering, 2012, 9 (4) : 915-935. doi: 10.3934/mbe.2012.9.915
[2]	Zhengyan Wang, Guanghua Xu, Peibiao Zhao, Zudi Lu. The optimal cash holding models for stochastic cash management of continuous time. Journal of Industrial and Management Optimization, 2018, 14 (1) : 1-17. doi: 10.3934/jimo.2017034
[3]	Roberto C. Alamino, Nestor Caticha. Bayesian online algorithms for learning in discrete hidden Markov models. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 1-10. doi: 10.3934/dcdsb.2008.9.1
[4]	Samuel N. Cohen. Uncertainty and filtering of hidden Markov models in discrete time. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 4-. doi: 10.1186/s41546-020-00046-x
[5]	Yuri Nechepurenko, Michael Khristichenko, Dmitry Grebennikov, Gennady Bocharov. Bistability analysis of virus infection models with time delays. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2385-2401. doi: 10.3934/dcdss.2020166
[6]	Heikki Haario, Leonid Kalachev, Marko Laine. Reduction and identification of dynamic models. Simple example: Generic receptor model. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 417-435. doi: 10.3934/dcdsb.2013.18.417
[7]	Lifeng Chen, Jifa Jiang. Stochastic epidemic models driven by stochastic algorithms with constant step. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 721-736. doi: 10.3934/dcdsb.2016.21.721
[8]	Manuela Caratozzolo, Santina Carnazza, Luigi Fortuna, Mattia Frasca, Salvatore Guglielmino, Giovanni Gurrieri, Giovanni Marletta. Self-organizing models of bacterial aggregation states. Mathematical Biosciences & Engineering, 2008, 5 (1) : 75-83. doi: 10.3934/mbe.2008.5.75
[9]	Thi Tuyet Trang Chau, Pierre Ailliot, Valérie Monbet, Pierre Tandeo. Comparison of simulation-based algorithms for parameter estimation and state reconstruction in nonlinear state-space models. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022054
[10]	S. Mohamad, K. Gopalsamy. Neuronal dynamics in time varying enviroments: Continuous and discrete time models. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 841-860. doi: 10.3934/dcds.2000.6.841
[11]	Wendi Wang. Epidemic models with nonlinear infection forces. Mathematical Biosciences & Engineering, 2006, 3 (1) : 267-279. doi: 10.3934/mbe.2006.3.267
[12]	Carlos Castillo-Chavez, Baojun Song. Dynamical Models of Tuberculosis and Their Applications. Mathematical Biosciences & Engineering, 2004, 1 (2) : 361-404. doi: 10.3934/mbe.2004.1.361
[13]	Joshua E.S. Socolar. Discrete models of force chain networks. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 601-618. doi: 10.3934/dcdsb.2003.3.601
[14]	Radek Erban, Hyung Ju Hwang. Global existence results for complex hyperbolic models of bacterial chemotaxis. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1239-1260. doi: 10.3934/dcdsb.2006.6.1239
[15]	Dianmo Li, Zengxiang Gao, Zufei Ma, Baoyu Xie, Zhengjun Wang. Two general models for the simulation of insect population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 623-628. doi: 10.3934/dcdsb.2004.4.623
[16]	Andriy Sokolov, Robert Strehl, Stefan Turek. Numerical simulation of chemotaxis models on stationary surfaces. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2689-2704. doi: 10.3934/dcdsb.2013.18.2689
[17]	Yilei Tang, Dongmei Xiao, Weinian Zhang, Di Zhu. Dynamics of epidemic models with asymptomatic infection and seasonal succession. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1407-1424. doi: 10.3934/mbe.2017073
[18]	Zhimin Li, Tailei Zhang, Xiuqing Li. Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China. Electronic Research Archive, 2021, 29 (1) : 1661-1679. doi: 10.3934/era.2020085
[19]	Francisco de la Hoz, Anna Doubova, Fernando Vadillo. Persistence-time estimation for some stochastic SIS epidemic models. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2933-2947. doi: 10.3934/dcdsb.2015.20.2933
[20]	Carlota Rebelo, Alessandro Margheri, Nicolas Bacaër. Persistence in some periodic epidemic models with infection age or constant periods of infection. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1155-1170. doi: 10.3934/dcdsb.2014.19.1155

2018 Impact Factor: 1.313

Tools

Metrics

Other articles
by authors

on AIMS
H. Thomas Banks Shuhua Hu Michele Joyner Anna Broido Brandi Canter Kaitlyn Gayvert Kathryn Link
on Google Scholar
H. Thomas Banks Shuhua Hu Michele Joyner Anna Broido Brandi Canter Kaitlyn Gayvert Kathryn Link

[Back to Top]