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Stochastic models for competing species with a shared pathogen
A comparison of computational efficiencies of stochastic algorithms in terms of two infection models
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 |
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. |
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