2012, 9(1): 1-25. doi: 10.3934/mbe.2012.9.1

Nonlinear stochastic Markov processes and modeling uncertainty in populations

1. 

Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, Raleigh, NC 27695-8212, United States, United States

Received  January 2011 Revised  July 2011 Published  December 2011

We consider an alternative approach to the use of nonlinear stochastic Markov processes (which have a Fokker-Planck or Forward Kolmogorov representation for density) in modeling uncertainty in populations. These alternate formulations, which involve imposing probabilistic structures on a family of deterministic dynamical systems, are shown to yield pointwise equivalent population densities. Moreover, these alternate formulations lead to fast efficient calculations in inverse problems as well as in forward simulations. Here we derive a class of stochastic formulations for which such an alternate representation is readily found.
Citation: H.Thomas Banks, Shuhua Hu. Nonlinear stochastic Markov processes and modeling uncertainty in populations. Mathematical Biosciences & Engineering, 2012, 9 (1) : 1-25. doi: 10.3934/mbe.2012.9.1
References:
[1]

G. Albano, V. Giorno, P. Roman-Roman and F. Torres-Ruiz, Inferring the effect of therapy on tumors showing stochastic Gompertzian growth, Journal of Theoretical Biology, 276 (2011), 67-77. doi: 10.1016/j.jtbi.2011.01.040.

[2]

L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," Second edition, CRC Press, Boca Raton, FL, 2011.

[3]

P. Bai, H. T. Banks, S. Dediu, A. Y. Govan, M. Last, A. Loyd, H. K. Nguyen, M. S. Olufsen, G. Rempala and B. D. Slenning, Stochastic and deterministic models for agricultural production networks, Math. Biosci. and Engr., 4 (2007), 373-402. doi: 10.3934/mbe.2007.4.373.

[4]

H. T. Banks and K. L. Bihari, Modelling and estimating uncertainty in parameter estimation, Inverse Problems, 17 (2001), 95-111. doi: 10.1088/0266-5611/17/1/308.

[5]

H. T. Banks, V. A. Bokil, S. Hu, A. K. Dhar, R. A. Bullis, C. L. Browdy and F. C. T. Allnutt, Modeling shrimp biomass and viral infection for production of biological countermeasures, Mathematical Biosciences and Engineering, 3 (2006), 635-660. doi: 10.3934/mbe.2006.3.635.

[6]

H. T. Banks, D. M. Bortz, G. A. Pinter and L. K. Potter, Modeling and imaging techniques with potential for application in bioterrorism, in "Bioterrorism" (eds. H. T. Banks and C. Castillo-Chavez), Frontiers in Applied Math., 28, SIAM, Philadelphia, PA, (2003), 129-154.

[7]

H. T. Banks, L. W. Botsford, F. Kappel and C. Wang, Modeling and estimation in size structured population models, in "Mathematical Ecology" (Trieste, 1986), World Sci. Publ., Teaneck, NJ, (1988), 521-541.

[8]

H. T. Banks and J. L. Davis, Quantifying uncertainty in the estimation of probability distributions, Math. Biosci. Engr., 5 (2008), 647-667. doi: 10.3934/mbe.2008.5.647.

[9]

H. T. Banks, J. L. Davis, S. L. Ernstberger, S. Hu, E. Artimovich, A. K. Dhar and C. L. Browdy, A comparison of probabilistic and stochastic formulations in modeling growth uncertainty and variability, Journal of Biological Dynamics, 3 (2009), 130-148. doi: 10.1080/17513750802304877.

[10]

H. T. Banks, J. L. Davis, S. L. Ernstberger, S. Hu, E. Artimovich and A. K. Dhar, Experimental design and estimation of growth rate distributions in size-structured shrimp populations, Inverse Problems, 25 (2009), 095003, 28 pp.

[11]

H. T. Banks, J. L. Davis and S. Hu, A computational comparison of alternatives to including uncertainty in structured population models, in "Three Decades of Progress in Systems and Control" (eds. X. Hu, U. Jonsson, B. Wahlberg and B. Ghosh), Springer, 2010.

[12]

H. T. Banks and B. G. Fitzpatrick, Estimation of growth rate distributions in size structured population models, Quarterly of Applied Mathematics, 49 (1991), 215-235.

[13]

H. T. Banks and S. Hu, "Nonlinear Stochastic Markov Processes and Modeling Uncertainty in Populations," Center for Research in Scientific Computation, North Carolina State University, January, 2011.

[14]

H. T. Banks, B. G. Fitzpatrick, L. K. Potter and Y. Zhang, Estimation of probability distributions for individual parameters using aggregate population data, in "Stochastic Analysis, Control, Optimization and Applications" (eds. W. McEneaney, G. Yin and Q. Zhang), Birkhäuser, Boston, (1989), 353-371.

[15]

H. T. Banks, P. M. Kareiva and L. Zia, Analyzing field studies of insect dispersal using two dimensional transport equations, Environmental Entomology, 17 (1988), 815-820.

[16]

H. T. Banks, K. L. Rehm and K. L. Sutton, Dynamic social network models incorporating stochasticity and delays, Quarterly Applied Math., 68 (2010), 783-802.

[17]

H. T. Banks, K. L. Sutton, W. C. Thompson, G. Bocharov, D. Roose, T. Schenkel and A. Meyerhans, Estimation of cell proliferation dynamics using CFSE data, Bull. Math. Biol., 73 (2011), 116-150. doi: 10.1007/s11538-010-9524-5.

[18]

H. T. Banks, K. L. Sutton, W. C. Thompson, G. Bocharov, M. Doumic, T. Schenkel, J. Argilaguet, S. Giest, C. Peligero and A. Meyerhans, A new model for the estimation of cell proliferation dynamics using CFSE data,, J. Immunological Methods, (). 

[19]

H. T. Banks and H. T. Tran, "Mathematical and Experimental Modeling of Physical and Biological Processes," With 1 CD-ROM (Windows, Macintosh and UNIX), Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2009.

[20]

H. T. Banks, H. T. Tran and D. E. Woodward, Estimation of variable coefficients in the Fokker-Planck equations using moving node finite elements, SIAM J. Numer. Anal., 30 (1993), 1574-1602. doi: 10.1137/0730082.

[21]

G. Casella and R. L. Berger, "Statistical Inference," The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990.

[22]

S. N. Ethier and T. G. Kurtz, "Markov Processes: Characterization and Convergence," Wiley Series in Probability and Statistics, New York, 1986.

[23]

L. Ferrante, S. Bompadre, L. Possati and L. Leone, Parameter estimation in a Gompertzian stochastic model for tumor growth, Biometrics, 56 (2000), 1076-1081. doi: 10.1111/j.0006-341X.2000.01076.x.

[24]

T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988.

[25]

G. W. Harrison, Numerical solution of the Fokker-Planck equation using moving finite elements, Numerical Methods for Partial Differential Equations, 4 (1988), 219-232. doi: 10.1002/num.1690040305.

[26]

I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991.

[27]

F. Klebaner, "Introduction to Stochastic Calculus with Applications," Second edition, Imperial College Press, London, 2005.

[28]

T. Luzyanina, D. Roose, T. Schenkel, M. Sester, S. Ehl, A. Meyerhans and G. Bocharov, Numerical modelling of label-structured cell population growth using CFSE distribution data, Theoretical Biology and Medical Modelling, 4 (2007), 1-26. doi: 10.1186/1742-4682-4-26.

[29]

J. A. J. Metz and O. Diekmann, eds., "The Dynamics of Physiologically Structured Populations," Papers from the colloquium held in Amsterdam, 1983, Lecture Notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986.

[30]

S. Michelson, K. Ito, H. T. Tran and J. T. Leith, Stochastic models for subpopulation emergence in heterogeneous tumors, Bulletin of Mathematical Biology, 51 (1989), 731-747.

[31]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," An extended version of the Japanese edition, Ecology and Diffusion, Translated by G. N. Parker, Biomathematics, 10, Springer-Verlag, Berlin-New York, 1980.

[32]

J. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. doi: 10.2307/1934533.

[33]

T. T. Soong, "Random Differential Equations in Science and Engineering," Mathematics in Science and Engineering, Vol. 103, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973.

show all references

References:
[1]

G. Albano, V. Giorno, P. Roman-Roman and F. Torres-Ruiz, Inferring the effect of therapy on tumors showing stochastic Gompertzian growth, Journal of Theoretical Biology, 276 (2011), 67-77. doi: 10.1016/j.jtbi.2011.01.040.

[2]

L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," Second edition, CRC Press, Boca Raton, FL, 2011.

[3]

P. Bai, H. T. Banks, S. Dediu, A. Y. Govan, M. Last, A. Loyd, H. K. Nguyen, M. S. Olufsen, G. Rempala and B. D. Slenning, Stochastic and deterministic models for agricultural production networks, Math. Biosci. and Engr., 4 (2007), 373-402. doi: 10.3934/mbe.2007.4.373.

[4]

H. T. Banks and K. L. Bihari, Modelling and estimating uncertainty in parameter estimation, Inverse Problems, 17 (2001), 95-111. doi: 10.1088/0266-5611/17/1/308.

[5]

H. T. Banks, V. A. Bokil, S. Hu, A. K. Dhar, R. A. Bullis, C. L. Browdy and F. C. T. Allnutt, Modeling shrimp biomass and viral infection for production of biological countermeasures, Mathematical Biosciences and Engineering, 3 (2006), 635-660. doi: 10.3934/mbe.2006.3.635.

[6]

H. T. Banks, D. M. Bortz, G. A. Pinter and L. K. Potter, Modeling and imaging techniques with potential for application in bioterrorism, in "Bioterrorism" (eds. H. T. Banks and C. Castillo-Chavez), Frontiers in Applied Math., 28, SIAM, Philadelphia, PA, (2003), 129-154.

[7]

H. T. Banks, L. W. Botsford, F. Kappel and C. Wang, Modeling and estimation in size structured population models, in "Mathematical Ecology" (Trieste, 1986), World Sci. Publ., Teaneck, NJ, (1988), 521-541.

[8]

H. T. Banks and J. L. Davis, Quantifying uncertainty in the estimation of probability distributions, Math. Biosci. Engr., 5 (2008), 647-667. doi: 10.3934/mbe.2008.5.647.

[9]

H. T. Banks, J. L. Davis, S. L. Ernstberger, S. Hu, E. Artimovich, A. K. Dhar and C. L. Browdy, A comparison of probabilistic and stochastic formulations in modeling growth uncertainty and variability, Journal of Biological Dynamics, 3 (2009), 130-148. doi: 10.1080/17513750802304877.

[10]

H. T. Banks, J. L. Davis, S. L. Ernstberger, S. Hu, E. Artimovich and A. K. Dhar, Experimental design and estimation of growth rate distributions in size-structured shrimp populations, Inverse Problems, 25 (2009), 095003, 28 pp.

[11]

H. T. Banks, J. L. Davis and S. Hu, A computational comparison of alternatives to including uncertainty in structured population models, in "Three Decades of Progress in Systems and Control" (eds. X. Hu, U. Jonsson, B. Wahlberg and B. Ghosh), Springer, 2010.

[12]

H. T. Banks and B. G. Fitzpatrick, Estimation of growth rate distributions in size structured population models, Quarterly of Applied Mathematics, 49 (1991), 215-235.

[13]

H. T. Banks and S. Hu, "Nonlinear Stochastic Markov Processes and Modeling Uncertainty in Populations," Center for Research in Scientific Computation, North Carolina State University, January, 2011.

[14]

H. T. Banks, B. G. Fitzpatrick, L. K. Potter and Y. Zhang, Estimation of probability distributions for individual parameters using aggregate population data, in "Stochastic Analysis, Control, Optimization and Applications" (eds. W. McEneaney, G. Yin and Q. Zhang), Birkhäuser, Boston, (1989), 353-371.

[15]

H. T. Banks, P. M. Kareiva and L. Zia, Analyzing field studies of insect dispersal using two dimensional transport equations, Environmental Entomology, 17 (1988), 815-820.

[16]

H. T. Banks, K. L. Rehm and K. L. Sutton, Dynamic social network models incorporating stochasticity and delays, Quarterly Applied Math., 68 (2010), 783-802.

[17]

H. T. Banks, K. L. Sutton, W. C. Thompson, G. Bocharov, D. Roose, T. Schenkel and A. Meyerhans, Estimation of cell proliferation dynamics using CFSE data, Bull. Math. Biol., 73 (2011), 116-150. doi: 10.1007/s11538-010-9524-5.

[18]

H. T. Banks, K. L. Sutton, W. C. Thompson, G. Bocharov, M. Doumic, T. Schenkel, J. Argilaguet, S. Giest, C. Peligero and A. Meyerhans, A new model for the estimation of cell proliferation dynamics using CFSE data,, J. Immunological Methods, (). 

[19]

H. T. Banks and H. T. Tran, "Mathematical and Experimental Modeling of Physical and Biological Processes," With 1 CD-ROM (Windows, Macintosh and UNIX), Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2009.

[20]

H. T. Banks, H. T. Tran and D. E. Woodward, Estimation of variable coefficients in the Fokker-Planck equations using moving node finite elements, SIAM J. Numer. Anal., 30 (1993), 1574-1602. doi: 10.1137/0730082.

[21]

G. Casella and R. L. Berger, "Statistical Inference," The Wadsworth & Brooks/Cole Statistics/Probability Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990.

[22]

S. N. Ethier and T. G. Kurtz, "Markov Processes: Characterization and Convergence," Wiley Series in Probability and Statistics, New York, 1986.

[23]

L. Ferrante, S. Bompadre, L. Possati and L. Leone, Parameter estimation in a Gompertzian stochastic model for tumor growth, Biometrics, 56 (2000), 1076-1081. doi: 10.1111/j.0006-341X.2000.01076.x.

[24]

T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988.

[25]

G. W. Harrison, Numerical solution of the Fokker-Planck equation using moving finite elements, Numerical Methods for Partial Differential Equations, 4 (1988), 219-232. doi: 10.1002/num.1690040305.

[26]

I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991.

[27]

F. Klebaner, "Introduction to Stochastic Calculus with Applications," Second edition, Imperial College Press, London, 2005.

[28]

T. Luzyanina, D. Roose, T. Schenkel, M. Sester, S. Ehl, A. Meyerhans and G. Bocharov, Numerical modelling of label-structured cell population growth using CFSE distribution data, Theoretical Biology and Medical Modelling, 4 (2007), 1-26. doi: 10.1186/1742-4682-4-26.

[29]

J. A. J. Metz and O. Diekmann, eds., "The Dynamics of Physiologically Structured Populations," Papers from the colloquium held in Amsterdam, 1983, Lecture Notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986.

[30]

S. Michelson, K. Ito, H. T. Tran and J. T. Leith, Stochastic models for subpopulation emergence in heterogeneous tumors, Bulletin of Mathematical Biology, 51 (1989), 731-747.

[31]

A. Okubo, "Diffusion and Ecological Problems: Mathematical Models," An extended version of the Japanese edition, Ecology and Diffusion, Translated by G. N. Parker, Biomathematics, 10, Springer-Verlag, Berlin-New York, 1980.

[32]

J. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. doi: 10.2307/1934533.

[33]

T. T. Soong, "Random Differential Equations in Science and Engineering," Mathematics in Science and Engineering, Vol. 103, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973.

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