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Propagation of Growth Uncertainty in a Physiologically Structured Population

Published online by Cambridge University Press:  17 October 2012

H.T. Banks  and
S. Hu
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Abstract

In this review paper we consider physiologically structured population models that have been widely studied and employed in the literature to model the dynamics of a wide variety of populations. However in a number of cases these have been found inadequate to describe some phenomena arising in certain real-world applications such as dispersion in the structure variables due to growth uncertainty/variability. Prompted by this, we described two recent approaches that have been investigated in the literature to describe this growth uncertainty/variability in a physiologically structured population. One involves formulating growth as a Markov diffusion process while the other entails imposing a probabilistic structure on the set of possible growth rates across the entire population. Both approaches lead to physiologically structured population models with nontrivial dispersion. Even though these two approaches are conceptually quite different, they were found in [17] to have a close relationship: in some cases with properly chosen parameters and coefficient functions, the resulting stochastic processes have the same probability density function at each time.

Keywords

uncertaintystructured population modelsMarkov diffusion processesItô stochastic differential equationsFokker-Planck equations
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 5: Immunology , 2012 , pp. 7 - 23
Copyright
© EDP Sciences, 2012

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References

Références

Ackleh, A.S., Banks, H.T., Deng, K.. A finite difference approximation for a coupled system of nonlinear size-structured populations. Nonlinear Analysis, 50 (2002), 727–748. CrossRefGoogle Scholar
Ackleh, A.S., Ito, K.. An implicit finite-difference scheme for the nonlinear size-structured population model. Numer. Funct. Anal. Optim. 18 (1997), 65–884. CrossRefGoogle Scholar
Ackleh, A.S., Deng, K.. A monotone approximation for the nonautonomous size-structured population model. Quart. Appl. Math., 57 (1999), 261–267. CrossRefGoogle Scholar
Angulo, O. J.C. López-Marcos, Numerical integration of fully nonlinear size-structured population models. Applied Numerical Mathematics, 50 (2004), 291–327. CrossRefGoogle Scholar
H.T. Banks. A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering, CRC Press/Taylor and Frances Publishing, Boca Raton, FL, June, 2012, (258 pages).
Banks, H.T., Bihari, K.L.. Modelling and estimating uncertainty in parameter estimation. Inverse Problems, 17 (2001), 95–111. CrossRefGoogle Scholar
Banks, H.T., Bokil, V.A., Hu, S., Dhar, A.K., Bullis, R.A., Browdy, C.L., Allnutt, F.C.T.. Modeling shrimp biomass and viral infection for production of biological countermeasures CRSC-TR05-45, NCSU, December, 2005 ; Mathematical Biosciences and Engineering, 3 (2006), 635–660. Google ScholarPubMed
H.T. Banks, L.W. Botsford, F. Kappel, C. Wang. Modeling and estimation in size structured population models. LCDS-CCS Report 87-13, Brown University ; Proceedings 2nd Course on Mathematical Ecology, (Trieste, December 8-12, 1986) World Press, Singapore, 1988, 521–541.
Banks, H.T., Charles, F., Doumic, M., Sutton, K.L., Thompson, W.C.. Label structured cell proliferation models. Appl. Math. Letters, 23 (2010), 1412–1415. CrossRefGoogle ScholarPubMed
Banks, H.T., Davis, J.L.. A comparison of approximation methods for the estimation of probability distributions on parameters. CRSC-TR05-38, October, 2005 ; Applied Numerical Mathematics, 57 (2007), 753–777. CrossRefGoogle Scholar
Banks, H.T., Davis, J.L.. Quantifying uncertainty in the estimation of probability distributions. CRSC-TR07-21, December, 2007 ; Math. Biosci. Engr., 5 (2008), 647–667. CrossRefGoogle ScholarPubMed
Banks, H.T., Davis, J.L., Ernstberger, S.L., Hu, S., Artimovich, E., Dhar, A.K., Browdy, C.L.. A comparison of probabilistic and stochastic differential equations in modeling growth uncertainty and variability. CRSC-TR08-03, NCSU, February, 2008 ; Journal of Biological Dynamics, 3 (2009), 130–148. CrossRefGoogle Scholar
H.T. Banks, J.L. Davis, S.L. Ernstberger, S. Hu, E. Artimovich, A.K. Dhar. Experimental design and estimation of growth rate distributions in size-structured shrimp populations. CRSC-TR08-20, NCSU, November, 2008 ; Inverse Problems, 25 (2009), 095003(28pp).
H.T. Banks, J.L. Davis, S. Hu. A computational comparison of alternatives to including uncertainty in structured population models. CRSC-TR09-14, June, 2009 ; Three Decades of Progress in Systems and Control, X. Hu, U. Jonsson, B. Wahlberg, B. Ghosh (Eds.), Springer, 2010, 19–33.
Banks, H.T., Fitzpatrick, B.G.. Estimation of growth rate distributions in size structured population models. Quarterly of Applied Mathematics, 49 (1991), 215–235. CrossRefGoogle Scholar
H.T. Banks, B.G. Fitzpatrick, L.K. Potter, Y. Zhang. Estimation of probability distributions for individual parameters using aggregate population data. CRSC-TR98-6, NCSU, January, 1998 ; Stochastic Analysis, Control, Optimization and Applications, (Edited by W. McEneaney, G. Yin and Q. Zhang), Birkhauser, Boston, 1998, 353–371.
Banks, H.T., Hu, S.. Nonlinear stochastic Markov processes and modeling uncertainty in populations. CRSC-TR11-02, NCSU, January, 2011 ; Mathematical Bioscience and Engineering, 9 (2012), 1–25. Google ScholarPubMed
Banks, H.T., Kappel, F.. Transformation semigroups and L1-approximation for size- structured population models. Semigroup Forum, 38 (1989), 141–155. CrossRefGoogle Scholar
Banks, H.T., Kappel, F., Wang, C.. Weak solutions and differentiability for size-structured population models. International Series of Numerical Mathematics, 100 (1991), 35–50. CrossRefGoogle Scholar
Banks, H.T., Pinter, G.A.. A probabilistic multiscale approach to hysteresis in shear wave propagation in biotissue, CRSC-TR04-03, January, 2004 ; SIAM J. Multiscale Modeling and Simulation, 3 (2005), 395–412. CrossRefGoogle Scholar
Banks, H.T., Sutton, K.L., Thompson, W.C., Bocharov, G., Doumic, M., Schenkel, T., Argilaguet, J., Giest, S., Peligero, C., Meyerhans, A.. A new model for the estimation of cell proliferation dynamics using CFSE data. Center for Research in Scientific Computation Technical Report CRSC-TR11-05, NCSU, July, 2011 ; J. Immunological Methods, 373 (2011), 143–160. Google ScholarPubMed
Banks, H.T., Sutton, K.L., Thompson, W.C., Bocharov, G., Roose, D., Schenkel, T., Meyerhans, A.. Estimation of cell proliferation dynamics using CFSE data, CRSC-TR09-17, NCSU, August, 2009 ; Bull. Math. Biol. 70 (2011), 116–150; doi :10.1007/s11538-010-9524-5. CrossRefGoogle Scholar
H.T. Banks, W.C. Thompson. Mathematical models of dividing cell populations : Application to CFSE data. CRSC-TR12-10, N. C. State University, Raleigh, NC, April, 2012 ; Journal on Mathematical Modelling of Natural Phenomena, to appear.
Banks, H.T., Thompson, W.C.. A division-dependent compartmental model with cyton and intracellular label dynamics. CRSC-TR12-12, N. C. State University, Raleigh, NC, May, 2012 ; Intl. J. of Pure and Applied Math., 77 (2012), 119–147. Google Scholar
H.T. Banks, W.C. Thompson, C. Peligero, S. Giest, J. Argilaguet, A. Meyerhans. A compartmental model for computing cell numbers in CFSE-based lymphocyte proliferation assays, CRSC-TR12-03, NCSU, January, 2012 ; Math. Biosci. Engr., to appear.
Banks, H.T., Tran, H.T., Woodward, D.E.. Estimation of variable coefficients in the Fokker-Planck equations using moving node finite elements. SIAM J. Numer. Anal., 30 (1993), 1574–1602. CrossRefGoogle Scholar
Bell, G.I., Anderson, E.C.. Cell growth and division I. a mathematical model with applications to cell volume distributions in mammalian suspension cultures. Biophysical Journal, 7 (1967), 329–351. CrossRefGoogle ScholarPubMed
Calsina, A., Saldana, J.. A model of physiologically structured population dynamics with a nonlinear individual growth rate. Journal of Mathematical Biology, 33 (1995), 335–364. CrossRefGoogle Scholar
G. Casella, R.L. Berger. Statistical Inference. Duxbury, California, 2002.
Castille, F.L., Samocha, T.M., Lawrence, A.L., He, H., Frelier, P., Jaenike, F.. Variability in growth and survival of early postlarval shrimp (Penaeus vannamei Boone 1931) Aquaculture, 113 (1993), 65–81. CrossRefGoogle Scholar
Chu, J., Ducrot, A., Magal, P., Ruan, S.. Hopf bifurcation in a size-structured population dynamic model with random growth. J. Differential Equations, 247 (2009), 956–1000. CrossRefGoogle Scholar
J.M. Cushing. An Introduction to Structured Population Dynamics. CMB-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1998.
T.C. Gard. Introduction to Stochastic Differential Equations. Marcel Dekker, New York, 1988.
C.W. Gardiner. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer-Verlag, Berlin, 1983.
Gyllenberg, M., Webb, G.F.. A nonlinear structured population model of tumor growth with quiescence. J. Math. Biol., 28 (1990), 671–694. CrossRefGoogle ScholarPubMed
Harrison, G.W.. Numerical solution of the Fokker-Planck equation using moving finite elements. Numerical Methods for Partial Differential Equations, 4 (1988), 219–232. CrossRefGoogle Scholar
J. Hasenauer, D. Schittler, F. Allgöer. A computational model for proliferation dynamics of division- and label-structured populations. February, 2012, preprint.
Hasenauer, J., Waldherr, S., Doszczak, M., Scheurich, P., Radde, N., Allgöer, F.. Analysis of heterogeneous cell populations : a density-based modeling and identification framework. J. Process Control, 21 (2011), 1417–1425. Google Scholar
K. Huang. Statistical Mechanics. J. Wiley & Sons, New York, NY, 1963.
M. Iannelli. Mathematical Theory of Age-Structured Population Dynamics. Applied Math. Monographs, CNR, Giardini Editori e Stampatori, Pisa, 1995.
Kimura, M.. Process leading to quasi-fixation of genes in natural populations due to random fluctuation of selection intensities. Genetics, 39 (1954), 280–295. Google ScholarPubMed
F. Klebaner. Introduction to Stochastic Calculus with Applications. 2nd ed., Imperial College Press, London, 2006.
I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Second Edition, Springer, New York, 1991.
Luzyanina, T., Roose, D., Bocharov, G.. Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data. J. Math. Biol., 59 (2009), 581–603. CrossRefGoogle ScholarPubMed
Luzyanina, T., Mrusek, M., Edwards, J.T., Roose, D., Ehl, S., Bocharov, G.. Computational analysis of CFSE proliferation assay. J. Math. Biol., 54 (2007), 57–89. CrossRefGoogle ScholarPubMed
T. Luzyanina, D. Roose, T. Schenkel, M. Sester, S. Ehl, A. Meyerhans, G. Bocharov. Numerical modelling of label-structured cell population growth using CFSE distribution data. Theoretical Biology and Medical Modelling, 4(2007), Published Online.
McKendrick, A.G.. Applications of mathematics to medical problems. Proceedings of the Edinburgh Mathematical Society, 44 (1926), 98–130. CrossRefGoogle Scholar
J.A.J. Metz, E.O. Diekmann, The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics, Vol. 68, Springer, Heidelberg, 1986.
Moyal, J.E.. Stochastic processes and statistical physics. Journal of the Royal Statistical Society. Series B (Methodological), 11 (1949), 150–210. Google Scholar
Prohorov, Yu. V.. Convergence of random processes and limit theorems in probability theory. Theor. Prob. Appl., 1 (1956), 157–214. CrossRefGoogle Scholar
B. Oksendal. Stochastic Differentail Equations. 5th edition, Springer, Berlin, 2000.
A. Okubo. Diffusion and Ecological Problems : Mathematical Models. Biomathematics, 10 (1980), Springer-Verlag, Berlin.
Oster, G., Takahashi, Y.. Models for age-specific interactions in a periodic environment. Ecological Monographs, 44 (1974), 483–501. CrossRefGoogle Scholar
B. Perthame. Transport Equations in Biology. Birkhauser Verlag, Basel, 2007.
R. Rudnicki. Models of population dynamics and genetics. From Genetics To Mathematics, (edited by M. Lachowicz and J. Miekisz), World Scientific, Singapore, 2009, 103–148.
H. Risken. The Fokker-Planck Equation : Methods of Solution and Applications. Springer, New York, 1996.
D. Schittler, J. Hasenauer, F. Allgöer. A generalized population model for cell proliferation : Integrating division numbers and label dynamics. Proceedings of Eighth International Workshop on Computational Systems Biology (WCSB 2011), June 2011, Zurich, Switzerland, 165–168.
D. Schittler, J. Hasenauer, F. Allgöer. A model for proliferating cell populations that accounts for cell types. Proc. of 9th International Workshop on Computational Systems Biology, 2012, 84–87.
Sinko, J., Streifer, W.. A new model for age-size structure of a population. Ecology, 48 (1967), 910–918. CrossRefGoogle Scholar
T.T. Soong. Random Differential Equations in Science and Engineering. Academic Press, New York and London, 1973.
Soong, T.T., Chuang, S.N.. Solutions of a class of random differential equations. SIAM J. Appl. Math., 24 (1973), 449–459. CrossRefGoogle Scholar
W.C. Thompson. Partial Differential Equation Modeling of Flow Cytometry Data from CFSE-based Proliferation Assays. Ph.D. Dissertation, North Carolina State University, December, 2011.
H. Von Foerster. Some remarks on changing populations. The Kinetics of Cellular Proliferation, F. Stohlman, Jr. (ed.), Grune and Stratton, New York, 1959.
G.F. Webb. Theory of Nonlinear Age-dependent Population Dynamics. Marcel Dekker, New York, 1985.
Weiße, A.Y., Middleton, R.H., Huisinga, W.. Quantifying uncertainty, variability and likelihood for ordinary differential equation models. BMC Syst. Bio., 4 (144), 2010. Google ScholarPubMed
Weiss, G.H. Equation for the age structure of growing populations. Bull. Math. Biophy., 30 (1968), 427–435. CrossRefGoogle Scholar
http://en.wikipedia.org/wiki/Probability_density_function.