Abstract
Diffusion processes enable realistic and convenient modelling of dynamic systems. They typically arise as approximations of exact but computationally expensive individual-based stochastic models. However, the correct derivation of an appropriate diffusion approximation is often complicated, and hence their utilisation is not widely spread in the applied sciences. Instead, practitioners often favour rather unrealistic deterministic models and their relatively simple analysis. This chapter motivates the application of diffusion approximations and explains their correct derivation. It reviews and develops different approaches and points out differences and correspondences between them. All methods are formulated for multi-dimensional processes and extended to an even more general framework where systems are characterised by multiple size parameters. The chapter addresses mathematicians who are interested in the theory of diffusion approximations and practitioners who wish to apply diffusion models for their specific problems.
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References
Allen L (2003) An introduction to stochastic processes with applications to biology. Pearson Prentice Hall, Upper Saddle River
Alonso D, McKane A, Pascual M (2007) Stochastic amplification in epidemics. J R Soc Interface 4:575–582
Anderson R, May R (1991) Infectious diseases of humans. Oxford University Press, Oxford
Andersson H, Britton T (2000) Stochastic epidemic models and their statistical analysis. Lecture notes in statistics, vol 151. Springer, New York
Arnold L (1973) Stochastische Differentialgleichungen. Oldenbourg, München
Bailey N (1975) The mathematical theory of infectious diseases, 2nd edn. Charles Griffin, London
Barbour A (1972) The principle of the diffusion of arbitrary constants. J Appl Probab 9:519–541
Barbour A (1974) On a functional central limit theorem for Markov population processes. Adv Appl Probab 6:21–39
Barbour A (1975a) The asymptotic behaviour of birth and death and some related processes. Adv Appl Probab 7:28–43
Barbour A (1975b) The duration of a closed stochastic epidemic. Biometrika 62:477–482
Barbour A (1975c) A note on the maximum size of a closed epidemic. J R Stat Soc Ser B 37:459–460
Billingsley P (1968) Convergence of probability measures. Wiley, New York
Bouchaud JP, Cont R (1998) A Langevin approach to stock market fluctuations and crashes. Eur Phys J B 6:543–550
Braumann C (2007) Itô versus Stratonovich calculus in random population growth. Math Biosci 206:81–107
Busenberg S, Martelli M (1990) Differential equations models in biology, epidemiology and ecology. Lecture notes in biomathematics, vol 92. Springer, Berlin
Capasso V, Morale D (2009) Stochastic modelling of tumour-induced angiogenesis. J Math Biol 58:219–233
Chaturvedi S, Gardiner C (1978) The Poisson representation. II. Two-time correlation functions. J Stat Phys 18:501–522
Chen WY, Bokka S (2005) Stochastic modeling of nonlinear epidemiology. J Theor Biol 234:455–470
Clancy D, French N (2001) A stochastic model for disease transmission in a managed herd, motivated by Neospora caninum amongst dairy cattle. Math Biosci 170:113–132
Clancy D, O’Neill P, Pollett P (2001) Approximations for the long-term behavior of an open-population epidemic model. Methodol Comput Appl Probab 3:75–95
Daley D, Gani J (1999) Epidemic modelling: an introduction. Cambridge studies in mathematical biology, vol 15. Cambridge University Press, Cambridge
Daley D, Kendall D (1965) Stochastic rumours. IMA J Appl Math 1:42–55
Daniels H (1974) The maximum size of a closed epidemic. Adv Appl Probab 6:607–621
de la Lama M, Szendro I, Iglesias J, Wio H (2006) Van Kampen’s expansion approach in an opinion formation model. Eur Phys J B 51:435–442
Drummond P, Gardiner C, Walls D (1981) Quasiprobability methods for nonlinear chemical and optical systems. Phys Rev A 24:914–926
Eigen M (1971) Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58:465–523
Elf J, Ehrenberg M (2003) Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Res 13:2475–2484
Ethier S, Kurtz T (1986) Markov processes. Characterization and convergence. Wiley, New York
Ewens W (1963) Numerical results and diffusion approximations in a genetic process. Biometrika 50:241–249
Feller W (1951) Diffusion processes in genetics. In: Proceedings of the second Berkeley symposium on mathematical statistics and probability. University of California Press, Berkeley, pp 227–246
Ferm L, Lötstedt P, Hellander A (2008) A hierarchy of approximations of the master equation scaled by a size parameter. J Sci Comput 34:127–151
Gardiner C (1983) Handbook of stochastic methods. Springer, Berlin/Heidelberg
Gardiner C, Chaturvedi S (1977) The Poisson representation. I. A new technique for chemical master equations. J Stat Phys 17:429–468
Gibson M, Mjolsness E (2001) Modeling of the activity of single genes. In: Bolouri H, Bower J (eds) Computational modeling of genetic and biochemical networks. Lecture notes in computer science, vol 4699. MIT, Cambridge, pp 1–48
Gillespie D (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22:403–434
Gillespie D (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361
Gillespie D (1980) Approximating the master equation by Fokker-Planck-type equations for single-variable chemical systems. J Chem Phys 72:5363–5370
Gitterman M, Weiss G (1991) Some comments on approximations to the master equation. Phys A 170:503–510
Goel N, Richter-Dyn N (1974) Stochastic models in biology. Academic, New York
Golightly A, Wilkinson D (2005) Bayesian inference for stochastic kinetic models using a diffusion approximation. Biometrics 61:781–788
Golightly A, Wilkinson D (2006) Bayesian sequential inference for stochastic kinetic biochemical network models. J Comput Biol 13:838–851
Golightly A, Wilkinson D (2008) Bayesian inference for nonlinear multivariate diffusion models observed with error. Comput Stat Data Anal 52:1674–1693
Golightly A, Wilkinson D (2010) Markov chain Monte Carlo algorithms for SDE parameter estimation. In: Lawrence N, Girolami M, Rattray M, Sanguinetti G (eds) Introduction to learning and inference for computational systems biology. MIT, Cambridge, pp 253–275
Grabert H, Green M (1979) Fluctuations and nonlinear irreversible processes. Phys Rev A 19:1747–1756
Grabert H, Graham R, Green M (1980) Fluctuations and nonlinear irreversible processes II. Phys Rev A 21:2136–2146
Grabert H, Hänggi P, Oppenheim I (1983) Fluctuations in reversible chemical reactions. Phys A 117:300–316
Grasman J, Ludwig D (1983) The accuracy of the diffusion approximation to the expected time to extinction for some discrete stochastic processes. J Appl Probab 20:305–321
Green M (1952) Markoff random processes and the statistical mechanics of time-dependent phenomena. J Chem Phys 20:1281–1295
Guess H, Gillespie J (1977) Diffusion approximations to linear stochastic difference equations with stationary coefficients. J Appl Probab 14:58–74
Hänggi P (1982) Nonlinear fluctuations: the problem of deterministic limit and reconstruction of stochastic dynamics. Phys Rev A 25:1130–1136
Hänggi P, Jung P (1988) Bistability in active circuits: application of a novel Fokker-Planck approach. IBM J Res Dev 32:119–126
Hänggi P, Grabert H, Talkner P, Thomas H (1984) Bistable systems: master equation versus Fokker-Planck modeling. Phys Rev A 29:371–378
Haskey H (1954) A general expression for the mean in a simple stochastic epidemic. Biometrika 41:272–275
Hayot F, Jayaprakash C (2004) The linear noise approximation for molecular fluctuations within cells. Phys Biol 1:205–210
Horsthemke W, Brenig L (1977) Non-linear Fokker-Planck equation as an asymptotic representation of the master equation. Z Phys B 27:341–348
Horsthemke W, Lefever R (1984) Noise-induced transitions: theory and applications in physics, chemistry, and biology. Springer, Berlin
Hsu JP, Wang HH (1987) Kinetics of bacterial adhesion – a stochastic analysis. J Theor Biol 124:405–413
Hufnagel L, Brockmann D, Geisel T (2004) Forecast and control of epidemics in a globalized world. Proc Natl Acad Sci USA 101:15124–15129
Karth M, Peinke J (2003) Stochastic modeling of fat-tailed probabilities of foreign exchange rates. Complexity 8:34–42
Keeling M, Rohani P (2008) Modeling infectious disease in humans and animals. Princeton University Press, Princeton
Kepler T, Elston T (2001) Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations. Biophys J 81:3116–3136
Kishida K, Kanemoto S, Sekiya T (1976) Reactor noise theory based on system size expansion. J Nucl Sci Technol 13:19–29
Kleinhans D, Friedrich R, Nawroth A, Peinke J (2005) An iterative procedure for the estimation of drift and diffusion coefficients of Langevin processes. Phys Lett A 346:42–46
Kloeden P, Platen E (1999) Numerical solution of stochastic differential equations, 3rd edn. Springer, Berlin/Heidelberg/New York
Kramers H (1940) Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7:284–304
Kubo R, Matsuo K, Kitahara K (1973) Fluctuation and relaxation of macrovariables. J Stat Phys 9:51–96
Kurtz T (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J Appl Probab 7:49–58
Kurtz T (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J Appl Probab 8:344–356
Kurtz T (1981) Approximation of population processes. Society for Industrial and Applied Mathematics, Philadelphia
Lande R, Engen S, Sæther B (2003) Stochastic population dynamics in ecology and conservation. Oxford University Press, New York
Leung H (1985) Expansion of the master equation for a biomolecular selection model. Bull Math Biol 47:231–238
McKane AJ, Newman T (2004) Stochastic models in population biology and their deterministic analogs. Phys Rev E 70:041902
McNeil D (1973) Diffusion limits for congestion models. J Appl Probab 10:368–376
McQuarrie D (1967) Stochastic approach to chemical kinetics. J Appl Probab 4:413–478
Moyal J (1949) Stochastic processes and statistical physics. J R Stat Soc Ser B 11:150–210
Muñoz M, Garrido P (1994) Fokker-Planck equation for nonequilibrium competing dynamic models. Phys Rev E 50:2458–2466
Naert A, Friedrich R, Peinke J (1997) Fokker-Planck equation for the energy cascade in turbulence. Phys Rev E 56:6719–6722
Nåsell I (2002) Stochastic models of some endemic infections. Math Biosci 179:1–19
Norman M (1974) A central limit theorem for Markov processes that move by small steps. Ann Probab 2:1065–1074
Norman M (1975) Diffusion approximation of non-Markovian processes. Ann Probab 3:358–364
Ohkubo J (2008) Approximation scheme for master equations: variational approach to multivariate case. J Chem Phys 129:044108
Paulsson J (2004) Summing up the noise in gene networks. Nature 427:415–418
Pawula R (1967a) Approximation of the linear Boltzmann equation by the Fokker-Planck equation. Phys Rev 162:186–188
Pawula R (1967b) Generalizations and extensions of the Fokker-Planck-Kolmogorov equations. IEEE Trans Inf Theory 13:33–41
Pielou (1969) An introduction to mathematical ecology. Wiley, New York
Pierobon P, Parmeggiani A, von Oppen F, Frey E (2005) Dynamic correlation functions and Boltzmann Langevin approach for a driven one dimensional lattice gas. Phys Rev E 72:036123
Pollard D (1984) Convergence of stochastic processes. Springer, New York
Pollett P (1990) On a model for interference between searching insect parasites. J Aust Math Soc Ser B 32:133–150
Pollett P (2001) Diffusion approximations for ecological models. Proceedings of the international congress of modelling and simulation, Australian National University, Canberra
Ramshaw J (1985) Augmented Langevin approach to fluctuations in nonlinear irreversible processes. J Stat Phys 38:669–680
Rao C, Wolf D, Arkin A (2002) Control, exploitation and tolerance of intracellular noise. Nature 420:231–237
Renshaw E (1991) Modelling biological populations in space and time. Cambridge University Press, Cambridge
Risken H (1984) The Fokker-Planck equation. Springer, Berlin
Risken H, Vollmer H (1987) On solutions of truncated Kramers-Moyal expansions; continuum approximations to the Poisson process. Condens Matter 66:257–262
Robertson S, Pilling M, Green N (1996) Diffusion approximations of the two-dimensional master equation. Mol Phys 88:1541–1561
Sancho J, San Miguel M (1984) Unified theory of internal and external fluctuations. In: Casas-Vázquez J, Jou D, Lebon G (eds) Recent developments in nonequilibrium thermodynamics. Lecture notes in physics, vol 199. Springer, Berlin, pp 337–352
Seifert U (2008) Stochastic thermodynamics: principles and perspectives. Eur Phys J B 64:423–431
Shizgal B, Barrett J (1989) Time dependent nucleation. J Chem Phys 91:6505–6518
Sjöberg P, Lötstedt P, Elf J (2009) Fokker-Planck approximation of the master equation in molecular biology. Comput Vis Sci 12:37–50
Song X, Wang H, van Voorhis T (2008) A Langevin equation approach to electron transfer reactions in the diabatic basis. J Chem Phys 129:144502
Strumik M, Macek W (2008) Statistical analysis of transfer of fluctuations in solar wind turbulence. Nonlinear Proc Geophys 15:607–613
Tian T, Burrage K, Burrage P, Carletti M (2007) Stochastic delay differential equations for genetic regulatory networks. J Comput Appl Math 205:696–707
van Kampen N (1961) A power series expansion of the master equation. Can J Phys 39:551–567
van Kampen N (1965) Fluctuations in nonlinear systems. In: Burgess R (ed) Fluctuation phenomena in solids. Academic, New York, pp 139–177
van Kampen N (1981a) Itô versus Stratonovich. J Stat Phys 24:175–187
van Kampen N (1981b) The validity of nonlinear Langevin equations. J Stat Phys 25:431–442
van Kampen N (1997) Stochastic processes in physics and chemistry, 2nd edn. Elsevier, Amsterdam
Walsh J (1981) Well-timed diffusion approximations. Adv Appl Probab 13:352–368
Wong E, Zakai M (1965) On the convergence of ordinary integrals to stochastic integrals. Ann Math Stat 36:1560–1564
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Fuchs, C. (2013). Approximation of Markov Jump Processes by Diffusions. In: Inference for Diffusion Processes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25969-2_4
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