2012, 7(1): 43-58. doi: 10.3934/nhm.2012.7.43

Differential equation approximations of stochastic network processes: An operator semigroup approach

1. 

Loránd Eötvös University, Institute of Mathematics, Pázmány Péter Sétány 1C, H-1117 Budapest, Hungary, Hungary, Hungary

2. 

School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF, United Kingdom

Received  July 2011 Revised  January 2012 Published  February 2012

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its master equation, which is a system of linear ODEs with large state space size ($N$). We derive a single non-linear ODE (called mean-field approximation) for the expected value that yields a good approximation as $N$ tends to infinity. Using only elementary semigroup theory we can prove the order $\mathcal{O}(1/N)$ convergence of the solution of the system to that of the mean-field equation. The proof holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach using a countable system of ODEs for proving convergence to the mean-field limit is proposed.
Citation: András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks & Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43
References:
[1]

J. Banasiak, M. Lachowicz and M. Moszyński, Semigroups for generalized birth-and-death equations in $ \l^p$ spaces,, Semigroup Forum, 73 (2006), 175. doi: 10.1007/s00233-006-0621-x.

[2]

F. Ball and P. Neal, Network epidemic models with two levels of mixing,, Math. Biosci., 212 (2008), 69. doi: 10.1016/j.mbs.2008.01.001.

[3]

A. Bátkai, P. Csomós and G. Nickel, Operator splittings and spatial approximations for evolution equations,, J. Evol. Equ., 9 (2009), 613. doi: 10.1007/s00028-009-0026-6.

[4]

A. Bobrowski, "Functional Analysis for Probability and Stochastic Processes. An Introduction,", Cambridge, (2005).

[5]

C. Chicone, "Ordinary Differential Equations with Applications,", Second edition, 34 (2006).

[6]

R. W. R. Darling and J. R. Norris, Differential equation approximations for Markov chains,, Probab. Surv., 5 (2008), 37. doi: 10.1214/07-PS121.

[7]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Graduate Texts in Math., 194 (2000).

[8]

S. N. Ethier and T. G. Kurtz, "Markov Processes: Characterization and Convergence,", John Wiley & Sons Ltd, (2005).

[9]

G. Grimmett and D. Stirzaker, "Probability and Random Processes,", Third edition, (2001).

[10]

T. Gross and B. Blasius, Adaptive coevolutionary networks: A review,, J. Roy. Soc. Interface, 5 (2008), 259. doi: 10.1098/rsif.2007.1229.

[11]

T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks,, J. R. Soc. Interface, 8 (2011), 67. doi: 10.1098/rsif.2010.0179.

[12]

T. Kato, On the semi-groups generated by Kolmogoroff's differential equations,, J. Math. Soc. Japan, 6 (1954), 1. doi: 10.2969/jmsj/00610001.

[13]

I. Z. Kiss, L. Berthouze, T. J. Taylor and P. L. Simon, Modelling approaches for simple dynamic networks and applications to disease transmission models,, Proc. Roy. Soc. A, ().

[14]

T. G. Kurtz, Extensions of Trotter's operator semigroup approximation theorems,, J. Functional Analysis, 3 (1969), 354. doi: 10.1016/0022-1236(69)90031-7.

[15]

T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes,, J. Appl. Prob., 7 (1970), 49. doi: 10.2307/3212147.

[16]

J. Lindquist, J. Ma, P. van den Driessche and F. H. Willeboordse, Effective degree network disease models,, J. Math. Biol., 62 (2011), 143. doi: 10.1007/s00285-010-0331-2.

[17]

R. McVinish and P. K. Pollett, The deterministic limit of heterogeneous density dependent Markov chains,, Ann. Appl., ().

[18]

P. L. Simon and I. Z. Kiss, From exact stochastic to mean-field ODE models: A case study of three different approaches to prove convergence results,, to appear., ().

[19]

P. L. Simon, M. Taylor and I. Z. Kiss, Exact epidemic models on graphs using graph-automorphism driven lumping,, J. Math. Biol., 62 (2011), 479. doi: 10.1007/s00285-010-0344-x.

show all references

References:
[1]

J. Banasiak, M. Lachowicz and M. Moszyński, Semigroups for generalized birth-and-death equations in $ \l^p$ spaces,, Semigroup Forum, 73 (2006), 175. doi: 10.1007/s00233-006-0621-x.

[2]

F. Ball and P. Neal, Network epidemic models with two levels of mixing,, Math. Biosci., 212 (2008), 69. doi: 10.1016/j.mbs.2008.01.001.

[3]

A. Bátkai, P. Csomós and G. Nickel, Operator splittings and spatial approximations for evolution equations,, J. Evol. Equ., 9 (2009), 613. doi: 10.1007/s00028-009-0026-6.

[4]

A. Bobrowski, "Functional Analysis for Probability and Stochastic Processes. An Introduction,", Cambridge, (2005).

[5]

C. Chicone, "Ordinary Differential Equations with Applications,", Second edition, 34 (2006).

[6]

R. W. R. Darling and J. R. Norris, Differential equation approximations for Markov chains,, Probab. Surv., 5 (2008), 37. doi: 10.1214/07-PS121.

[7]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Graduate Texts in Math., 194 (2000).

[8]

S. N. Ethier and T. G. Kurtz, "Markov Processes: Characterization and Convergence,", John Wiley & Sons Ltd, (2005).

[9]

G. Grimmett and D. Stirzaker, "Probability and Random Processes,", Third edition, (2001).

[10]

T. Gross and B. Blasius, Adaptive coevolutionary networks: A review,, J. Roy. Soc. Interface, 5 (2008), 259. doi: 10.1098/rsif.2007.1229.

[11]

T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks,, J. R. Soc. Interface, 8 (2011), 67. doi: 10.1098/rsif.2010.0179.

[12]

T. Kato, On the semi-groups generated by Kolmogoroff's differential equations,, J. Math. Soc. Japan, 6 (1954), 1. doi: 10.2969/jmsj/00610001.

[13]

I. Z. Kiss, L. Berthouze, T. J. Taylor and P. L. Simon, Modelling approaches for simple dynamic networks and applications to disease transmission models,, Proc. Roy. Soc. A, ().

[14]

T. G. Kurtz, Extensions of Trotter's operator semigroup approximation theorems,, J. Functional Analysis, 3 (1969), 354. doi: 10.1016/0022-1236(69)90031-7.

[15]

T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes,, J. Appl. Prob., 7 (1970), 49. doi: 10.2307/3212147.

[16]

J. Lindquist, J. Ma, P. van den Driessche and F. H. Willeboordse, Effective degree network disease models,, J. Math. Biol., 62 (2011), 143. doi: 10.1007/s00285-010-0331-2.

[17]

R. McVinish and P. K. Pollett, The deterministic limit of heterogeneous density dependent Markov chains,, Ann. Appl., ().

[18]

P. L. Simon and I. Z. Kiss, From exact stochastic to mean-field ODE models: A case study of three different approaches to prove convergence results,, to appear., ().

[19]

P. L. Simon, M. Taylor and I. Z. Kiss, Exact epidemic models on graphs using graph-automorphism driven lumping,, J. Math. Biol., 62 (2011), 479. doi: 10.1007/s00285-010-0344-x.

[1]

Jacek Banasiak, Marcin Moszyński. Dynamics of birth-and-death processes with proliferation - stability and chaos. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 67-79. doi: 10.3934/dcds.2011.29.67

[2]

Jacek Banasiak, Mustapha Mokhtar-Kharroubi. Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 529-542. doi: 10.3934/dcdsb.2005.5.529

[3]

Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475

[4]

Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure & Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457

[5]

Manuela Giampieri, Stefano Isola. A one-parameter family of analytic Markov maps with an intermittency transition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 115-136. doi: 10.3934/dcds.2005.12.115

[6]

Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877

[7]

Stephen C. Preston, Alejandro Sarria. One-parameter solutions of the Euler-Arnold equation on the contactomorphism group. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123

[8]

Jun Hu, Oleg Muzician, Yingqing Xiao. Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3189-3221. doi: 10.3934/dcds.2018139

[9]

Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks & Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263

[10]

Hilla Behar, Alexandra Agranovich, Yoram Louzoun. Diffusion rate determines balance between extinction and proliferation in birth-death processes. Mathematical Biosciences & Engineering, 2013, 10 (3) : 523-550. doi: 10.3934/mbe.2013.10.523

[11]

Jacques Henry. For which objective is birth process an optimal feedback in age structured population dynamics?. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 107-114. doi: 10.3934/dcdsb.2007.8.107

[12]

Xianlong Fu, Dongmei Zhu. Stability results for a size-structured population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 109-131. doi: 10.3934/dcdsb.2013.18.109

[13]

Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735

[14]

Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086

[15]

Stephen Thompson, Thomas I. Seidman. Approximation of a semigroup model of anomalous diffusion in a bounded set. Evolution Equations & Control Theory, 2013, 2 (1) : 173-192. doi: 10.3934/eect.2013.2.173

[16]

Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279

[17]

Dongxue Yan, Xianlong Fu. Asymptotic analysis of a spatially and size-structured population model with delayed birth process. Communications on Pure & Applied Analysis, 2016, 15 (2) : 637-655. doi: 10.3934/cpaa.2016.15.637

[18]

Dongxue Yan, Yu Cao, Xianlong Fu. Asymptotic analysis of a size-structured cannibalism population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1975-1998. doi: 10.3934/dcdsb.2016032

[19]

Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003

[20]

Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235

2016 Impact Factor: 1.2

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (5)

[Back to Top]