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

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

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

András Bátkai ^1, , Istvan Z. Kiss ^2, , Eszter Sikolya ^1, and Péter L. Simon ^1,

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

Abstract
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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.

Keywords: one-parameter operator semigroup, birth-and-death process, mean field approximation., Dynamic network.

Mathematics Subject Classification: Primary: 47D06; Secondary: 60J28, 92A1.

Citation: András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks and 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), 175-193. 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), 69-87. 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), 613-636. 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, Texts in Applied Mathematics, 34, Springer, New York, 2006.
[6]	R. W. R. Darling and J. R. Norris, Differential equation approximations for Markov chains, Probab. Surv., 5 (2008), 37-79. doi: 10.1214/07-PS121.
[7]	K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Math., 194, Springer-Verlag, New York, 2000.
[8]	S. N. Ethier and T. G. Kurtz, "Markov Processes: Characterization and Convergence," John Wiley & Sons Ltd, USA, 2005.
[9]	G. Grimmett and D. Stirzaker, "Probability and Random Processes," Third edition, Oxford University Press, New York, 2001.
[10]	T. Gross and B. Blasius, Adaptive coevolutionary networks: A review, J. Roy. Soc. Interface, 5 (2008), 259-271. 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), 67-73. 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), 1-15. 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, to appear.
[14]	T. G. Kurtz, Extensions of Trotter's operator semigroup approximation theorems, J. Functional Analysis, 3 (1969), 354-375. 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), 49-58. 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), 143-164. 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., Prob., submitted.
[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), 479-508. 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), 175-193. 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), 69-87. 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), 613-636. 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, Texts in Applied Mathematics, 34, Springer, New York, 2006.
[6]	R. W. R. Darling and J. R. Norris, Differential equation approximations for Markov chains, Probab. Surv., 5 (2008), 37-79. doi: 10.1214/07-PS121.
[7]	K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Math., 194, Springer-Verlag, New York, 2000.
[8]	S. N. Ethier and T. G. Kurtz, "Markov Processes: Characterization and Convergence," John Wiley & Sons Ltd, USA, 2005.
[9]	G. Grimmett and D. Stirzaker, "Probability and Random Processes," Third edition, Oxford University Press, New York, 2001.
[10]	T. Gross and B. Blasius, Adaptive coevolutionary networks: A review, J. Roy. Soc. Interface, 5 (2008), 259-271. 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), 67-73. 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), 1-15. 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, to appear.
[14]	T. G. Kurtz, Extensions of Trotter's operator semigroup approximation theorems, J. Functional Analysis, 3 (1969), 354-375. 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), 49-58. 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), 143-164. 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., Prob., submitted.
[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), 479-508. doi: 10.1007/s00285-010-0344-x.

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