-
Previous Article
A sufficient condition for classified networks to possess complex network features
- NHM Home
- This Issue
-
Next Article
From the Newton equation to the wave equation in some simple cases
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 |
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. |
[1] |
Jacek Banasiak, Marcin Moszyński. Dynamics of birth-and-death processes with proliferation - stability and chaos. Discrete and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123 |
[8] |
Marco Di Francesco, Serikbolsyn Duisembay, Diogo Aguiar Gomes, Ricardo Ribeiro. Particle approximation of one-dimensional Mean-Field-Games with local interactions. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3569-3591. doi: 10.3934/dcds.2022025 |
[9] |
Jun Hu, Oleg Muzician, Yingqing Xiao. Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3189-3221. doi: 10.3934/dcds.2018139 |
[10] |
Sung-Seok Ko. A nonhomogeneous quasi-birth-death process approach for an $ (s, S) $ policy for a perishable inventory system with retrial demands. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1415-1433. doi: 10.3934/jimo.2019009 |
[11] |
Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080 |
[12] |
Francesco Sanna Passino, Nicholas A. Heard. Modelling dynamic network evolution as a Pitman-Yor process. Foundations of Data Science, 2019, 1 (3) : 293-306. doi: 10.3934/fods.2019013 |
[13] |
Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks and Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263 |
[14] |
Zhenquan Zhang, Meiling Chen, Jiajun Zhang, Tianshou Zhou. Analysis of non-Markovian effects in generalized birth-death models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3717-3735. doi: 10.3934/dcdsb.2020254 |
[15] |
Tigran Bakaryan, Rita Ferreira, Diogo Gomes. A potential approach for planning mean-field games in one dimension. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2147-2187. doi: 10.3934/cpaa.2022054 |
[16] |
Qianqian Wang, Minan Tang, Aimin An, Jiawei Lu, Yingying Zhao. Parameter optimal identification and dynamic behavior analysis of nonlinear model for the solution purification process of zinc hydrometallurgy. Journal of Industrial and Management Optimization, 2022, 18 (1) : 693-712. doi: 10.3934/jimo.2021159 |
[17] |
Joachim Crevat. Mean-field limit of a spatially-extended FitzHugh-Nagumo neural network. Kinetic and Related Models, 2019, 12 (6) : 1329-1358. doi: 10.3934/krm.2019052 |
[18] |
Fabio Bagagiolo, Rosario Maggistro, Raffaele Pesenti. Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach. Journal of Dynamics and Games, 2021, 8 (4) : 359-380. doi: 10.3934/jdg.2021007 |
[19] |
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 |
[20] |
Diogo Gomes, Marc Sedjro. One-dimensional, forward-forward mean-field games with congestion. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 901-914. doi: 10.3934/dcdss.2018054 |
2020 Impact Factor: 1.213
Tools
Metrics
Other articles
by authors
[Back to Top]