nach oben

Erschienen in:

2020 | OriginalPaper | Buchkapitel

Deterministic and Stochastic Mean-Field SIRS Models on Heterogeneous Networks

verfasst von : Stefano Bonaccorsi, Silvia Turri

Erschienen in: Discrete and Continuous Models in the Theory of Networks

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this paper we study a model for the spread of an SIRS-type epidemics on a network, both in a deterministic setting and under the presence of a random environment, that enters in the definition of the infection rates of the nodes. Accordingly, we model the infection rates in the form of independent stochastic processes. To analyze the problem, we apply a mean field approximation that is known in the literature as NIMFA model, which allows to get a differential equation for the probability of infection in each node. We discover a sufficient condition which guarantees the extinction of the epidemics both in the deterministic and in the stochastic setting.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Many-Particle Quantum Graphs: A Review

Nächstes Kapitel Kreı̆n Formula and Convergence of Hamiltonians with Scaled Potentials in Dimension One

In demography, this quantity is referred to as the basic reproduction number, [7], and in classical SIS models without spatial structure the critical threshold is equal to 1

L.J.Allen, An introduction to stochastic epidemic models, Mathematical Epidemiology, Springer, 2008, pp. 81–130MATH

R.M. Anderson and R.M. May, Infectious Diseases in Humans: Dynamics and Control. Oxford University Press, Oxford, 1991.

Ludwig Arnold, Stochastic differential equations: theory and applications. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

S. Bansal, B.T. Grenfell and L.A. Meyers, When individual behaviour matters: homogeneous and network models in epidemiology, Journal of the Royal Society Interface, 4, 879–891 (2007)CrossRef

Luís M.A. Bettencourt, Ariel Cintrón-Arias, David I. Kaiser and Carlos Castillo-Chávez, The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models, Physica A: Statistical Mechanics and its Applications 364, 513–536, (2006)CrossRef

Stefano Bonaccorsi and Stefania Ottaviano, Epidemics on networks with heterogeneous population and stochastic infection rates, Mathematical Biosciences, 279, 43–52 (2016). https://doi.org/10.1016/j.mbs.2016.07.002 MathSciNetCrossRef

Tom Britton, Stochastic epidemic models: a survey, arXiv:0910.4443, (2009)

W. Hethcote, Qualitative analyses of communicable disease models, Mathematical Biosciences 28(3–4), 335–356 (1976). https://doi.org/10.1016/0025-5564(76)90132-2 MathSciNetCrossRef

L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differential Equations, 217, 26–53 (2005)MathSciNetCrossRef

10.

Daqing Jiang, Jiajia Yu, Chunyan Ji, and Ningzhong Shi, Asymptotic behavior of global positive solution to a stochastic SIR model, Mathematical and Computer Modelling 54(1–2), 221–232 (2011). https://doi.org/10.1016/j.mcm.2011.02.004 MathSciNetCrossRef

11.

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115, 700–721 (1927)CrossRef

12.

A. Lahrouz, L. Omari, and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Analysis: Modelling and Control 16(1), 59–76 (2011)MathSciNetCrossRef

13.

Cong Li, Ruud van de Bovenkamp, and Piet Van Mieghem, Susceptible-infected-susceptible model: A comparison of N-intertwined and heterogeneous mean-field approximations, Physical Review E 86, 026116 (2012)

14.

Chun-Hsien Li, Chiung-Chiou Tsai and Suh-Yuh Yang, Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks, Communications in Nonlinear Science and Numerical Simulation, 19(4), 1042–1054 (2014). https://doi.org/10.1016/j.cnsns.2013.08.033 MathSciNetCrossRef

15.

Dan Li, Shengqiang Liu, and Jing’an Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, Journal of Differential Equations, 263(12), 8873–8915 (2017). https://doi.org/10.1016/j.jde.2017.08.066 MathSciNetCrossRef

16.

Qiuying Lu, Stability of SIRS system with random perturbations, Physica A: Statistical Mechanics and its Applications 388(18), 3677–3686 (2009). https://doi.org/10.1016/j.physa.2009.05.036 MathSciNetCrossRef

17.

Mao, X., Stochastic Differential Equations and Applications, 2nd Edition, Horwood Publishing, Chichester, 2007.

18.

R. Pastor-Satorras, A. Vespignani Epidemic dynamics in scale-free networks, Phys Rev Lett, 86, 3200 (2001)CrossRef

19.

F. Sánchez, X. Wang, C. Castillo-Chávez, D. M. Gorman and P. J. Gruenewald, Drinking as an Epidemic-A Simple Mathematical Model with Recovery and Relapse, in: Therapist’s Guide to Evidence-Based Relapse Prevention. Elsevier Inc., 353–368 (2007). https://doi.org/10.1016/B978-012369429-4/50046-X

20.

P. Van Mieghem, J., Omic and R., Kooij, Virus Spread in Networks, IEEE/ACM Transactions on Networking 17(1), 1–14 (2009). https://doi.org/10.1109/TNET.2008.925623

21.

Piet Van Mieghem and Jasmina Omic, In-homogeneous Virus Spread in Networks, arXiv:1306.2588 (2008)

22.

Piet van Mieghem, Graph Spectra for Complex Networks, Cambridge University Press (2011)

23.

Piet Van Mieghem, The viral conductance of a network, Computer Communications 35, 1494–1506 (2012)CrossRef

24.

Piet Van Mieghem, Exact Markovian SIR and SIS epidemics on networks and an upper bound for the epidemic threshold, arXiv:1402.1731 (2014)

25.

Duncan J. Watts, A simple model of global cascades on random networks, P. Natl. Acad. Sci. USA 99(9), 5766–5771 (2002). https://doi.org/10.1073/pnas.082090499 CrossRef

Titel: Deterministic and Stochastic Mean-Field SIRS Models on Heterogeneous Networks
verfasst von: Stefano Bonaccorsi
Silvia Turri
Verlag: Springer International Publishing
Buch: Discrete and Continuous Models in the Theory of Networks
Print ISBN: 978-3-030-44096-1

Electronic ISBN: 978-3-030-44097-8

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-030-44097-8_3

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner