Abstract
This chapter presents a simple and intuitive stochastic model of a temporal network and investigate how a simulated infection co-evolves with the temporal structures, focusing on the growth dynamics of the epidemics. The model assumes no underlying topological structure and is only constrained by the time between two consecutive events of vertex activation, hereafter called vertex inter-event time. The network model consists of random activations of a vertex according to a pre-defined vertex inter-event time distribution. The vertices active at a given time are randomly connected in pairs during one time unit. The link is then destroyed and the vertices set to the inactive state. The infection event occurs through this link if one of the vertices is in an infective state. This model has been studied by using a susceptible infective dynamics with one (SI) and two (SII) infective stages. The first dynamics is motivated for being an upper limit case, where once infected, the vertex continues infecting at every contact. The second dynamics is more realistic and corresponds to a model of HIV spreading including an acute (high infectivity) and chronic (low infectivity) stages of infection with different periods. If the second stage is set to zero in the SII model, we recover the susceptible infected recovered (SIR).
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Notes
- 1.
The geometric is a discrete time equivalent to the exponential distribution.
References
L.J.S. Allen, Some discrete-time SI, SIR and SIS epidemic models. Math. Biosci. 124, 83–105 (1994)
A.-L. Barabási, The origin of bursts and heavy tails in human dynamics. Nature 435, 207–210 (2005)
P. Bajardi, A. Barrat, F. Natale, L. Savini, et al., Dynamical patterns of cattle trade movements. PLoS ONE 6, 5 e19869 (2011)
A. Barrat, M. Barthélemy, A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, Cambridge, 2008)
S. Bansal, J. Read, B. Pourbohloul, L.A. Meyers, The dynamic nature of contact networks in infectious disease epidemiology. J. Biol. Dyn. 4, 5 478–489 (2010)
A. Clauset, C.R. Shalizi, M.E.J. Newman, Power law distributions in empirical data. SIAM Rev. 51(4), 661–703 (2009)
H.W. Hethcote, The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000)
T.D. Hollingsworth, R.M. Anderson, C. Fraser, HIV-1 transmission, by stage of infection. JID 198, 687–693 (2008)
P. Holme, J. Saramäki, Temporal networks. To appear in Phys. Rep. 519(3), 97–125 (2012)
J.L. Iribarren, E. Moro, Impact of human activity patterns on the dynamics of information diffusion. PRL 103, 038702 (2009)
J.S. Koopman, J.A. Jacquez, G.W. Welch, et al., The role of early HIV infection in the spread of HIV through populations. JAIDS 14(3), 249–258 (1997)
M. Karsai, M. Kivelä, R.K. Pan, K. Kaski, et al., Small but slow world: How network topology and burstiness slow down spreading. PRE 83, 025102 (2011)
R.D. Malmgren, D.B. Stouffer, A.E. Motter, L.A.N. Amaral, A poissonian explanation for heavy tails in e-mail communication. PNAS 105(47), 18153–18158 (2008)
M. Newman, Networks: An Introduction (Oxford University Press, Oxford, 2010)
R.K. Pan, J. Saramäki, Path lengths, correlations, and centrality in temporal networks. PRE 84, 016105 (2011)
L.E.C. Rocha, V.D. Blondel, Bursts of vertex activation and epidemics in evolving networks. To appear in PLoS Comput. Bio. arXiv:1206.6036 (2013)
J.M. Read, K.T.D. Eames, W.J. Edmunds, Dynamic social networks and the implications for the spread of infectious disease. J. R. Soc. Interface 5, 26 1001–1007 (2008)
L.E.C. Rocha, F. Liljeros, P. Holme, Information dynamics shape the sexual networks of internet-mediated prostitution. PNAS 107(13), 5706–5711 (2010)
L.E.C. Rocha, F. Liljeros, P. Holme, Simulated epidemics in an empirical spatiotemporal network of 50,185 sexual contacts. PLoS Comput. Biol. 7(3), e1001109 (2011)
M. Salathé, M. Kazandjieva, J.W. Leeb, et al., A high-resolution human contact network for infectious disease transmission. PNAS 107(51), 22020–22025 (2010)
J. Stehlé, N. Voirin, A. Barrat, C. Cattuto, et al., Simulation of a SEIR infectious disease model on the dynamic contact network of conference attendees. BMC Med. 9(87), 1–15 (2011)
J. Tang, S. Scellato, M. Musolesi, et al., Small-world behavior in time-varying graphs. PRE 81, 055101(R) (2010)
A. Vazquez, B. Rácz, A. Lukács, A.-L. Barabási, Impact of non-Poisson activity patterns on spreading processes. PRL 98, 158702 (2007)
M. Vernon, M.J. Keeling, Representing the UK cattle herd as static and dynamic networks. Proc. R. Soc. Lond. B Bio. 276, 469–476 (2009)
E. Volz, L.A. Meyers, Susceptible-infected-recovered epidemics in dynamic contact networks. Proc. R. Soc. B 274 2925–2933 (2007)
Acknowledgement
LECR is beneficiary of a FSR incoming postdoctoral fellowship of the Académie universitaire Louvain, co-funded by the Marie Curie Actions of the European Commission. AD is a research fellow with the Fonds National de la Recherche Scientifique (FRS-FNRS). Computational resources have been provided by the supercomputing facilities of the Université catholique de Louvain (CISM/UCL) and the Consortium des Équipements de Calcul Intensif en Fédération Wallonie Bruxelles (CECI) funded by FRS-FNRS.
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Rocha, L.E.C., Decuyper, A., Blondel, V.D. (2013). Epidemics on a Stochastic Model of Temporal Network. In: Mukherjee, A., Choudhury, M., Peruani, F., Ganguly, N., Mitra, B. (eds) Dynamics On and Of Complex Networks, Volume 2. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6729-8_15
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