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The front of the epidemic spread and first passage percolation

Part of: Graph theory Combinatorial probability Operations research and management science

Published online by Cambridge University Press:  30 March 2016

Shankar Bhamidi ,
Remco van der Hofstad  and
Júlia Komjáthy
Shankar Bhamidi
Affiliation:
Department of Statistics, University of North Carolina, Chapel Hill, USA. Email address: bhamidi@email.unc.edu.
Remco van der Hofstad
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: rhofstad@win.tue.nl.
Júlia Komjáthy
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: j.komjathy@tue.nl.
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Abstract

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We establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert (2013) and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad and Hooghiemstra (2012), when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour and Reinert (2013) from bounded degree graphs to general sparse random graphs with degrees having finite second moments as n → ∞, with an appropriate X2log+X condition. We also study the epidemic trail between the source and typical vertices in the graph.

Keywords

Flowrandom graphrandom networkepidemics on random graphsfirst passage percolationhop countinteracting particle system

MSC classification

Primary: 60C05: Combinatorial probability 05C80: Random graphs 90B15: Network models, stochastic
Type
Part 4. Random graphs and particle systems
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 101 - 121
Copyright
Copyright © Applied Probability Trust 2014 

References

Aldous, D. (2013). Interacting particle systems as stochastic social dynamics. Bernoulli 19, 1122–1149.Google Scholar
Ball, F., and Donnelly, P. (1995). Strong approximations for epidemic models. Stoch. Process. Appl. 55, 1–21.CrossRefGoogle Scholar
Barbour, A. D., and Reinert, G. (2013). Approximating the epidemic curve. Electron. J. Prob. 18, 30 pp.Google Scholar
Barrat, A., Barthélemy, M., and Vespignani, A. (2008). Dynamical Processes on Complex Networks. Cambridge University Press.Google Scholar
Bartlett, M. S. (1955). An Introduction to Stochastic Processes, with Special Reference to Methods and Applications. Cambridge University Press.Google Scholar
Bhamidi, S., van der Hofstad, R., and Hooghiemstra, G. (2010). First passage percolation on random graphs with finite mean degrees. Ann. Appl. Prob. 20, 1907–1965.Google Scholar
Bhamidi, S., van der Hofstad, R., and Hooghiemstra, G. (2011). First passage percolation on the Erd Hos-Rényi random graph. Combinatorics Prob. Comput. 20, 683–707.CrossRefGoogle Scholar
Bhamidi, S., van der Hofstad, R., and Hooghiemstra, G. (2012). Universality for first passage percolation on sparse random graphs. Preprint. Available at http://arxiv.org/abs/1210.6839v1.Google Scholar
Bohman, T., and Picollelli, M. (2012). S{IR} epidemics on random graphs with a fixed degree sequence. Random Structures Algorithms 41, 179–214.Google Scholar
Bollobás, B. (2001). Random Graphs (Camb. Stud. Adv. Math. 73), 2nd edn. Cambridge University Press.Google Scholar
Daley, D. J., and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.Google Scholar
Decreusefond, L., Dhersin, J.-S., Moyal, P., and Tran, V. C. (2012). Large graph limit for an {SIR} process in random network with heterogeneous connectivity. Ann. Appl. Prob. 22, 541–575.Google Scholar
Draief, M. and Massoulié, L. (2010). Epidemics and Rumours in Complex Networks (London Math. Soc. Lecture Note Series 369). Cambridge University Press.Google Scholar
Durrett, R. (2007). Random Graph Dynamics. Cambridge University Press.Google Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.Google Scholar
Jagers, P., and Nerman, O. (1984). The growth and composition of branching populations. Adv. Appl. Prob. 16, 221–259.Google Scholar
Janson, S. (2009). The probability that a random multigraph is simple. Combinatorics Prob. Comput. 18, 205–225.CrossRefGoogle Scholar
Janson, S., and Luczak, M. J. (2009). A new approach to the giant component problem. Random Structures Algorithms 34, 197–216.Google Scholar
Janson, S., Luczak, M., and Windridge, P. (2014). Law of large numbers for the SIR epidemic on a random graph with given degrees. Preprint. Available at http://arxiv.org/abs/1308.5493v3.Google Scholar
Kallenberg, O. (1976). Random Measures. Akademie, Berlin.Google Scholar
Kendall, D. G. (1956). Deterministic and stochastic epidemics in closed populations. In Proc. 3rd Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, Vol. IV, University of California Press, Berkeley, pp. 149–165.Google Scholar
Kermack, W. O., and McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proc. R. Soc. London. A 115, 700–721.Google Scholar
Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeitsth. 57, 365–395.Google Scholar
Newman, M., Barabási, A.-L., and Watts, D. J. (eds) (2006). The Structure and Dynamics of Networks. Princeton University Press, Princeton, NJ.Google Scholar
Van der Hofstad, R. (2014). Random Graphs and Complex Networks. In preparation.Google Scholar
Volz, E. (2008). SIR dynamics in random networks with heterogeneous connectivity. J. Math. Biol. 56, 293–310.CrossRefGoogle ScholarPubMed