The Annals of Applied Probability

Large graph limit for an SIR process in random network with heterogeneous connectivity

Laurent Decreusefond, Jean-Stéphane Dhersin, Pascal Moyal, and Viet Chi Tran

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We consider an SIR epidemic model propagating on a configuration model network, where the degree distribution of the vertices is given and where the edges are randomly matched. The evolution of the epidemic is summed up into three measure-valued equations that describe the degrees of the susceptible individuals and the number of edges from an infectious or removed individual to the set of susceptibles. These three degree distributions are sufficient to describe the course of the disease. The limit in large population is investigated. As a corollary, this provides a rigorous proof of the equations obtained by Volz [Mathematical Biology 56 (2008) 293–310].

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Ann. Appl. Probab. Volume 22, Number 2 (2012), 541-575.

First available in Project Euclid: 2 April 2012

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C80: Random graphs [See also 60B20] 92D30: Epidemiology 60F99: None of the above, but in this section

Configuration model graph SIR model mathematical model for epidemiology measure-valued process large network limit


Decreusefond, Laurent; Dhersin, Jean-Stéphane; Moyal, Pascal; Tran, Viet Chi. Large graph limit for an SIR process in random network with heterogeneous connectivity. Ann. Appl. Probab. 22 (2012), no. 2, 541--575. doi:10.1214/11-AAP773.

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  • [1] Andersson, H. (1998). Limit theorems for a random graph epidemic model. Ann. Appl. Probab. 8 1331–1349.
  • [2] Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis. Lecture Notes in Statistics 151. Springer, New York.
  • [3] Ball, F. and Neal, P. (2008). Network epidemic models with two levels of mixing. Math. Biosci. 212 69–87.
  • [4] Barthélemy, M., Barrat, A., Pastor-Satorras, R. and Vespignani, A. (2005). Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. J. Theoret. Biol. 235 275–288.
  • [5] Bartlett, M. S. (1960). Stochastic Population Models in Ecology and Epidemiology. Methuen, London.
  • [6] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [7] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [8] Clémençon, S., De Arazoza, H., Rossi, F. and Tran, V. C. A network analysis of the HIV–AIDS epidemic in Cuba. Unpublished manuscript.
  • [9] Clémençon, S., Tran, V. C. and De Arazoza, H. (2008). A stochastic SIR model with contact-tracing: Large population limits and statistical inference. J. Biol. Dyn. 2 392–414.
  • [10] Durrett, R. (2007). Random Graph Dynamics. Cambridge Univ. Press, Cambridge.
  • [11] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processus, Characterization and Convergence. Wiley, New York.
  • [12] Evans, L. C. (1998). Partial Differential Equations. Graduate Studies in Mathematics 19. Amer. Math. Soc., Providence, RI.
  • [13] Fournier, N. and Méléard, S. (2004). A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 1880–1919.
  • [14] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [15] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • [16] Joffe, A. and Métivier, M. (1986). Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. in Appl. Probab. 18 20–65.
  • [17] Kermack, W. O. and McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 115 700–721.
  • [18] Miller, J. C. (2011). A note on a paper by Erik Volz: SIR dynamics in random networks. J. Math. Biol. 62 349–358.
  • [19] Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6 161–180.
  • [20] Newman, M. E. J. (2002). The spread of epidemic disease on networks. Phys. Rev. E (3) 66 016128, 11.
  • [21] Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45 167–256.
  • [22] Newman, M. E. J., Strogatz, S. H. and Watts, D. J. (2001). Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E (3) 64.
  • [23] Pastor-Satorras, R. and Vespignani, A. (2002). Epidemics and immunization in scale-free networks. In Handbook of Graphs and Networks: From the Genome to the Internet 113–132. Wiley-VCH, Berlin.
  • [24] Roelly-Coppoletta, S. (1986). A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics 17 43–65.
  • [25] Tran, V. C. (2007). Modèles particulaires stochastiques pour des problèmes d’évolution adaptative et pour l’approximation de solutions statistiques. Ph.D. thesis, Univ. Paris X—Nanterre. Available at
  • [26] van der Hofstad, R. (2011). Random graphs and complex networks. Lecture Notes. To appear. Available at
  • [27] Volz, E. (2008). SIR dynamics in random networks with heterogeneous connectivity. J. Math. Biol. 56 293–310.
  • [28] Whitt, W. (1985). Blocking when service is required from several facilities simultaneously. AT&T Tech. J. 64 1807–1856.