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State space collapse for critical multistage epidemics

Part of: Limit theorems

Published online by Cambridge University Press:  21 March 2016

Florian Simatos
Florian Simatos*
Affiliation:
Eindhoven University of Technology
*
∗ Current address: ISAE Supaero, Département DISC, 10 avenue Edouard Belin, BP 54032, 31055 TOULOUSE CEDEX 4, France. Email address: florian.simatos@isae.fr
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Abstract

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We study a multistage epidemic model which generalizes the SIR model and where infected individuals go through K ≥ 1 stages of the epidemic before being removed. An infected individual in stage k ∈ {1, …, K} may infect a susceptible individual, who directly goes to stage k of the epidemic; or it may go to the next stage k + 1 of the epidemic. For this model, we identify the critical regime in which we establish diffusion approximations. Surprisingly, the limiting diffusion exhibits an unusual form of state space collapse which we analyze in detail.

Keywords

Multistage epidemicstate space collapsescaling limits

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 715 - 740
Copyright
Copyright © Applied Probability Trust 2015 

References

Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Prob. 25, 812–854.CrossRefGoogle Scholar
Antal, T. and Krapivsky, P. L. (2012). Outbreak size distributions in epidemics with multiple stages. J. Statist. Mech. Theory Exp. 2012, P07018.CrossRefGoogle Scholar
Barbour, A. and Mollison, D. (1990). Epidemics and random graphs. In Stochastic Processes in Epidemic Theory (Lecture Notes Biomath. 86). Springer, Berlin, pp. 86–89.CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Borst, S. and Simatos, F. (2013). A stochastic network with mobile users in heavy traffic. Queueing Systems 74, 1–40.CrossRefGoogle Scholar
Bramson, M. (1998). State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Systems Theory Appl. 30, 89–148.CrossRefGoogle Scholar
Dolgoarshinnykh, R. G. and Lalley, S. P. (2006). Critical scaling for the SIS stochastic epidemic. J. Appl. Prob. 43, 892–898.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
Helland, I. S. (1978). Continuity of a class of random time transformations. Stoch. Process. Appl. 7, 79–99.CrossRefGoogle Scholar
Hyman, J. M., Li, J. and Stanley, E. A. (1999). The differential infectivity and staged progression models for the transmission of HIV. Math. Biosci. 155, 77–109.CrossRefGoogle ScholarPubMed
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Janson, S. and Luczak, M. J. (2008). Susceptibility in subcritical random graphs. J. Math. Phys. 49, 125207.CrossRefGoogle Scholar
Kendall, D. G. (1956). Deterministic and stochastic epidemics in closed populations. In Proc. Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. IV, University of California Press, Berkeley, pp. 149–165.Google Scholar
Lambert, A. and Simatos, F. (2015). Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case. J. Theoret. Prob. 28, 41–91.CrossRefGoogle Scholar
Lambert, A., Simatos, F. and Zwart, B. (2013). Scaling limits via excursion theory: interplay between Crump–Mode–Jagers branching processes and processor-sharing queues. Ann. Appl. Prob. 23, 2357–2381.CrossRefGoogle Scholar
Lamperti, J. (1967). Continuous state branching processes. Bull. Amer. Math. Soc. 73, 382–386.CrossRefGoogle Scholar
Martin-Löf, A. (1986). Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. J. Appl. Prob. 23, 265–282.CrossRefGoogle Scholar
Martin-Löf, A. (1998). The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Prob. 35, 671–682.CrossRefGoogle Scholar
Reiman, M. I. (1984). Some diffusion approximations with state space collapse. In Modelling and Performance Evaluation Methodology (Paris, 1983; Lecture Notes Control Inf. Sci. 60), Springer, Berlin, pp. 209-240.CrossRefGoogle Scholar
Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales, Vol. 2, Itô Calculus. John Wiley, New York.Google Scholar
Stolyar, A. L. (2004). Maxweight scheduling in a generalized switch: state space collapse and workload minimization in heavy traffic. Ann. Appl. Prob. 14, 1–53.CrossRefGoogle Scholar
Verloop, I. M., Ayesta, U. and Núñez-Queija, R. (2011). Heavy-traffic analysis of a multiple-phase network with discriminatory processor sharing. Operat. Res. 59, 648–660.CrossRefGoogle Scholar
Von Bahr, B. and Martin-Löf, A. (1980). Threshold limit theorems for some epidemic processes. Adv. Appl. Prob. 12, 319–349.CrossRefGoogle Scholar
Williams, R. J. (1998). Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse. Queueing Systems Theory Appl. 30, 27–88.CrossRefGoogle Scholar