Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-13T00:18:23.591Z Has data issue: false hasContentIssue false
  • English
  • Français

A Markov Model for the Spread of Viruses in an Open Population

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

L. Coutin ,
L. Decreusefond  and
J. S. Dhersin
L. Coutin*
Affiliation:
Université Paul Sabatier
L. Decreusefond*
Affiliation:
Institut Telecom
J. S. Dhersin*
Affiliation:
University of Paris 13
*
∗Postal address: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 9, France. Email address: laure.coutin@math.univ-toulouse.fr
∗∗Postal address: Telecom ParisTech, Institut Telecom, 46 rue Barrault, 75634 Paris Cedex 13, France. Email address: laurent.decreusefond@telecom-paristech.fr
∗∗∗Postal address: Department of Mathematics, Institute Galilée, University of Paris 13, 99 Avenue Jean-Baptiste Clément, F-93430 Villetaneuse, France. Email address: dhersin@math.univ-paris13.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Inspired by methods of queueing theory, we propose a Markov model for the spread of viruses in an open population with an exogenous flow of infectives. We apply it to the diffusion of AIDS and hepatitis C diseases among drug users. From a mathematical point of view, the difference between the two viruses is shown in two parameters: the probability of curing the disease (which is 0 for AIDS but positive for hepatitis C) and the infection probability, which seems to be much higher for hepatitis. This model bears some resemblance to the M/M/∞ queueing system and is thus rather different from the models based on branching processes commonly used in the epidemiological literature. We carry out an asymptotic analysis (large initial population) and show that the Markov process is close to the solution of a nonlinear autonomous differential system. We prove both a law of large numbers and a functional central limit theorem to determine the speed of convergence towards the limiting system. The deterministic system itself converges, as time tends to ∞, to an equilibrium point. We then show that the sequence of stationary probabilities of the stochastic models shrinks to a Dirac measure at this point. This means that in a large population and for long-term analysis, we may replace the individual-based microscopic stochastic model with the macroscopic deterministic system without loss of precision. Moreover, we show how to compute the sensitivity of any functional of the Markov process with respect to a slight variation of any parameter of the model. This approach is applied to the spread of diseases among drug users, but could be applied to many other case studies in epidemiology.

Keywords

EpidemiologyMarkov processmean-field approximation

MSC classification

Secondary: 60F17: Functional limit theorems; invariance principles 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 92D30: Epidemiology
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 976 - 996
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Andersson, H. and Britton, T. (2000). Stochastic Epidemic Models and Their Statistical Analysis. Springer, New York.Google Scholar
[2] Andersson, P. and Lindenstrand, D. (2010). A stochastic SIS infection model incorporating indirect transmission epidemic with demography: initial stages and time to extinction. Submitted.Google Scholar
[3] Bell, J. et al. (1990). Hepatitis C virus in intravenous drug users. Med. J. Austral. 153, 274–276.CrossRefGoogle ScholarPubMed
[4] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
[5] Clancy, D. (2005). A stochastic SIS infection model incorporating indirect transmission. J. Appl. Prob. 42, 726–737.Google Scholar
[6] Clancy, D. and Pollett, P. K. (2003). A note on quasi-stationary distributions of birth–death processes and the SIS logistic epidemic. J. Appl. Prob. 40, 821–825.Google Scholar
[7] Decreusefond, L. (1998). Perturbation analysis and Malliavin calculus. Ann. Appl. Prob. 8, 496–523.CrossRefGoogle Scholar
[8] Esposito, N. and Rossi, C. (2004). A nested-epidemic model for the spread of hepatitis C among injecting drug users. Math. Biosci. 188, 29–45.CrossRefGoogle ScholarPubMed
[9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
[10] Garfein, R. S. et al. (1996). Viral infections in short-term injection drug users: the prevalence of the hepatitis {C}, hepatitis B, human immunodeficiency, and human T-lymphotropic viruses. Amer. J. Public Health 86, 655–661.CrossRefGoogle ScholarPubMed
[11] Graham, C. (2000). Kinetic limits for large communication networks. In Modeling in Applied Sciences, Birkhäuser, Boston, MA, pp. 317–370.Google Scholar
[12] Health Protection Agency (2004). Shooting up: infection among injecting drug users in the United Kingdom 2003. Tech. Rep., London.Google Scholar
[13] Jacob, C. and Viet, A. F. (2003). Epidemiological modeling in a branching population. Particular case of a general SIS model with two age classes. Math. Biosci. 182, 93–111.CrossRefGoogle Scholar
[14] Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Springer, Berlin.Google Scholar
[15] Jauffret-Roustide, M. et al. (2006). Impact of a harm-reduction policy on HIV and hepatitis C virus transmission among drug users: recent French data–the ANRS-Coquelicot study. Subst. Use Misuse 41, 1603–1621.CrossRefGoogle ScholarPubMed
[16] Métivier, M. (1988). Stochastic Partial Differential Equations in Infinite-Dimensional Spaces. Scuola Normale Superiore, Pisa.Google Scholar
[17] Robert, P. (2003). Stochastic Networks and Queues (Appl. Math. 52). Springer, Berlin.Google Scholar
[18] Verhulst, F. (1996). Nonlinear Differential Equations and Dynamical Systems, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
[19] Vickerman, P., Hickman, M. and Judd, A. (2007). Modelling the impact on Hepatitis C transmission of reducing syringe sharing: London case study. Internat. J. Epidemiology 36, 396–405.CrossRefGoogle ScholarPubMed
[20] Whitt, W. (1985). Blocking when service is required from several facilities simultaneously. Bell Syst. Tech. J. 64, 1807–1856.Google Scholar