Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T22:45:53.716Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE QUASI-STATIONARY DISTRIBUTION OF SIS MODELS

Published online by Cambridge University Press:  16 September 2016

Gaofeng Da ,
Maochao Xu  and
Shouhuai Xu
Gaofeng Da
Affiliation:
College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu Province, China E-mail: [email protected]
Maochao Xu
Affiliation:
Department of Mathematics, Illinois State University, Illinois, USA E-mail: [email protected]
Shouhuai Xu
Affiliation:
Department of Computer Science, University of Texas at San Antonio, San Antonio, Texas, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we propose a novel method for constructing upper bounds of the quasi-stationary distribution of SIS processes. Using this method, we obtain an upper bound that is better than the state-of-the-art upper bound. Moreover, we prove that the fixed point map Φ [7] actually preserves the equilibrium reversed hazard rate order under a certain condition. This allows us to further improve the upper bound. Some numerical results are presented to illustrate the results.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 4 , October 2016 , pp. 622 - 639
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bartholomew, D.J. (1976). Continuous time diffusion models with random duration of interest. The Journal of Mathematical Sociology 4: 187–199.Google Scholar
2. Cavender, J.A. (1978). Quasi-stationary distributions of birth-and-death processes. Advanced in Applied Probability 10: 570–586.Google Scholar
3. Clancy, D. (2012). Approximating quasistationary distributions of birth-death processes. Journal of Applied Probability 49: 1036–1051.Google Scholar
4. Clancy, D. & Mendy, S.T. (2011). Approximating the quasi-stationary distribution of the sis model for endemic infection. Methodology and Computing in Applied Probability 13: 603–618.Google Scholar
5. Clancy, D. & Pollett, P.K. (2003). A note on quasi-stationary distributions of birth-death processes and the SIS logistic epidemic. Journal of Applied Probability 40: 821–825.CrossRefGoogle Scholar
6. Darroch, J.N. & Seneta, E. (1967). On quasi-stationary distributions in absorbing continuous-time finite markov chains. Journal of Applied Probability 4: 192–196.Google Scholar
7. Ferrari, P., Kesten, H., Martínez, S., & Picco, P. (1995). Existence of quasi-stationary distributions: a renewal dynamical approach. Annals of Probability 23: 501–521.Google Scholar
8. Ganesh, A., Massoulie, L., & Towsley, D. (2005). The effect of network topology on the spread of epidemics. In Proceedings of 24th IEEE Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM 2005), vol. 2, pp. 1455–1466.Google Scholar
9. Hill, A.L., Rand, D.G., Nowak, M.A., & Christakis, N.A. Emotions as infectious diseases in a large social network: the SISA model. Proceedings of the Royal Society B 2: 3827–3835.Google Scholar
10. Hu, T., Kundu, A., & Nanda, A.K. (2001). On generalized orderings and aging properties with their implications. In Hayakawa, Y., Irony, T., & Xie, M. (eds.), System and Bayesian Reliability. Singapore: World Scientific.Google Scholar
11. Karlin, S. (1968). Total Positivity. Stanford: Stanford University Press.Google Scholar
12. Keilson, J. & Ramaswamy, R. (1984). Convergence of quasi-stationary distributions in birth–death processes. Stochastic Processes and their Applications 5: 231–241.Google Scholar
13. Kephart, J.O. & White, S.R. (1991). Directed-graph epidemiological models of computer viruses. In IEEE Computer Society Symposium on Research in Security and Privacy, pp. 343–359.Google Scholar
14. Kijima, M. (1995). Bounds for the quasi-stationary distribution of some specialized Markov chains. Mathematical and Computer Modelling 22: 141–147.Google Scholar
15. Kijima, M. & Seneta, E. (1991). Some results for quasi-stationary distributions of birth and death processes. Journal of Applied Probability 28: 503–511.Google Scholar
16. Kryscio, R.J. & Lefèvre, C. (1989). On the extinction of the sis stochastic logistic epidemic. Journal of Applied Probability 27: 685–694.CrossRefGoogle Scholar
17. Li, X. & Xu, M. (2008). Reversed hazard rate order of equilibrium distributions and a related aging notion. Statistical Papers 49: 749–767.Google Scholar
18. Li, X., Parker, P., & Xu, S. (2007). Towards quantifying the (in)security of networked systems. In 21st IEEE International Conference on Advanced Information Networking and Applications, pp. 420–427.Google Scholar
19. Matis, J.H. & Kiffe, T.R. (1996). On approximating the moments of the equilibrium distribution of a stochastic logistic model. Biometrics 52: 980–991.Google Scholar
20. Nåsell, I. (1996). The quasi-stationary distribution of the closed endemic sis model. Advances of Applied Probability 28: 895–932.CrossRefGoogle Scholar
21. Nåsell, I. (1999). On the quasi-stationary distribution of the stochastic logistic epidemic. Mathematical Bioscience 156: 21–40.CrossRefGoogle ScholarPubMed
22. Nåsell, I. (2001). Extinction and quasi-stationary in the Verhulst logistic model. Journal of Theoretical Biology 211: 11–27.Google Scholar
23. Nåsell, I. (2003). An extension of the moment closure method. Theoretical Population Biology 64: 233–239.Google Scholar
24. Nåsell, I. (2011). Extinction and Quasi-stationarity in the Stochastic Logistic SIS Model. New York: Springer, vol. 2022.CrossRefGoogle Scholar
25. Oppenheim, I., Shuler, K.E., & Weiss, G.H. (1977). Stochastic theory of nonlinear rate processes with multiple stationary states. Physica A 88: 191–214.Google Scholar
26. Pastor-Satorras, R. & Vespignani, A. (2001). Epidemic spreading in scale free networks. Physics Review Letters 86(14): 3200–3203.Google Scholar
27. Pinsky, M. & Karlin, S. (2010). An Introduction to Stochastic Modeling. Burlington: Academic Press.Google Scholar
28. Pollett, P.K. & Vassallo, A. (1992). Diffusion approximations for some simple chemical reaction schemes. Advances in Applied Probability 24: 875–893.Google Scholar
29. Shaked, M. & Shanthikumar, J.G. (2007). Stochastic Orders. New York: Springer.Google Scholar
30. van Doorn, E.A. & Pollett, P.K. (2013). Quasi-stationary distributions for discrete-state models. European Journal of Operational Research 230: 1–14.Google Scholar
31. Van Mieghem, P. & Cator, E. (2012). Epidemics in networks with nodal self-infection and the epidemic threshold. Physical Review E 86: 016116.Google Scholar
32. Van Mieghem, P., Omic, J., & Kooij, R. (2009). Virus spread in networks. IEEE/ACM Transactions on Networking 17(1): 1–14.Google Scholar
33. Xu, S., Lu, W., & Xu, L. (2012). Push- and pull-based epidemic spreading in networks: thresholds and deeper insights. TAAS 7(3): 32.Google Scholar
34. Xu, S., Lu, W., & Zhan, Z. (2012). A stochastic model of multivirus dynamics. IEEE Transactions on Dependable Secure Computing 9(1): 30–45.Google Scholar
35. Xu, S., Lu, W., Xu, L., & Zhan, Z. (2014). Adaptive epidemic dynamics in networks: thresholds and control. TAAS 8(4): 19.Google Scholar