Article contents
ON THE QUASI-STATIONARY DISTRIBUTION OF SIS MODELS
Published online by Cambridge University Press: 16 September 2016
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
References
1.
Bartholomew, D.J. (1976). Continuous time diffusion models with random duration of interest. The Journal of Mathematical Sociology
4: 187–199.CrossRefGoogle Scholar
2.
Cavender, J.A. (1978). Quasi-stationary distributions of birth-and-death processes. Advanced in Applied Probability
10: 570–586.CrossRefGoogle 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.CrossRefGoogle 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.Google 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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
12.
Keilson, J. & Ramaswamy, R. (1984). Convergence of quasi-stationary distributions in birth–death processes. Stochastic Processes and their Applications
5: 231–241.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
16.
Kryscio, R.J. & Lefèvre, C. (1989). On the extinction of the sis stochastic logistic epidemic. Journal of Applied Probability
27: 685–694.Google Scholar
17.
Li, X. & Xu, M. (2008). Reversed hazard rate order of equilibrium distributions and a related aging notion. Statistical Papers
49: 749–767.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
23.
Nåsell, I. (2003). An extension of the moment closure method. Theoretical Population Biology
64: 233–239.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
26.
Pastor-Satorras, R. & Vespignani, A. (2001). Epidemic spreading in scale free networks. Physics Review Letters
86(14): 3200–3203.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
29.
Shaked, M. & Shanthikumar, J.G. (2007). Stochastic Orders. New York: Springer.CrossRefGoogle 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.CrossRefGoogle Scholar
31.
Van Mieghem, P. & Cator, E. (2012). Epidemics in networks with nodal self-infection and the epidemic threshold. Physical Review E
86: 016116.CrossRefGoogle ScholarPubMed
32.
Van Mieghem, P., Omic, J., & Kooij, R. (2009). Virus spread in networks. IEEE/ACM Transactions on Networking
17(1): 1–14.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
- 1
- Cited by