Advertisement

The Journal of Supercomputing

, Volume 62, Issue 3, pp 1385–1403 | Cite as

Efficient solution of a stochastic SI epidemic system

  • Samiur ArifEmail author
  • Stephan Olariu
Article

Abstract

One of the side-effects of the climate changes that are upon us is that infectious diseases are adapting, evolving and spreading to new geographic regions. It is, therefore, imperious to develop epidemic models that shed light on the interplay between the dynamics of the spread of infectious diseases and the combined effects of various vaccination and prevention regimens. With this in mind, in this work we propose a epidemic model operating on a large population; we restrict our attention to strains of infectious diseases that resist treatment. The time-dependent epidemic accounts, among others, for the effects of improved sanitation, education and vaccination. Our first main contribution is to derive the time-dependent probability mass function of the number of infected individuals in such a system. Our derivation does not use probability generating functions and partial differential equations. Instead, we develop an iterative solution that is conceptually simple and easy to implement. Somewhat surprisingly, the epidemic model also provides insight into various stochastic phenomena noticed in sociology, macroeconomics, marketing, transportation and computer science. Our second main contribution is to show, by extensive simulations, that suitably instantiated, our epidemic model be used to model phenomena describing the adoption of durable consumer goods, the spread of AIDS and the dissemination of mobile worm spread.

Keywords

Stochastic epidemic models Time-dependent probabilities Waiting-time probabilities Compartmental epidemics model 

Notes

Acknowledgements

We would like to thank six anonymous referees for their helpful comments that have greatly contributed to improve the presentation of the paper. Last, but certainly not least, we wish to extend our thanks to Professor H. Arabnia for his professional handling of our submission.

References

  1. 1.
    Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University Press, Oxford Google Scholar
  2. 2.
    Arif S, Khalil I, Olariu S (2012) On a versatile stochastic growth model. Int J Comput Intel Syst. doi: 10.1080/18756891.2012.696911 Google Scholar
  3. 3.
    Bailey NTJ (1950) A simple stochastic epidemic. Biometrika 37:193–2007 MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bailey NTJ (1975) The mathematical theory of infectious diseases and its applications. Haffner, New York zbMATHGoogle Scholar
  5. 5.
    Bass FM (1969) A new product growth for model consumer durables. Manag Sci 15(5):215–227 zbMATHCrossRefGoogle Scholar
  6. 6.
    Becker NG (2005) How does mass immunization affect disease incidence? In: Proc workshop on mathematical modeling of infectious diseases: dynamics and control, Singapore, October Google Scholar
  7. 7.
    Billings L, Schwartz IB (2002) A unified prediction of computer virus spread in connected networks. Phys Lett A 297:261–262 MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Boland PJ, Singh H (2003) A birth-process approach to Moranda’s geometric software-reliability model. IEEE Trans Reliab 22(2):168–174 CrossRefGoogle Scholar
  9. 9.
    Daley DJ, Gani J (1999) Epidemic modelling: an introduction. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  10. 10.
    Hammond BJ, Tyrrell DAJ (1971) A mathematical model of common-cold epidemics on Tristan da Cunha. J Hyg 69:423 CrossRefGoogle Scholar
  11. 11.
    Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42(4):599–652 MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Mantle J, Tyrrell DAJ (1973) An epidemic of influenza on Tristan da Cunha. J Hyg 71(1):89–95 CrossRefGoogle Scholar
  13. 13.
    Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond A 115(772):700–721 zbMATHCrossRefGoogle Scholar
  14. 14.
    Nicholl KL, Tummers K, Hoyer-Leitzel A, Marsh J, Moynihan M et al (2010) Modeling seasonal influenza outbreak in a closed college campus: impact of pre-season vaccination, in-season vaccination and holidays/breaks. PLoS ONE 5(3):e9548 CrossRefGoogle Scholar
  15. 15.
    McInnes CW, Druyts E, Harvard SS, Gilbert M, Tyndall MW, Lima VD, Wood E, Montaner JS, Hogg RS (2009) HIV/AIDS in Vancouver, British Columbia: a growing epidemic. Harm Reduct J 6:5. doi: 10.1186/1477-7517-6-5 CrossRefGoogle Scholar
  16. 16.
    Murray JD (2003) Mathematical biology: I. An introduction. Springer, Berlin Google Scholar
  17. 17.
    Nee S (2006) Birth-death models in macroeconomics. Annu Rev Ecol Evol Syst 37:1–17 CrossRefGoogle Scholar
  18. 18.
    Yan G, Rizvi S, Olariu S (2010) A time-critical information diffusion model in vehicular ad-hoc networks. In: Proc. ACM MOMM’2010, Paris, France, November Google Scholar
  19. 19.
    Bulygin Y (2007) Epidemics of mobile worms. In: Proc IEEE international performance, computing, and communications conference (IPCCC 2007), pp 475–478, April. doi: 10.1109/PCCC.2007.358929

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Computer Science DepartmentOld Dominion UniversityNorfolkUSA

Personalised recommendations