Abstract
An important issue in theoretical epidemiology is the epidemic thresholdphenomenon, which specify the conditions for an epidemic to grow or die out.In standard (mean-field-like) compartmental models the concept of the basic reproductive number, R0, has been systematically employed as apredictor for epidemic spread and as an analytical tool to study thethreshold conditions. Despite the importance of this quantity, there are nogeneral formulation of R0 when one considers the spread of a disease ina generic finite population, involving, for instance, arbitrary topology ofinter-individual interactions and heterogeneous mixing of susceptible andimmune individuals. The goal of this work is to study this concept in ageneralized stochastic system described in terms of global and localvariables. In particular, the dependence of R0 on the space ofparameters that define the model is investigated; it is found that near ofthe `classical' epidemic threshold transition the uncertainty about thestrength of the epidemic process still is significantly large. Theforecasting attributes of R0 for a discrete finite system is discussedand generalized; in particular, it is shown that, for a discrete finitesystem, the pretentious predictive power of R0 is significantlyreduced.
Keywords: Cellular-Automata, Epidemics, Monte Carlo, R0
Full Text
The Full Text of this article is available as a PDF (149.6 KB).
References
- 1.Ross R. Report on the Prevention of Malaria in Mauritius. London: J and A Churchill; 1909. [Google Scholar]
- 2.Kermack W.O., McKendrick A.G. A Contribution to the Mathematical Theory of Epidemics. Proc. R. Soc. Lond. A. 1927;115:700–721. [Google Scholar]
- 3.Bartlett M.S. Measles Periodicity and Community Size. J. R. Statist. Soc. Ser. A. 1957;120:48–70. [Google Scholar]
- 4.Keeling M.J., Grenfell B.T. Individual-Based Perspectives on R0. J. Theor. Biol. 2000;203:51–61. doi: 10.1006/jtbi.1999.1064. [DOI] [PubMed] [Google Scholar]
- 5.Anderson R.M., May R.M. Infectious Diseases of Humans. Oxford: Oxford University Press; 1991. [Google Scholar]
- 6.Cardy J.L., Grassberger P. Epidemic Models and Percolation. J. Phys. A-Math. Gen. 1985;18:L267–L271. [Google Scholar]
- 7.Watts D.J., Strogats S.H. Collective Dynamics of Small-Word Networks. Nature. 1988;393:440. doi: 10.1038/30918. [DOI] [PubMed] [Google Scholar]
- 8.Grassberger P. On the Critical-Behavior of the General Epidemic Process and Dynamical Percolation. Math. Biosci. 1983;63:157. [Google Scholar]
- 9.dos Santos C.B., Barbin D., Caliri A. Epidemic Phase and Site Percolation with Distant-Neighbor Interactions. Phys. Lett. A. 1998;238:54. [Google Scholar]
- 10.Grassberger P., Chaté H., Rouseau G. Spreading in Media with Long-Time Memory. Phys. Review E. 1997;55:2488. [Google Scholar]
- 11.Mollison, D.: Epidemic Models: Their Structure and Relation to Data, Cambridge University Press, 1995.
- 12.Haas V.J., Caliri A., da Silva M.A.A. Temporal Duration and Event Size Distribution at the Epidemic Threshold. J. Biol. Phys. 1999;25:309–324. doi: 10.1023/A:1005115117228. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 13.Aiello O.E., Haas V.J., da Silva M.A.A., Caliri A. Solution of Deterministic-Stochastic Epidemic Models by Dynamical Monte Carlo Method. Physica A. 2000;282:546–558. [Google Scholar]
- 14.Gani R., Leach S. Transmission Potential of Smallpox in Contemporary Populations. Nature. 2001;414:748. doi: 10.1038/414748a. [DOI] [PubMed] [Google Scholar]