Abstract
This chapter introduces and studies discrete epidemic modeling. The chapter begins with single species discrete models of population growth. Tools for the analysis of single equation discrete models are introduced and applied to the population models. The concepts of 2-cycle, 4-cycle, and period doubling are introduced and illustrated. The chapter includes the mathematical tools for studying higher dimensional models, such as Jury conditions. It then applied these tools to study a discrete SIS and a discrete SEIS epidemic models. A generalization of the next-generation approach to discrete models is given and applied to the computation of the reproduction number of the SEIS and a two-patch SIS models. The chapter concludes with the introduction and analysis of a discrete SARS epidemic model.
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References
L. J. S. Allen and P. van den Driessche, The basic reproduction number in some discrete-time epidemic models, J. Difference Equ. Appl., 14 (2008), pp. 1127–1147.
R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, vol. XIX of Fishery Investigations Series II, Ministry of Agriculture, Fisheries and Food, 1957.
F. Brauer and C. Castillo-Chavez, Mathematical models in population biology and epidemiology, vol. 40 of Texts in Applied Mathematics, Springer, New York, second ed., 2012.
C. Castillo-Chavez and A.-A. Yakubu, Discrete-time S-I-S models with complex dynamics, in Proceedings of the Third World Congress of Nonlinear Analysts, Part 7 (Catania, 2000), vol. 47, 2001, pp. 4753–4762.
M. P. Hassell, Density-dependence is single species populations, The Journal of Animal Ecology, 44 (1975), pp. 283–295.
M. Kot, Elements of mathematical ecology, Cambridge University Press, Cambridge, 2001.
W. E. Ricker, Stock and recruitment, Journal of the Fisheries Research Board of Canada, 11 (1954).
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Martcheva, M. (2015). Discrete Epidemic Models. In: An Introduction to Mathematical Epidemiology. Texts in Applied Mathematics, vol 61. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7612-3_16
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DOI: https://doi.org/10.1007/978-1-4899-7612-3_16
Publisher Name: Springer, Boston, MA
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