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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 508794, 7 pages
http://dx.doi.org/10.1155/2013/508794
Research Article

Computation of the Domain of Attraction for Suboptimal Immunity Epidemic Models Using the Maximal Lyapunov Function Method

1Faculty of Science, Technology and Human Development, Universiti Tun Hussein Onn Malaysia, Batu Pahat, 86400 Johor, Malaysia
2Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia
3Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand

Received 24 October 2012; Accepted 12 December 2012

Academic Editor: Xinguang Zhang

Copyright © 2013 Chang Phang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We are concerned with the estimation of the domain of attraction (DOA) for suboptimal immunity epidemic models. We establish a procedure to determine the maximal Lyapunov function in the form of rational functions. Based on the definition of DOA and the maximal Lyapunov function, a theorem and subsequently a numerical procedure are established to determine the maximal Lyapunov function and the DOA. Determination of the domain of attraction for epidemic models is very important for understanding the dynamic behaviour of the disease transmission as a function of the state of population distribution in different categories of disease states. We focus on suboptimal immunity epidemic models with saturated treatment rate and nonlinear incidence rate. Different from classical models, suboptimal immunity models are more realistic to explain the microparasite infection diseases such as Pertussis and Influenza A. We show that, for certain values of the parameter, larger k value (i.e., the model is more toward the SIR model) leads to a smaller DOA.