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Abstract and Applied Analysis
Volume 2013, Article ID 405258, 17 pages
http://dx.doi.org/10.1155/2013/405258
Research Article

Mathematical Analysis of a Malaria Model with Partial Immunity to Reinfection

1Department of Mathematics, Xinyang Normal University, Xinyang 464000, China
2Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan
3Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
4Department of Mathematics, Vaal University of Technology, Andries Potgieter Boulevard, X021, Vanderbijlpark 1900, South Africa
5National Fisheries Research and Development Institute, Busan 619-705, Republic of Korea

Received 5 October 2012; Accepted 7 December 2012

Academic Editor: Sanyi Tang

Copyright © 2013 Li-Ming Cai 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

A deterministic model with variable human population for the transmission dynamics of malaria disease, which allows transmission by the recovered humans, is first developed and rigorously analyzed. The model reveals the presence of the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated reproduction number is less than unity. This phenomenon may arise due to the reinfection of host individuals who recovered from the disease. The model in an asymptotical constant population is also investigated. This results in a model with mass action incidence. A complete global analysis of the model with mass action incidence is given, which reveals that the global dynamics of malaria disease with reinfection is completely determined by the associated reproduction number. Moreover, it is shown that the phenomenon of backward bifurcation can be removed by replacing the standard incidence function with a mass action incidence. Graphical representations are provided to study the effect of reinfection rate and to qualitatively support the analytical results on the transmission dynamics of malaria.