Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 679613, 12 pages
doi:10.1155/2010/679613
Research Article

Analysis of a Simple Vector-Host Epidemic Model with Direct Transmission

Liming Cai1,2 and Xuezhi Li1

1College of Mathematics and Information Science, Xinyang Normal University, Henan, Xinyang 464000, China
2Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, China

Received 7 December 2009; Accepted 3 March 2010

Academic Editor: Leonid Berezansky

Copyright © 2010 Liming Cai and Xuezhi Li. 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

Vector-host epidemic models with direct transmission are proposed and analyzed. It is shown that the stability of the equilibria in the proposed models can be controlled by the basic reproduction number of the disease transmission. One model considers that the dynamics of human hosts and vectors are described by SIS and SI model, respectively, where the global asymptotical stability for the equilibria of the model is analyzed by constructing Lyapunov function, respectively. The other model considers that the dynamics of the human hosts and vectors are described by SIRS and SI model, respectively, where the global stability of the disease-free equilibrium and the persistence of the disease in the model are also analyzed, respectively.