Journal of Applied Mathematics
Volume 2012 (2012), Article ID 824192, 20 pages
Research Article

Dynamical Models for Infectious Diseases with Varying Population Size and Vaccinations

Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China

Received 24 July 2011; Accepted 13 October 2011

Academic Editor: Venky Krishnan

Copyright © 2012 Peilin Shi and Lingzhen Dong. 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.


We formulate and discuss models for the spread of infectious diseases with variable population sizes and vaccinations on the susceptible individuals. First, we assume that the susceptible individuals are vaccinated continuously. We establish the threshold-like results for the existence and global stability of the disease-free and the endemic equilibriums for these systems. Especially, we prove the global stability of the endemic equilibriums by converting the systems into integrodifferential equations. Second, we suppose that vaccinations occur once per time period. We obtain the existence and global stability of the disease-free periodic solutions for such systems with impulsive effects. By a useful bifurcation theorem, we acquire the existence of the endemic periodic solutions when the disease-related deaths do not occur. At last, we compare the results with vaccinations and without vaccinations and illustrate our results by numerical simulations.