《Permanence of an SIR epidemic model with density dependent birth rate and distributed time delay》.pdf

Applied Mathematics and Computation 218 (2011) 1682–1693 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: /locate/amc Permanence of an SIR epidemic model with density dependent birth rate and distributed time delay Chun-Hsien Li a, Chiung-Chiou Tsai b, Suh-Yuh Yang a,⇑ a Department of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan b Department of Computer Science and Information Engineering, Nanya Institute of Technology, Jhongli City, Taoyuan County 32091, Taiwan a r t i c l e i n f o a b s t r a c t Keywords: In this paper, we investigate the permanence of an SIR epidemic model with a density- SIR epidemic model dependent birth rate and a distributed time delay. We first consider the attractivity of Time delay the disease-free equilibrium and then show that for any time delay, the delayed SIR epi- Asymptotic stability demic model is permanent if and only if an endemic equilibrium exists. Numerical exam- Permanence ples are given to illustrate the theoretical analysis. The results obtained are also compared with those from the analog system with a discrete time delay. 2011 Elsevier Inc. All rights reserved. 1. Introduction and preliminaries In modeling the spread process of infectious diseases, many classical epidemic models have been proposed and studied, such as SIR, SEIR and SIRS m