Nonlinear Analysis 63 (2005) e779 – e788 www.elsevier.com/locate/na Recurrent epidemic cycles in an infectious disease model with a time delay in loss of vaccine immunity D. Greenhalgh a , ∗ , Q.J.A. Khan b , F.I. Lewis c , 1 a Department of Statistics and Modelling Science, University of Strathclyde, LivingstoneTower, 26 Richmond Street, Glasgow G1 1XH, UK b Department of Mathematics and Statistics, College of Science, Sultan Quaboos University, P.O.Box 36, Al-Khod, Postal Code 123, Muscat, Oman c Institute of Evolutionary Biology, Ashworth Laboratories, School of Biological Sciences, University of Edinburgh,West Mains Road, Edinburgh, EH9 3JT, UK Abstract In this paper we look at an SIRS epidemiological model with vaccination. Although immunity gained by experiencing the disease is permanent, vaccine-induced immunity is only temporary and a fixed time after vaccination individuals return to the susceptible class. The model is described and equilibrium and stability results are shown. There are three possible equilibria: one where the population is extinct; a disease-free equilibrium where the population is at a steady level and a unique endemic equilibrium. Simulation confirms that although Hopf bifurcation is theoretically possible for realistic parameter values the simulations approach the unique endemic equilibrium.  2005 Elsevier Ltd. All rights reserved. Keywords: SIRS epidemic model; Time delay; Vaccination; Equilibrium and Stability analysis; Hopf bifurcation 1. Introduction This paper discusses an SIRS epidemiological model with a time delay in vaccination. A typical individual starts off susceptible, at some stage enters the infectious class and after a ∗ Corresponding author. Tel.: +44 141 5524400x3653; fax: +44 141 552 2079. E-mail address: david@stams.strath.ac.uk (D. Greenhalgh). 1 Fraser Lewis wishes to thank the EPSRC for a studentship for this work. 0362-546X/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.12.018