Hopf bifurcation in two SIRS density dependent epidemic models

D. Greenhalgh, Q. J.A. Khan, F. I. Lewis

Research output: Contribution to journalArticlepeer-review

28 Citations (Scopus)


This paper uses two SIRS type epidemiological models to examine the impact on the spread of disease caused by vaccination when the immunity gained from such an intervention is not lifelong. This occurs, for example, in vaccination against influenza. We assume that susceptible individuals become immune immediately after vaccination and that immune individuals become susceptible to infection after a sufficient lapse of time. In our first model, we consider a constant contact rate between infectious and susceptible individuals, whereas in our second model this depends on the current size of the population. The death rate in both models depends on population density. We examine the different types of dynamic and long term behaviour possible in our models and in particular examine the existence and stability of equilibrium solutions. We find that Hopf bifurcation is theoretically possible but appears not to occur for realistic parameter values. Numerical simulations confirm the analytical results. The paper concludes with a brief discussion.

Original languageEnglish
Pages (from-to)1261-1283
Number of pages23
JournalMathematical and Computer Modelling
Issue number11-12
Publication statusPublished - Jun 2004


  • Contact rates
  • Density dependence
  • Hopf bifurcation
  • Immunization
  • SIRS epidemic models
  • Time delay

ASJC Scopus subject areas

  • Modelling and Simulation
  • Computer Science Applications


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