Mathematical Modelling of Natural Phenomena

Research Article

Bifurcation Thresholds in an SIR Model with Information-Dependent Vaccination

A. d'Onofrioa1, P. Manfredia2 and P. Manfredia3

a1 Division of Epidemiology and Biostatistics, European Institute of Oncology Via Ripamonti 435, 20141 Milano, Italy

a2 Dipartimento di Statistica e Matematica Applicata all' Economia, Università di Pisa Via Ridolfi 10, 5612 Pisa, Italy

a3 Dipartimento di Scienze Economiche e Metodi Quantitativi, Università del Piemonte Orientale “A. Avogadro”, Via Perrone 18, 28100 Novara, Italy


Simple epidemiological models with information dependent vaccination functions can generate sustained oscillations via Hopf bifurcation of the endemic state. The onset of these oscillations depend on the shape of the vaccination function. A “global” approach is used to characterize the instability condition and identify classes of functions that always lead to stability/instability. The analysis allows the identification of an analytically determined “threshold vaccination function” having a simple interpretation: coverage functions lying always above the threshold always lead to oscillations, whereas coverage functions always below never lead to instability.

(Online publication June 15 2008)

Key Words:

  • SIR epidemiological models;
  • information-dependent vaccination;
  • stability;
  • Hopf bifurcations

Mathematics Subject Classification:

  • 92D30