Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Modelling the effect of imperfect vaccines on disease epidemiology

Pages: 999 - 1012, Volume 4, Issue 4, November 2004      doi:10.3934/dcdsb.2004.4.999

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S.M. Moghadas - Institute for Biodiagnostics, National Research Council Canada, Winnipeg, Manitoba, Canada, R3B 1Y6, Canada (email)

Abstract: We develop a mathematical model to monitor the effect of imperfect vaccines on the transmission dynamics of infectious diseases. It is assumed that the vaccine efficacy is not $100\%$ and may wane with time. The model will be analyzed using a new technique based on some results related to the Poincaré index of a piecewise smooth Jordan curve defined as the boundary of a positively invariant region for the model. Using global analysis of the model, it is shown that reducing the basic reproductive number ($\mathcal{R}_0$) to values less than one no longer guarantees disease eradication. This analysis is extended to determine the threshold level of vaccination coverage that guarantees disease eradication.

Keywords:  Epidemic models, basic reproductive number, PoincarĂ© index, equilibria, stability, positively invariant.
Mathematics Subject Classification:  Primary: 92B05; Secondary: 37N25.

Received: March 2003;      Revised: February 2004;      Available Online: August 2004.