Modelling the effect of imperfect vaccines on disease epidemiology

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November 2004, 4(4): 999-1012. doi: 10.3934/dcdsb.2004.4.999

Modelling the effect of imperfect vaccines on disease epidemiology

1.	Institute for Biodiagnostics, National Research Council Canada, Winnipeg, Manitoba, Canada, R3B 1Y6, Canada

Received March 2003 Revised February 2004 Published August 2004

Abstract
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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: equilibria, basic reproductive number, Epidemic models, Poincaré index, stability, positively invariant..

Mathematics Subject Classification: Primary: 92B05; Secondary: 37N2.

Citation: S.M. Moghadas. Modelling the effect of imperfect vaccines on disease epidemiology. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 999-1012. doi: 10.3934/dcdsb.2004.4.999

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