# American Institute of Mathematical Sciences

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

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.
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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