We develop a mathematical model to monitor the effect of imperfectvaccines on the transmission dynamics of infectious diseases. It isassumed that the vaccine efficacy is not $100%$ and may wane withtime. The model will be analyzed using a new technique based on someresults related to the Poincaré index of a piecewise smooth Jordancurve defined as the boundary of a positively invariant region forthe model. Using global analysis of the model, it is shown thatreducing the basic reproductive number (R₀) to values less thanone no longer guarantees disease eradication. This analysis isextended to determine the threshold level of vaccination coveragethat guarantees disease eradication.
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Seyed M. Moghadas (2004) studied this question.