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Modelling the effect of imperfect vaccines on disease epidemiology
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.