Modelling the effect of imperfect vaccines on disease epidemiology

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 it may wane with time. The model will be analysed 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 the global analysis of the model, it is shown that reducing the basic reproductive number (ℛ0) 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.