Effect of a preventive vaccine on the dynamics of HIV transmission

A deterministic mathematical model for the transmission dynamics of HIV infection in the presence of a preventive vaccine is considered. Although the equilibria of the model could not be expressed in closed form, their existence and threshold conditions for their stability are theoretically investigated. It is shown that the disease-free equilibrium is locally-asymptotically stable if the basic reproductive number ?<1 (thus, HIV disease can be eradicated from the community) and unstable if ?>1 (leading to the persistence of HIV within the community). A robust, positivity-preserving, non-standard finite-difference method is constructed and used to solve the model equations. In addition to showing that the anti-HIV vaccine coverage level and the vaccine-induced protection are critically important in reducing the threshold quantity ?, our study predicts the minimum threshold values of vaccine coverage and efficacy levels needed to eradicate HIV from the community.