Modelling the spread of AIDS epidemic with vertical transmission

This paper considers a non-linear mathematical model for HIV epidemic that spreads in a variable size population through both horizontal and vertical transmission. The equilibrium points of the model system are found and their stability is investigated. The model exhibits two equilibria namely, the disease-free and the endemic equilibrium. It is found that if the basic reproduction number R0 < 1, the disease-free equilibrium is always locally asymptotically stable and in such a case the endemic equilibrium does not exist. If R0 > 1, a unique endemic equilibrium exists which is locally asymptotically stable and becomes globally asymptotically stable under certain conditions showing that the disease becomes endemic due to constant immigration of population into the community. Using stability theory and computer simulation, it is shown that by controlling the rate of vertical transmission, the spread of the disease can be reduced significantly and consequently the equilibrium values of infective and AIDS population can be maintained at desired levels. A numerical study of the model is also used to investigate the influence of certain other key parameters on the spread of the disease.