A qualitative study of a vaccination model with non-linear incidence

We propose a new deterministic model for the dynamics of an infectious disease in the presence of a preventive (prophylactic) vaccine and an effective therapeutic treatment. The three-dimensional model, which assumes a non-linear incidence rate, is analysed qualitatively to determine the stability of its equilibria. The optimal vaccine coverage threshold needed for disease control and eradication is determined analytically (and verified using numerical simulations). The case where no vaccination is given (vaccination-free model) is also investigated. Using a Dulac function, it is shown that the vaccination-free model has no limit cycles.