The effects of vaccination and treatment on the spread of HIV/AIDS

This study investigates the effects of vaccination and treatment on the spread of HIV/AIDS. The objectives are (i) to derive conditions for the success of vaccination and treatment programs and (ii) to derive threshold conditions for the existence and stability of equilibria in terms of the effective reproduction number R. It is found, firstly, that the success of a vaccination and treatment program is achieved when R_0t < R₀, R _0t < R_0v and γ_eR_VT(σ) < R_UT(α), where R_0t and R_0v are respectively the reproduction numbers for populations consisting entirely of treated and vaccinated individuals, R₀ is the basic reproduction number in the absence of any intervention, R_UT(α) and R _VT(σ) are respectively the reproduction numbers in the presence of a treatment (α) and a combination of vaccination and treatment (σ) strategies. Secondly, that if R < 1, there exists a unique disease free equilibrium point which is locally asymptotically stable, while if R > 1 there exists a unique locally asymptotically stable endemic equilibrium point, and that the two equilibrium points coalesce at R = 1. Lastly, it is concluded heuristically that the stable disease free equilibrium point exists when the conditions R_0t < R₀, R_0t < R_0v and γ_eR_VT(σ) < R_UT(α) are satisfied.