Bulletin of Mathematical Biology

, Volume 68, Issue 8, pp 2105–2128

# Mathematical Study of a Staged-Progression HIV Model with Imperfect Vaccine

• A. B. Gumel
• Connell C. McCluskey
• P. van den Driessche
Original Article

## Abstract

A staged-progression HIV model is formulated and used to investigate the potential impact of an imperfect vaccine. The vaccine is assumed to have several desirable characteristics such as protecting against infection, causing bypass of the primary infection stage, and offering a disease-altering therapeutic effect (so that the vaccine induces reversal from the full blown AIDS stage to the asymptomatic stage). The model, which incorporates HIV transmission by individuals in the AIDS stage, is rigorously analyzed to gain insight into its qualitative features. Using a comparison theorem, the model with mass action incidence is shown to have a globally-asymptotically stable disease-free equilibrium whenever a certain threshold, known as the vaccination reproduction number, is less than unity. Furthermore, the model with mass action incidence has a unique endemic equilibrium whenever this threshold exceeds unity. Using the Li-Muldowney techniques for a reduced version of the mass action model, this endemic equilibrium is shown to be globally-asymptotically stable, under certain parameter restrictions. The epidemiological implications of these results are that an imperfect vaccine can eliminate HIV in a given community if it can reduce the reproduction number to a value less than unity, but the disease will persist otherwise. Furthermore, a future HIV vaccine that induces the bypass of primary infection amongst vaccinated individuals (who become infected) would decrease HIV prevalence, whereas a vaccine with therapeutic effect could have a positive or negative effect at the community level.

## Keywords

HIV/AIDS Staged progression Vaccination reproduction number Global stability

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© Society for Mathematical Biology 2006

## Authors and Affiliations

• A. B. Gumel
• 1
• Connell C. McCluskey
• 2
• P. van den Driessche
• 3
1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada
2. 2.Department of MathematicsWilfrid Laurier UniversityWaterlooCanada
3. 3.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada