Bulletin of Mathematical Biology

, Volume 68, Issue 8, pp 2105–2128 | Cite as

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

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


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.


HIV/AIDS Staged progression Vaccination reproduction number Global stability 


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  1. Anderson, R.M., May, R.M., 1991. Infectious Diseases of Humans. Oxford University Press, London/New York.Google Scholar
  2. Arino, J., McCluskey, C.C., van den Driessche, P., 2003. Global results for an epidemic model with vaccination that exhibits backward bifurcation. SIAM J. Appl. Math. 64, 260–276.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Baggeley, R.F., Ferguson, N.M., Garnett, G.P., 2005. The epidemiological impact of antiretroviral use predicted by mathematical models: A review. Emerg. Themes Epidemiol. 2, 9. ( Accessed September 20, 2005.Google Scholar
  4. Blower, S.M., McLean, A.R., 1994. Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco. Science 265, 1451–1454.CrossRefGoogle Scholar
  5. Brauer, F., Castillo-Chavez, C., 2000. Mathematical Models in Population Biology and Epidemiology. Texts in Applied Mathematics Series, vol. 40. Springer-Verlag, New York.Google Scholar
  6. Burton, D.R., et al., 2004. A sound rationale needed for Phase III HIV-1 vaccine trials. Science 303, 316.Google Scholar
  7. Castillo-Chavez, C., Feng, Z., Huang, W., 2002. On the computation of \(\cal R\) and its role on global stability. In: Castillo-Chavez, C., with Blower, S., van den Driessche, P., Kirschner, D., Yakubu, A.-A. (Eds.), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction. The IMA Volumes in Mathematics and its Applications, vol. 125. Springer, New York, pp. 229–250.Google Scholar
  8. Chang, M.L., Vitek, C., Esparza, J., 2003. Public health considerations for the use of a first generation HIV vaccine. Report from a WHO-UNAIDS-CDC consultation, Geneva, 20–21 November 2002. AIDS 17, W1–10.Google Scholar
  9. Corbett, B.D., Moghadas, S.M., Gumel, A.B., 2003. Subthreshold domain of bistable equilibria for a model of HIV epidemiology. IJMMS 58, 3679–3698.CrossRefMathSciNetGoogle Scholar
  10. Del Valle, S., Evangelista, A.M., Velasco, M.C., Kribs Zaleta, C.M., Schmitz, S.H., 2004. Effects of education, vaccination and treatment on HIV transmission in homosexuals with genetic heterogeneity. Math. Bios. 187, 111–133.zbMATHCrossRefGoogle Scholar
  11. Elbasha, E.H., Gumel, A.B., 2006. Theoretical assessment of public health impact of imperfect prophylactic HIV-1 vaccines with therapeutic benefits. Bull. Math. Biol. DOI 10.1007/s11538-005-9057-5.Google Scholar
  12. Esparza, J., Osmanov, S., 2003. HIV vaccines: A global perspective. Curr. Mol. Med. 3, 183–193.CrossRefGoogle Scholar
  13. Fleck, F., 2004. Developing economies shrink as AIDS reduces workforce. BMJ 329, 129.Google Scholar
  14. Gray, R., Li, X., Wawer, M., Gange, S., Serwadda, D., et al., 2003. Stochastic simulation of the impact of antiretroviral therapy and HIV vaccines on HIV transmission; Rakai, Uganda. AIDS 17, 1941–1951.CrossRefGoogle Scholar
  15. Hethcote, H.W., 1989. Three basic epidemiological models. In: Gross, L., Hallam, T.G., Levin, S.A. (Eds.), Applied Mathematical Ecology. Springer, Berlin, pp. 119–144.Google Scholar
  16. Hethcote, H.W., 2000. The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653.zbMATHCrossRefMathSciNetGoogle Scholar
  17. Hyman, J.M., Li, J., Stanley, E.A., 1999. The differential infectivity and staged progression models for the transmission of HIV. Math. Biosci. 208, 227–249.Google Scholar
  18. Klausner, R.D., et al., 2003. The need for a global HIV vaccine enterprise. Science 300, 2036–2039.CrossRefGoogle Scholar
  19. Kribs-Zaleta, C., Velasco-Hernández, J., 2000. A simple vaccination model with multiple endemic states. Math. Biosci. 164, 183–201.zbMATHCrossRefGoogle Scholar
  20. Lakshmikantham, V., Leela, S., Martynyuk, A.A., 1989. Stability Analysis of Nonlinear Systems. Marcel Dekker, Inc., New York and Basel.zbMATHGoogle Scholar
  21. Lansky, A., Nakashima, A., Jones, J., 2000. Risk behaviors related to heterosexual transmission from HIV-infected persons. Sex. Transm. Dis. 27, 483–489.CrossRefGoogle Scholar
  22. Lee, D., et al., 2004. Breakthrough infections during phase 1 and 2 prime-boost HIV-1 vaccine trials with canarypox vectors (ALVAC) and booster dose of recombinant gp120 or gp160. J. Infect. Dis. 190(5), 903–907.CrossRefGoogle Scholar
  23. Li, M.Y., Muldowney, J.S., 1995. On R.A. Smith's autonomous convergence theorem. Rocky Mount. J. Math. 25, 365–379.zbMATHMathSciNetCrossRefGoogle Scholar
  24. Li, M.Y., Muldowney, J.S., 1996. A geometric approach to global-stability problems. SIAM J. Math. Anal. 21, 1070–1083.CrossRefMathSciNetGoogle Scholar
  25. McCluskey, C.C., 2003. A Model for HIV/AIDS with staged progression and amelioration. Math. Bios. 181, 1–16.zbMATHCrossRefMathSciNetGoogle Scholar
  26. McCluskey, C.C., 2005. A strategy for constructing Lyapunov functions for non-autonomous linear differential equations. Linear Algebra Appl. 409, 100–110.zbMATHCrossRefMathSciNetGoogle Scholar
  27. McLean, A.R., Blower, S.M., 1993. Imperfect vaccines and herd immunity to HIV. Proc. R. Soc. Lond. Series B 253, 9–13.CrossRefGoogle Scholar
  28. Nicolosi, A., Musicco, M., Saracco, A., Lazzarin, A., 1994. Risk factors for woman-to-man sexual transmission of the human immunodeficiency virus. J. Acq. Immun. Def. Synd. 7, 296–300.Google Scholar
  29. O'Brien, T., Busch, M., Donegan, E., Ward, J., Wong, L., et al., 1994. Heterosexual transmission of human immunodeficiency virus type 1 from transfusion recipients to their sex partners. J. Acquir. Immune. Defic. Syndr. 7, 705–710.Google Scholar
  30. Perelson, A., Nelson, P., 1999. Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41, 3–44.zbMATHCrossRefMathSciNetGoogle Scholar
  31. Shiver, J., Emini, E., 2004. Recent advances in the development of HIV-1 vaccines using replication-incompetent adenovirus vectors. Ann. Rev. Med. 55, 355–372.CrossRefGoogle Scholar
  32. Shiver, J., Fu, T.-M., Chen, L., Casimiro, D., Davies, M.-A., et al., 2002. Replication-incompetent adenoviral vaccine vector elicits effective anti-immunodeficiency-virus immunity. Nature 415, 331–335.CrossRefGoogle Scholar
  33. Smith, R.J., Blower, S.M., 2004. Could disease-modifying HIV vaccines cause population-level pervasity? Lancet Infec. Dis. 4, 636–639.Google Scholar
  34. Smith, H.L., Waltman, P., 1995. The Theory of the Chemostat. Cambridge University Press.Google Scholar
  35. van den Driessche, P., Watmough, J., 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Bios. 180, 29–48.zbMATHCrossRefMathSciNetGoogle Scholar
  36. World Bank, 1997. Confronting AIDS: Public Priorities in a Global Epidemic. Oxford University Press, Oxford.Google Scholar
  37. WHO, 2004. The World Health Report: Changing History. World Health Organization, 1211 Geneva 27, Switzerland.Google Scholar
  38. Zinkernagel, R.M., 2004. The challenges of an HIV vaccine enterprise. Science 303, 1294–1297.CrossRefGoogle Scholar

Copyright information

© 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

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