Abstract
A two strain HIV/AIDS model with treatment which allows AIDS patients with sensitive HIV-strain to undergo amelioration is presented as a system of non-linear ordinary differential equations. The disease-free equilibrium is shown to be globally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than unity. The centre manifold theory is used to show that the sensitive HIV-strain only and resistant HIV-strain only endemic equilibria are locally asymptotically stable when the associated reproduction numbers are greater than unity. Qualitative analysis of the model including positivity, boundedness and persistence of solutions are presented. The model is numerically analysed to assess the effects of treatment with amelioration on the dynamics of a two strain HIV/AIDS model. Numerical simulations of the model show that the two strains co-exist whenever the reproduction numbers exceed unity. Further, treatment with amelioration may result in an increase in the total number of infective individuals (asymptomatic) but results in a decrease in the number of AIDS patients. Further, analysis of the reproduction numbers show that antiretroviral resistance increases with increase in antiretroviral use.
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References
Ackleh AS, Allen LJS (2004) Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality. Discret Contin Dyn Syst B (in press)
Allen LJS, Kirupaharan N (2005) Asymptotic dynamics of deterministic and stochastic epidemic models with multiple pathogens. Int J Numer Anal Model 2(3):329–344
Allen LJS, Langais M, Phillips CJ (2003) The dynamics of two viral infections in a single host population with applications to hantavirus. Math Biosci 186:191–217
Anderson RM, Gupta S, May RM (1991) Potential of community-wide chemotherapy or immunotherapy to control the spread of HIV-1. Nature 350:356–359
Baggaley RF, Ferguson NM, Garnett GP (2005) The epidemiological impact of antiretroviral use predicted by mathematical models: a review. Emerg Themes Epidemiol
Barbalat I (1959) Syst\(\grave e\) me d’\(\acute e\) quationdiff \(\acute e\) rentielle d’Oscillation, Nonlin \(\acute e\) aires. Rev Roum Math Pures Appl 4:267–270
Birkoff G, Rota GC (1982) Ordinary differential equations. Ginn, Boston
Blower SM, Gerberding JL (1998) Understanding, predicting and controlling the emergence of drug-resistant tuberculosis: a theoretical framework. J Mol Med 76:624–636
Blyuss KB, Kyrychko YN (2005) On a basic model of a two-disease epidemic. Appl Math Comput 160:177–187
Carr J (1981) Applications of centre manifold theory. Springer-Verlag, New York
Castillo-Chavez C, Feng Z (1997) To treat or not to treat: the case of tuberculosis. J Math Biol 35: 629–656
Castillo-Chavez C, Feng Z (1998) Mathematical models for the disease dynamics of tuberculosis. World Scientific Press, Singapore, pp 629–656
Castillo-Chavez C, Song B (2004) Dynamical models of tuberculosis and their applications. Math Biosci Eng 1(2):361–404
Castillo-Chavez C, Feng Z, Huang W (2002) On the computation of R 0 and its role on global stability. http://math.la.asu.edu/chavez/2002/JB276.pdf
Dye C, William BG (2000) Criteria for the control of drug resistant tuberculosis. Proc Natl Acad Sci 97:8180–8185
Farmer P, Walton DA, Furin JJ (2000) The changing face of AIDS: implications for policy and practice. In: Mayer K, Pizer H (ed) The emergence of AIDS: the impact on immunology, microbiology,and public health. American Public Health Association, Washington, DC
Harrington PR, Lader BA (2002) Extent of cross resistance between agents used to treat HIV-1 infection in clinically derived isolates. Antimicrob Agents Chemother 46:909–912
Hsu Schmizt SF (2000) Effects of treatment or/and vaccination on HIV transmission in homosexuals with genetic heterogeneity. Math Biosci 167:1–18
Hsu Schmizt SF (2002) Effects of genetic heterogeneity on HIV transmission in homosexuals populations. In: Castillo-Chavez C, Blower S, van den Driessche P, Kirshner D, Yakubu A-A (eds) Mathematical approaches for emerging and re-emerging infectious diseases: models, methods and theory. Springer-Verlag, Berlin, pp 245–260
Hofbauer J, So JWH (1989) Uniform persistence and repellers for maps. Proc Am Math Soc 107:1137–1142
Hyman JM, Li J, Stanley EA (1999) The differential infectivity and staged progression models for the transmission of HIV. Math Biosci 155:77–109
Korobeinikov A, Maini PK (2004) A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Math Biosci Eng 1(1):57–60
Lader BA, Darby G, Richman DD (1989) HIV with reduced sensitivity to zidovudine isolated during prolonged therapy. Science 243:1731–1734
Levy SB (1992) The antibiotic paradox: how miracle drugs are destroying the miracle. Plenum Press, New York
Martcheva M, Ianelli M, Xue-Zhi-Li (2007) Subthreshold coexistence of strains: the impact of vaccination and mutation. Math Biosci Eng 4(2):287–317
May R, Nowak M (1995) Coinfection and the evolution of parasite virulence. Proc R Soc Lond 261:209–215
Mukandavire Z, Garira W (2006) HIV/AIDS model for assessing the effects of prophylactic sterilizing vaccines, condoms and treatment with amelioration. J Biol Syst 14(3):323–355
Naresh R, Tripathi A (2005) Modelling and analysis of HIV-TB co-infection in a variable population size. Math Model Anal 10(3):275–286
Nowak MA, Sigmund K (2002) Super-and coinfection: the two extremes. In: Dieckmann U, Metz JAJ, Sabelis M, Sigmund K (eds) Adaptive dynamics of infectious diseases: in pursuit of virulence management. Cambridge University Press, Cambridge
Operskalski EA, Stram DO, Busch MP, Huang et al (1997) Role of viral load in heterosexual transmission of human immunodeciency virus type-1 by blood transfusion recipients. Am J Epidemiol 146(8):655–661
Quinones-Mateu ME, Arts EJ (2002) Fitness of drug resistant HIV-1: methodology and clinical implications. Drug Resist Updat 5:224–233
Rossberry W, Seal A, Mphuka S (2005) Assessing the application of the three ones in Zambia. DFID Health System Centre, London
Schinazi RF, Lader BA, Mellow JW (2000) Mutations in retroviral genes associated with drug resistance: 2000–2001 update. Int Antiviral News 6:65–91
Sharomi O, Podder CN, Gumel AB, Song B (2007) Mathematical analysis of the transmission dynamics of HIV/TB co-infection in the presence of treatment. Math Biosci Eng (accepted)
Steven W, Kaye S, Corrah T (2004) Antiretroviral therapy in Africa. Br Med J 328:280–282
UNAIDS/WHO (2005) AIDS epidemic update. UNAIDS/WHO, Geneva
van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for the compartmental models of disease transmission. Math Biosci 180:29–48
Acknowledgements
C. P. Bhunu would like to acknowledge the financial support given to him by International Clinical Operational Health Services Research Training Award (ICOHRTA) through the Biomedical Research Training Institute (BRTI). The authors would like thank the anonymous referrees whose commends and suggestions really improved our work.
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Bhunu, C.P., Garira, W. & Magombedze, G. Mathematical Analysis of a Two Strain HIV/AIDS Model with Antiretroviral Treatment. Acta Biotheor 57, 361–381 (2009). https://doi.org/10.1007/s10441-009-9080-2
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DOI: https://doi.org/10.1007/s10441-009-9080-2