Analysis of a Nonautonomous HIV/AIDS Model

G. P. Samanta

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 6, page 70-95
  • ISSN: 0973-5348

Abstract

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In this paper we have considered a nonlinear and nonautonomous stage-structured HIV/AIDS epidemic model with an imperfect HIV vaccine, varying total population size and distributed time delay to become infectious due to intracellular delay between initial infection of a cell by HIV and the release of new virions. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values R0 and R∗ and further obtained that the disease will be permanent when R0 > 1 and the disease will be going to extinct when R∗ < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

How to cite

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Samanta, G. P.. "Analysis of a Nonautonomous HIV/AIDS Model." Mathematical Modelling of Natural Phenomena 5.6 (2010): 70-95. <http://eudml.org/doc/197637>.

@article{Samanta2010,
abstract = {In this paper we have considered a nonlinear and nonautonomous stage-structured HIV/AIDS epidemic model with an imperfect HIV vaccine, varying total population size and distributed time delay to become infectious due to intracellular delay between initial infection of a cell by HIV and the release of new virions. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values R0 and R∗ and further obtained that the disease will be permanent when R0 > 1 and the disease will be going to extinct when R∗ < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.},
author = {Samanta, G. P.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {HIV/AIDS; time delay; permanence; extinction; Lyapunov functional; global stability},
language = {eng},
month = {4},
number = {6},
pages = {70-95},
publisher = {EDP Sciences},
title = {Analysis of a Nonautonomous HIV/AIDS Model},
url = {http://eudml.org/doc/197637},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Samanta, G. P.
TI - Analysis of a Nonautonomous HIV/AIDS Model
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/4//
PB - EDP Sciences
VL - 5
IS - 6
SP - 70
EP - 95
AB - In this paper we have considered a nonlinear and nonautonomous stage-structured HIV/AIDS epidemic model with an imperfect HIV vaccine, varying total population size and distributed time delay to become infectious due to intracellular delay between initial infection of a cell by HIV and the release of new virions. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values R0 and R∗ and further obtained that the disease will be permanent when R0 > 1 and the disease will be going to extinct when R∗ < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.
LA - eng
KW - HIV/AIDS; time delay; permanence; extinction; Lyapunov functional; global stability
UR - http://eudml.org/doc/197637
ER -

References

top
  1. R. M. Anderson. The role of mathematical models in the study of HIV transmission and the epidemiology of AIDS. J. AIDS, 1 (1988), 241–256. 
  2. R. M. Anderson R. M. May. Population Biology of Infectious Diseases. Part I. Nature, 280 (1979), 361–367. 
  3. R. M. Anderson, G. F. Medly, R. M. May A. M. Johnson. A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS. IMA J. Math. Appl. Med. Biol., 3 (1986), 229–263. 
  4. M. Bachar A. Dorfmayr. HIV treatment models with time delay. C.R. Biologies, 327 (2004), 983–994. 
  5. BBC News (BBC). HIV reduces infection. September 2009, http://news.bbc.co.uk/2/hi/health/8272113.stm.  
  6. S. Blower. Calculating the consequences: HAART and risky sex. AIDS, 15 (2001), 1309–1310. 
  7. F. Brauer. Models for the disease with vertical transmission and nonlinear population dynamics. Math. Biosci., 128 (1995), 13–24. 
  8. S. Busenberg, K. Cooke. Vertically transmitted diseases. Springer, Berlin, 1993.  
  9. L. M. Cai, X. Li, M. Ghosh B. Guo. Stability of an HIV/AIDS epidemic model with treatment. J. Comput. Appl. Math., 229 (2009), 313–323. 
  10. V. Capasso. Mathematical structures of epidemic systems, Lectures Notes in Biomathematics, Vol. 97. Springer-Verlag, Berlin, 1993.  
  11. Centers for Disease Control and Prevention. HIV and its transmission. Divisions of HIV/AIDS Prevention, 2003.  
  12. C. Connell McCluskey. A model of HIV/AIDS with staged progression and amelioration. Math. Biosci., 181 (2003), 1–16. 
  13. R. V. Culshaw S. Ruan. A delay-differential equation model of HIV infection of CD4 +T-cells. Math. Biosci., 165 (2000), 27–39. 
  14. J. M. Cushing. Integrodifferential equations and delay models in population dynamics. Spring, Heidelberg, 1977.  
  15. O. Diekmann, J. A. P. Heesterbeek. Mathematical epidemiology of infectious diseases: model building, analysis, and interpretation. John Wiley and Sons Ltd., Chichester, New York, 2000.  
  16. E. H. Elbasha A. B. Gumel. Theoretical assessment of public health impact of imperfect prophylactic HIV-1 vaccines with therapeutic benefits. Bull. Math. Biol., 68 (2006), 577–614. 
  17. J. Esparza S. Osmanov. HIV vaccine: a global perspective. Curr. Mol. Med., 3 (2003), 183–193. 
  18. K. Gopalsamy. Stability and oscillations in delay-differential equations of population dynamics. Kluwer, Dordrecht, 1992.  
  19. D. Greenhalgh, M. Doyle F. Lewis. A mathematical treatment of AIDS and condom use . IMA J. Math. Appl. Med. Biol., 18 (2001), 225–262. 
  20. A. B. Gumel, C. C. McCluskey P. van den Driessche. Mathematical study of a staged-progression HIV model with imperfect vaccine. Bull. Math. Biol., 68 (2006), 2105–2128. 
  21. J. K. Hale, S. M. V. Lunel. Introduction to functional differential equations. Springer-Verlag, New York, 1993.  
  22. A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May M. A. Nowak. Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. Proc. Nat. Acad. Sci. USA, 93 (1996), 7247–7251. 
  23. G. Herzong R. Redheffer. Nonautonomous SEIRS and Thron models for epidemiology and cell biology. Nonlinear Anal.: RWA, 5 (2004), 33–44. 
  24. H. W. Hethcote, J. W. Van Ark. Modelling HIV transmission and AIDS in the United States, in: Lect. Notes Biomath., vol. 95. Springer, Berlin, 1992.  
  25. Y. H. Hsieh C. H. Chen. Modelling the social dynamics of a sex industry: Its implications for spread of HIV/AIDS. Bull. Math. Biol., 66 (2004), 143–166. 
  26. Y. H. Hsieh K. Cooke. Behaviour change and treatment of core groups: its effect on the spread of HIV/AIDS. IMA J. Math. Appl. Med. Biol., 17 (2000), 213–241. 
  27. D. W. Jordan, P. Smith. Nonlinear ordinary differential equations. Oxford University Press, New York, 2004.  
  28. W. O. Kermack A. G. Mckendrick.Contributions to the mathematical theory of epidemics. Part I. Proc. R. Soc. A, 115 (1927), No. 5, 700–721. 
  29. Y. Kuang. Delay-differential equations with applications in population dynamics. Academic Press, New York, 1993.  
  30. P. D. Leenheer H. L. Smith. Virus dynamics: a global analysis. SIAM. J. Appl. Math., 63 (2003), 1313–1327. 
  31. M. Y. Li, H. L. Smith L. Wang. Global dynamics of an SEIR epidemic with vertical transmission. SIAM. J. Appl. Math., 62 (2001), No. 1, 58–69. 
  32. M. C. I. Lipman, R. W. Baker, M. A. Johnson. An atlas of differential diagnosis in HIV disease. CRC Press-Parthenon Publishers, pp. 22-27, 2003.  
  33. Z. Ma, Y. Zhou, W. Wang, Z. Jin. Mathematical modelling and research of epidemic dynamical systems. Science Press, Beijing, 2004.  
  34. R. M. May R. M. Anderson. Transmission dynamics of HIV infection. Nature, 326 (1987), 137–142. 
  35. Medical News Today, dated 9th February, 2007, East Sussex, TN 40 9BA, United Kingdom.  
  36. X. Meng, L. Chen H. Cheng. Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination. Appl. Math. Comput., 186 (2007), 516–529. 
  37. R. Naresh, A. Tripathi S. Omar. Modelling the spread of AIDS epidemic with vertical transmission. Appl. Math. Comput., 178 (2006), 262–272. 
  38. A. S. Perelson P. W. Nelson. Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev., 41 (1999), No. 1, 3–44. 
  39. A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard D. D. Ho. HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science, 271 (1996), 1582–1586. 
  40. G. P. Samanta. Dynamic behaviour for a nonautonomous heroin epidemic model with time delay. J. Appl. Math. Comput., 2009, DOI .  URI10.1007/s12190-009-0349-z
  41. J. Shiver E. Emini. Recent advances in the development of HIV-1 vaccines using replication-incompetant adenovirus vectors. Ann. Rev. Med., 55 (2004), 355–372. 
  42. C. A. Stoddart R. A. Reyes. Models of HIV-1 disease: A review of current status. Drug Discovery Today: Disease Models, 3 (2006), No. 1, 113–119. 
  43. Z. Teng L. Chen. The positive periodic solutions of periodic Kolmogorov type systems with delays. Acta Math. Appl. Sin., 22 (1999), 446–456. 
  44. H. R. Thieme. Uniform weak implies uniform strong persistence for non-autonomous semiflows. Proc. Am. Math. Soci., 127 (1999), 2395–2403. 
  45. H. R. Thieme. Uniform persistence and permanence for nonautonomous semiflows in population biology. Math. Biosci., 166 (2000), 173–201. 
  46. UNAIDS. 2007 AIDS epidemic update. WHO, December 2007. 
  47. L. Wang M. Y. Li. Mathematical analysis of the global dynamics of a model for HIV infection of CD4+T-cells. Math. Biosci., 200 (2006), 44–57. 
  48. K. Wang, W. Wang X. Liu. Viral infection model with periodic lytic immune response. Chaos Solitons Fractals, 28 (2006), No. 1, 90–99. 
  49. Wikipedia. HIV vaccine. September, 2009, . URIhttp://en.wikipedia.org/wiki/HIV_vaccine
  50. T. Zhang Z. Teng. On a nonautonomous SEIRS model in epidemiology. Bull. Math. Biol., 69 (2007), 2537–2559. 
  51. T. Zhang Z. Teng. Permanence and extinction for a nonautonomous SIRS epidemic model with time delay. Appl. Math. Model., 33 (2009), 1058–1071. 
  52. R. M. Zinkernagel. The challenges of an HIV vaccine enterprise. Science, 303 (2004), 1294–1297. 

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