Journal of Mathematical Biology

, Volume 60, Issue 5, pp 711–726 | Cite as

An accurate two-phase approximate solution to an acute viral infection model

  • Amber M. SmithEmail author
  • Frederick R. Adler
  • Alan S. Perelson


During an acute viral infection, virus levels rise, reach a peak and then decline. Data and numerical solutions suggest the growth and decay phases are linear on a log scale. While viral dynamic models are typically nonlinear with analytical solutions difficult to obtain, the exponential nature of the solutions suggests approximations can be found. We derive a two-phase approximate solution to the target cell limited influenza model and illustrate its accuracy using data and previously established parameter values of six patients infected with influenza A. For one patient, the fall in virus concentration from its peak was not consistent with our predictions during the decay phase and an alternate approximation is derived. We find expressions for the rate and length of initial viral growth in terms of model parameters, the extent each parameter is involved in viral peaks, and the single parameter responsible for virus decay. We discuss applications of this analysis in antiviral treatments and in investigating host and virus heterogeneities.


Acute virus infection Influenza Virus dynamics model Approximation of nonlinear differential equations 

Mathematics Subject Classification (2000)

92B05 37N25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aoki FY, Macleod MD, Paggiaro P, Carewicz O, El Sawy A, Wat C, Griffiths M, Waalberg E, Ward P (2003) Early administration of oral oseltamivir increases the benefits of influenza treatment. J Antimicrob Chemother 51(1): 123–129CrossRefGoogle Scholar
  2. Baccam P, Beauchemin C, Macken CA, Hayden FG, Perelson AS (2006) Kinetics of influenza A virus infection in humans. J Virol 80(15): 7590–7599CrossRefGoogle Scholar
  3. Bonhoeffer S, May RM, Shaw GM, Nowak MA (1997) Virus dynamics and drug therapy. Proc Natl Acad Sci USA 94(13): 6971–6976CrossRefGoogle Scholar
  4. Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley, New YorkGoogle Scholar
  5. Dixit NM, Markowitz M, Ho DD, Perelson AS (2004) Estimates of intracellular delay and average drug efficacy from viral load data of HIV-infected individuals under antiretroviral therapy. Antivir Ther 9: 237–246Google Scholar
  6. Handel A, Longini IM Jr, Antia R (2007) Neuraminidase inhibitor resistance in influenza: assessing the danger of its generation and spread. PLoS Comput Biol 3(12): e240CrossRefMathSciNetGoogle Scholar
  7. Ho DD, Neumann AU, Perelson AS, Chen W, Leonard JM, Markowitz M (1995) Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 373(6510): 123–126CrossRefGoogle Scholar
  8. Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond Ser A 115(772): 700–721CrossRefGoogle Scholar
  9. Lee HY, Topham DJ, Park SY, Hollenbaugh J, Treanor J, Mosmann TR, Jin X, Ward B, Miao H, Holden-Wiltse J, Perelson AS, Zand M, Wu H (2009) Simulation and prediction of the adaptive immune response to influenza A virus infection. J Virol 83(14): 7151–7165CrossRefGoogle Scholar
  10. Lewin S, Ribeiro RM, Walters T, Lau GK, Bowden S, Locarnini S, Perelson AS (2001) Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed. Hepatol 34(5): 1012–1020CrossRefGoogle Scholar
  11. Nelson PW, Mittler JE, Perelson AS (2001) Effect of Drug efficacy and the eclipse phase of the viral life cycle on estimates of HIV viral dynamic parameters. J Acquir Immune Defic Syndr 26(5): 405–412Google Scholar
  12. Neumann AU, Lam NP, Dahari H, Gretch DR, Wiley TE, Layden TJ, Perelson AS (1998) Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-therapy. Science 282(5386): 103–107CrossRefGoogle Scholar
  13. Nowak MA, Bonhoeffer S, Hill AM, Boehme R, Thomas HC, McDade H (1996) Viral dynamics in hepatitis B virus infection. Proc Natl Acad Sci USA 93(9): 4398–4402CrossRefGoogle Scholar
  14. Nowak MA, Lloyd AL, Vasquez GM, Wiltrout TA, Wahl LM, Bischofberger N, Williams J, Kinter A, Fauci AS, Hirsch VM, Lifson JD (1997) Viral dynamics of primary viremia and antiretroviral therapy in simian immunodeficiency virus infection. J Virol 71(10): 7518–7525Google Scholar
  15. Nowak MA, May R (2001) Virus dynamics: mathematical principles of immunology and virology. Oxford University Press, New YorkGoogle Scholar
  16. Perelson AS (2002) Modelling viral and immune system dynamics. Nat Rev Immunol 2(1): 28–36CrossRefGoogle Scholar
  17. Perelson AS, Essunger P, Cao Y, Vesanen M, Hurley A, Saksela K, Markowitz M, Ho DD (1997) Decay characteristics of HIV-1-infected compartments during combination therapy. Nature 387(6629): 188–191CrossRefGoogle Scholar
  18. Perelson AS, Kirschner DE, De Boer R (1993) Dynamics of HIV infection in CD4+ T cells. Math Biosci 114: 81–81zbMATHCrossRefGoogle Scholar
  19. Perelson AS, Neumann AU, Markowitz M, Leonard JM, Ho DD (1996) HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271(5255): 1582–1586CrossRefGoogle Scholar
  20. Stafford MA, Corey L, Cao Y, Daar ES, Ho DD, Perelson AS (2000) Modeling plasma virus concentration during primary HIV infection. J Theor Biol 203(3): 285–301CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Amber M. Smith
    • 1
    Email author
  • Frederick R. Adler
    • 2
  • Alan S. Perelson
    • 3
  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.Departments of Mathematics and BiologyUniversity of UtahSalt Lake CityUSA
  3. 3.Theoretical Biology and Biophysics Group, Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA

Personalised recommendations