Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses

SIAM Journal on Applied Mathematics

Volumn 73, Issue 3, 2013, Pages 1280-1302

  Shu, Hongying ^a   Wang, Lin ^a   Watmough, James ^a  

^a UNIVERSITY OF NEW BRUNSWICK   (Canada)

Author keywords

Distributed delay; Global stability; Immune response; Lyapunov functional; Viral infection

Indexed keywords

DISTRIBUTED DELAYS; GLOBAL STABILITY; IMMUNE RESPONSE; LYAPUNOV FUNCTIONALS; VIRAL INFECTIONS;

ENZYME ACTIVITY; IMMUNE SYSTEM; LYAPUNOV FUNCTIONS; VIRUSES;

BIOLOGICAL SYSTEMS;

EID: 84884128689     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/120896463     Document Type: Article

References (37)

1. C. L. ALTHAUS AND R. J. DE BOER, Dynamics of immune escape during HIV/SIV infection, PLoS Comput. Biol., 4(7), e1000103.
2. F. V. ATKINSON AND J. R. HADDOCK, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31(1988), pp. 331-347.
3. S. BONHOEFFER, R. M. MAY, G. M. SHAW, AND M. A. NOWAK, Virus dynamics and drup therapy, Proc. Natl. Acad. Sci. USA, 94(1997), pp. 6971-6976.
4. A. A. CANABARRO, I. M. GLÉRIA, AND M. L. LYRA, Periodic solutions and chaos in a nonlinear model for the delayed cellular immune response, Phys. A, 342(2004), pp. 234-241.
5. P. DE LEENHEER AND H. L. SMITH, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63(2003), pp. 1313-1327.
6. V. V. GANUSOV, N. GOONETILLEKE, M. K. P. LIU, G. FERRARI, G. M. SHAW, A. J. MCMICHAEL, P. BORROW, B. T. KORBER, AND A. S. PERELSON, Fitness costs and diversity of the Cytotoxic T Lymphocyte (CTL) response determine the rate of CTL escape during acute and chronic phases of HIV infection, J. Virol., 85(2011), pp. 10518-10528.
7. P. GEORGESCU AND Y. H. HSIEH, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J. Appl. Math., 67(2006), pp. 337-353. (Pubitemid 46674841)
8. Z. GROSSMAN, M. POLIS, M. B. FEINBERG, Z. GROSSMAN, I. LEVI, S. JANKELEVICH, R. YARCHOAN, J. BOON, F. DE WOLF, J. M. A. LANGE, J. GOUDSMIT, D. S. DIMITROV, AND W. E. PAUL, Ongoing HIV dissemination during HAART, Nat. Med., 5(1999), pp. 1099-1104. (Pubitemid 29474403)
9. J. R. HADDOCK AND J. TERJÉKI, Liapunov-Razumikhin functions and an invariance principle for functional-differential equations, J. Differential Equations, 48(1983), pp. 95-122.
10. J. R. HADDOCK, T. KRISZTIN, AND J. TERJÉKI, Invariance principles for autonomous functional-differential equations, J. Integral Equations, 10(1985), pp. 123-136.
11. J. K. HALE AND S. M. VERDUYN LUNEL, Introduction to Functional Differential Equations, Appl. Math. Sci. 99, Springer, New York, 1993.
12. V. HERZ, S. BONHOEFFER, R. ANDERSON, R. MAY, AND M. NOWAK, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA, 93(1996), pp. 7247-7251. (Pubitemid 26243534)
13. G. HUANG, Y. TAKEUCHI, AND W. MA, Lyapunov functional for delay differential equations model of viral infections, SIAM J. Appl. Math., 70(2010), pp. 2693-2708.
14. N. L. KOMAROVA, E. BARNES, P. KLENERMAN, AND D. WODARZ, Boosting immunity by antiviral drug therapy: A simple relationship among timing, efficacy, and success, Proc. Natl. Acad. Sci. USA, 100(2003), pp. 1855-1860. (Pubitemid 36254538)
15. A. KOROBEINIKOV, Global properties of basic virus dynamics models, Bull. Math. Biol., 66(2004), pp. 879-883. (Pubitemid 38800956)
16. Y. KUANG, Delay Diffential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
17. D. LI AND W. MA, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl., 335(2007), pp. 683-691.
18. M. Y. LI AND H. SHU, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72(2010), pp. 1492-1505.
19. M. Y. LI AND H. SHU, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70(2010), pp. 2434-2448.
20. M. Y. LI AND H. SHU, Multiple stable periodic oscillations in a mathematical model of CTL-response to HTLV-I infection, Bull. Math. Biol., 73(2011), pp. 1774-1793.
21. M. Y. LI AND H. SHU, Joint effects of mitosis and intracellular delay on viral dynamics: Twoparameter bifurcation analysis, J. Math. Biol., 64(2012), pp. 1005-1020.
22. Y. LI, R. XU, Z. LI, AND S. MAO, Global dynamics of a delayed HIV-1 infection model with CTL immune response, Discrete Dyn. Nat. Soc., 2011 (2011).
23. S. LIU AND L. WANG, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Math. Biosci. Eng., 7(2010), pp. 675-685.
24. A. L. LLOYD, Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods, Proc. Roy. Soc. Lond. B, 268(2001), pp. 985-993. (Pubitemid 32453390)
25. C. C. MCCLUSKEY, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal., 11(2010), pp. 55-59.
26. A. MURASE, T. SASAKI, AND T. KAJIWARA, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51(2005), pp. 247-267. (Pubitemid 41253195)
27. Y. NAKATA, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375(2011), pp. 14-27.
28. P. W. NELSON AND A. S. PERELSON, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179(2002), pp. 73-94. (Pubitemid 34686428)
29. M. A. NOWAK AND C. R. M. BANGHAM, Population dynamics of immune responses to persistent viruses, Science, 272(1996), pp. 74-79. (Pubitemid 26110757)
30. M. A. NOWAK, S. BONHOEFFER, A. M. HILL, R. BOEHME, AND H. C. THOMAS, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93(1996), pp. 4398-4402. (Pubitemid 26144299)
31. A. S. PERELSON AND P. W. NELSON, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41(1999), pp. 3-44. (Pubitemid 129723746)
32. H. L. SMITH, Monotone Dynamical Systems, AMS, Providence, RI, 1995.
33. + T cells, Math. Biosci., 200(2006), pp. 44-57.
34. X. WANG, Y. TAO, AND X. SONG, Global stability of a virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response, Nonlinear Dyn., 66(2011), pp. 825-830.
35. X. WANG AND Y. TAO, Lyapunov function and global properties of virus dynamics with CTL immune response, Int. J. Biomath., 1(2008), pp. 443-448.
36. K. WANG, W. WANG, H. PANG, AND X. LIU, Complex dynamic behavior in a viral model with delayed immune response, Phys. D, 226(2007), pp. 197-208. (Pubitemid 46185615)
37. H. ZHU AND X. ZOU, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12(2009), pp. 511-524.