American Institute of Mathematical Sciences

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May  2014, 19(3): 735-745. doi: 10.3934/dcdsb.2014.19.735

Analysis of a CD4$^+$ T cell viral infection model with a class of saturated infection rate

Zhixing Hu 1, , Weijuan Pang 1, , Fucheng Liao 1, and  Wanbiao Ma 2,

1. 

Department of Applied Mathematics, University of Science and Technology Beijing, Beijing, 100083, China, China, China
2. 

Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083

Received  October 2012 Revised  October 2013 Published  February 2014

This paper formulates and analyzes an HIV-1 infection model with saturated infection rate. We first discuss the boundedness of the solution and the existence of the equilibrium. The local stability of the virus-free equilibrium and infected equilibrium is established by analyzing the roots of the characteristic equations. Furthermore, we study the global stability of the virus-free equilibrium and infected equilibrium by using suitable Lyapunov function and LaSalle's invariance principle, and obtain sufficient conditions for the global stability of the infected equilibrium. Finally, numerical simulations are presented to illustrate the main results.
Keywords: Lyapunov function, saturated infection rate., HIV, global asymptotical stability.
Mathematics Subject Classification: Primary: 92D30; Secondary: 34K2.
Citation: Zhixing Hu, Weijuan Pang, Fucheng Liao, Wanbiao Ma. Analysis of a CD4$^+$ T cell viral infection model with a class of saturated infection rate. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 735-745. doi: 10.3934/dcdsb.2014.19.735
References:
[1]

S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad.,, Sci. USA., 94 (1997), 6971.   Google Scholar

[2]

B. Buonomo and C. Vargas-De-León, Global stability for an HIV-1 infection model including an eclipse stage of infected cells,, J. Math. Anal. Appl., 385 (2012), 709.  doi: 10.1016/j.jmaa.2011.07.006.  Google Scholar

[3]

D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density- dependent regulation of two microparasites of Daphnia magna,, Oecologia, 122 (2000), 200.  doi: 10.1007/PL00008847.  Google Scholar

[4]

D. Ho, A. Neumann, A. Perelson, W. Chen, J. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4+ lymphocytes in HIV-1 infection,, Nature, 373 (1995), 123.  doi: 10.1038/373123a0.  Google Scholar

[5]

D. Kirschner, Using mathematics to understand HIV immune dynamics,, Notices Amer. Math. Soc., 43 (1996), 191.   Google Scholar

[6]

A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models,, Appl. Math. Lett., 15 (2002), 955.  doi: 10.1016/S0893-9659(02)00069-1.  Google Scholar

[7]

M. Y. Li, H. L. Smith and L. Wang, Global dynamics of an SEIR epidemic model with vertical transmission,, SIAM J. Math. Anal., 62 (2001), 58.  doi: 10.1137/S0036139999359860.  Google Scholar

[8]

C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis,, J. Math. Anal. Appl., 338 (2008), 518.  doi: 10.1016/j.jmaa.2007.05.012.  Google Scholar

[9]

M. Nowak, R. Anderson, M. Boerlijst, S. Bonhoeffer, R. May and A. McMichael, HIV-1 evolution and disease progression,, Science, 274 (1996), 1008.  doi: 10.1126/science.274.5289.1008.  Google Scholar

[10]

M. Nowak, S. Bonhoeffer, G. Shaw and R. May, Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell populations,, J. Theoret. Biol., 184 (1997), 203.  doi: 10.1006/jtbi.1996.0307.  Google Scholar

[11]

M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology,, Oxford University Press, (2000).   Google Scholar

[12]

A. S. Perelson, Modelling viral and immune system dynamics,, Nat. Rev. Immunol, 2 (2002), 28.  doi: 10.1038/nri700.  Google Scholar

[13]

A. Perelson, D. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells,, Math. Biosci., 114 (1993), 81.  doi: 10.1016/0025-5564(93)90043-A.  Google Scholar

[14]

A. Perelson and P. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3.  doi: 10.1137/S0036144598335107.  Google Scholar

[15]

A. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time,, Science, 271 (1996), 1582.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[16]

R. R. Regoes, D. Ebert and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology,, Proc. R. Soc. Lond. B., 269 (2002), 271.  doi: 10.1098/rspb.2001.1816.  Google Scholar

[17]

L. Rong, M. A. Gilchrist, Z. Feng and A. S. Perelson, Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility,, J. Theoret. Biol., 247 (2007), 804.  doi: 10.1016/j.jtbi.2007.04.014.  Google Scholar

[18]

X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics,, J. Math. Anal. Appl., 329 (2007), 281.  doi: 10.1016/j.jmaa.2006.06.064.  Google Scholar

[19]

J. Tumwiine, J. Y. T. Mugisha and L. S. Luboobi, A host-vector model for malaria with infective immigrants,, J. Math. Anal. Appl., 361 (2010), 139.  doi: 10.1016/j.jmaa.2009.09.005.  Google Scholar

[20]

C. Vargas De León, Constructions of Lyapunov functions for classic SIS, SIR and SIRS epidemic models with variable population size,, Foro-Red-Mat: Revista Electrónica de Contenido Matemático, 26 (2009).   Google Scholar

[21]

L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells,, Math. Biosci., 200 (2006), 44.  doi: 10.1016/j.mbs.2005.12.026.  Google Scholar

show all references

References:
[1]

S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad.,, Sci. USA., 94 (1997), 6971.   Google Scholar

[2]

B. Buonomo and C. Vargas-De-León, Global stability for an HIV-1 infection model including an eclipse stage of infected cells,, J. Math. Anal. Appl., 385 (2012), 709.  doi: 10.1016/j.jmaa.2011.07.006.  Google Scholar

[3]

D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density- dependent regulation of two microparasites of Daphnia magna,, Oecologia, 122 (2000), 200.  doi: 10.1007/PL00008847.  Google Scholar

[4]

D. Ho, A. Neumann, A. Perelson, W. Chen, J. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4+ lymphocytes in HIV-1 infection,, Nature, 373 (1995), 123.  doi: 10.1038/373123a0.  Google Scholar

[5]

D. Kirschner, Using mathematics to understand HIV immune dynamics,, Notices Amer. Math. Soc., 43 (1996), 191.   Google Scholar

[6]

A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models,, Appl. Math. Lett., 15 (2002), 955.  doi: 10.1016/S0893-9659(02)00069-1.  Google Scholar

[7]

M. Y. Li, H. L. Smith and L. Wang, Global dynamics of an SEIR epidemic model with vertical transmission,, SIAM J. Math. Anal., 62 (2001), 58.  doi: 10.1137/S0036139999359860.  Google Scholar

[8]

C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis,, J. Math. Anal. Appl., 338 (2008), 518.  doi: 10.1016/j.jmaa.2007.05.012.  Google Scholar

[9]

M. Nowak, R. Anderson, M. Boerlijst, S. Bonhoeffer, R. May and A. McMichael, HIV-1 evolution and disease progression,, Science, 274 (1996), 1008.  doi: 10.1126/science.274.5289.1008.  Google Scholar

[10]

M. Nowak, S. Bonhoeffer, G. Shaw and R. May, Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell populations,, J. Theoret. Biol., 184 (1997), 203.  doi: 10.1006/jtbi.1996.0307.  Google Scholar

[11]

M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology,, Oxford University Press, (2000).   Google Scholar

[12]

A. S. Perelson, Modelling viral and immune system dynamics,, Nat. Rev. Immunol, 2 (2002), 28.  doi: 10.1038/nri700.  Google Scholar

[13]

A. Perelson, D. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells,, Math. Biosci., 114 (1993), 81.  doi: 10.1016/0025-5564(93)90043-A.  Google Scholar

[14]

A. Perelson and P. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3.  doi: 10.1137/S0036144598335107.  Google Scholar

[15]

A. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time,, Science, 271 (1996), 1582.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[16]

R. R. Regoes, D. Ebert and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology,, Proc. R. Soc. Lond. B., 269 (2002), 271.  doi: 10.1098/rspb.2001.1816.  Google Scholar

[17]

L. Rong, M. A. Gilchrist, Z. Feng and A. S. Perelson, Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility,, J. Theoret. Biol., 247 (2007), 804.  doi: 10.1016/j.jtbi.2007.04.014.  Google Scholar

[18]

X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics,, J. Math. Anal. Appl., 329 (2007), 281.  doi: 10.1016/j.jmaa.2006.06.064.  Google Scholar

[19]

J. Tumwiine, J. Y. T. Mugisha and L. S. Luboobi, A host-vector model for malaria with infective immigrants,, J. Math. Anal. Appl., 361 (2010), 139.  doi: 10.1016/j.jmaa.2009.09.005.  Google Scholar

[20]

C. Vargas De León, Constructions of Lyapunov functions for classic SIS, SIR and SIRS epidemic models with variable population size,, Foro-Red-Mat: Revista Electrónica de Contenido Matemático, 26 (2009).   Google Scholar

[21]

L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells,, Math. Biosci., 200 (2006), 44.  doi: 10.1016/j.mbs.2005.12.026.  Google Scholar

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