American Institute of Mathematical Sciences

Advanced
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

[1]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441
[2]

Yolanda Guerrero–Sánchez, Muhammad Umar, Zulqurnain Sabir, Juan L. G. Guirao, Muhammad Asif Zahoor Raja. Solving a class of biological HIV infection model of latently infected cells using heuristic approach. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020431
[3]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362
[4]

Xueli Bai, Fang Li. Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3075-3092. doi: 10.3934/dcds.2020035
[5]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002
[6]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[7]

Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020  doi: 10.3934/jcd.2021006
[8]

Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020378
[9]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304
[10]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331
[11]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041
[12]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
[13]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106
[14]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443
[15]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117
[16]

Alex P. Farrell, Horst R. Thieme. Predator – Prey/Host – Parasite: A fragile ecoepidemic system under homogeneous infection incidence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 217-267. doi: 10.3934/dcdsb.2020328
[17]

Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261
[18]

Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332
[19]

Skyler Simmons. Stability of broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015
[20]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences