# American Institute of Mathematical Sciences

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

 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.
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 and 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), 6971-6976. [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), 709-720. doi: 10.1016/j.jmaa.2011.07.006. [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), 200-209. doi: 10.1007/PL00008847. [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), 123-126. doi: 10.1038/373123a0. [5] D. Kirschner, Using mathematics to understand HIV immune dynamics, Notices Amer. Math. Soc., 43 (1996), 191-202. [6] A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1. [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), 58-69. doi: 10.1137/S0036139999359860. [8] C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis, J. Math. Anal. Appl., 338 (2008), 518-535. doi: 10.1016/j.jmaa.2007.05.012. [9] M. Nowak, R. Anderson, M. Boerlijst, S. Bonhoeffer, R. May and A. McMichael, HIV-1 evolution and disease progression, Science, 274 (1996), 1008-1011. doi: 10.1126/science.274.5289.1008. [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), 203-217. doi: 10.1006/jtbi.1996.0307. [11] M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. [12] A. S. Perelson, Modelling viral and immune system dynamics, Nat. Rev. Immunol, 2 (2002), 28-36. doi: 10.1038/nri700. [13] A. Perelson, D. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A. [14] A. Perelson and P. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107. [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), 1582-1586. doi: 10.1126/science.271.5255.1582. [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), 271-279. doi: 10.1098/rspb.2001.1816. [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), 804-818. doi: 10.1016/j.jtbi.2007.04.014. [18] X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. doi: 10.1016/j.jmaa.2006.06.064. [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), 139-149. doi: 10.1016/j.jmaa.2009.09.005. [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). [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), 44-57. doi: 10.1016/j.mbs.2005.12.026.

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), 6971-6976. [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), 709-720. doi: 10.1016/j.jmaa.2011.07.006. [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), 200-209. doi: 10.1007/PL00008847. [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), 123-126. doi: 10.1038/373123a0. [5] D. Kirschner, Using mathematics to understand HIV immune dynamics, Notices Amer. Math. Soc., 43 (1996), 191-202. [6] A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960. doi: 10.1016/S0893-9659(02)00069-1. [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), 58-69. doi: 10.1137/S0036139999359860. [8] C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis, J. Math. Anal. Appl., 338 (2008), 518-535. doi: 10.1016/j.jmaa.2007.05.012. [9] M. Nowak, R. Anderson, M. Boerlijst, S. Bonhoeffer, R. May and A. McMichael, HIV-1 evolution and disease progression, Science, 274 (1996), 1008-1011. doi: 10.1126/science.274.5289.1008. [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), 203-217. doi: 10.1006/jtbi.1996.0307. [11] M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. [12] A. S. Perelson, Modelling viral and immune system dynamics, Nat. Rev. Immunol, 2 (2002), 28-36. doi: 10.1038/nri700. [13] A. Perelson, D. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A. [14] A. Perelson and P. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107. [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), 1582-1586. doi: 10.1126/science.271.5255.1582. [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), 271-279. doi: 10.1098/rspb.2001.1816. [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), 804-818. doi: 10.1016/j.jtbi.2007.04.014. [18] X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. doi: 10.1016/j.jmaa.2006.06.064. [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), 139-149. doi: 10.1016/j.jmaa.2009.09.005. [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). [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), 44-57. doi: 10.1016/j.mbs.2005.12.026.
 [1] Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525 [2] Jaouad Danane. Optimal control of viral infection model with saturated infection rate. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 363-375. doi: 10.3934/naco.2020031 [3] Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483 [4] Yu Yang, Yueping Dong, Yasuhiro Takeuchi. Global dynamics of a latent HIV infection model with general incidence function and multiple delays. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 783-800. doi: 10.3934/dcdsb.2018207 [5] Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053 [6] Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297 [7] A. M. Elaiw, N. H. AlShamrani. Global stability of HIV/HTLV co-infection model with CTL-mediated immunity. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1725-1764. doi: 10.3934/dcdsb.2021108 [8] Qi Wang. Global solutions of a Keller--Segel system with saturated logarithmic sensitivity function. Communications on Pure and Applied Analysis, 2015, 14 (2) : 383-396. doi: 10.3934/cpaa.2015.14.383 [9] Ting Guo, Haihong Liu, Chenglin Xu, Fang Yan. Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4223-4242. doi: 10.3934/dcdsb.2018134 [10] Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133 [11] Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859 [12] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [13] Songbai Guo, Wanbiao Ma. Global behavior of delay differential equations model of HIV infection with apoptosis. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 103-119. doi: 10.3934/dcdsb.2016.21.103 [14] Yoichi Enatsu, Yukihiko Nakata. Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate. Mathematical Biosciences & Engineering, 2014, 11 (4) : 785-805. doi: 10.3934/mbe.2014.11.785 [15] Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255 [16] Patrick W. Nelson, Michael A. Gilchrist, Daniel Coombs, James M. Hyman, Alan S. Perelson. An Age-Structured Model of HIV Infection that Allows for Variations in the Production Rate of Viral Particles and the Death Rate of Productively Infected Cells. Mathematical Biosciences & Engineering, 2004, 1 (2) : 267-288. doi: 10.3934/mbe.2004.1.267 [17] Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences & Engineering, 2014, 11 (3) : 449-469. doi: 10.3934/mbe.2014.11.449 [18] Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101 [19] Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57 [20] 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 and Continuous Dynamical Systems - S, 2021, 14 (10) : 3541-3556. doi: 10.3934/dcdss.2020441

2021 Impact Factor: 1.497