2013, 10(2): 483-498. doi: 10.3934/mbe.2013.10.483

Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays

1. 

College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China

2. 

Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7

Received  January 2012 Revised  September 2012 Published  January 2013

We consider a mathematical model that describes the interactions of the HIV virus, CD4 cells and CTLs within host, which is a modification of some existing models by incorporating (i) two distributed kernels reflecting the variance of time for virus to invade into cells and the variance of time for invaded virions to reproduce within cells; (ii) a nonlinear incidence function $f$ for virus infections, and (iii) a nonlinear removal rate function $h$ for infected cells. By constructing Lyapunov functionals and subtle estimates of the derivatives of these Lyapunov functionals, we shown that the model has the threshold dynamics: if the basic reproduction number (BRN) is less than or equal to one, then the infection free equilibrium is globally asymptotically stable, meaning that HIV virus will be cleared; whereas if the BRN is larger than one, then there exist an infected equilibrium which is globally asymptotically stable, implying that the HIV-1 infection will persist in the host and the viral concentration will approach a positive constant level. This together with the dependence/independence of the BRN on $f$ and $h$ reveals the effect of the adoption of these nonlinear functions.
Citation: 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
References:
[1]

L. K. Andrea and S. Ranjan, Evaluation of HIV-1 kinetic models using quantitative discrimination analysis,, Bioinformatics, 21 (2005), 1668. Google Scholar

[2]

R. Arnaout, M. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?,, Proc. Roy. Soc. Lond. B, 265 (2000), 1347. Google Scholar

[3]

S. Bonhoeffer, J. M. Coffin and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load,, J. Virol., 71 (1997), 3275. Google Scholar

[4]

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

[5]

N. Burić, M. Mudrinic and N. Vasović, Time delay in a basic model of the immune response,, Chaos, 12 (2001), 483. Google Scholar

[6]

T. A. Burton, Volterra integral and differential equations,, in, 202 (2005). Google Scholar

[7]

L. Cai and J. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence,, Chaos, 41 (2009), 175. doi: 10.1016/j.chaos.2007.11.023. Google Scholar

[8]

D. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads,, Bulletin of Mathematical Biology, 64 (2002), 29. Google Scholar

[9]

A. A. Canabarro, I. M. Gléria and M. L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response,, Physica A, 342 (2004), 234. Google Scholar

[10]

M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models,, Math. Biosci., 200 (2006), 1. doi: 10.1016/j.mbs.2005.12.006. Google Scholar

[11]

R. Culshaw, S. Ruan and R. Spiteri, Optimal HIV treatment by maximising immune response,, J. Math. Biol., 48 (2004), 545. doi: 10.1007/s00285-003-0245-3. Google Scholar

[12]

R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation,, J. Theoret. Biol., 175 (1995), 567. Google Scholar

[13]

R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison,, J. Theoret. Biol., 190 (1998), 201. Google Scholar

[14]

P. Georgescu and Y. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal,, SIAM J. Appl. Math., 67 (2006), 337. doi: 10.1137/060654876. Google Scholar

[15]

T. Kajiwara and T. Sasaki, A note on the stability analysis of pathogen-immune interaction dynamics,, Discrete Continuous Dynam. Systems-B, 4 (2004), 615. doi: 10.3934/dcdsb.2004.4.615. Google Scholar

[16]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993). Google Scholar

[17]

Y. Li, R. Xu, Z. Li and S. Mao, Global dynamics of a delayed HIV-1 infection model with CTL immune response,, Discrete Dynamics in Nature and Society, 2011 (2011). doi: 10.1155/2011/673843. Google Scholar

[18]

S. Liu and L. Wang, Global stability of an HIV-1 model with dstributed intracellular delays and a combination therapy,, Math. Biosci. and Eng., 7 (2010), 675. doi: 10.3934/mbe.2010.7.675. Google Scholar

[19]

W. Liu, Nonlinear oscillation in models of immune response to persistent viruses,, Theor. Popul. Biol., 52 (1997), 224. Google Scholar

[20]

C. Lv and Z. Yuan, Stability analysis of delay differential equation models of HIV-1 therapy for fighting a virus with another virus,, J. Math. Anal. Appl., 352 (2009), 672. doi: 10.1016/j.jmaa.2008.11.026. Google Scholar

[21]

J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients,, Math. Biosci., 152 (1998), 143. Google Scholar

[22]

Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays,, J. Math. Anal. Appl., 375 (2011), 14. doi: 10.1016/j.jmaa.2010.08.025. Google Scholar

[23]

P. Nelson, J. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay,, Math. Biosci., 163 (2000), 201. doi: 10.1016/S0025-5564(99)00055-3. Google Scholar

[24]

P. Nelson and A. S. Perelson, Mathematica analysis of delay differential equation models of HIV-1 infection,, Math. Biosci., 179 (2002), 73. doi: 10.1016/S0025-5564(02)00099-8. Google Scholar

[25]

M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses,, Science, 272 (1996), 74. Google Scholar

[26]

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

[27]

K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data,, Math. Biosci., 235 (2012), 98. doi: 10.1016/j.mbs.2011.11.002. Google Scholar

[28]

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

[29]

A. S. 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. Google Scholar

[30]

A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response,, Science, 271 (1996), 497. Google Scholar

[31]

J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays,, Mathematical Medicine and Biology, (). doi: 10.1093/imammb/dqr009. Google Scholar

[32]

K. Wang, W. Wang and X. Liu, Global Stability in a viral infection model with lytic and nonlytic immune response,, Comput. Math. Appl., 51 (2006), 1593. doi: 10.1016/j.camwa.2005.07.020. Google Scholar

[33]

K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response,, Physica D, 226 (2007), 197. doi: 10.1016/j.physd.2006.12.001. Google Scholar

[34]

R. Xu, Global dynamics of an HIV-1 infection model with distributed intracellular delays,, Comput. Math. Appl., 61 (2011), 2799. doi: 10.1016/j.camwa.2011.03.050. Google Scholar

[35]

R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay,, J. Math. Anal. Appl., 375 (2011), 75. doi: 10.1016/j.jmaa.2010.08.055. Google Scholar

[36]

H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics,, Mathematical Medicine and Biology, 25 (2008), 99. Google Scholar

[37]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay,, Discrete Continuous Dynam. Systems-B, 12 (2009), 511. Google Scholar

[38]

H. Zhu, Y. Luo and M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay,, Comput. Math. Appl., 62 (2011), 3091. doi: 10.1016/j.camwa.2011.08.022. Google Scholar

show all references

References:
[1]

L. K. Andrea and S. Ranjan, Evaluation of HIV-1 kinetic models using quantitative discrimination analysis,, Bioinformatics, 21 (2005), 1668. Google Scholar

[2]

R. Arnaout, M. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?,, Proc. Roy. Soc. Lond. B, 265 (2000), 1347. Google Scholar

[3]

S. Bonhoeffer, J. M. Coffin and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load,, J. Virol., 71 (1997), 3275. Google Scholar

[4]

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

[5]

N. Burić, M. Mudrinic and N. Vasović, Time delay in a basic model of the immune response,, Chaos, 12 (2001), 483. Google Scholar

[6]

T. A. Burton, Volterra integral and differential equations,, in, 202 (2005). Google Scholar

[7]

L. Cai and J. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence,, Chaos, 41 (2009), 175. doi: 10.1016/j.chaos.2007.11.023. Google Scholar

[8]

D. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads,, Bulletin of Mathematical Biology, 64 (2002), 29. Google Scholar

[9]

A. A. Canabarro, I. M. Gléria and M. L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response,, Physica A, 342 (2004), 234. Google Scholar

[10]

M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models,, Math. Biosci., 200 (2006), 1. doi: 10.1016/j.mbs.2005.12.006. Google Scholar

[11]

R. Culshaw, S. Ruan and R. Spiteri, Optimal HIV treatment by maximising immune response,, J. Math. Biol., 48 (2004), 545. doi: 10.1007/s00285-003-0245-3. Google Scholar

[12]

R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation,, J. Theoret. Biol., 175 (1995), 567. Google Scholar

[13]

R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison,, J. Theoret. Biol., 190 (1998), 201. Google Scholar

[14]

P. Georgescu and Y. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal,, SIAM J. Appl. Math., 67 (2006), 337. doi: 10.1137/060654876. Google Scholar

[15]

T. Kajiwara and T. Sasaki, A note on the stability analysis of pathogen-immune interaction dynamics,, Discrete Continuous Dynam. Systems-B, 4 (2004), 615. doi: 10.3934/dcdsb.2004.4.615. Google Scholar

[16]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press, (1993). Google Scholar

[17]

Y. Li, R. Xu, Z. Li and S. Mao, Global dynamics of a delayed HIV-1 infection model with CTL immune response,, Discrete Dynamics in Nature and Society, 2011 (2011). doi: 10.1155/2011/673843. Google Scholar

[18]

S. Liu and L. Wang, Global stability of an HIV-1 model with dstributed intracellular delays and a combination therapy,, Math. Biosci. and Eng., 7 (2010), 675. doi: 10.3934/mbe.2010.7.675. Google Scholar

[19]

W. Liu, Nonlinear oscillation in models of immune response to persistent viruses,, Theor. Popul. Biol., 52 (1997), 224. Google Scholar

[20]

C. Lv and Z. Yuan, Stability analysis of delay differential equation models of HIV-1 therapy for fighting a virus with another virus,, J. Math. Anal. Appl., 352 (2009), 672. doi: 10.1016/j.jmaa.2008.11.026. Google Scholar

[21]

J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients,, Math. Biosci., 152 (1998), 143. Google Scholar

[22]

Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays,, J. Math. Anal. Appl., 375 (2011), 14. doi: 10.1016/j.jmaa.2010.08.025. Google Scholar

[23]

P. Nelson, J. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay,, Math. Biosci., 163 (2000), 201. doi: 10.1016/S0025-5564(99)00055-3. Google Scholar

[24]

P. Nelson and A. S. Perelson, Mathematica analysis of delay differential equation models of HIV-1 infection,, Math. Biosci., 179 (2002), 73. doi: 10.1016/S0025-5564(02)00099-8. Google Scholar

[25]

M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses,, Science, 272 (1996), 74. Google Scholar

[26]

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

[27]

K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data,, Math. Biosci., 235 (2012), 98. doi: 10.1016/j.mbs.2011.11.002. Google Scholar

[28]

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

[29]

A. S. 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. Google Scholar

[30]

A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response,, Science, 271 (1996), 497. Google Scholar

[31]

J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays,, Mathematical Medicine and Biology, (). doi: 10.1093/imammb/dqr009. Google Scholar

[32]

K. Wang, W. Wang and X. Liu, Global Stability in a viral infection model with lytic and nonlytic immune response,, Comput. Math. Appl., 51 (2006), 1593. doi: 10.1016/j.camwa.2005.07.020. Google Scholar

[33]

K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response,, Physica D, 226 (2007), 197. doi: 10.1016/j.physd.2006.12.001. Google Scholar

[34]

R. Xu, Global dynamics of an HIV-1 infection model with distributed intracellular delays,, Comput. Math. Appl., 61 (2011), 2799. doi: 10.1016/j.camwa.2011.03.050. Google Scholar

[35]

R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay,, J. Math. Anal. Appl., 375 (2011), 75. doi: 10.1016/j.jmaa.2010.08.055. Google Scholar

[36]

H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics,, Mathematical Medicine and Biology, 25 (2008), 99. Google Scholar

[37]

H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay,, Discrete Continuous Dynam. Systems-B, 12 (2009), 511. Google Scholar

[38]

H. Zhu, Y. Luo and M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay,, Comput. Math. Appl., 62 (2011), 3091. doi: 10.1016/j.camwa.2011.08.022. Google Scholar

[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]

Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297

[3]

Songbai Guo, Wanbiao Ma. Global behavior of delay differential equations model of HIV infection with apoptosis. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 103-119. doi: 10.3934/dcdsb.2016.21.103

[4]

Jaime Angulo, Carlos Matheus, Didier Pilod. Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system. Communications on Pure & Applied Analysis, 2009, 8 (3) : 815-844. doi: 10.3934/cpaa.2009.8.815

[5]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

[6]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations & Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193

[7]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689

[8]

Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133

[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 & Continuous Dynamical Systems - B, 2018, 23 (10) : 4223-4242. doi: 10.3934/dcdsb.2018134

[10]

Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184

[11]

Franca Franchi, Barbara Lazzari, Roberta Nibbi. Uniqueness and stability results for non-linear Johnson-Segalman viscoelasticity and related models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2111-2132. doi: 10.3934/dcdsb.2014.19.2111

[12]

Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : ⅰ-ⅳ. doi: 10.3934/dcdss.201702i

[13]

Dmitry Dolgopyat. Bouncing balls in non-linear potentials. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 165-182. doi: 10.3934/dcds.2008.22.165

[14]

Dorin Ervin Dutkay and Palle E. T. Jorgensen. Wavelet constructions in non-linear dynamics. Electronic Research Announcements, 2005, 11: 21-33.

[15]

Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems & Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675

[16]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435

[17]

Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239

[18]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121

[19]

Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems & Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004

[20]

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

2018 Impact Factor: 1.313

Metrics

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

Other articles
by authors

[Back to Top]