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On latencies in malaria infections and their impact on the disease dynamics
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 
References:
[1] 
L. K. Andrea and S. Ranjan, Evaluation of HIV1 kinetic models using quantitative discrimination analysis, Bioinformatics, 21 (2005), 16681677. 
[2] 
R. Arnaout, M. Nowak and D. Wodarz, HIV1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. Roy. Soc. Lond. B, 265 (2000), 13471354. 
[3] 
S. Bonhoeffer, J. M. Coffin and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virol., 71 (1997), 32753278. 
[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), 69716976. 
[5] 
N. Burić, M. Mudrinic and N. Vasović, Time delay in a basic model of the immune response, Chaos, Solitons and Fractals, 12 (2001), 483489 
[6] 
T. A. Burton, Volterra integral and differential equations, in "Mathematics In Science And Engineering" $2^{nd}$ edition, Elsevier, AmsterdamBoston, 202 (2005). 
[7] 
L. Cai and J. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence, Chaos, Solitons and Fractals, 41 (2009), 175182. doi: 10.1016/j.chaos.2007.11.023. 
[8] 
D. Callaway and A. S. Perelson, HIV1 infection and low steady state viral loads, Bulletin of Mathematical Biology, 64 (2002), 2964. 
[9] 
A. A. Canabarro, I. M. Gléria and M. L. Lyra, Periodic solutions and chaos in a nonlinear model for the delayed cellular immune response, Physica A, 342 (2004), 234241. 
[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), 127. doi: 10.1016/j.mbs.2005.12.006. 
[11] 
R. Culshaw, S. Ruan and R. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545562. doi: 10.1007/s0028500302453. 
[12] 
R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theoret. Biol., 175 (1995), 567576. 
[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), 201214. 
[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), 337353. doi: 10.1137/060654876. 
[15] 
T. Kajiwara and T. Sasaki, A note on the stability analysis of pathogenimmune interaction dynamics, Discrete Continuous Dynam. SystemsB, 4 (2004), 615622. doi: 10.3934/dcdsb.2004.4.615. 
[16] 
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993. 
[17] 
Y. Li, R. Xu, Z. Li and S. Mao, Global dynamics of a delayed HIV1 infection model with CTL immune response, Discrete Dynamics in Nature and Society, 2011 (2011), Art. ID 673843, 13 pp. doi: 10.1155/2011/673843. 
[18] 
S. Liu and L. Wang, Global stability of an HIV1 model with dstributed intracellular delays and a combination therapy, Math. Biosci. and Eng., 7 (2010), 675685. doi: 10.3934/mbe.2010.7.675. 
[19] 
W. Liu, Nonlinear oscillation in models of immune response to persistent viruses, Theor. Popul. Biol., 52 (1997), 224230. 
[20] 
C. Lv and Z. Yuan, Stability analysis of delay differential equation models of HIV1 therapy for fighting a virus with another virus, J. Math. Anal. Appl., 352 (2009), 672683. doi: 10.1016/j.jmaa.2008.11.026. 
[21] 
J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed viral production on viral dynamics in HIV1 infected patients, Math. Biosci., 152 (1998), 143163. 
[22] 
Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 1427. doi: 10.1016/j.jmaa.2010.08.025. 
[23] 
P. Nelson, J. Murray and A. S. Perelson, A model of HIV1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201215. doi: 10.1016/S00255564(99)000553. 
[24] 
P. Nelson and A. S. Perelson, Mathematica analysis of delay differential equation models of HIV1 infection, Math. Biosci., 179 (2002), 7394. doi: 10.1016/S00255564(02)000998. 
[25] 
M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 7479. 
[26] 
M. A. Nowak, S. Bonhoeffer, G. M. Shaw and R. M. May, Antiviral drug treatment: Dynamics of resistance in free virus and infected cell populations, J. Theor. Biol., 184 (1997), 203217. 
[27] 
K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98109. doi: 10.1016/j.mbs.2011.11.002. 
[28] 
A. S. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 344. doi: 10.1137/S0036144598335107. 
[29] 
A. S. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV1 dynamics in vivo: Virion clearance rate, infected cell lifespan, and viral generation time, Science, 271 (1996), 15821586. 
[30] 
A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response, Science, 271 (1996), 497499. 
[31] 
J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV1 dynamics with two distributed delays, Mathematical Medicine and Biology, IMA. doi: 10.1093/imammb/dqr009. 
[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), 15931610. doi: 10.1016/j.camwa.2005.07.020. 
[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), 197208. doi: 10.1016/j.physd.2006.12.001. 
[34] 
R. Xu, Global dynamics of an HIV1 infection model with distributed intracellular delays, Comput. Math. Appl., 61 (2011), 27992805. doi: 10.1016/j.camwa.2011.03.050. 
[35] 
R. Xu, Global stability of an HIV1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 7581. doi: 10.1016/j.jmaa.2010.08.055. 
[36] 
H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV1 dynamics, Mathematical Medicine and Biology, IMA, 25 (2008), 99112. 
[37] 
H. Zhu and X. Zou, Dynamics of a HIV1 infection model with cellmediated immune response and intracellular delay, Discrete Continuous Dynam. SystemsB, 12 (2009), 511524. 
[38] 
H. Zhu, Y. Luo and M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTLresponse delay, Comput. Math. Appl., 62 (2011), 30913102. doi: 10.1016/j.camwa.2011.08.022. 
show all references
References:
[1] 
L. K. Andrea and S. Ranjan, Evaluation of HIV1 kinetic models using quantitative discrimination analysis, Bioinformatics, 21 (2005), 16681677. 
[2] 
R. Arnaout, M. Nowak and D. Wodarz, HIV1 dynamics revisited: Biphasic decay by cytotoxic lymphocyte killing?, Proc. Roy. Soc. Lond. B, 265 (2000), 13471354. 
[3] 
S. Bonhoeffer, J. M. Coffin and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virol., 71 (1997), 32753278. 
[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), 69716976. 
[5] 
N. Burić, M. Mudrinic and N. Vasović, Time delay in a basic model of the immune response, Chaos, Solitons and Fractals, 12 (2001), 483489 
[6] 
T. A. Burton, Volterra integral and differential equations, in "Mathematics In Science And Engineering" $2^{nd}$ edition, Elsevier, AmsterdamBoston, 202 (2005). 
[7] 
L. Cai and J. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence, Chaos, Solitons and Fractals, 41 (2009), 175182. doi: 10.1016/j.chaos.2007.11.023. 
[8] 
D. Callaway and A. S. Perelson, HIV1 infection and low steady state viral loads, Bulletin of Mathematical Biology, 64 (2002), 2964. 
[9] 
A. A. Canabarro, I. M. Gléria and M. L. Lyra, Periodic solutions and chaos in a nonlinear model for the delayed cellular immune response, Physica A, 342 (2004), 234241. 
[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), 127. doi: 10.1016/j.mbs.2005.12.006. 
[11] 
R. Culshaw, S. Ruan and R. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545562. doi: 10.1007/s0028500302453. 
[12] 
R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation, J. Theoret. Biol., 175 (1995), 567576. 
[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), 201214. 
[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), 337353. doi: 10.1137/060654876. 
[15] 
T. Kajiwara and T. Sasaki, A note on the stability analysis of pathogenimmune interaction dynamics, Discrete Continuous Dynam. SystemsB, 4 (2004), 615622. doi: 10.3934/dcdsb.2004.4.615. 
[16] 
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993. 
[17] 
Y. Li, R. Xu, Z. Li and S. Mao, Global dynamics of a delayed HIV1 infection model with CTL immune response, Discrete Dynamics in Nature and Society, 2011 (2011), Art. ID 673843, 13 pp. doi: 10.1155/2011/673843. 
[18] 
S. Liu and L. Wang, Global stability of an HIV1 model with dstributed intracellular delays and a combination therapy, Math. Biosci. and Eng., 7 (2010), 675685. doi: 10.3934/mbe.2010.7.675. 
[19] 
W. Liu, Nonlinear oscillation in models of immune response to persistent viruses, Theor. Popul. Biol., 52 (1997), 224230. 
[20] 
C. Lv and Z. Yuan, Stability analysis of delay differential equation models of HIV1 therapy for fighting a virus with another virus, J. Math. Anal. Appl., 352 (2009), 672683. doi: 10.1016/j.jmaa.2008.11.026. 
[21] 
J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed viral production on viral dynamics in HIV1 infected patients, Math. Biosci., 152 (1998), 143163. 
[22] 
Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 1427. doi: 10.1016/j.jmaa.2010.08.025. 
[23] 
P. Nelson, J. Murray and A. S. Perelson, A model of HIV1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201215. doi: 10.1016/S00255564(99)000553. 
[24] 
P. Nelson and A. S. Perelson, Mathematica analysis of delay differential equation models of HIV1 infection, Math. Biosci., 179 (2002), 7394. doi: 10.1016/S00255564(02)000998. 
[25] 
M. Nowak and C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 7479. 
[26] 
M. A. Nowak, S. Bonhoeffer, G. M. Shaw and R. M. May, Antiviral drug treatment: Dynamics of resistance in free virus and infected cell populations, J. Theor. Biol., 184 (1997), 203217. 
[27] 
K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98109. doi: 10.1016/j.mbs.2011.11.002. 
[28] 
A. S. Perelson and P. Nelson, Mathematical models of HIV dynamics in vivo, SIAM Rev., 41 (1999), 344. doi: 10.1137/S0036144598335107. 
[29] 
A. S. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV1 dynamics in vivo: Virion clearance rate, infected cell lifespan, and viral generation time, Science, 271 (1996), 15821586. 
[30] 
A. N. Phillips, Reduction of HIV concentration during acute infection: Independence from a specific immune response, Science, 271 (1996), 497499. 
[31] 
J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV1 dynamics with two distributed delays, Mathematical Medicine and Biology, IMA. doi: 10.1093/imammb/dqr009. 
[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), 15931610. doi: 10.1016/j.camwa.2005.07.020. 
[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), 197208. doi: 10.1016/j.physd.2006.12.001. 
[34] 
R. Xu, Global dynamics of an HIV1 infection model with distributed intracellular delays, Comput. Math. Appl., 61 (2011), 27992805. doi: 10.1016/j.camwa.2011.03.050. 
[35] 
R. Xu, Global stability of an HIV1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 7581. doi: 10.1016/j.jmaa.2010.08.055. 
[36] 
H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV1 dynamics, Mathematical Medicine and Biology, IMA, 25 (2008), 99112. 
[37] 
H. Zhu and X. Zou, Dynamics of a HIV1 infection model with cellmediated immune response and intracellular delay, Discrete Continuous Dynam. SystemsB, 12 (2009), 511524. 
[38] 
H. Zhu, Y. Luo and M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTLresponse delay, Comput. Math. Appl., 62 (2011), 30913102. doi: 10.1016/j.camwa.2011.08.022. 
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