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Impact of delay on HIV1 dynamics of fighting a virus with another virus
1.  Department of Applied Mathematics, Western University, London, Ontario N6A 5B7, Canada, Canada, Canada 
References:
[1] 
E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 11441165. doi: 10.1137/S0036141000376086. 
[2] 
S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics, Springer, New York, 1993. doi: 10.1007/9783642753015. 
[3] 
F. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York,1959. 
[4] 
J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, SpringerVerlag, New York, 1993. doi: 10.1007/9781461243427. 
[5] 
B. D. Hassard, N. D. Kazarinoff and Y.H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. 
[6] 
X. Jiang, P. Yu, Z. Yuan and X. Zou, Dynamics of an HIV1 therapy model of fighting a virus with another virus, Journal of Biological Dynamics, 3 (2009), 387409. doi: 10.1080/17513750802485007. 
[7] 
T. Kajiwara, T. Saraki and Y. Takeuchi, Construction of lyapunov functionals for delay differential equations in virology and epidemiology, Nonlinear Analysis: Real World Applications, 13 (2012), 18021826. doi: 10.1016/j.nonrwa.2011.12.011. 
[8] 
J. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. 
[9] 
C. Michie, A. McLean, C. Alcock and P. Beverly, Lifespan of human lymphocyte subsets defined by cd45 isoforms, Nature, 360 (1992), 264265. doi: 10.1038/360264a0. 
[10] 
J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed virus production on viral dynamics in HIV1 infected patients, Math. Biosci., 152 (1998), 143163. 
[11] 
P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV1 pathogenesis that includes an intracellular delay, Mathematical Biosciences, 163 (2000), 201215. doi: 10.1016/S00255564(99)000553. 
[12] 
G. Nolan, Harnessing viral devices as pharmaceuticals: Fighting HIV1s fire with fire, Cell, 90 (1997), 821824. 
[13] 
T. Revilla and G. GarcíaRamos, Fighting a virus with a virus: A dynamic model for HIV1 therapy, Math. Biosci., 185 (2003), 191203. doi: 10.1016/S00255564(03)000919. 
[14] 
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41, American Mathematical Socienty, Providence, RI, 1995. 
[15] 
E. Wagner and M. Hewlett, Basic Virology, Blackwell, New York, 1999. 
[16] 
P. Yu, Y. Ding and W. Jiang, Equivalence of MTS method and CMR method for delay differential equations associated with semisimple singularity, Int. J. Bifurcation and Chaos, 24 (2014), 1450003, 49 pp. doi: 10.1142/S0218127414500035. 
[17] 
P. Yu and X. Zou, Bifurcation analysis on an HIV1 Model with constant injection of recombinant, Int. J. Bifurcation and Chaos, 22 (2012), 1250062, 21 pp. doi: 10.1142/S0218127412500629. 
[18] 
H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV1 dynamics, Math. Medic. Bio., 25 (2008), 99112. doi: 10.1093/imammb/dqm010. 
[19] 
H. Zhu and X. Zou, Dynamics of a HIV1 infection model with cellmediated immune response and intracellular delay, Disc. Cont. Dyan. Syst. B., 12 (2009), 511524. doi: 10.3934/dcdsb.2009.12.511. 
show all references
References:
[1] 
E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 11441165. doi: 10.1137/S0036141000376086. 
[2] 
S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics, Springer, New York, 1993. doi: 10.1007/9783642753015. 
[3] 
F. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York,1959. 
[4] 
J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, SpringerVerlag, New York, 1993. doi: 10.1007/9781461243427. 
[5] 
B. D. Hassard, N. D. Kazarinoff and Y.H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. 
[6] 
X. Jiang, P. Yu, Z. Yuan and X. Zou, Dynamics of an HIV1 therapy model of fighting a virus with another virus, Journal of Biological Dynamics, 3 (2009), 387409. doi: 10.1080/17513750802485007. 
[7] 
T. Kajiwara, T. Saraki and Y. Takeuchi, Construction of lyapunov functionals for delay differential equations in virology and epidemiology, Nonlinear Analysis: Real World Applications, 13 (2012), 18021826. doi: 10.1016/j.nonrwa.2011.12.011. 
[8] 
J. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. 
[9] 
C. Michie, A. McLean, C. Alcock and P. Beverly, Lifespan of human lymphocyte subsets defined by cd45 isoforms, Nature, 360 (1992), 264265. doi: 10.1038/360264a0. 
[10] 
J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed virus production on viral dynamics in HIV1 infected patients, Math. Biosci., 152 (1998), 143163. 
[11] 
P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV1 pathogenesis that includes an intracellular delay, Mathematical Biosciences, 163 (2000), 201215. doi: 10.1016/S00255564(99)000553. 
[12] 
G. Nolan, Harnessing viral devices as pharmaceuticals: Fighting HIV1s fire with fire, Cell, 90 (1997), 821824. 
[13] 
T. Revilla and G. GarcíaRamos, Fighting a virus with a virus: A dynamic model for HIV1 therapy, Math. Biosci., 185 (2003), 191203. doi: 10.1016/S00255564(03)000919. 
[14] 
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41, American Mathematical Socienty, Providence, RI, 1995. 
[15] 
E. Wagner and M. Hewlett, Basic Virology, Blackwell, New York, 1999. 
[16] 
P. Yu, Y. Ding and W. Jiang, Equivalence of MTS method and CMR method for delay differential equations associated with semisimple singularity, Int. J. Bifurcation and Chaos, 24 (2014), 1450003, 49 pp. doi: 10.1142/S0218127414500035. 
[17] 
P. Yu and X. Zou, Bifurcation analysis on an HIV1 Model with constant injection of recombinant, Int. J. Bifurcation and Chaos, 22 (2012), 1250062, 21 pp. doi: 10.1142/S0218127412500629. 
[18] 
H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV1 dynamics, Math. Medic. Bio., 25 (2008), 99112. doi: 10.1093/imammb/dqm010. 
[19] 
H. Zhu and X. Zou, Dynamics of a HIV1 infection model with cellmediated immune response and intracellular delay, Disc. Cont. Dyan. Syst. B., 12 (2009), 511524. doi: 10.3934/dcdsb.2009.12.511. 
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