# American Institute of Mathematical Sciences

2014, 11(5): 1181-1198. doi: 10.3934/mbe.2014.11.1181

## Impact of delay on HIV-1 dynamics of fighting a virus with another virus

Received  January 2014 Revised  April 2014 Published  June 2014

In this paper, we propose a mathematical model for HIV-1 infection with intracellular delay. The model examines a viral-therapy for controlling infections through recombining HIV-1 virus with a genetically modified virus. For this model, the basic reproduction number $\mathcal{R}_0$ are identified and its threshold properties are discussed. When $\mathcal{R}_0 < 1$, the infection-free equilibrium $E_0$ is globally asymptotically stable. When $\mathcal{R}_0 > 1$, $E_0$ becomes unstable and there occurs the single-infection equilibrium $E_s$, and $E_0$ and $E_s$ exchange their stability at the transcritical point $\mathcal{R}_0 =1$. If $1< \mathcal{R}_0 < R_1$, where $R_1$ is a positive constant explicitly depending on the model parameters, $E_s$ is globally asymptotically stable, while when $\mathcal{R}_0 > R_1$, $E_s$ loses its stability to the double-infection equilibrium $E_d$. There exist a constant $R_2$ such that $E_d$ is asymptotically stable if $R_1<\mathcal R_0 < R_2$, and $E_s$ and $E_d$ exchange their stability at the transcritical point $\mathcal{R}_0 =R_1$. We use one numerical example to determine the largest range of $\mathcal R_0$ for the local stability of $E_d$ and existence of Hopf bifurcation. Some simulations are performed to support the theoretical results. These results show that the delay plays an important role in determining the dynamic behaviour of the system. In the normal range of values, the delay may change the dynamic behaviour quantitatively, such as greatly reducing the amplitudes of oscillations, or even qualitatively changes the dynamical behaviour such as revoking oscillating solutions to equilibrium solutions. This suggests that the delay is a very important fact which should not be missed in HIV-1 modelling.
Citation: Yun Tian, Yu Bai, Pei Yu. Impact of delay on HIV-1 dynamics of fighting a virus with another virus. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1181-1198. doi: 10.3934/mbe.2014.11.1181
##### 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), 1144. doi: 10.1137/S0036141000376086. Google Scholar [2] S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics,, Springer, (1993). doi: 10.1007/978-3-642-75301-5. Google Scholar [3] F. Gantmacher, The Theory of Matrices,, Vol. 2, (). Google Scholar [4] J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar [5] B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcation,, Cambridge University Press, (1981). Google Scholar [6] X. Jiang, P. Yu, Z. Yuan and X. Zou, Dynamics of an HIV-1 therapy model of fighting a virus with another virus,, Journal of Biological Dynamics, 3 (2009), 387. doi: 10.1080/17513750802485007. Google Scholar [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), 1802. doi: 10.1016/j.nonrwa.2011.12.011. Google Scholar [8] J. LaSalle, The Stability of Dynamical Systems,, SIAM, (1976). Google Scholar [9] C. Michie, A. McLean, C. Alcock and P. Beverly, Lifespan of human lymphocyte subsets defined by cd45 isoforms,, Nature, 360 (1992), 264. doi: 10.1038/360264a0. Google Scholar [10] J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed virus production on viral dynamics in HIV-1 infected patients,, Math. Biosci., 152 (1998), 143. Google Scholar [11] P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay,, Mathematical Biosciences, 163 (2000), 201. doi: 10.1016/S0025-5564(99)00055-3. Google Scholar [12] G. Nolan, Harnessing viral devices as pharmaceuticals: Fighting HIV-1s fire with fire,, Cell, 90 (1997), 821. Google Scholar [13] T. Revilla and G. García-Ramos, Fighting a virus with a virus: A dynamic model for HIV-1 therapy,, Math. Biosci., 185 (2003), 191. doi: 10.1016/S0025-5564(03)00091-9. Google Scholar [14] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). Google Scholar [15] E. Wagner and M. Hewlett, Basic Virology,, Blackwell, (1999). Google Scholar [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). doi: 10.1142/S0218127414500035. Google Scholar [17] P. Yu and X. Zou, Bifurcation analysis on an HIV-1 Model with constant injection of recombinant,, Int. J. Bifurcation and Chaos, 22 (2012). doi: 10.1142/S0218127412500629. Google Scholar [18] H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics,, Math. Medic. Bio., 25 (2008), 99. doi: 10.1093/imammb/dqm010. Google Scholar [19] H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay,, Disc. Cont. Dyan. Syst. B., 12 (2009), 511. doi: 10.3934/dcdsb.2009.12.511. Google Scholar

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##### 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), 1144. doi: 10.1137/S0036141000376086. Google Scholar [2] S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics,, Springer, (1993). doi: 10.1007/978-3-642-75301-5. Google Scholar [3] F. Gantmacher, The Theory of Matrices,, Vol. 2, (). Google Scholar [4] J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar [5] B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcation,, Cambridge University Press, (1981). Google Scholar [6] X. Jiang, P. Yu, Z. Yuan and X. Zou, Dynamics of an HIV-1 therapy model of fighting a virus with another virus,, Journal of Biological Dynamics, 3 (2009), 387. doi: 10.1080/17513750802485007. Google Scholar [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), 1802. doi: 10.1016/j.nonrwa.2011.12.011. Google Scholar [8] J. LaSalle, The Stability of Dynamical Systems,, SIAM, (1976). Google Scholar [9] C. Michie, A. McLean, C. Alcock and P. Beverly, Lifespan of human lymphocyte subsets defined by cd45 isoforms,, Nature, 360 (1992), 264. doi: 10.1038/360264a0. Google Scholar [10] J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed virus production on viral dynamics in HIV-1 infected patients,, Math. Biosci., 152 (1998), 143. Google Scholar [11] P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay,, Mathematical Biosciences, 163 (2000), 201. doi: 10.1016/S0025-5564(99)00055-3. Google Scholar [12] G. Nolan, Harnessing viral devices as pharmaceuticals: Fighting HIV-1s fire with fire,, Cell, 90 (1997), 821. Google Scholar [13] T. Revilla and G. García-Ramos, Fighting a virus with a virus: A dynamic model for HIV-1 therapy,, Math. Biosci., 185 (2003), 191. doi: 10.1016/S0025-5564(03)00091-9. Google Scholar [14] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). Google Scholar [15] E. Wagner and M. Hewlett, Basic Virology,, Blackwell, (1999). Google Scholar [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). doi: 10.1142/S0218127414500035. Google Scholar [17] P. Yu and X. Zou, Bifurcation analysis on an HIV-1 Model with constant injection of recombinant,, Int. J. Bifurcation and Chaos, 22 (2012). doi: 10.1142/S0218127412500629. Google Scholar [18] H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics,, Math. Medic. Bio., 25 (2008), 99. doi: 10.1093/imammb/dqm010. Google Scholar [19] H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay,, Disc. Cont. Dyan. Syst. B., 12 (2009), 511. doi: 10.3934/dcdsb.2009.12.511. Google Scholar
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