# American Institute of Mathematical Sciences

2015, 12(3): 525-536. doi: 10.3934/mbe.2015.12.525

## Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection

 1 Department of Mathematics, Beijing Technology and Business University, Beijing, 100048, China

Received  October 2014 Revised  December 2014 Published  January 2015

In this paper, the dynamical behavior of a viral infection model with general incidence rate and two time delays is studied. By using the Lyapunov functional and LaSalle invariance principle, the global stabilities of the infection-free equilibrium and the endemic equilibrium are obtained. We obtain a threshold of the global stability for the uninfected equilibrium, which means the disease will be under control eventually. These results can be applied to a variety of viral infections of disease that would make it possible to devise optimal treatment strategies. Numerical simulations with application to HIV infection are given to verify the analytical results.
Citation: 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
##### References:
 [1] A. A. Canabarro, I. M. Gleria and M. L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response, Physica A, 342 (2004), 234-241. doi: 10.1016/S0378-4371(04)00503-5. [2] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [3] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibra for compartmental models of disease trensmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [4] S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential model of hepatitis B virus infection, J. Biol. Dynam., 2 (2008), 140-153. doi: 10.1080/17513750701769873. [5] K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal-Real, 13 (2012), 1866-1872. doi: 10.1016/j.nonrwa.2011.12.015. [6] Z. X. Hu, J. J. Zhang and H. Wang, et al., Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment, Appl. Math. Model., 38 (2014), 524-534. doi: 10.1016/j.apm.2013.06.041. [7] G. Huang, W. B. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203. doi: 10.1016/j.aml.2011.02.007. [8] Y. Ji, L. Q. Min and Y. A. Ye, Global analysis of a viral infection model with application to HBV infection, J. Biol. Syst., 18 (2010), 325-337. doi: 10.1142/S0218339010003299. [9] S. Lewin, T. Walters and S. Locarnini, Hepatitis B treatment: Rational combination chemotherapy based on viral kinetic and animal model studies, Antivir. Res., 55 (2002), 381-396. doi: 10.1016/S0166-3542(02)00071-2. [10] D. Li and W. B. Ma, Asymptotic properties of an HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691. doi: 10.1016/j.jmaa.2007.02.006. [11] L. Q. Min, Y. M. Su and Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection, Rocky Mt. J. Math., 38 (2008), 1573-1585. doi: 10.1216/RMJ-2008-38-5-1573. [12] Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response, Nonlinear Anal-Theor., 74 (2011), 2929-2940. doi: 10.1016/j.na.2010.12.030. [13] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, New York, 2000. [14] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107. [15] X. Y. 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. [16] X. Y. Song, X. Y. Zhou and X. Zhao, Properties of stability and Hopf bifurcaion for a HIV infection model with time delay, Appl. Math. Model., 34 (2010), 1511-1523. doi: 10.1016/j.apm.2009.09.006. [17] Y. N. Tian and X. N. Liu, Global dynamics of a virus dynamical model with general incidence rate and cure rate, Nonlinear Anal-Real, 16 (2014), 17-26. doi: 10.1016/j.nonrwa.2013.09.002. [18] K. F. Wang, A. J. Fan and A. Torres, Global properties of an improved hepatitis B virus model, Nonlinear Anal-Real, 11 (2010), 3131-3138. doi: 10.1016/j.nonrwa.2009.11.008. [19] T. L. Wang, Z. X. Hu and F. C. Liao, et al., Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulat., 89 (2013), 13-22. doi: 10.1016/j.matcom.2013.03.004. [20] S. L. Wang, X. Y. Song and Z. H. Ge, Dynamics analysisi of a delayed viral infection model with immune impairment, Appl. Math. Model., 35 (2011), 4877-4885. doi: 10.1016/j.apm.2011.03.043. [21] Z. P. Wang and R. Xu, Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 964-978. doi: 10.1016/j.cnsns.2011.06.024. [22] R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81. doi: 10.1016/j.jmaa.2010.08.055.

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##### References:
 [1] A. A. Canabarro, I. M. Gleria and M. L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response, Physica A, 342 (2004), 234-241. doi: 10.1016/S0378-4371(04)00503-5. [2] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324. [3] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibra for compartmental models of disease trensmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [4] S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential model of hepatitis B virus infection, J. Biol. Dynam., 2 (2008), 140-153. doi: 10.1080/17513750701769873. [5] K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate, Nonlinear Anal-Real, 13 (2012), 1866-1872. doi: 10.1016/j.nonrwa.2011.12.015. [6] Z. X. Hu, J. J. Zhang and H. Wang, et al., Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment, Appl. Math. Model., 38 (2014), 524-534. doi: 10.1016/j.apm.2013.06.041. [7] G. Huang, W. B. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203. doi: 10.1016/j.aml.2011.02.007. [8] Y. Ji, L. Q. Min and Y. A. Ye, Global analysis of a viral infection model with application to HBV infection, J. Biol. Syst., 18 (2010), 325-337. doi: 10.1142/S0218339010003299. [9] S. Lewin, T. Walters and S. Locarnini, Hepatitis B treatment: Rational combination chemotherapy based on viral kinetic and animal model studies, Antivir. Res., 55 (2002), 381-396. doi: 10.1016/S0166-3542(02)00071-2. [10] D. Li and W. B. Ma, Asymptotic properties of an HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691. doi: 10.1016/j.jmaa.2007.02.006. [11] L. Q. Min, Y. M. Su and Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection, Rocky Mt. J. Math., 38 (2008), 1573-1585. doi: 10.1216/RMJ-2008-38-5-1573. [12] Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response, Nonlinear Anal-Theor., 74 (2011), 2929-2940. doi: 10.1016/j.na.2010.12.030. [13] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, New York, 2000. [14] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107. [15] X. Y. 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. [16] X. Y. Song, X. Y. Zhou and X. Zhao, Properties of stability and Hopf bifurcaion for a HIV infection model with time delay, Appl. Math. Model., 34 (2010), 1511-1523. doi: 10.1016/j.apm.2009.09.006. [17] Y. N. Tian and X. N. Liu, Global dynamics of a virus dynamical model with general incidence rate and cure rate, Nonlinear Anal-Real, 16 (2014), 17-26. doi: 10.1016/j.nonrwa.2013.09.002. [18] K. F. Wang, A. J. Fan and A. Torres, Global properties of an improved hepatitis B virus model, Nonlinear Anal-Real, 11 (2010), 3131-3138. doi: 10.1016/j.nonrwa.2009.11.008. [19] T. L. Wang, Z. X. Hu and F. C. Liao, et al., Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simulat., 89 (2013), 13-22. doi: 10.1016/j.matcom.2013.03.004. [20] S. L. Wang, X. Y. Song and Z. H. Ge, Dynamics analysisi of a delayed viral infection model with immune impairment, Appl. Math. Model., 35 (2011), 4877-4885. doi: 10.1016/j.apm.2011.03.043. [21] Z. P. Wang and R. Xu, Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 964-978. doi: 10.1016/j.cnsns.2011.06.024. [22] R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81. doi: 10.1016/j.jmaa.2010.08.055.
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