# 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. doi: 10.1016/S0378-4371(04)00503-5. Google Scholar [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. doi: 10.1007/BF00178324. Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [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. doi: 10.1080/17513750701769873. Google Scholar [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. doi: 10.1016/j.nonrwa.2011.12.015. Google Scholar [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. doi: 10.1016/j.apm.2013.06.041. Google Scholar [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. doi: 10.1016/j.aml.2011.02.007. Google Scholar [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. doi: 10.1142/S0218339010003299. Google Scholar [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. doi: 10.1016/S0166-3542(02)00071-2. Google Scholar [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. doi: 10.1016/j.jmaa.2007.02.006. Google Scholar [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. doi: 10.1216/RMJ-2008-38-5-1573. Google Scholar [12] Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response,, Nonlinear Anal-Theor., 74 (2011), 2929. doi: 10.1016/j.na.2010.12.030. Google Scholar [13] M. A. Nowak and R. M. May, Virus Dynamics,, Oxford University Press, (2000). Google Scholar [14] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar [15] X. Y. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics,, J. Math. Anal. Appl., 329 (2007), 281. doi: 10.1016/j.jmaa.2006.06.064. Google Scholar [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. doi: 10.1016/j.apm.2009.09.006. Google Scholar [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. doi: 10.1016/j.nonrwa.2013.09.002. Google Scholar [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. doi: 10.1016/j.nonrwa.2009.11.008. Google Scholar [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. doi: 10.1016/j.matcom.2013.03.004. Google Scholar [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. doi: 10.1016/j.apm.2011.03.043. Google Scholar [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. doi: 10.1016/j.cnsns.2011.06.024. Google Scholar [22] 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

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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. doi: 10.1016/S0378-4371(04)00503-5. Google Scholar [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. doi: 10.1007/BF00178324. Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [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. doi: 10.1080/17513750701769873. Google Scholar [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. doi: 10.1016/j.nonrwa.2011.12.015. Google Scholar [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. doi: 10.1016/j.apm.2013.06.041. Google Scholar [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. doi: 10.1016/j.aml.2011.02.007. Google Scholar [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. doi: 10.1142/S0218339010003299. Google Scholar [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. doi: 10.1016/S0166-3542(02)00071-2. Google Scholar [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. doi: 10.1016/j.jmaa.2007.02.006. Google Scholar [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. doi: 10.1216/RMJ-2008-38-5-1573. Google Scholar [12] Y. Nakata, Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response,, Nonlinear Anal-Theor., 74 (2011), 2929. doi: 10.1016/j.na.2010.12.030. Google Scholar [13] M. A. Nowak and R. M. May, Virus Dynamics,, Oxford University Press, (2000). Google Scholar [14] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar [15] X. Y. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics,, J. Math. Anal. Appl., 329 (2007), 281. doi: 10.1016/j.jmaa.2006.06.064. Google Scholar [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. doi: 10.1016/j.apm.2009.09.006. Google Scholar [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. doi: 10.1016/j.nonrwa.2013.09.002. Google Scholar [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. doi: 10.1016/j.nonrwa.2009.11.008. Google Scholar [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. doi: 10.1016/j.matcom.2013.03.004. Google Scholar [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. doi: 10.1016/j.apm.2011.03.043. Google Scholar [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. doi: 10.1016/j.cnsns.2011.06.024. Google Scholar [22] 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
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