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

show all references

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

[1]

Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133

[2]

Ting Guo, Haihong Liu, Chenglin Xu, Fang Yan. Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4223-4242. doi: 10.3934/dcdsb.2018134

[3]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121

[4]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689

[5]

Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483

[6]

Yu Yang, Yueping Dong, Yasuhiro Takeuchi. Global dynamics of a latent HIV infection model with general incidence function and multiple delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 783-800. doi: 10.3934/dcdsb.2018207

[7]

C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837

[8]

Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082

[9]

Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749

[10]

Shouying Huang, Jifa Jiang. Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate. Mathematical Biosciences & Engineering, 2016, 13 (4) : 723-739. doi: 10.3934/mbe.2016016

[11]

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

[12]

Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297

[13]

Ariel D. Weinberger, Alan S. Perelson. Persistence and emergence of X4 virus in HIV infection. Mathematical Biosciences & Engineering, 2011, 8 (2) : 605-626. doi: 10.3934/mbe.2011.8.605

[14]

Pierre Gabriel. Global stability for the prion equation with general incidence. Mathematical Biosciences & Engineering, 2015, 12 (4) : 789-801. doi: 10.3934/mbe.2015.12.789

[15]

Steffen Eikenberry, Sarah Hews, John D. Nagy, Yang Kuang. The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth. Mathematical Biosciences & Engineering, 2009, 6 (2) : 283-299. doi: 10.3934/mbe.2009.6.283

[16]

Hong Yang, Junjie Wei. Global behaviour of a delayed viral kinetic model with general incidence rate. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1573-1582. doi: 10.3934/dcdsb.2015.20.1573

[17]

Songbai Guo, Wanbiao Ma. Global behavior of delay differential equations model of HIV infection with apoptosis. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 103-119. doi: 10.3934/dcdsb.2016.21.103

[18]

Shouying Huang, Jifa Jiang. Epidemic dynamics on complex networks with general infection rate and immune strategies. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2071-2090. doi: 10.3934/dcdsb.2018226

[19]

Yoichi Enatsu, Yukihiko Nakata. Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate. Mathematical Biosciences & Engineering, 2014, 11 (4) : 785-805. doi: 10.3934/mbe.2014.11.785

[20]

Patrick W. Nelson, Michael A. Gilchrist, Daniel Coombs, James M. Hyman, Alan S. Perelson. An Age-Structured Model of HIV Infection that Allows for Variations in the Production Rate of Viral Particles and the Death Rate of Productively Infected Cells. Mathematical Biosciences & Engineering, 2004, 1 (2) : 267-288. doi: 10.3934/mbe.2004.1.267

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]