# American Institute of Mathematical Sciences

November  2015, 20(9): 3215-3233. doi: 10.3934/dcdsb.2015.20.3215

## Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells

 1 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China, China 2 School of Mathematical Science, Heilongjiang University, Harbin 150080, China

Received  August 2014 Revised  January 2015 Published  September 2015

In this paper, we formulate a viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. The model can describe the in vivo infection dynamics of many viruses such as HIV-I, HCV, and HBV, where the infected cells of eclipse stage can revert to the uninfected class. Under certain parameters range, we establish that the global stability of equilibria is completely determined by the basic reproduction number $\mathfrak{R}_0$, which give us a complete picture on their global dynamics.
Citation: Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215
##### References:
 [1] S. Bonhoeffer, J. Coffin and M. Nowak, Human immunodeficiency virus drug therapy and virus load,, J. Virol., 71 (1997), 3275. Google Scholar [2] B. Buonomo and C. Vargas-De-León, Global stability for an HIV-1 infection model including an eclipse stage of infected cells,, J. Math. Anal. Appl., 385 (2012), 709. doi: 10.1016/j.jmaa.2011.07.006. Google Scholar [3] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency,, J. Animal Ecol., 44 (1975), 331. doi: 10.2307/3866. Google Scholar [4] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction,, Ecology, 56 (1975), 881. Google Scholar [5] P. Essunger and A. S. Perelson, Modeling HIV infection of CD4$^+$ T-cell subpopulations,, J. Theoret. Biol., 170 (1994), 367. doi: 10.1006/jtbi.1994.1199. Google Scholar [6] D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna,, Oecologia, 122 (2000), 200. doi: 10.1007/PL00008847. Google Scholar [7] V. Herz, S. Bonhoeffer, R. Anderson, R. May and M. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus decay,, Proc. Nat. Acad. Sci., 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247. Google Scholar [8] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 22 (2009), 1690. doi: 10.1016/j.aml.2009.06.004. Google Scholar [9] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 22 (2009), 1690. doi: 10.1016/j.aml.2009.06.004. Google Scholar [10] Z. Hu, W. Pang, F. Liao and W. Ma, Analysis of a CD$4^{+}$ T Cell viral infection model with a class of saturated infection rate,, Disc. Cont. Dyn. Sys. B, 19 (2014), 735. doi: 10.3934/dcdsb.2014.19.735. Google Scholar [11] A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate,, Math. Med. Bio., 26 (2009), 225. Google Scholar [12] A. Korobeinikov, Global properties of basic virus dynamics models,, Bull. Math. Biol., 66 (2004), 879. doi: 10.1016/j.bulm.2004.02.001. Google Scholar [13] A. Korobeinikov, Global properties of infectious disease model with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871. doi: 10.1007/s11538-007-9196-y. Google Scholar [14] T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functionals for delay differential equations in virology and epidemiology,, Nonlinear Anal.: Real World Appl., 13 (2012), 1802. doi: 10.1016/j.nonrwa.2011.12.011. Google Scholar [15] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993). Google Scholar [16] X. Liu, H. Wang, Z. Hu and W. Ma, Global stability of an HIV pathogenesis model with cure rate,, Nonlinear Anal.: Real World Appl., 12 (2011), 2947. doi: 10.1016/j.nonrwa.2011.04.016. Google Scholar [17] S. Lewin, T. Walters and S. Locarnini, Hepatitis B treatment: Rational combination chemotherapy based on viral kinetic and animal model studies,, Antiviral Research, 55 (2002), 381. doi: 10.1016/S0166-3542(02)00071-2. Google Scholar [18] M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492. doi: 10.1007/s11538-010-9503-x. Google Scholar [19] M. Y. Li and H. Shu, Impact of intracellular delay and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434. doi: 10.1137/090779322. Google Scholar [20] D. Li and W. 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 [21] C. C. McCluskey, Global stability of an SIR epidemic model with delay and general non linear incidence,, Math. Biosci. Eng., 7 (2010), 837. doi: 10.3934/mbe.2010.7.837. Google Scholar [22] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete,, Nonlinear Anal.: Real World Appl., 11 (2010), 55. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar [23] J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients,, Math. Biosci., 152 (1998), 143. doi: 10.1016/S0025-5564(98)10027-5. Google Scholar [24] M. Nowak and R. M. May, Virus Dynamics,, Cambridge University Press, (2000). Google Scholar [25] M. Nowak, S. Bonhoeffer, A. Hill, R. Boehme, H. Thomas and H. Mcdade, Viral dynamics in hepatitis B virus infection,, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398. doi: 10.1073/pnas.93.9.4398. Google Scholar [26] Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays,, J. Math. Anal. Appl., 375 (2011), 14. doi: 10.1016/j.jmaa.2010.08.025. Google Scholar [27] A. S. Perelson, D. E. Kirschner and R. de Boer, Dynamics of HIV infection of CD4 T cells,, Math. Biosci., 114 (1993), 81. doi: 10.1016/0025-5564(93)90043-A. Google Scholar [28] A. S. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time,, Science, 271 (1996), 1582. doi: 10.1126/science.271.5255.1582. Google Scholar [29] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar [30] L. Rong, M. A. Gilchrist, Z. Feng and A. S. Perelson, Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility,, J. Theoret. Biol., 247 (2007), 804. doi: 10.1016/j.jtbi.2007.04.014. Google Scholar [31] R. R. Regoes, D. Ebert and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology,, Proc. R. Soc. Lond. Ser., 269 (2002), 271. doi: 10.1098/rspb.2001.1816. Google Scholar [32] X. 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 [33] P. K. Srivastava and P. Chandra, Modeling the dynamics of HIV and CD4+ T cells during primary infection,, Nonlinear Anal.: Real World Appl., 11 (2010), 612. doi: 10.1016/j.nonrwa.2008.10.037. Google Scholar [34] Y. Tian and X. Liu, Global dynamics of a virus dynamical model with general incidence rate and cure rate,, Nonlinear Anal.: Real World Appl., 16 (2014), 17. doi: 10.1016/j.nonrwa.2013.09.002. Google Scholar [35] J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays,, Math. Med. Biol., 29 (2012), 283. doi: 10.1093/imammb/dqr009. Google Scholar [36] J. Wang and L. Guan, Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay,, Disc. Cont. Dyn. Sys. B, 17 (2012), 297. doi: 10.3934/dcdsb.2012.17.297. Google Scholar [37] J. Wang, J. Pang and T. Kuniya, A note on global stability for malaria infections model with latencies,, Math. Biosci. Eng., 11 (2014), 995. doi: 10.3934/mbe.2014.11.995. Google Scholar [38] J. Wang, J. Pang, T. Kuniya and Y. Enatsu, Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays,, Appl. Math. Comput., 241 (2014), 298. doi: 10.1016/j.amc.2014.05.015. Google Scholar [39] J. Wang and S. Liu, The stability analysis of a general viral infection model with distributed delays and multi-staged infected progression,, Commun. Nonlinear Sci. Numer. Simulat., 20 (2015), 263. doi: 10.1016/j.cnsns.2014.04.027. Google Scholar [40] R. Xu, Global stability ofan 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 [41] Z. Yuan and X. Zou, Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays,, Math. Biosci. Eng., 10 (2013), 483. doi: 10.3934/mbe.2013.10.483. Google Scholar [42] C. Lv, L. Huang and Z. Yuan, Global stability for an HIV-1 infection model with Beddington-DeAngelis incidence rate and CTL immune response,, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 121. doi: 10.1016/j.cnsns.2013.06.025. Google Scholar [43] H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics,, Math. Med. Biol., 25 (2008), 99. doi: 10.1093/imammb/dqm010. Google Scholar [44] J. A. Zack, S. J. Arrigo, S. R. Weitsman, A. S. Go, A. Haislip and I. S. Chen, HIV-1 entry into quiescent primary lymphocytes: Molecular analysis reveals a labile latent viral structure,, Cell, 61 (1990), 213. doi: 10.1016/0092-8674(90)90802-L. Google Scholar [45] J. A. Zack, A. M. Haislip, P. Krogstad and I. S. Chen, Incompletely reverse-transcribed human immunodeficiency virus type 1 genomes in quiescent cells can function as intermediates in the retroviral cycle,, J. Virol., 66 (1992), 1717. Google Scholar [46] X. Y. Zhou, X. Y. Song and X. Y. Shi, A differential equation model of HIV infection of CD4$^+$ T-cells with cure rate,, J. Math. Anal. Appl., 342 (2008), 1342. doi: 10.1016/j.jmaa.2008.01.008. Google Scholar

show all references

##### References:
 [1] S. Bonhoeffer, J. Coffin and M. Nowak, Human immunodeficiency virus drug therapy and virus load,, J. Virol., 71 (1997), 3275. Google Scholar [2] B. Buonomo and C. Vargas-De-León, Global stability for an HIV-1 infection model including an eclipse stage of infected cells,, J. Math. Anal. Appl., 385 (2012), 709. doi: 10.1016/j.jmaa.2011.07.006. Google Scholar [3] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency,, J. Animal Ecol., 44 (1975), 331. doi: 10.2307/3866. Google Scholar [4] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction,, Ecology, 56 (1975), 881. Google Scholar [5] P. Essunger and A. S. Perelson, Modeling HIV infection of CD4$^+$ T-cell subpopulations,, J. Theoret. Biol., 170 (1994), 367. doi: 10.1006/jtbi.1994.1199. Google Scholar [6] D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna,, Oecologia, 122 (2000), 200. doi: 10.1007/PL00008847. Google Scholar [7] V. Herz, S. Bonhoeffer, R. Anderson, R. May and M. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus decay,, Proc. Nat. Acad. Sci., 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247. Google Scholar [8] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 22 (2009), 1690. doi: 10.1016/j.aml.2009.06.004. Google Scholar [9] G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 22 (2009), 1690. doi: 10.1016/j.aml.2009.06.004. Google Scholar [10] Z. Hu, W. Pang, F. Liao and W. Ma, Analysis of a CD$4^{+}$ T Cell viral infection model with a class of saturated infection rate,, Disc. Cont. Dyn. Sys. B, 19 (2014), 735. doi: 10.3934/dcdsb.2014.19.735. Google Scholar [11] A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate,, Math. Med. Bio., 26 (2009), 225. Google Scholar [12] A. Korobeinikov, Global properties of basic virus dynamics models,, Bull. Math. Biol., 66 (2004), 879. doi: 10.1016/j.bulm.2004.02.001. Google Scholar [13] A. Korobeinikov, Global properties of infectious disease model with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871. doi: 10.1007/s11538-007-9196-y. Google Scholar [14] T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functionals for delay differential equations in virology and epidemiology,, Nonlinear Anal.: Real World Appl., 13 (2012), 1802. doi: 10.1016/j.nonrwa.2011.12.011. Google Scholar [15] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993). Google Scholar [16] X. Liu, H. Wang, Z. Hu and W. Ma, Global stability of an HIV pathogenesis model with cure rate,, Nonlinear Anal.: Real World Appl., 12 (2011), 2947. doi: 10.1016/j.nonrwa.2011.04.016. Google Scholar [17] S. Lewin, T. Walters and S. Locarnini, Hepatitis B treatment: Rational combination chemotherapy based on viral kinetic and animal model studies,, Antiviral Research, 55 (2002), 381. doi: 10.1016/S0166-3542(02)00071-2. Google Scholar [18] M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492. doi: 10.1007/s11538-010-9503-x. Google Scholar [19] M. Y. Li and H. Shu, Impact of intracellular delay and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434. doi: 10.1137/090779322. Google Scholar [20] D. Li and W. 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 [21] C. C. McCluskey, Global stability of an SIR epidemic model with delay and general non linear incidence,, Math. Biosci. Eng., 7 (2010), 837. doi: 10.3934/mbe.2010.7.837. Google Scholar [22] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete,, Nonlinear Anal.: Real World Appl., 11 (2010), 55. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar [23] J. Mittler, B. Sulzer, A. Neumann and A. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients,, Math. Biosci., 152 (1998), 143. doi: 10.1016/S0025-5564(98)10027-5. Google Scholar [24] M. Nowak and R. M. May, Virus Dynamics,, Cambridge University Press, (2000). Google Scholar [25] M. Nowak, S. Bonhoeffer, A. Hill, R. Boehme, H. Thomas and H. Mcdade, Viral dynamics in hepatitis B virus infection,, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398. doi: 10.1073/pnas.93.9.4398. Google Scholar [26] Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays,, J. Math. Anal. Appl., 375 (2011), 14. doi: 10.1016/j.jmaa.2010.08.025. Google Scholar [27] A. S. Perelson, D. E. Kirschner and R. de Boer, Dynamics of HIV infection of CD4 T cells,, Math. Biosci., 114 (1993), 81. doi: 10.1016/0025-5564(93)90043-A. Google Scholar [28] A. S. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time,, Science, 271 (1996), 1582. doi: 10.1126/science.271.5255.1582. Google Scholar [29] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar [30] L. Rong, M. A. Gilchrist, Z. Feng and A. S. Perelson, Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility,, J. Theoret. Biol., 247 (2007), 804. doi: 10.1016/j.jtbi.2007.04.014. Google Scholar [31] R. R. Regoes, D. Ebert and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology,, Proc. R. Soc. Lond. Ser., 269 (2002), 271. doi: 10.1098/rspb.2001.1816. Google Scholar [32] X. 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 [33] P. K. Srivastava and P. Chandra, Modeling the dynamics of HIV and CD4+ T cells during primary infection,, Nonlinear Anal.: Real World Appl., 11 (2010), 612. doi: 10.1016/j.nonrwa.2008.10.037. Google Scholar [34] Y. Tian and X. Liu, Global dynamics of a virus dynamical model with general incidence rate and cure rate,, Nonlinear Anal.: Real World Appl., 16 (2014), 17. doi: 10.1016/j.nonrwa.2013.09.002. Google Scholar [35] J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays,, Math. Med. Biol., 29 (2012), 283. doi: 10.1093/imammb/dqr009. Google Scholar [36] J. Wang and L. Guan, Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay,, Disc. Cont. Dyn. Sys. B, 17 (2012), 297. doi: 10.3934/dcdsb.2012.17.297. Google Scholar [37] J. Wang, J. Pang and T. Kuniya, A note on global stability for malaria infections model with latencies,, Math. Biosci. Eng., 11 (2014), 995. doi: 10.3934/mbe.2014.11.995. Google Scholar [38] J. Wang, J. Pang, T. Kuniya and Y. Enatsu, Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays,, Appl. Math. Comput., 241 (2014), 298. doi: 10.1016/j.amc.2014.05.015. Google Scholar [39] J. Wang and S. Liu, The stability analysis of a general viral infection model with distributed delays and multi-staged infected progression,, Commun. Nonlinear Sci. Numer. Simulat., 20 (2015), 263. doi: 10.1016/j.cnsns.2014.04.027. Google Scholar [40] R. Xu, Global stability ofan 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 [41] Z. Yuan and X. Zou, Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays,, Math. Biosci. Eng., 10 (2013), 483. doi: 10.3934/mbe.2013.10.483. Google Scholar [42] C. Lv, L. Huang and Z. Yuan, Global stability for an HIV-1 infection model with Beddington-DeAngelis incidence rate and CTL immune response,, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 121. doi: 10.1016/j.cnsns.2013.06.025. Google Scholar [43] H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics,, Math. Med. Biol., 25 (2008), 99. doi: 10.1093/imammb/dqm010. Google Scholar [44] J. A. Zack, S. J. Arrigo, S. R. Weitsman, A. S. Go, A. Haislip and I. S. Chen, HIV-1 entry into quiescent primary lymphocytes: Molecular analysis reveals a labile latent viral structure,, Cell, 61 (1990), 213. doi: 10.1016/0092-8674(90)90802-L. Google Scholar [45] J. A. Zack, A. M. Haislip, P. Krogstad and I. S. Chen, Incompletely reverse-transcribed human immunodeficiency virus type 1 genomes in quiescent cells can function as intermediates in the retroviral cycle,, J. Virol., 66 (1992), 1717. Google Scholar [46] X. Y. Zhou, X. Y. Song and X. Y. Shi, A differential equation model of HIV infection of CD4$^+$ T-cells with cure rate,, J. Math. Anal. Appl., 342 (2008), 1342. doi: 10.1016/j.jmaa.2008.01.008. Google Scholar
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