Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission

Zhikun She; Xin Jiang

doi:10.3934/dcdsb.2020259

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July 2021, 26(7): 3835-3861. doi: 10.3934/dcdsb.2020259

Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission

Zhikun She ^1,, and Xin Jiang ^1,2,

1.	School of Mathematical Sciences, Beihang University, Beijing 100191, China
2.	College of Science, North China University of Technology, Beijing 100144, China

^* Corresponding author: Zhikun She

Received December 2019 Revised July 2020 Published July 2021 Early access August 2020

Fund Project: The first author is supported by Beijing Natural Science Foundation (Z180005) and National Natural Science Foundation of China (11422111)

In this paper, a general viral infection model with humoral immunity is investigated. The model describes the interaction of uninfected target cells, infected cells, free viruses and humoral immune response, incorporating two virus transmission modes and intracellular delay. Some reasonable hypothesises are made for the general incidence rates. Through stability analysis of equilibria under these hypothesises, the model exhibits threshold dynamics with respect to the immune-inactivated reproduction rate $ \mathfrak{R}_0 $ and the immune-activated reproduction rate $ \mathfrak{R}_1 $. The theoretical results and corresponding numerical simulations show that the intracellular latency, both of virus-to-cell infection and cell-to-cell infection have direct effects on the global dynamics of the general viral infection model. Our results improve and generalize some known results on within-host virus dynamics.

Keywords: Threshold dynamics, intracellular delay, virus-to-cell transmission, cell-to-cell transmission, humoral immunity.

Mathematics Subject Classification: 92D25, 34K20, 34K25.

Citation: Zhikun She, Xin Jiang. Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3835-3861. doi: 10.3934/dcdsb.2020259

References:

[1]	A. Alshorman, X. Wang, J. Meyer and L. Rong, Analysis of HIV models with two time delays, J. Biol. Dyn., 11 (2017), 40-64. doi: 10.1080/17513758.2016.1148202.
[2]	C. R. M. Bangam, The immune control and cell-to-cell spread of human T-lymphotropic virus type 1, J. Gen. Virol., 84 (2003), 3177-3189. doi: 10.1099/vir.0.19334-0.
[3]	N. Barretto, B. Sainz, S. Hussain and S. L. Uprichard, Determining the involvement and therapeutic implications of host cellular factors in hepatitis C virus cell-to-cell spread, J. Virol., 88 (2014), 5050-5061. doi: 10.1128/JVI.03241-13.
[4]	S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976. doi: 10.1073/pnas.94.13.6971.
[5]	S. M. Ciupe, Modeling the dynamics of hepatitis B infection, immunity, and drug therapy, Immunol. Rev., 285 (2018), 38-54. doi: 10.1111/imr.12686.
[6]	S. M. Ciupe, R. M. Ribeiro, P. W. Nelson and A. S. Perelson, Modeling the mechanisms of acute hepatitis B virus infection, J. Theor. Biol., 247 (2007), 23-35. doi: 10.1016/j.jtbi.2007.02.017.
[7]	R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7.
[8]	R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. Math. Biol., 46 (2003), 425-444. doi: 10.1007/s00285-002-0191-5.
[9]	A. Del Portillo, J. Tripodi, V. Najfeld, D. Wodarz, D. N. Levy and B. K. Chen, Multiploid inheritance of HIV-1 during cell-to-cell infection, J. Virol., 85 (2011), 7169-7176. doi: 10.1128/JVI.00231-11.
[10]	D. S. Dimitrov, R. L. Willey, H. Sato, L. Chang, R. Blumenthal and M. A. Martin, Quantitation of human immunodeficiency virus type 1 infection kinetics, J. Virol., 67 (1993), 2182-2190.
[11]	H. Dahari, A. Lo, R. M. Ribeiro and A. S. Perelson, Modeling hepatitis C virus dynamics: Liver regeneration and critical drug efficacy, J. Theor. Biol., 247 (2007), 371-381. doi: 10.1016/j.jtbi.2007.03.006.
[12]	S. Gummuluru, C. M. Kinsey and M. Emerman, An in vitro rapid-turnover assay for human immunodeficiency virus type 1 replication selects for cell-to-cell spread of virua, J. Virol., 74 (2000), 10882-10891. doi: 10.1128/JVI.74.23.10882-10891.2000.
[13]	J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
[14]	D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126. doi: 10.1038/373123a0.
[15]	G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. doi: 10.1016/j.aml.2009.06.004.
[16]	G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.
[17]	W. H$\ddot{u}$bner, G. P. McNerney, P. Chen, B. M. Dale, R. E. Gordon, F. Y. S. Chuang, X. Li, D. M. Asmuth, T. Huser and B. K. Chen, Quantitative 3D video microscopy of HIV transfer across T cell virological synapses, Science, 323 (2009), 1743-1747.
[18]	D. E. Kirschner and G. F. Webb, A mathematical model of combined drug therapy of HIV infection, J. Theor. Med., 1 (1997), Article ID 293715, 10 pages. doi: 10.1080/10273669708833004.
[19]	P. Katri and S. Ruan, Dynamics of human T-cell lymphotropic virus i (HTLV-I) infection of CD4+ T-cells, Comptes Rendus Biol., 327 (2004), 1009-1016. doi: 10.1016/j.crvi.2004.05.011.
[20]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
[21]	X. Lai and X. Zou, Modelling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898-917. doi: 10.1137/130930145.
[22]	D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691. doi: 10.1016/j.jmaa.2007.02.006.
[23]	J. Lin, R. Xu and X. Tian, Threshold dynamics of an HIV-1 virus model with both virus-to-cell and cell-to-cell transmissions, intracellular delay, and humoral immunity, Appl. Math. Comput., 315 (2017), 516-530. doi: 10.1016/j.amc.2017.08.004.
[24]	J. H. MacLachlan and B. C. Cowie, Hepatitis B virus epidemiology, Cold Spring Harb. Perspect. Med., 5 (2015), a021410. doi: 10.1101/cshperspect.a021410.
[25]	C. C. McCluskey, Using Lyapunov functions to construct Lyapunov functionals for delay differential equations, SIAM J. Appl. Dyn. Syst., 14 (2015), 1-24. doi: 10.1137/140971683.
[26]	F. E. McKenzie and W. H. Bossert, A target for intervention in plasmodium falciparum infections, Am. J. Trop. Med. Hyg., 58 (1998), 763-767. doi: 10.4269/ajtmh.1998.58.763.
[27]	F. Merwaiss, C. Czibener and D. E. Alvarez, Cell-to-cell transmission is the main mechanism supporting bovine viral diarrhea virus spread in cell culture, J. Virol., 93 (2019), e01776–18. doi: 10.1128/JVI.01776-18.
[28]	W. Mothes, N. M. Sherer, J. Jin and P. Zhong, Virus cell-to-cell transmission, J. Virol., 84 (2010), 8360-8368. doi: 10.1128/JVI.00443-10.
[29]	M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas and H. McDade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398-4402. doi: 10.1073/pnas.93.9.4398.
[30]	M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.
[31]	A. S. Perelson, D. E. Kirschner and R. J. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A.
[32]	A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582.
[33]	H. Pourbashash, S. S. Pilyugin, P. D. Leenheer and C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3341-3357. doi: 10.3934/dcdsb.2014.19.3341.
[34]	B. Ramratnam, S. Bonhoeffer, J. Binley, A. Hurley, L. Zhang, J. E. Mittler, M. Markowitz, J. P. Moore, A. S. Perelson and D. D. Ho, Rapid production and clearance of HIV-1 and hepatitis C virus assessed by large volume plasma apheresis, Lancet, 354 (1999), 1782-1785. doi: 10.1016/S0140-6736(99)02035-8.
[35]	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. B, 269 (2002), 271-279. doi: 10.1098/rspb.2001.1816.
[36]	M. Roederer, B. F. Keele, S. D. Schmidt, et al., Immunological and virological mechanisms of vaccine-mediated protection against SIV and HIV, Nature, 505 (2014), 502-508. doi: 10.1038/nature12893.
[37]	L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bull. Math. Biol., 69 (2007), 2027-2060. doi: 10.1007/s11538-007-9203-3.
[38]	L. Rong and A.S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theoret. Biol., 260 (2009), 308-331. doi: 10.1016/j.jtbi.2009.06.011.
[39]	S. Ruan, Absolute stability, conditional stability and bifurcaiton in Kolmogorov-type predator-prey systems with discrete delays, Quant. Appl. Math., 59 (2001), 159-173. doi: 10.1090/qam/1811101.
[40]	S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.
[41]	S. Ruan, J. Wei and D. Xiao, On the distribution of zeros of a third-degree exponiential polynomial with applications to delayed biological models, J. Nanjing Univ. Information Sci., 9 (2017), 381-390.
[42]	Q. Sattentau, The direct passage of animal viruses between cells, Curr. Opin. Virol., 1 (2011), 396-402. doi: 10.1016/j.coviro.2011.09.004.
[43]	H. Shu, Y. Chen and L. Wang, Impacts of the cell-free and cell-to-cell infection modes on viral dynamics, J. Dyn. Diff. Equat., 30 (2018), 1817-1836. doi: 10.1007/s10884-017-9622-2.
[44]	A. Sigal, J. T. Kim, A. B. Balazs, E. Dekel, A. Mayo, R. Milo and D. Baltimore, Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy, Nature, 477 (2011), 95-98. doi: 10.1038/nature10347.
[45]	G. L. Smith, B. J. Murphy and M. Law, Vaccinia virus motility, Annu. Rev. Microbiol., 57 (2003), 323-342. doi: 10.1146/annurev.micro.57.030502.091037.
[46]	H. Sun and J. Wang, Dynamics of a diffusive virus model with general incidence function, cell-to-cell transmission and time delay, Comput. Math. Appl., 77 (2019), 284-301. doi: 10.1016/j.camwa.2018.09.032.
[47]	R. Thimme, J. Bukh, H. C. Spangenberg, et al., Viral and immunological determinants of hepatitis C virus clearance, persistence, and disease. Proc. Natl. Acad. Sci. USA, 99 (2002), 15661-15668. doi: 10.1073/pnas.202608299.
[48]	T. Wang, Z. Hu, F. Liao and W. Ma, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simul., 89 (2013), 13-22. doi: 10.1016/j.matcom.2013.03.004.
[49]	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.
[50]	Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183-191. doi: 10.1016/j.mbs.2015.05.001.
[51]	R. Zhang and S. Liu, Global dynamics of an age-structured within-host viral infection model with cell-to-cell transmission and general humoral immunity response, Math. Biosci. Eng., 17 (2020), 1450-1478. doi: 10.1016/j.mbs.2015.05.001.

show all references

References:

[1]	A. Alshorman, X. Wang, J. Meyer and L. Rong, Analysis of HIV models with two time delays, J. Biol. Dyn., 11 (2017), 40-64. doi: 10.1080/17513758.2016.1148202.
[2]	C. R. M. Bangam, The immune control and cell-to-cell spread of human T-lymphotropic virus type 1, J. Gen. Virol., 84 (2003), 3177-3189. doi: 10.1099/vir.0.19334-0.
[3]	N. Barretto, B. Sainz, S. Hussain and S. L. Uprichard, Determining the involvement and therapeutic implications of host cellular factors in hepatitis C virus cell-to-cell spread, J. Virol., 88 (2014), 5050-5061. doi: 10.1128/JVI.03241-13.
[4]	S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976. doi: 10.1073/pnas.94.13.6971.
[5]	S. M. Ciupe, Modeling the dynamics of hepatitis B infection, immunity, and drug therapy, Immunol. Rev., 285 (2018), 38-54. doi: 10.1111/imr.12686.
[6]	S. M. Ciupe, R. M. Ribeiro, P. W. Nelson and A. S. Perelson, Modeling the mechanisms of acute hepatitis B virus infection, J. Theor. Biol., 247 (2007), 23-35. doi: 10.1016/j.jtbi.2007.02.017.
[7]	R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7.
[8]	R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. Math. Biol., 46 (2003), 425-444. doi: 10.1007/s00285-002-0191-5.
[9]	A. Del Portillo, J. Tripodi, V. Najfeld, D. Wodarz, D. N. Levy and B. K. Chen, Multiploid inheritance of HIV-1 during cell-to-cell infection, J. Virol., 85 (2011), 7169-7176. doi: 10.1128/JVI.00231-11.
[10]	D. S. Dimitrov, R. L. Willey, H. Sato, L. Chang, R. Blumenthal and M. A. Martin, Quantitation of human immunodeficiency virus type 1 infection kinetics, J. Virol., 67 (1993), 2182-2190.
[11]	H. Dahari, A. Lo, R. M. Ribeiro and A. S. Perelson, Modeling hepatitis C virus dynamics: Liver regeneration and critical drug efficacy, J. Theor. Biol., 247 (2007), 371-381. doi: 10.1016/j.jtbi.2007.03.006.
[12]	S. Gummuluru, C. M. Kinsey and M. Emerman, An in vitro rapid-turnover assay for human immunodeficiency virus type 1 replication selects for cell-to-cell spread of virua, J. Virol., 74 (2000), 10882-10891. doi: 10.1128/JVI.74.23.10882-10891.2000.
[13]	J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
[14]	D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126. doi: 10.1038/373123a0.
[15]	G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. doi: 10.1016/j.aml.2009.06.004.
[16]	G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.
[17]	W. H$\ddot{u}$bner, G. P. McNerney, P. Chen, B. M. Dale, R. E. Gordon, F. Y. S. Chuang, X. Li, D. M. Asmuth, T. Huser and B. K. Chen, Quantitative 3D video microscopy of HIV transfer across T cell virological synapses, Science, 323 (2009), 1743-1747.
[18]	D. E. Kirschner and G. F. Webb, A mathematical model of combined drug therapy of HIV infection, J. Theor. Med., 1 (1997), Article ID 293715, 10 pages. doi: 10.1080/10273669708833004.
[19]	P. Katri and S. Ruan, Dynamics of human T-cell lymphotropic virus i (HTLV-I) infection of CD4+ T-cells, Comptes Rendus Biol., 327 (2004), 1009-1016. doi: 10.1016/j.crvi.2004.05.011.
[20]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
[21]	X. Lai and X. Zou, Modelling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898-917. doi: 10.1137/130930145.
[22]	D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691. doi: 10.1016/j.jmaa.2007.02.006.
[23]	J. Lin, R. Xu and X. Tian, Threshold dynamics of an HIV-1 virus model with both virus-to-cell and cell-to-cell transmissions, intracellular delay, and humoral immunity, Appl. Math. Comput., 315 (2017), 516-530. doi: 10.1016/j.amc.2017.08.004.
[24]	J. H. MacLachlan and B. C. Cowie, Hepatitis B virus epidemiology, Cold Spring Harb. Perspect. Med., 5 (2015), a021410. doi: 10.1101/cshperspect.a021410.
[25]	C. C. McCluskey, Using Lyapunov functions to construct Lyapunov functionals for delay differential equations, SIAM J. Appl. Dyn. Syst., 14 (2015), 1-24. doi: 10.1137/140971683.
[26]	F. E. McKenzie and W. H. Bossert, A target for intervention in plasmodium falciparum infections, Am. J. Trop. Med. Hyg., 58 (1998), 763-767. doi: 10.4269/ajtmh.1998.58.763.
[27]	F. Merwaiss, C. Czibener and D. E. Alvarez, Cell-to-cell transmission is the main mechanism supporting bovine viral diarrhea virus spread in cell culture, J. Virol., 93 (2019), e01776–18. doi: 10.1128/JVI.01776-18.
[28]	W. Mothes, N. M. Sherer, J. Jin and P. Zhong, Virus cell-to-cell transmission, J. Virol., 84 (2010), 8360-8368. doi: 10.1128/JVI.00443-10.
[29]	M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas and H. McDade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398-4402. doi: 10.1073/pnas.93.9.4398.
[30]	M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.
[31]	A. S. Perelson, D. E. Kirschner and R. J. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A.
[32]	A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582.
[33]	H. Pourbashash, S. S. Pilyugin, P. D. Leenheer and C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3341-3357. doi: 10.3934/dcdsb.2014.19.3341.
[34]	B. Ramratnam, S. Bonhoeffer, J. Binley, A. Hurley, L. Zhang, J. E. Mittler, M. Markowitz, J. P. Moore, A. S. Perelson and D. D. Ho, Rapid production and clearance of HIV-1 and hepatitis C virus assessed by large volume plasma apheresis, Lancet, 354 (1999), 1782-1785. doi: 10.1016/S0140-6736(99)02035-8.
[35]	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. B, 269 (2002), 271-279. doi: 10.1098/rspb.2001.1816.
[36]	M. Roederer, B. F. Keele, S. D. Schmidt, et al., Immunological and virological mechanisms of vaccine-mediated protection against SIV and HIV, Nature, 505 (2014), 502-508. doi: 10.1038/nature12893.
[37]	L. Rong, Z. Feng and A. S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bull. Math. Biol., 69 (2007), 2027-2060. doi: 10.1007/s11538-007-9203-3.
[38]	L. Rong and A.S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theoret. Biol., 260 (2009), 308-331. doi: 10.1016/j.jtbi.2009.06.011.
[39]	S. Ruan, Absolute stability, conditional stability and bifurcaiton in Kolmogorov-type predator-prey systems with discrete delays, Quant. Appl. Math., 59 (2001), 159-173. doi: 10.1090/qam/1811101.
[40]	S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.
[41]	S. Ruan, J. Wei and D. Xiao, On the distribution of zeros of a third-degree exponiential polynomial with applications to delayed biological models, J. Nanjing Univ. Information Sci., 9 (2017), 381-390.
[42]	Q. Sattentau, The direct passage of animal viruses between cells, Curr. Opin. Virol., 1 (2011), 396-402. doi: 10.1016/j.coviro.2011.09.004.
[43]	H. Shu, Y. Chen and L. Wang, Impacts of the cell-free and cell-to-cell infection modes on viral dynamics, J. Dyn. Diff. Equat., 30 (2018), 1817-1836. doi: 10.1007/s10884-017-9622-2.
[44]	A. Sigal, J. T. Kim, A. B. Balazs, E. Dekel, A. Mayo, R. Milo and D. Baltimore, Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy, Nature, 477 (2011), 95-98. doi: 10.1038/nature10347.
[45]	G. L. Smith, B. J. Murphy and M. Law, Vaccinia virus motility, Annu. Rev. Microbiol., 57 (2003), 323-342. doi: 10.1146/annurev.micro.57.030502.091037.
[46]	H. Sun and J. Wang, Dynamics of a diffusive virus model with general incidence function, cell-to-cell transmission and time delay, Comput. Math. Appl., 77 (2019), 284-301. doi: 10.1016/j.camwa.2018.09.032.
[47]	R. Thimme, J. Bukh, H. C. Spangenberg, et al., Viral and immunological determinants of hepatitis C virus clearance, persistence, and disease. Proc. Natl. Acad. Sci. USA, 99 (2002), 15661-15668. doi: 10.1073/pnas.202608299.
[48]	T. Wang, Z. Hu, F. Liao and W. Ma, Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Math. Comput. Simul., 89 (2013), 13-22. doi: 10.1016/j.matcom.2013.03.004.
[49]	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.
[50]	Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183-191. doi: 10.1016/j.mbs.2015.05.001.
[51]	R. Zhang and S. Liu, Global dynamics of an age-structured within-host viral infection model with cell-to-cell transmission and general humoral immunity response, Math. Biosci. Eng., 17 (2020), 1450-1478. doi: 10.1016/j.mbs.2015.05.001.

Figure 1. Graph trajectories of system (23) with respect to different values of $ \Lambda $. The time delay $ \tau $ is increased from $ \tau = 4\; days $ $ (\mathfrak{R}_0 = 1.3558,\; \mathfrak{R}_1 = 1.2030) $ to $ \tau = 4.5\; days $ $ (\mathfrak{R}_0 = 1.1101,\; \mathfrak{R}_1 = 0.9790) $ and finally to $ \tau = 6\; days $

$ (\mathfrak{R}_0 = 0.7441,\; \mathfrak{R}_1 = 0.6453) $

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Figure 2. Graph trajectories of system (23) with respect to different values of $ \beta_1 $. $ \beta_1 $ is decreased from $ \beta_1 = 0.0000004\; ml\cdot day^{-1} $ $ (\mathfrak{R}_0 = 2.4913,\; \mathfrak{R}_1 = 1.0639) $ to $ \beta_1 = 0.00000015\; ml\cdot day^{-1} $ $ (\mathfrak{R}_0 = 1.5027,\; \mathfrak{R}_1 = 0.9469) $ and finally to $ \beta_1 = 0.00000001\; ml\cdot day^{-1} $

$ (\mathfrak{R}_0 = 0.9491,\; \mathfrak{R}_1 = 0.8814) $

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Figure 3. Graph trajectories of system (23) with respect to different values of $ \beta_2 $. $ \beta_2 $ is decreased from $ \beta_2 = 0.0000015\; ml\cdot day^{-1} $ $ (\mathfrak{R}_0 = 2.3134,\; \mathfrak{R}_1 = 1.4273) $ to $ \beta_2 = 0.000001\; ml\cdot day^{-1} $ $ (\mathfrak{R}_0 = 1.8586,\; \mathfrak{R}_1 = 0.9890) $ and finally to $ \beta_2 = 0.00000001\; ml\cdot day^{-1} $

$ (\mathfrak{R}_0 = 0.9582,\; \mathfrak{R}_1 = 0.1211) $

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American Institute of Mathematical Sciences

Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission

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American Institute of Mathematical Sciences

Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission

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