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 & Continuous Dynamical Systems - B, 2021, 26 (7) : 3835-3861. doi: 10.3934/dcdsb.2020259

References:

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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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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