Global stability of a diffusive humoral immunity viral infection model with time delays and two modes of transmission

Hui Miao; Hui Miao

doi:10.3934/math.2025636

AIMS Mathematics

2025, Volume 10, Issue 6: 14122-14139. doi: 10.3934/math.2025636

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Research article

Global stability of a diffusive humoral immunity viral infection model with time delays and two modes of transmission

Hui Miao ^,

School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China

Received: 22 April 2025 Revised: 07 June 2025 Accepted: 10 June 2025 Published: 19 June 2025
MSC : 34D40, 35K57, 92D30

In this paper, the dynamical behaviors for a diffusive and delayed viral infection model with two modes of transmission were investigated. The uninfected cells dynamics, two infection modes for both virus-to-cell infection and cell-to-cell transmission, and concentration dependence for the latently infected cells, infected cells, viruses, and B cells were modeled by seven general nonlinear functions along with some assumptions. The basic reproduction number was calculated and demonstrated the global properties of the virus model. The theoretical results were illustrated by numerical simulations.
- virus infection model,
- cell-to-cell transmission,
- diffusion,
- latent infection,
- global stability
Citation: Hui Miao. Global stability of a diffusive humoral immunity viral infection model with time delays and two modes of transmission[J]. AIMS Mathematics, 2025, 10(6): 14122-14139. doi: 10.3934/math.2025636

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Abstract

In this paper, the dynamical behaviors for a diffusive and delayed viral infection model with two modes of transmission were investigated. The uninfected cells dynamics, two infection modes for both virus-to-cell infection and cell-to-cell transmission, and concentration dependence for the latently infected cells, infected cells, viruses, and B cells were modeled by seven general nonlinear functions along with some assumptions. The basic reproduction number was calculated and demonstrated the global properties of the virus model. The theoretical results were illustrated by numerical simulations.

References

[1]	X. Yang, Y. M. Su, X. J. Zhuo, T. H. Gao, Global analysis for a delayed HCV model with saturation incidence and two target cells, Chaos Soliton. Fract., 166 (2023), 112950. https://doi.org/10.1016/j.chaos.2022.112950 doi: 10.1016/j.chaos.2022.112950
[2]	A. M. Elaiw, N. H. AlShamrani, Stability of an adaptive immunity pathogen dynamics model with latency and multiple delays, Math. Method. Appl. Sci., 41 (2018), 6645–6672. https://doi.org/10.1002/mma.5182 doi: 10.1002/mma.5182
[3]	S. F. Wang, D. Y. Zou, Global stability of in host viral models with humoral immunity and intracellular delays, Appl. Math. Model., 36 (2012), 1313–1322. https://doi.org/10.1016/j.apm.2011.07.086 doi: 10.1016/j.apm.2011.07.086
[4]	J. L. Wang, S. Q. Liu, The stability analysis of a general viral infection model with distributed delays and multi-staged infected progression, Commun. Nonlinear Sci., 20 (2015), 263–272. https://doi.org/10.1016/j.cnsns.2014.04.027 doi: 10.1016/j.cnsns.2014.04.027
[5]	T. Q. Zhang, X. N. Xu, X. Z. Wang, Dynamic analysis of a cytokine-enhanced viral infection model with time delays and CTL immune response, Chaos Soliton. Fract., 170 (2023), 113357. https://doi.org/10.1016/j.chaos.2023.113357 doi: 10.1016/j.chaos.2023.113357
[6]	P. Georgescu, Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J. Appl. Math., 67 (2006), 337–353. https://doi.org/10.1137/060654876 doi: 10.1137/060654876
[7]	Z. H. Yuan, X. F. Zou, Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays, Math. Biosci. Eng., 10 (2013), 483–498. https://doi.org/10.3934/mbe.2013.10.483 doi: 10.3934/mbe.2013.10.483
[8]	K. Hattaf, Spatiotemporal dynamics of a generalized viral infection model with distributed delays and CTL immune response, Computation, 7 (2019), 21. https://doi.org/10.3390/computation7020021 doi: 10.3390/computation7020021
[9]	K. A. Pawelek, S. Q. Liu, F. Pahlevani, L. B. Rong, A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98–109. https://doi.org/10.1016/j.mbs.2011.11.002 doi: 10.1016/j.mbs.2011.11.002
[10]	P. Balasubramaniam, P. Tamilalagan, M. Prakash, Bifurcation analysis of HIV infection model with antibody and cytotoxic T-lymphocyte immune responses and Beddington-DeAngelis functional response, Math. Method. Appl. Sci., 38 (2015), 1330–1341. https://doi.org/10.1002/mma.3148 doi: 10.1002/mma.3148
[11]	J. L. Wang, M. Guo, X. N. Liu, Z. T. Zhao, Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay, Appl. Math. Comput., 291 (2016), 149–161. https://doi.org/10.1016/j.amc.2016.06.032 doi: 10.1016/j.amc.2016.06.032
[12]	N. L. Komarova, D. Wodarz, Virus dynamics in the presence of synaptic transmission, Math. Biosci., 242 (2013), 161–171. https://doi.org/10.1016/j.mbs.2013.01.003 doi: 10.1016/j.mbs.2013.01.003
[13]	A. Sigal, J. T. Kim, A. B. Balazs, E. Dekel, A. Mayo, R. Milo, et al., Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy, Nature, 477 (2011), 95–98. https://doi.org/10.1038/nature10347 doi: 10.1038/nature10347
[14]	S. Iwami, J. S. Takeuchi, S. Nakaoka, F. Mammano, F. Clavel, H. Inaba, et al., Cell-to-cell infection by HIV contributes over half of virus infection, eLife, 4 (2015), e08150. https://doi.org/10.7554/eLife.08150 doi: 10.7554/eLife.08150
[15]	X. Wang, S. Y. Tang, X. Y. Song, L. B. Rong, Mathematical analysis of an HIV latent infection model including both virus-to-cell infection and cell-to-cell transmission, J. Biol. Dynam., 11 (2017), 455–483. https://doi.org/10.1080/17513758.2016.1242784 doi: 10.1080/17513758.2016.1242784
[16]	H. Y. Shu, Y. M. Chen, L. Wang, Impacts of the cell-free and cell-to-cell infection modes on viral dynamics, J. Dyn. Diff. Equat., 30 (2018), 1817–1836. https://doi.org/10.1007/s10884-017-9622-2 doi: 10.1007/s10884-017-9622-2
[17]	X. L. Lai, X. F. 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. https://doi.org/10.1137/130930145 doi: 10.1137/130930145
[18]	Y. Yang, L. Zou, S. G. 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. https://doi.org/10.1016/j.mbs.2015.05.001 doi: 10.1016/j.mbs.2015.05.001
[19]	F. Li, J. L. Wang, Analysis of an HIV infection model with logistic target-cell growth and cell-to-cell transmission, Chaos Soliton. Fract., 81 (2015), 136–145. https://doi.org/10.1016/j.chaos.2015.09.003 doi: 10.1016/j.chaos.2015.09.003
[20]	A. M. Elaiw, N. H. AlShamrani, Global stability of a delayed adaptive immunity viral infection with two routes of infection and multi-stages of infected cells, Commun. Nonlinear Sci., 86 (2020), 105259. https://doi.org/10.1016/j.cnsns.2020.105259 doi: 10.1016/j.cnsns.2020.105259
[21]	A. M. Elaiw, N. H. AlShamrani, Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal, Nonlinear Anal.-Real, 26 (2015), 161–190. https://doi.org/10.1016/j.nonrwa.2015.05.007 doi: 10.1016/j.nonrwa.2015.05.007
[22]	J. Z. Lin, R. Xu, X. H. 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. https://doi.org/10.1016/j.amc.2017.08.004 doi: 10.1016/j.amc.2017.08.004
[23]	K. F. Wang, W. D. Wang, S. P. Song, Dynamics of an HBV model with diffusion and delay, J. Theor. Biol., 253 (2008), 36–44. https://doi.org/10.1016/j.jtbi.2007.11.007 doi: 10.1016/j.jtbi.2007.11.007
[24]	Y. Gao, J. L. Wang, Threshold dynamics of a delayed nonlocal reaction-diffusion HIV infection model with both cell-free and cell-to-cell transmissions, J. Math. Anal. Appl., 488 (2020), 124047. https://doi.org/10.1016/j.jmaa.2020.124047 doi: 10.1016/j.jmaa.2020.124047
[25]	Y. Yang, Y. C. Xu, Global stability of a diffusive and delayed virus dynamics model with Beddington-DeAngelies function and CTL immune response, Comput. Math. Appl., 71 (2016), 922–930. https://doi.org/10.1016/j.camwa.2016.01.009 doi: 10.1016/j.camwa.2016.01.009
[26]	H. Q. Sun, J. L. 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. https://doi.org/10.1016/j.camwa.2018.09.032 doi: 10.1016/j.camwa.2018.09.032
[27]	C. C. Travis, G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395–418. https://doi.org/10.2307/1997265 doi: 10.2307/1997265
[28]	W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differ. Equations, 29 (1978), 1–14. https://doi.org/10.1016/0022-0396(78)90037-2 doi: 10.1016/0022-0396(78)90037-2
[29]	R. H. Martin, H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1–44. https://doi.org/10.1090/S0002-9947-1990-0967316-X doi: 10.1090/S0002-9947-1990-0967316-X
[30]	R. H. Martin, H. L. Smith, Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1–35. https://doi.org/10.1515/crll.1991.413.1 doi: 10.1515/crll.1991.413.1
[31]	J. H. Wu, Theory and applications of partial functional differential equations, New York: Springer, 1996. https://doi.org/10.1007/978-1-4612-4050-1
[32]	D. Henry, Geometric theory of semilinear parabolic equations, Berlin: Springer, 1981. https://doi.org/10.1007/BFb0089647
[33]	O. Diekmann, J. A. P. Heesterbeek, 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), 365–382. https://doi.org/10.1007/BF00178324 doi: 10.1007/BF00178324
[34]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[35]	H. Q. Sun, J. Li, A numerical method for a diffusive virus model with general incidence function, cell-to-cell transmission and time delay, Physica A, 545 (2020), 123477. https://doi.org/10.1016/j.physa.2019.123477 doi: 10.1016/j.physa.2019.123477
[36]	O. Nave, Modification of semi-analytical method applied system of ODE, Modern Applied Science, 14 (2020), 75–81. https://doi.org/10.5539/mas.v14n6p75 doi: 10.5539/mas.v14n6p75
[37]	Y. Wang, Y. C. Zhou, F. Brauer, J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901–934. https://doi.org/10.1007/s00285-012-0580-3 doi: 10.1007/s00285-012-0580-3
[38]	D. Wodarz, Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743–1750. https://doi.org/10.1099/vir.0.19118-0 doi: 10.1099/vir.0.19118-0

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