Note on a class of time-delay virus models with virus-to-cell infection, cell-to-cell transmission, CTL and antibody immune responses

Yan Li; Rui Zhu; Xiaoqun Li; Xianshan Yang; Yong Li; Ruixia Yuan; Yan Li; Rui Zhu; Xiaoqun Li; Xianshan Yang; Yong Li; Ruixia Yuan

doi:10.3934/math.20251162

AIMS Mathematics

2025, Volume 10, Issue 11: 26446-26458. doi: 10.3934/math.20251162

Previous Article Next Article

Research article

Note on a class of time-delay virus models with virus-to-cell infection, cell-to-cell transmission, CTL and antibody immune responses

1.
School of Information Engineering, Wuhan Huaxia Institute of Technology, Wuhan 430223, China
2.
School of Information and Mathematics, Yangtze University, Jingzhou 434023, China
3.
Research School of Finance, Actuarial Studies and Statistics, The Australian National University, Canberra 2601, Australia
4.
College of Business and Economics, Shanghai Business School, Shanghai 201400, China

Received: 03 September 2025 Revised: 03 November 2025 Accepted: 05 November 2025 Published: 17 November 2025
MSC : 34D23, 92B05

This paper investigates the coexistence of equilibria and local and global asymptotic stability of a class of time-delay virus models involving virus-to-cell infection, cell-to-cell transmission, Cytotoxic T Lymphocytes (CTL), and antibody immune responses. These models typically exhibit five equilibria; however, the existing literature lacks comprehensive studies on the uniqueness and coexistence of these equilibria. Our study fills the gaps and rectifies the shortcomings or lack of rigor in certain prior research.
- virus-to-cell infection,
- cell-to-cell transmission,
- CTL and antibody immune responses,
- co-existence,
- stability analysis
Citation: Yan Li, Rui Zhu, Xiaoqun Li, Xianshan Yang, Yong Li, Ruixia Yuan. Note on a class of time-delay virus models with virus-to-cell infection, cell-to-cell transmission, CTL and antibody immune responses[J]. AIMS Mathematics, 2025, 10(11): 26446-26458. doi: 10.3934/math.20251162

Related Papers:

Abstract

This paper investigates the coexistence of equilibria and local and global asymptotic stability of a class of time-delay virus models involving virus-to-cell infection, cell-to-cell transmission, Cytotoxic T Lymphocytes (CTL), and antibody immune responses. These models typically exhibit five equilibria; however, the existing literature lacks comprehensive studies on the uniqueness and coexistence of these equilibria. Our study fills the gaps and rectifies the shortcomings or lack of rigor in certain prior research.

References

[1]	A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard, D. D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, 27 (1996), 1582–1586. http://dx.doi.org/10.1126/science.271.5255.1582 doi: 10.1126/science.271.5255.1582
[2]	A. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, et al., Decay characteristics of HIV-1-infected compartments during combination therapy, Nature, 387 (1997), 188–191. http://dx.doi.org/10.1038/387188a0 doi: 10.1038/387188a0
[3]	N. L. Komarova, E. Barnes, P. Klenerman, D. Wodarz, Boosting immunity by antiviral drug therapy: a simple relationship among timing, efficacy, and success, PNAS, 100 (2003), 1855–1860. http://dx.doi.org/10.1073/pnas.0337483100 doi: 10.1073/pnas.0337483100
[4]	H. Shu, L. Wang, J. Watmough, Sustained and transient oscillations and chaos induced by delayed antiviral immune response in an immunosuppressive infection model, J. Math. Biol., 68 (2014), 488–503. http://dx.doi.org/10.1007/s00285-012-0639-1 doi: 10.1007/s00285-012-0639-1
[5]	X. Lai, X. Zou, Modeling 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
[6]	W. Wang, T. Zhang, Caspase-1-mediated pyroptosis of the predominance for driving CD4$^{+}$ T cells death: a nonlocal spatial mathematical model, Bull. Math. Biol., 80 (2018), 540–582. https://doi.org/10.1007/s11538-017-0389-8 doi: 10.1007/s11538-017-0389-8
[7]	W. Wang, X. Wang, X. Fan, On the global attractivity of a diffusive viral infection model with spatial heterogeneity, Math. Method. Appl. Sci., 48 (2025), 15656–15660. https://doi.org/10.1002/mma.70040 doi: 10.1002/mma.70040
[8]	W. Wang, G. Wu, X. Fan, Global dynamics of a novel viral infection model mediated by pattern recognition receptors, Appl. Math. Lett., 173 (2026), 109757. https://doi.org/10.1016/j.aml.2025.109757 doi: 10.1016/j.aml.2025.109757
[9]	Y. Li, L. Zhang, J. Zhang, S. Liu, Z. Peng, Dynamical modeling and data analysis of HIV infection with infection-age, CTLs immune response and delayed antibody immune response, J. Math. Biol., 91 (2025), 57. https://doi.org/10.1007/s00285-025-02285-y doi: 10.1007/s00285-025-02285-y
[10]	Y. Yan, W. Wang, Global stability of a five-dimensional model with immune responses and delay, Discrete Cont. Dyn. B, 17 (2012), 401–416. https://doi.org/10.3934/dcdsb.2012.17.401 doi: 10.3934/dcdsb.2012.17.401
[11]	A. M. Elaiw, N. H. AlShamrani, Stability of a delayed adaptive immunity HIV infection model with silent infected cells and cellular infection, J. Appl. Anal. Comput., 11 (2021), 964–1005. https://doi.org/10.11948/20200124 doi: 10.11948/20200124
[12]	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
[13]	J. Li, X. Ma, J. Li, D. Zhang, Dynamics of a chronic virus infection model with viral stimulation delay, Appl. Math. Lett., 122 (2021), 107547. https://doi.org/10.1016/j.aml.2021.107547 doi: 10.1016/j.aml.2021.107547
[14]	T. Guo, Z. Qiu, L. Rong, Analysis of an HIV model with immune responses and cell-to-cell transmission, Bull. Malays. Math. Sci. Soc., 43 (2020), 581–607. https://doi.org/10.1007/s40840-018-0699-5 doi: 10.1007/s40840-018-0699-5
[15]	Y. Li, Y. Wei, S. Li, Y. Li, Z. Peng, Stochastic HIV-1 infection model with time delay: case study of clinical data, J. Biol. Dynam., 19 (2025), 2553766. http://dx.doi.org/10.1080/17513758.2025.2553766 doi: 10.1080/17513758.2025.2553766
[16]	C. Jiang, W. Wang, Complete classification of global dynamics of a virus model with immune responses, Discrete Cont. Dyn. B, 19 (2014), 1087–1103. https://doi.org/10.3934/dcdsb.2014.19.1087 doi: 10.3934/dcdsb.2014.19.1087
[17]	J. Lin, R. Xu, X. Tian, Threshold dynamics of an HIV-1 model with both viral and cellular infections, cell-mediated and humoral immune responses, Math. Biosci. Eng., 16 (2018), 292–319. https://doi.org/ 10.3934/mbe.2019015 doi: 10.3934/mbe.2019015
[18]	K. Hattaf, M. Khabouze, N. Yousfi, Dynamics of a generalized viral infection model with adaptive immune response, Int. J. Dynam. Control, 3 (2015), 253–261. https://doi.org/10.1007/s40435-014-0130-5 doi: 10.1007/s40435-014-0130-5
[19]	Y. Su, D. Sun, L. Zhao, Global analysis of a humoral and cellular immunity virus dynamics model with the Beddington-DeAngelis incidence rate, Math. Method. Appl. Sci., 38 (2015), 2984–2993. https://doi.org/10.1002/mma.3274 doi: 10.1002/mma.3274
[20]	K. Hattaf, N. Yousfi, A class of delayed viral infection models with general incidence rate and adaptive immune response, Int. J. Dynam. Control, 4 (2016), 254–265. https://doi.org/10.1007/s40435-015-0158-1 doi: 10.1007/s40435-015-0158-1
[21]	J. Wang, J. Pang, T. Kuniya, 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), 298–316. https://doi.org/10.1016/j.amc.2014.05.015 doi: 10.1016/j.amc.2014.05.015
[22]	N. Yousfi, K. Hattaf, A. Tridane, Modeling the adaptive immune response in HBV infection, J. Math. Biol., 63 (2011), 933–957. https://doi.org/10.1007/s00285-010-0397-x doi: 10.1007/s00285-010-0397-x
[23]	J. Wang, M. Guo, X. Liu, Z. 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
[24]	X. Wang, X. Meng, L. Rong, Global dynamics of a multiscale model for hepatitis C virus infection, Appl. Math. Lett., 149 (2024), 108904. https://doi.org/10.1016/j.aml.2023.108904 doi: 10.1016/j.aml.2023.108904

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)