In this study, we formulate a within-host mathematical model of human papillomavirus (HPV) infection that captures the dynamics of susceptible epithelial cells, infected cells, free virions, cytotoxic T lymphocytes (CTLs), neutralizing antibodies, and cancerous cells. The model incorporates immune mechanisms, including CTL-mediated killing of infected cells and antibody-driven neutralization of virions, as well as therapeutic interventions. Building on this framework, we apply optimal control theory using three treatment strategies: (i) chemotherapy intensity, which enhances infected-cell death and reduces virion production, (ii) CTL stimulation (immunotherapy) intensity, which augments the CTL source term, and (iii) antibody therapy, which strengthens humoral immunity. We first identify the equilibria of the system and test their stability. The disease-free equilibrium is stable when the basic reproduction number is $ R_0 < 1 $, meaning the infection cannot persist in the population. When $ R_0 > 1 $, the endemic equilibrium (EE) becomes feasible and is globally asymptotically stable. Finally, the optimal control approach shows that a combined treatment strategy-chemotherapy, immune cell (CTL) boosting, and antibody therapy can significantly minimize viral load and cancer cell growth. This integrated approach demonstrates how traditional chemotherapy and immune-based therapies can complement each other, offering a more effective pathway for managing HPV.
Citation: Fahad Al Basir, Khalid Aldawsari, Konstantin B. Blyuss. A model for HPV-related cervical cancer dynamics with saturated terms: an optimal control problem using chemotherapy, immunotherapy, and antibody drugs[J]. AIMS Mathematics, 2026, 11(7): 21293-21322. doi: 10.3934/math.2026864
In this study, we formulate a within-host mathematical model of human papillomavirus (HPV) infection that captures the dynamics of susceptible epithelial cells, infected cells, free virions, cytotoxic T lymphocytes (CTLs), neutralizing antibodies, and cancerous cells. The model incorporates immune mechanisms, including CTL-mediated killing of infected cells and antibody-driven neutralization of virions, as well as therapeutic interventions. Building on this framework, we apply optimal control theory using three treatment strategies: (i) chemotherapy intensity, which enhances infected-cell death and reduces virion production, (ii) CTL stimulation (immunotherapy) intensity, which augments the CTL source term, and (iii) antibody therapy, which strengthens humoral immunity. We first identify the equilibria of the system and test their stability. The disease-free equilibrium is stable when the basic reproduction number is $ R_0 < 1 $, meaning the infection cannot persist in the population. When $ R_0 > 1 $, the endemic equilibrium (EE) becomes feasible and is globally asymptotically stable. Finally, the optimal control approach shows that a combined treatment strategy-chemotherapy, immune cell (CTL) boosting, and antibody therapy can significantly minimize viral load and cancer cell growth. This integrated approach demonstrates how traditional chemotherapy and immune-based therapies can complement each other, offering a more effective pathway for managing HPV.
| [1] |
J. Wolf, L. F. Kist, S. B. Pereira, M. A. Quessada, H. Petek, A. Pille, et al., Human papillomavirus infection: epidemiology, biology, host interactions, cancer development, prevention, and therapeutics, Rev. Med. Virol., 34 (2024), e2537. https://doi.org/10.1002/rmv.2537 doi: 10.1002/rmv.2537
|
| [2] |
D. Hanahan, R. A. Weinberg, Hallmarks of cancer: the next generation, Cell, 144 (2011), 646–674. https://doi.org/10.1016/j.cell.2011.02.013 doi: 10.1016/j.cell.2011.02.013
|
| [3] |
E. H. Elbasha, E. J. Dasbach, R. P. Insinga, Model for assessing human papillomavirus vaccination strategies, Emerg. Infect. Dis., 13 (2007), 28–41. https://doi.org/10.3201/eid1301.060438 doi: 10.3201/eid1301.060438
|
| [4] |
L. Kumar, S. Gupta, Integrating chemotherapy in the management of cervical cancer: a critical appraisal, Oncology, 91 (2016), 8–17. https://doi.org/10.1159/000447576 doi: 10.1159/000447576
|
| [5] |
M. Xu, C. Cao, P. Wu, X. Huang, D. Ma, Advances in cervical cancer: current insights and future directions, Cancer Commun., 45 (2025), 77–109. https://doi.org/10.1002/cac2.12629 doi: 10.1002/cac2.12629
|
| [6] | S. Dilruba, G. V. Kalayda, Platinum-based drugs: past, present and future, Cancer Chemother. Pharmacol., 77 (2016), 1103–1124. https://doi.org/10.1007/s00280-016-2976-z |
| [7] |
M. Robati, D. Holtz, C. J. Dunton, A review of topotecan in combination chemotherapy for advanced cervical cancer, Ther. Clin. Risk Manag., 4 (2008), 213–218. https://doi.org/10.2147/TCRM.S1771 doi: 10.2147/TCRM.S1771
|
| [8] |
C. A. Burmeister, S. F. Khan, G. Schäfer, N. Mbatani, T. Adams, J. Moodley, et al., Cervical cancer therapies: current challenges and future perspectives, Tumour Virus Res., 13 (2022), 200238. https://doi.org/10.1016/j.tvr.2022.200238 doi: 10.1016/j.tvr.2022.200238
|
| [9] |
S. Movva, L. Rodriguez, H. Arias-Pulido, C. Verschraegen, Novel chemotherapy approaches for cervical cancer, Cancer, 115 (2009), 3166–3180. https://doi.org/10.1002/cncr.24364 doi: 10.1002/cncr.24364
|
| [10] |
A. K. Bag, S. Ghosh, A. N. Chatterjee, P. K. Roy, A mathematical framework investigating the impact of Chemo-iPSC therapy for the dynamics of cervical cancer, J. Appl. Math. Comput., 71 (2025), 6061–6093. https://doi.org/10.1007/s12190-025-02476-2 doi: 10.1007/s12190-025-02476-2
|
| [11] |
R. O. Dillman, Monoclonal antibodies in the treatment of malignancy: basic concepts and recent developments, Cancer Invest., 19 (2001), 833–841. https://doi.org/10.1081/CNV-100107745 doi: 10.1081/CNV-100107745
|
| [12] |
G. J. Weiner, Building better monoclonal antibody-based therapeutics, Nat. Rev. Cancer, 15 (2015), 361–370. https://doi.org/10.1038/nrc3930 doi: 10.1038/nrc3930
|
| [13] |
D. Zahavi, L. Weiner, Monoclonal antibodies in cancer therapy, Antibodies, 9 (2020), 34. https://doi.org/10.3390/antib9030034 doi: 10.3390/antib9030034
|
| [14] |
K. Allali, Stability analysis and optimal control of HPV infection model with early-stage cervical cancer, BioSystems, 199 (2021), 104321. https://doi.org/10.1016/j.biosystems.2020.104321 doi: 10.1016/j.biosystems.2020.104321
|
| [15] |
A. Boukhouima, E. M. Lotfi, M. Mahrouf, S. Rosa, D. F. M. Torres, N. Yousfi, Stability analysis and optimal control of a fractional HIV-AIDS epidemic model with memory and general incidence rate, Eur. Phys. J. Plus, 136 (2021), 103. https://doi.org/10.1140/epjp/s13360-020-01013-3 doi: 10.1140/epjp/s13360-020-01013-3
|
| [16] |
J. P. Mateus, P. Rebelo, S. Rosa, C. M. Silva, D. F. M. Torres, Optimal control of non-autonomous SEIRS models with vaccination and treatment, Discrete Contin. Dyn. Syst. Ser. S, 11 (2017), 1179–1199. https://doi.org/10.3934/dcdss.2018067 doi: 10.3934/dcdss.2018067
|
| [17] |
S. Chakraborty, X. Z. Li, P. K. Roy, How can HPV-induced cervical cancer be controlled by a combination of drug therapy? A mathematical study, Int. J. Biomath., 12 (2019), 1950070. https://doi.org/10.1142/S1793524519500700 doi: 10.1142/S1793524519500700
|
| [18] |
A. S. Perelson, P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3–44. https://doi.org/10.1137/S0036144598335107 doi: 10.1137/S0036144598335107
|
| [19] |
R. J. De Boer, A. S. Perelson, Target cell limited and immune control models of HIV infection: a comparison, J. Theor. Biol., 190 (1998), 391–402. https://doi.org/10.1006/jtbi.1997.0548 doi: 10.1006/jtbi.1997.0548
|
| [20] |
C. D. Murin, I. A. Wilson, A. B. Ward, Antibody responses to viral infections: a structural perspective across three different enveloped viruses, Nat. Microbiol., 4 (2019), 734–747. https://doi.org/10.1038/s41564-019-0392-y doi: 10.1038/s41564-019-0392-y
|
| [21] |
G. E. Mahlbacher, K. C. Reihmer, H. B. Frieboes, Mathematical modeling of tumor-immune cell interactions, J. Theor. Biol., 469 (2019), 47–60. https://doi.org/10.1016/j.jtbi.2019.03.002 doi: 10.1016/j.jtbi.2019.03.002
|
| [22] |
P. Dubey, U. S. Dubey, B. Dubey, Modeling the dynamics of viral–host interaction during treatment of productively infected cells and free virus involving total immune response, Nonlinear Anal. Model. Control, 26 (2021), 678–701. https://doi.org/10.15388/namc.2021.26.21434 doi: 10.15388/namc.2021.26.21434
|
| [23] |
C. Jiang, W. Wang, Complete classification of global dynamics of a virus model with immune responses, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1087–1103. https://doi.org/10.3934/dcdsb.2014.19.1087 doi: 10.3934/dcdsb.2014.19.1087
|
| [24] |
O. Diekmann, J. A. P. Heesterbeek, M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2010), 873–885. https://doi.org/10.1098/rsif.2009.0386 doi: 10.1098/rsif.2009.0386
|
| [25] |
F. Al Basir, K. B. Blyuss, E. Venturino, Stability and bifurcation analysis of a multi-delay model for mosaic disease transmission, AIMS Math., 8 (2023), 24545–24567. https://doi.org/10.3934/math.20231252 doi: 10.3934/math.20231252
|
| [26] | S. Lenhart, J. T. Workman, Optimal control applied to biological models, Boca Raton, FL: Chapman & Hall/CRC, 2007. |
| [27] | L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The mathematical theory of optimal processes, New York: Interscience Publishers, 1962. |
| [28] |
M. Verma, S. Erwin, V. Abedi, R. Hontecillas, S. Hoops, A. Leber, et al., Modeling the mechanisms by which HIV-associated immunosuppression influences HPV persistence at the oral mucosa, PLoS ONE, 12 (2017), e0168133. https://doi.org/10.1371/journal.pone.0168133 doi: 10.1371/journal.pone.0168133
|
| [29] |
M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74–79. https://doi.org/10.1126/science.272.5258.74 doi: 10.1126/science.272.5258.74
|
| [30] |
F. Al Basir, K. B. Blyuss, A. A. Raezah, B. Rahman, H. J. Alsakaji, Mathematical modelling of the role of Langerhans cells in the dynamics of HPV infection, Sci. Rep., 15 (2025), 17505. https://doi.org/10.1038/s41598-025-02608-5 doi: 10.1038/s41598-025-02608-5
|
| [31] |
C. Liu, Z. Gong, C. Yu, S. Wang, K. L. Teo, Optimal control computation for nonlinear fractional time-delay systems with state inequality constraints, J. Optim. Theory Appl., 191 (2021), 83–117. https://doi.org/10.1007/s10957-021-01926-8 doi: 10.1007/s10957-021-01926-8
|
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Fahad Al Basir, Khalid Aldawsari, Konstantin B. Blyuss. A model for HPV-related cervical cancer dynamics with saturated terms: an optimal control problem using chemotherapy, immunotherapy, and antibody drugs[J]. AIMS Mathematics, 2026, 11(7): 21293-21322. doi: 10.3934/math.2026864
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