Mathematical modeling of HIV/HTLV co-infection with CTL-mediated immunity

A. M. Elaiw; N. H. AlShamrani; A. D. Hobiny; A. M. Elaiw; N. H. AlShamrani; A. D. Hobiny

doi:10.3934/math.2021098

AIMS Mathematics

2021, Volume 6, Issue 2: 1634-1676. doi: 10.3934/math.2021098

Previous Article Next Article

Research article

Mathematical modeling of HIV/HTLV co-infection with CTL-mediated immunity

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut Branch, Assiut, Egypt
3.
Department of Mathematics, Faculty of Science, University of Jeddah, P. O. Box 80327, Jeddah 21589, Saudi Arabia

Received: 06 September 2020 Accepted: 16 November 2020 Published: 26 November 2020
MSC : 34D20, 34D23, 37N25, 92B05

In the literature, a great number of HIV and HTLV-I mono-infection models has been formulated and analyzed. However, the within-host dynamics of HIV/HTLV-I co-infection has not been modeled. In the present paper we formulate and analyze a new HIV/HTLV-I co-infection model with latency and Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD$4^{+}$T cells, latently HIV-infected cells, actively HIV-infected cells, latently HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell and cell-to-cell transmissions, while the HTLV-I can only spread via cell-to-cell transmission. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We derive the threshold parameters which govern the existence and stability of all equilibria of the model. We prove the global asymptotic stability of all equilibria by utilizing Lyapunov function and Lyapunov-LaSalle asymptotic stability theorem. We have presented numerical simulations to illustrate the effectiveness of our main results. In addition, we have discussed the effect of HTLV-I infection on the HIV-infected patients and vice versa. We have pointed out the influence of CTL immune response on the co-infection dynamics.
- HIV/HTLV-I co-infection,
- global stability,
- CTL-mediated immune response,
- Lyapunov function
Citation: A. M. Elaiw, N. H. AlShamrani, A. D. Hobiny. Mathematical modeling of HIV/HTLV co-infection with CTL-mediated immunity[J]. AIMS Mathematics, 2021, 6(2): 1634-1676. doi: 10.3934/math.2021098

Related Papers:

Abstract

In the literature, a great number of HIV and HTLV-I mono-infection models has been formulated and analyzed. However, the within-host dynamics of HIV/HTLV-I co-infection has not been modeled. In the present paper we formulate and analyze a new HIV/HTLV-I co-infection model with latency and Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD$4^{+}$T cells, latently HIV-infected cells, actively HIV-infected cells, latently HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell and cell-to-cell transmissions, while the HTLV-I can only spread via cell-to-cell transmission. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We derive the threshold parameters which govern the existence and stability of all equilibria of the model. We prove the global asymptotic stability of all equilibria by utilizing Lyapunov function and Lyapunov-LaSalle asymptotic stability theorem. We have presented numerical simulations to illustrate the effectiveness of our main results. In addition, we have discussed the effect of HTLV-I infection on the HIV-infected patients and vice versa. We have pointed out the influence of CTL immune response on the co-infection dynamics.

References

[1]	WHO, Global Health Observatory (GHO) data. HIV/AIDS, 2018. Available from: http://www.who.int/gho/hiv/en/.
[2]	M. A. Nowak, R. M. May, Virus dynamics: Mathematical principles of immunology and virology, Oxford University Press, 2001.
[3]	M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74
[4]	K. D. Pedro, A. J. Henderson, L. M. Agosto, Mechanisms of HIV-1 cell-to-cell transmission and the establishment of the latent reservoir, Virus Res., 265 (2019), 115-121. doi: 10.1016/j.virusres.2019.03.014
[5]	L. Rong, A. S. Perelson, Modeling latently infected cell activation: viral and latent reservoir persistence, and viral blips in HIV-infected patients on potent therapy, PLoS Comput. Biol., 5 (2009), 1-18.
[6]	D. S. Callaway, A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64. doi: 10.1006/bulm.2001.0266
[7]	A. M. Elaiw, S. A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Method. Appl. Sci., 36 (2013), 383-394. doi: 10.1002/mma.2596
[8]	A. M. Elaiw, M. A. Alshaikh, Stability of discrete-time HIV dynamics models with three categories of infected CD4⁺ T-cells, Adv. Differ. Equ., 2019 (2019), 1-24. doi: 10.1186/s13662-018-1939-6
[9]	A. M. Elaiw, N. H. AlShamrani, Stability of a general CTL-mediated immunity HIV infection model with silent infected cell-to-cell spread, Adv. Differ. Equ., 2020 (2020), 1-25. doi: 10.1186/s13662-019-2438-0
[10]	A. M. Elaiw, E. K. Elnahary, A. A. Raezah, Effect of cellular reservoirs and delays on the global dynamics of HIV, Adv. Differ. Equ., 2018 (2018), 1-36. doi: 10.1186/s13662-017-1452-3
[11]	B. Buonomo, C. Vargas-De-Leon, Global stability for an HIV-1 infection model including an eclipse stage of infected cells, J. Math. Anal. Appl., 385 (2012), 709-720. doi: 10.1016/j.jmaa.2011.07.006
[12]	H. Liu, J. F. Zhang, Dynamics of two time delays differential equation model to HIV latent infection, Physica A, 514 (2019), 384-395. doi: 10.1016/j.physa.2018.09.087
[13]	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), 1-16.
[14]	M. Sourisseau, N. Sol-Foulon, F. Porrot, F. Blanchet, O. Schwartz, Inefficient human immunodeficiency virus replication in mobile lymphocytes, J. Virol., 81 (2007), 1000-1012. doi: 10.1128/JVI.01629-06
[15]	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. doi: 10.1038/nature10347
[16]	Y. Gao, J. 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. doi: 10.1016/j.jmaa.2020.124047
[17]	A. Mojaver, H. Kheiri, Mathematical analysis of a class of HIV infection models of CD4+T-cells with combined antiretroviral therapy, Appl. Math. Comput., 259 (2015), 258-270.
[18]	X. Wang, S. Tang, X. Song, L. 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. doi: 10.1080/17513758.2016.1242784
[19]	H. R. Norrgren, S. Bamba, O. Larsen, Z. Da Silva, P. Aaby, T. Koivula, et al. Increased prevalence of HTLV-1 in patients with pulmonary tuberculosis coinfected with HIV, but not in HIV-negative patients with tuberculosis, J. Acq. Imm. Def., 48 (2008), 607-610. doi: 10.1097/QAI.0b013e31817efb83
[20]	A. G. Lim, P. K. Maini, HTLV-Iinfection: A dynamic struggle between viral persistence and host immunity, J. Theor. Biol., 352 (2014), 92-108. doi: 10.1016/j.jtbi.2014.02.022
[21]	X. Pan, Y. Chen, H. Shu, Rich dynamics in a delayed HTLV-I infection model: Stability switch, multiple stable cycles, and torus, J. Math. Anal. Appl., 479 (2019), 2214-2235. doi: 10.1016/j.jmaa.2019.07.051
[22]	M. Y. Li, A. G. Lim, Modelling the role of Tax expression in HTLV-1 persisence in vivo, Bull. Math. Biol., 73 (2011), 3008-3029. doi: 10.1007/s11538-011-9657-1
[23]	B. Asquith, C. R. M. Bangham, The dynamics of T-cell fratricide: application of a robust approach to mathematical modeling in immunology, J. Theor. Biol., 222 (2003), 53-69. doi: 10.1016/S0022-5193(03)00013-4
[24]	S. Tokudome et al., Incidence of adult T cell leukemia/lymphoma among human T lymphotropic virus type 1 carriers in Saga, Japan, Cancer Res., 49 (1989), 226-228.
[25]	N. I. Stilianakis, J. Seydel, Modeling the T-cell dynamics and pathogenesis of HTLV-I infection, Bull. Math. Biol., 61 (1999), 935-947. doi: 10.1006/bulm.1999.0117
[26]	H. Gomez-Acevedo, M. Y. Li, Backward bifurcation in a model for HTLV-I infection of CD4⁺ T cells, Bull. Math. Biol., 67 (2005), 101-114. doi: 10.1016/j.bulm.2004.06.004
[27]	C. Vargas-De-Leon, The complete classification for global dynamics of amodel for the persistence of HTLV-1 infection, Appl. Math. Comput., 237 (2014), 489-493.
[28]	X. Song, Y. Li, Global stability and periodic solution of a model for HTLV-1 infection and ATL progression, Appl. Math. Comput., 180 (2006), 401-410.
[29]	E. S. Rosenberg, M. Altfeld, S. H. Poon, M. N. Phillips, B. M. Wilkes, R. L. Eldridge, et al., Immune control of HIV-1 following early treatment of acute infection, Nature, 407 (2000), 523-526. doi: 10.1038/35035103
[30]	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. doi: 10.1016/j.cnsns.2020.105259
[31]	B. Asquith, C. R. M. Bangham, Quantifying HTLV-I dynamics, Immunol. Cell Biol., 85 (2007), 280-286. doi: 10.1038/sj.icb.7100050
[32]	C. Bartholdy, J. P. Christensen, D. Wodarz, A. R. Thomsen, Persistent virus infection despite chronic cytotoxic T-lymphocyte activation in gamma interferon-deficient mice infected with lymphocytic choriomeningitis virus, J. Virol., 74 (2000), 10304-10311. doi: 10.1128/JVI.74.22.10304-10311.2000
[33]	H. Gomez-Acevedo, M. Y. Li, S. Jacobson, Multi-stability in a model for CTL response to HTLV-I infection and its consequences Bull. Math. Biol., 72 (2010), 681-696. doi: 10.1007/s11538-009-9465-z
[34]	M. Y. Li, H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection, Bull. Math. Biol., 73 (2011), 1774-1793. doi: 10.1007/s11538-010-9591-7
[35]	L. Wang, Z. Liu, Y. Li, D. Xu, Complete dynamical analysis for a nonlinear HTLV-I infection model with distributed delay, CTL response and immune impairment, Discrete Cont. Dyn. B, 25 (2020), 917-933.
[36]	Y. Wang, J. Liu, J. M. Heffernan, Viral dynamics of an HTLV-I infection model with intracellular delay and CTL immune response delay, J. Math. Anal. Appl., 459 (2018), 506-527. doi: 10.1016/j.jmaa.2017.10.027
[37]	F. Li, W. Ma, Dynamics analysis of an HTLV-1 infection model with mitotic division of actively infected cells and delayed CTL immune response, Math. Method. Appl. Sci., 41 (2018), 3000-3017. doi: 10.1002/mma.4797
[38]	S. Li, Y. Zhou, Backward bifurcation of an HTLV-I model with immune response, Discrete Cont. Dyn. B, 21 (2016), 863-881. doi: 10.3934/dcdsb.2016.21.863
[39]	C. Casoli, E. Pilotti, U. Bertazzoni, Molecular and cellular interactions of HIV-1/HTLV coinfection and impact on AIDS progression, AIDS Rev., 9 (2007), 140-149.
[40]	E. Pilotti, M. V. Bianchi, A. De Maria, F. Bozzano, M. G. Romanelli, U. Bertazzoni, et al. HTLV-1/-2 and HIV-1 co-infections: retroviral interference on host immune status, Front. Microbiol., 4 (2013), 1-13.
[41]	M. A. Beilke, K. P. Theall, M. O'Brien, J. L. Clayton, S. M. Benjamin, E. L. Winsor, et al. Clinical outcomes and disease progression among patients coinfected with HIV and human T lymphotropic virus types 1 and 2, Clin. Infect. Dis., 39 (2004), 256-263. doi: 10.1086/422146
[42]	C. Isache, M. Sands, N. Guzman, D. Figueroa, HTLV-1 and HIV-1 co-infection: A case report and review of the literature, IDCases, 4 (2016), 53-55. doi: 10.1016/j.idcr.2016.03.002
[43]	M. Tulius Silva, O. de Melo Espíndola, A. C. Bezerra Leite, A. Araújo, Neurological aspects of HIV/human T lymphotropic virus coinfection, AIDS Rev., 11 (2009), 71-78.
[44]	N. Rockwood, L. Cook, H. Kagdi, S. Basnayake, C. R. M. Bangham, A. L. Pozniak, et al. Immune compromise in HIV-1/HTLV-1 coinfection with paradoxical resolution of CD4 lymphocytosis during antiretroviral therapy: A case report, Medicine, 94 (2015), 1-4.
[45]	C. Brites, J. Sampalo, A. Oliveira, HIV/human T-cell lymphotropic virus coinfection revisited: impact on AIDS progression, AIDS Rev., 11 (2009), 8-16.
[46]	V. E. V. Geddes, D. P. José, F. E. Leal, D. F. Nixond, A. Tanuri, R. S. Aguiar, HTLV-1 Tax activates HIV-1 transcription in latency models, Virology, 504 (2017), 45-51. doi: 10.1016/j.virol.2017.01.014
[47]	R. Bingham, E. Dykeman, R. Twarock, RNA virus evolution via a quasispecies-based model reveals a drug target with a high barrier to resistance, Viruses, 9 (2017), 347. doi: 10.3390/v9110347
[48]	E. Ticona, M. A. Huaman, O. Yanque, J. R. Zunt, HIV and HTLV-1 coinfection: the need to initiate antiretroviral therapy, J. Int. Assoc. Provid. AIDS Care, 12 (2013), 373-374. doi: 10.1177/2325957413500988
[49]	N. Bellomo, R. Bingham, M. A. J. Chaplain, G. Dosi, G. Forni, D. A. Knopoff, et al. A multi-scale model of virus pandemic: Heterogeneous interactive entities in a globally connected world, Math. Mod. Meth. Appl. S., 30 (2000), 1591-1651.
[50]	A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001
[51]	E. A. Barbashin, Introduction to the theory of stability, Wolters-Noordhoff, 1970.
[52]	J. P. LaSalle, The stability of dynamical systems, Philadelphia, SIAM, 1976.
[53]	A. M. Lyapunov, The general problem of the stability of motion, Int. J. Control, 55 (1992), 531-534. doi: 10.1080/00207179208934253
[54]	A. Vandormael, F. Rego, S. Danaviah, L. Carlos Junior Alcantara, D. R. Boulware, T. de Oliveira, CD4+ T-cell count may not be a useful strategy to monitor antiretroviral therapy response in HTLV-1/HIV co-infected patients, Curr. HIV Res., 15 (2017), 225-231.
[55]	T. Inoue, T. Kajiwara, T. Saski, Global stability of models of humoral immunity against multiple viral strains, J. Biol. Dynam., 4 (2010), 282-295. doi: 10.1080/17513750903180275
[56]	G. Huang, Y. Takeuchi, W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821
[57]	A. M. Elaiw, I. A. Hassanien, S. A. Azoz, Global stability of HIV infection models with intracellular delays, J. Korean Math. Soc., 49 (2012), 779-794. doi: 10.4134/JKMS.2012.49.4.779
[58]	A. M. Elaiw, S. F. Alshehaiween, Global stability of delay-distributed viral infection model with two modes of viral transmission and B-cell impairment, Math. Method. Appl. Sci., 43 (2020), 6677-6701. doi: 10.1002/mma.6408
[59]	P. W. Nelson, J. D. Murray, A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215. doi: 10.1016/S0025-5564(99)00055-3
[60]	A. M. Elaiw, A. A. Raezah, Stability of general virus dynamics models with both cellular and viral infections and delays, Math. Method. Appl. Sci., 40 (2017), 5863-5880. doi: 10.1002/mma.4436
[61]	A. M. Elaiw, N. A. Almuallem, Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells, Math. Method. Appl. Sci., 39 (2016), 4-31. doi: 10.1002/mma.3453
[62]	A. M. Elaiw, S. F. Alshehaiween, A. D. Hobiny, Global properties of a delay-distributed HIV dynamics model including impairment of B-cell functions, Mathematics, 7 (2019), 1-27.
[63]	A. M. Elaiw, E. K. Elnahary, Analysis of general humoral immunity HIV dynamics model with HAART and distributed delays, Mathematics, 7 (2019), 1-35.
[64]	R. V. Culshaw, S. Ruan, 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
[65]	N. Bellomo, K. J. Painter, Y. Tao, M. Winkler, Occurrence vs. Absence of taxis-driven instabilities in a May-Nowak model for virus infection, SIAM J. Appl. Math., 79 (2019), 1990-2010. doi: 10.1137/19M1250261
[66]	A. M. Elaiw, A. D. AlAgha, Global analysis of a reaction-diffusion within-host malaria infection model with adaptive immune response, Mathematics, 8 (2020), 1-32. doi: 10.3390/math8101793

Reader Comments

Your name:*

Email:*
© 2021 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)