# American Institute of Mathematical Sciences

March  2022, 27(3): 1725-1764. doi: 10.3934/dcdsb.2021108

## Global stability of HIV/HTLV co-infection model 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

* Corresponding author: A. M. Elaiw

Received  July 2020 Revised  November 2020 Published  March 2022 Early access  April 2021

Mathematical modeling of human immunodeficiency virus (HIV) and human T-lymphotropic virus type Ⅰ (HTLV-I) mono-infections has received considerable attention during the last decades. These two viruses share the same way of transmission between individuals; through direct contact with certain contaminated body fluids. Therefore, a person can be co-infected with both viruses. In the present paper, we construct and analyze a new HIV/HTLV-I co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD$4^{+}$T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing (active) HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by two routes of transmission, virus-to-cell (VTC) and cell-to-cell (CTC). Both active and silent HIV-infected cells can infect the susceptible CD$4^{+}$T cells by CTC mechanism. On the other side, HTLV-I has only one mode of transmission via direct cell-to-cell contact. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We calculate all possible equilibria and define the key threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and LaSalle's invariance principle. We have discussed the influence of CTL immune response on the co-infection dynamics. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.

Citation: A. M. Elaiw, N. H. AlShamrani. Global stability of HIV/HTLV co-infection model with CTL-mediated immunity. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1725-1764. doi: 10.3934/dcdsb.2021108
##### References:
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Perelson, HIV-1 infection and low steady state viral loads, Bulletin of Mathematical Biology, 64 (2002), 29-64. [7] C. Casoli, E. Pilotti and U. Bertazzoni, Molecular and cellular interactions of HIV-1/HTLV coinfection and impact on AIDS progression, AIDS Reviews, 9 (2007), 140-149. [8] A. G. Cervantes-Perez and E. Avila-Vales, Dynamical analysis of multipathways and multidelays of general virus dynamics model, International Journal of Bifurcation and Chaos, 29 (2019), 1950031. doi: 10.1142/S0218127419500317. [9] T.-W. Chun, L. Stuyver, S. B. Mizell, L. A. Ehler, J. A. M. Mican, M. Baseler, A. L. Lloyd, M. A. Nowak and A. S. Fauci, Presence of an inducible HIV-1 latent reservoir during highly active antiretroviral therapy, Proceedings of the National Academy of Sciences of the USA, 94 (1997), 13193-13197. [10] A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Analysis: Real World Applications, 11 (2010), 2253-2263.  doi: 10.1016/j.nonrwa.2009.07.001. [11] A. M. 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Azoz, Global stability of HIV infection models with intracellular delays, Journal of the Korean Mathematical Society, 49 (2012), 779-794.  doi: 10.4134/JKMS.2012.49.4.779. [16] A. M. Elaiw, A. A. Raezah and S. A. Azoz, Stability of delayed HIV dynamics models with two latent reservoirs and immune impairment, Advances in Difference Equations, 2018 (2018), 25 pp. doi: 10.1186/s13662-018-1869-3. [17] V. E. V. Geddes, D. P. José, F. E. Leal, D. F. Nixond, A. Tanuri and R. S. Aguiar, HTLV-1 Tax activates HIV-1 transcription in latency models, Virology, 504 (2017), 45-51. [18] H. Gomez-Acevedo and M. Y. Li, Backward bifurcation in a model for HTLV-I infection of CD4$^{+}$ T cells, Bulletin of Mathematical Biology, 67 (2005), 101-114.  doi: 10.1016/j.bulm.2004.06.004. [19] H. Gomez-Acevedo, M. Y. Li and S. Jacobson, Multi-stability in a model for CTL response to HTLV-I infection and its consequences in HAM/TSP development, and prevention, Bulletin of Mathematical Biology, 72 (2010), 681-696.  doi: 10.1007/s11538-009-9465-z. [20] T. Guo, Z. Qiu and L. Rong, Analysis of an HIV model with immune responses and cell-to-cell transmission, Bulletin of the Malaysian Mathematical Sciences Society, 43 (2020), 581-607.  doi: 10.1007/s40840-018-0699-5. [21] T. Inoue, T. Kajiwara and T. Saski, Global stability of models of humoral immunity against multiple viral strains, Journal of Biological Dynamics, 4 (2010), 282-295.  doi: 10.1080/17513750903180275. [22] C. Isache, M. Sands, N. Guzman and D. Figueroa, HTLV-1 and HIV-1 co-infection: A case report and review of the literature, IDCases, 4 (2016), 53-55. [23] S. Iwami, J. S. Takeuchi, S. Nakaoka, F. Mammano, F. Clavel, H. Inaba, T. Kobayashi, N. Misawa, K. Aihara, Y. Koyanagi and K. 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##### References:
 [1] L. Agosto, M. Herring, W. Mothes and A. Henderson, HIV-1-infected CD4+ T cells facilitate latent infection of resting CD4+ T cells through cell-cell contact, Cell, 24 (2018), 2088-2100. [2] C. R. M. Bangham, CTL quality and the control of human retroviral infections, European Journal of Immunology, 39 (2009), 1700-1712. [3] E. A. Barbashin, Introduction to the Theory of Stability, Wolters-Noordhoff, Groningen, 1970. [4] M. A. Beilke, K. P. Theall, M. O'Brien, J. L. Clayton, S. M. Benjamin, E. L. Winsor and P. J. Kissinger, Clinical outcomes and disease progression among patients coinfected with HIV and human T lymphotropic virus types 1 and 2, Clinical Infectious Diseases, 39 (2004), 256-263. [5] C. Brites, J. Sampalo and A. Oliveira, HIV/human T-cell lymphotropic virus coinfection revisited: Impact on AIDS progression, AIDS Reviews, 11 (2009), 8-16. [6] D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bulletin of Mathematical Biology, 64 (2002), 29-64. [7] C. Casoli, E. Pilotti and U. Bertazzoni, Molecular and cellular interactions of HIV-1/HTLV coinfection and impact on AIDS progression, AIDS Reviews, 9 (2007), 140-149. [8] A. G. Cervantes-Perez and E. Avila-Vales, Dynamical analysis of multipathways and multidelays of general virus dynamics model, International Journal of Bifurcation and Chaos, 29 (2019), 1950031. doi: 10.1142/S0218127419500317. [9] T.-W. Chun, L. Stuyver, S. B. Mizell, L. A. Ehler, J. A. M. Mican, M. Baseler, A. L. Lloyd, M. A. Nowak and A. S. Fauci, Presence of an inducible HIV-1 latent reservoir during highly active antiretroviral therapy, Proceedings of the National Academy of Sciences of the USA, 94 (1997), 13193-13197. [10] A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Analysis: Real World Applications, 11 (2010), 2253-2263.  doi: 10.1016/j.nonrwa.2009.07.001. [11] A. M. Elaiw and N. A. Almuallem, Global dynamics of delay-distributed HIV infection models with differential drug efficacy in cocirculating target cells, Mathematical Methods in the Applied Sciences, 39 (2016), 4-31.  doi: 10.1002/mma.3453. [12] A. M. Elaiw and N. H. AlShamrani, Stability of a general CTL-mediated immunity HIV infection model with silent infected cell-to-cell spread, Advances in Difference Equations, 2020 (2020), Art. No. 355. doi: 10.1186/s13662-020-02818-3. [13] A. M. Elaiw and S. A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Mathematical Methods in the Applied Sciences, 36 (2013), 383-394.  doi: 10.1002/mma.2596. [14] A. M. Elaiw and E. K. Elnahary, Analysis of general humoral immunity HIV dynamics model with HAART and distributed delays, Mathematics, 7 (2019), Art. No. 157. [15] A. M. Elaiw, I. A. Hassanien and S. A. Azoz, Global stability of HIV infection models with intracellular delays, Journal of the Korean Mathematical Society, 49 (2012), 779-794.  doi: 10.4134/JKMS.2012.49.4.779. [16] A. M. Elaiw, A. A. Raezah and S. A. Azoz, Stability of delayed HIV dynamics models with two latent reservoirs and immune impairment, Advances in Difference Equations, 2018 (2018), 25 pp. doi: 10.1186/s13662-018-1869-3. [17] V. E. V. Geddes, D. P. José, F. E. Leal, D. F. Nixond, A. Tanuri and R. S. Aguiar, HTLV-1 Tax activates HIV-1 transcription in latency models, Virology, 504 (2017), 45-51. [18] H. Gomez-Acevedo and M. Y. Li, Backward bifurcation in a model for HTLV-I infection of CD4$^{+}$ T cells, Bulletin of Mathematical Biology, 67 (2005), 101-114.  doi: 10.1016/j.bulm.2004.06.004. [19] H. Gomez-Acevedo, M. Y. Li and S. Jacobson, Multi-stability in a model for CTL response to HTLV-I infection and its consequences in HAM/TSP development, and prevention, Bulletin of Mathematical Biology, 72 (2010), 681-696.  doi: 10.1007/s11538-009-9465-z. [20] T. Guo, Z. Qiu and L. Rong, Analysis of an HIV model with immune responses and cell-to-cell transmission, Bulletin of the Malaysian Mathematical Sciences Society, 43 (2020), 581-607.  doi: 10.1007/s40840-018-0699-5. [21] T. Inoue, T. Kajiwara and T. Saski, Global stability of models of humoral immunity against multiple viral strains, Journal of Biological Dynamics, 4 (2010), 282-295.  doi: 10.1080/17513750903180275. [22] C. Isache, M. Sands, N. Guzman and D. Figueroa, HTLV-1 and HIV-1 co-infection: A case report and review of the literature, IDCases, 4 (2016), 53-55. [23] S. Iwami, J. S. Takeuchi, S. Nakaoka, F. Mammano, F. Clavel, H. Inaba, T. Kobayashi, N. Misawa, K. Aihara, Y. Koyanagi and K. Sato, Cell-to-cell infection by HIV contributes over half of virus infection, Elife, 4 (2015), e08150. [24] C. Jolly and Q. Sattentau, Retroviral spread by induction of virological synapses, Traffic, 5 (2004), 643-650. [25] N. L. Komarova and D. Wodarz, Virus dynamics in the presence of synaptic transmission, Mathematical Biosciences, 242 (2013), 161-171.  doi: 10.1016/j.mbs.2013.01.003. [26] A. Korobeinikov, Global properties of basic virus dynamics models, Bulletin of Mathematical Biology, 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001. [27] J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. [28] F. Li and W. Ma, Dynamics analysis of an HTLV-1 infection model with mitotic division of actively infected cells and delayed CTL immune response, Mathematical Methods in the Applied Sciences, 41 (2018), 3000-3017.  doi: 10.1002/mma.4797. [29] M. Y. Li and A. G. Lim, Modelling the role of tax expression in HTLV-1 persisence in vivo, Bulletin of Mathematical Biology, 73 (2011), 3008-3029.  doi: 10.1007/s11538-011-9657-1. [30] M. Y. Li and H. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4+ T cells with delayed CTL response, Nonlinear Analysis: Real World Applications, 13 (2012), 1080-1092.  doi: 10.1016/j.nonrwa.2011.02.026. [31] S. Li and Y. Zhou, Backward bifurcation of an HTLV-I model with immune response, Discrete and Continuous Dynamical Systems Series B, 21 (2016), 863-881.  doi: 10.3934/dcdsb.2016.21.863. [32] X. Li and S. Fu, Global stability of a virus dynamics model with intracellular delay and CTL immune response, Mathematical Methods in the Applied Sciences, 38 (2015), 420-430.  doi: 10.1002/mma.3078. [33] A. G. Lim and P. K. Maini, HTLV-Iinfection: A dynamic struggle between viral persistence and host immunity, Journal of Theoretical Biology, 352 (2014), 92-108.  doi: 10.1016/j.jtbi.2014.02.022. [34] Y. Liu and X. Liu, Global properties and bifurcation analysis of an HIV-1 infection model with two target cells, Computational and Applied Mathematics, 37 (2018), 3455-3472.  doi: 10.1007/s40314-017-0523-0. [35] C. Lv, L. Huang and Z. Yuan, Global stability for an HIV-1 infection model with Beddington-DeAngelis incidence rate and CTL immune response, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 121-127.  doi: 10.1016/j.cnsns.2013.06.025. [36] A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, Ltd., London, 1992. [37] Y. Muroya, Y. Enatsu and H. Li, Global stability of a delayed HTLV-I infection model with a class of nonlinear incidence rates and CTLs immune response, Applied Mathematics and Computation, 219 (2013), 10559-10573.  doi: 10.1016/j.amc.2013.03.081. [38] M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. [39] M. A. Nowak and C. R. M. Bangham., Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. [40] X. Pan, Y. Chen and H. Shu, Rich dynamics in a delayed HTLV-I infection model: Stability switch, multiple stable cycles, and torus, Journal of Mathematical Analysis and Applications, 479 (2019), 2214-2235.  doi: 10.1016/j.jmaa.2019.07.051. [41] A. S. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz and D. D. Ho, Decay characteristics of HIV-1-infected compartments during combination therapy, Nature, 387 (1997), 188-191. [42] H. Sato, J. Orenstein, D. Dimitrov and M. Martin, Cell-to-cell spread of HIV-1 occurs within minutes and may not involve the participation of virus particles, Virology, 186 (1992), 712-724. [43] X. Shi, X. Zhou and X. Song, Dynamical behavior of a delay virus dynamics model with CTL immune response, Nonlinear Analysis: Real World Applications, 11 (2010), 1795-1809.  doi: 10.1016/j.nonrwa.2009.04.005. [44] H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL imune responses, SIAM Journal of Applied Mathematics, 73 (2013), 1280-1302.  doi: 10.1137/120896463. [45] 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. [46] X. Song and Y. Li, Global stability and periodic solution of a model for HTLV-1 infection and ATL progression, Applied Mathematics and Computation, 180 (2006), 401-410.  doi: 10.1016/j.amc.2005.12.022. [47] M. Sourisseau, N. Sol-Foulon, F. Porrot, F. Blanchet and O. Schwartz, Inefficient human immunodeficiency virus replication in mobile lymphocytes, Journal of Virology, 81 (2007), 1000-1012. [48] M. Tulius Silva, O. de Melo Espíndola, A. C. Bezerra Leite and A. Araújo, Neurological aspects of HIV/human T lymphotropic virus coinfection, AIDS Reviews, 11 (2009), 71-78. [49] A. Vandormael, F. Rego, S. Danaviah, L. C. J. Alcantara, D. R. Boulware and 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, Current HIV Research, 15 (2017), 225-231. [50] C. Vargas-De-Leon, The complete classification for global dynamics of amodel for the persistence of HTLV-1 infection, Applied Mathematics and Computation, 237 (2014), 489-493.  doi: 10.1016/j.amc.2014.03.138. [51] J. Wang, M. Guo, X. Liu and Z. Zhao, Threshold dynamics of HIV-1 virus model with cell-to-cell transmission, cell-mediated immune responses and distributed delay, Applied Mathematics and Computation, 291 (2016) 149–161. doi: 10.1016/j.amc.2016.06.032. [52] L. Wang, M. Y. Li and D. 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The schematic diagram of the HIV/HTLV-I co-infection dynamics in vivo
The behavior of solution trajectories of system (4) when $\Re_{1}\leq1$ and $\Re_{2}\leq1$
The behavior of solution trajectories of system (4) when $\Re_{1} > 1$, $\Re_{2}/\Re_{1}\leq1$ and $\Re_{3}\leq1$
The behavior of solution trajectories of system (4) when $\Re_{2} > 1$, $\Re_{1}/\Re_{2}\leq1$ and $\Re_{4}\leq1$
The behavior of solution trajectories of system (4) when $\Re_{3} > 1$ and $\Re_{5}\leq1$
The behavior of solution trajectories of system (4) when $\Re_{4} > 1$ and $\Re_{6}\leq1$
The behavior of solution trajectories of system (4) when $\Re_{5}^{\ast} > 1$, $\Re_{5} > 1$, $\Re_{8}\leq1$ and $\Re_{1}/\Re_{2} > 1$
The behavior of solution trajectories of system (4) when $\Re_{6} > 1$, $\Re_{7}\leq1$ and $\Re_{2}/\Re_{1} > 1$
The behavior of solution trajectories of system (4) when $\Re_{7} > 1$ and $\Re_{8} > 1$
The influence of HTLV-I infection rate ($\eta_{4}\neq0$) on HIV mono-infection dynamics (31) will cause a chronic HIV/HTLV-I co-infection
The influence of HIV infection rates ($\eta_{1}, \eta_{2}, \eta_{3} \neq 0$) on HTLV mono-infection dynamics (32) will cause a chronic HIV/HTLV-I co-infection
Parameter description
 Parameter Description $\rho$ Recruitment rate for the susceptible CD4+T cells $\alpha$ Natural mortality rate constant for the susceptible CD4+T cells $\eta_{1}$ Virus-cell incidence rate constant between free HIV particles and susceptible CD4+T cells $\eta_{2}$ Cell-cell incidence rate constant between silent HIV-infected cells and susceptible CD4+T cells $\eta_{3}$ Cell-cell incidence rate constant between active HIV-infected cells and susceptible CD4+T cells $\eta_{4}$ Cell-cell incidence rate constant between Tax-expressing HTLV-infected cells and susceptible CD4+T cells $\beta \in \left( 0,1\right)$ Fraction coefficient accounts for the probability of new HIV-infected cells could be active, and the remaining part 1-β will be latent $\gamma$ Death rate constant of silent HIV-infected cells $a$ Death rate constant of active HIV-infected cells $\mu_{1}$ Killing rate constant of active HIV-infected cells due to HIV-specific CTLs $\mu_{2}$ Killing rate constant of Tax-expressing HTLV-infected cells due to HTLV-specific CTLs $\varphi \in \left( 0,1\right)$ Probability of new HTLV infections could be enter a silent period $\lambda$ Transmission rate constant of silent HIV-infected cells that become active HIV-infected cells $\psi$ Transmission rate constant of silent HTLV-infected cells that become Tax-expressing HTLV-infected cells $\omega$ Death rate constant of silent HTLV-infected cells $\delta$ Death rate constant of Tax-expressing HTLV-infected cells $b$ Generation rate constant of new HIV particles $\varepsilon$ Death rate constant of free HIV particles $\sigma_{1}$ Proliferation rate constant of HIV-specific CTLs $\sigma_{2}$ Proliferation rate constant of HTLV-specific CTLs $\pi_{1}$ Decay rate constant of HIV-specific CTLs $\pi_{2}$ Decay rate constant of HTLV-specific CTLs
 Parameter Description $\rho$ Recruitment rate for the susceptible CD4+T cells $\alpha$ Natural mortality rate constant for the susceptible CD4+T cells $\eta_{1}$ Virus-cell incidence rate constant between free HIV particles and susceptible CD4+T cells $\eta_{2}$ Cell-cell incidence rate constant between silent HIV-infected cells and susceptible CD4+T cells $\eta_{3}$ Cell-cell incidence rate constant between active HIV-infected cells and susceptible CD4+T cells $\eta_{4}$ Cell-cell incidence rate constant between Tax-expressing HTLV-infected cells and susceptible CD4+T cells $\beta \in \left( 0,1\right)$ Fraction coefficient accounts for the probability of new HIV-infected cells could be active, and the remaining part 1-β will be latent $\gamma$ Death rate constant of silent HIV-infected cells $a$ Death rate constant of active HIV-infected cells $\mu_{1}$ Killing rate constant of active HIV-infected cells due to HIV-specific CTLs $\mu_{2}$ Killing rate constant of Tax-expressing HTLV-infected cells due to HTLV-specific CTLs $\varphi \in \left( 0,1\right)$ Probability of new HTLV infections could be enter a silent period $\lambda$ Transmission rate constant of silent HIV-infected cells that become active HIV-infected cells $\psi$ Transmission rate constant of silent HTLV-infected cells that become Tax-expressing HTLV-infected cells $\omega$ Death rate constant of silent HTLV-infected cells $\delta$ Death rate constant of Tax-expressing HTLV-infected cells $b$ Generation rate constant of new HIV particles $\varepsilon$ Death rate constant of free HIV particles $\sigma_{1}$ Proliferation rate constant of HIV-specific CTLs $\sigma_{2}$ Proliferation rate constant of HTLV-specific CTLs $\pi_{1}$ Decay rate constant of HIV-specific CTLs $\pi_{2}$ Decay rate constant of HTLV-specific CTLs
Model (4) equilibria and their existence conditions
 Equilibrium point Definition Existence conditions Ð$_{0}=(S_{0},0,0,0,0,0,0,0)$ Infection-free equilibrium None Ð$_{1}=(S_{1},L_{1},I_{1},0,0,V_{1},0,0)$ Chronic HIV mono-infection equilibrium with inactive immune response $\Re_{1}>1$ Ð$_{2}=(S_{2},0,0,E_{2},Y_{2},0,0,0)$ Chronic HTLV mono-infection equilibrium with inactive immune response $\Re_{2}>1$ Ð$_{3}=(S_{3},L_{3},I_{3},0,0,V_{3},C_{3}^{I},0)$ Chronic HIV mono-infection equilibrium with only active HIV-specific CTL $\Re_{3}>1$ Ð$_{4}=(S_{4},0,0,E_{4},Y_{4},0,0,C_{4}^{Y})$ Chronic HTLV mono-infection equilibrium with only active HTLV-specific CTL $\Re_{4}>1$ Ð$_{5}=(S_{5},L_{5},I_{5},E_{5},Y_{5},V_{5},C_{5}^{I},0)$ Chronic HIV/HTLV co-infection equilibrium with only active HIV-specific CTL $\Re_{5}^{\ast},$ $\Re_{5}>1$ and $\Re_{1}/\Re_{2}>1$ Ð$_{6}=(S_{6},L_{6},I_{6},E_{6},Y_{6},V_{6},0,C_{6}^{Y})$ Chronic HIV/HTLV co-infection equilibrium with only active HTLV-specific CTL $\Re_{6}>1$ and $\Re_{2}/\Re_{1}>1$ Ð$_{7}=(S_{7},L_{7},I_{7},E_{7},Y_{7},V_{7},C_{7}^{I},C_{7}^{Y})$ Chronic HIV/HTLV co-infection equilibrium with active HIV-specific CTL and HTLV-specific CTL $\Re_{7}>1$ and $\Re_{8}>1$
 Equilibrium point Definition Existence conditions Ð$_{0}=(S_{0},0,0,0,0,0,0,0)$ Infection-free equilibrium None Ð$_{1}=(S_{1},L_{1},I_{1},0,0,V_{1},0,0)$ Chronic HIV mono-infection equilibrium with inactive immune response $\Re_{1}>1$ Ð$_{2}=(S_{2},0,0,E_{2},Y_{2},0,0,0)$ Chronic HTLV mono-infection equilibrium with inactive immune response $\Re_{2}>1$ Ð$_{3}=(S_{3},L_{3},I_{3},0,0,V_{3},C_{3}^{I},0)$ Chronic HIV mono-infection equilibrium with only active HIV-specific CTL $\Re_{3}>1$ Ð$_{4}=(S_{4},0,0,E_{4},Y_{4},0,0,C_{4}^{Y})$ Chronic HTLV mono-infection equilibrium with only active HTLV-specific CTL $\Re_{4}>1$ Ð$_{5}=(S_{5},L_{5},I_{5},E_{5},Y_{5},V_{5},C_{5}^{I},0)$ Chronic HIV/HTLV co-infection equilibrium with only active HIV-specific CTL $\Re_{5}^{\ast},$ $\Re_{5}>1$ and $\Re_{1}/\Re_{2}>1$ Ð$_{6}=(S_{6},L_{6},I_{6},E_{6},Y_{6},V_{6},0,C_{6}^{Y})$ Chronic HIV/HTLV co-infection equilibrium with only active HTLV-specific CTL $\Re_{6}>1$ and $\Re_{2}/\Re_{1}>1$ Ð$_{7}=(S_{7},L_{7},I_{7},E_{7},Y_{7},V_{7},C_{7}^{I},C_{7}^{Y})$ Chronic HIV/HTLV co-infection equilibrium with active HIV-specific CTL and HTLV-specific CTL $\Re_{7}>1$ and $\Re_{8}>1$
Global stability conditions of the equilibria of model (4)
 Equilibrium point Global stability conditions Ð$_{0}=(S_{0},0,0,0,0,0,0,0)$ $\Re_{1}\leq1$ and $\Re_{2}\leq1$ Ð$_{1}=(S_{1},L_{1},I_{1},0,0,V_{1},0,0)$ $\Re_{1}>1$, $\Re_{2}/\Re _{1}\leq1$ and $\Re_{3}\leq1$ Ð$_{2}=(S_{2},0,0,E_{2},Y_{2},0,0,0)$ $\Re_{2}>1$, $\Re_{1}/\Re_{2}\leq1$ and $\Re_{4}\leq1$ Ð$_{3}=(S_{3},L_{3},I_{3},0,0,V_{3},C_{3}^{I},0)$ $\Re_{3}>1$ and $\Re_{5}\leq1$ Ð$_{4}=(S_{4},0,0,E_{4},Y_{4},0,0,C_{4}^{Y})$ $\Re_{4}>1$ and $\Re _{6}\leq1$ Ð$_{5}=(S_{5},L_{5},I_{5},E_{5},Y_{5},V_{5},C_{5}^{I},0)$ $\Re_{5}^{\ast},$ $\Re_{5}>1$, $\Re_{8}\leq1$ and $\Re_{1}/\Re_{2}>1$ Ð$_{6}=(S_{6},L_{6},I_{6},E_{6},Y_{6},V_{6},0,C_{6}^{Y})$ $\Re_{6}>1$, $\Re_{7}\leq1$ and $\Re_{2}/\Re_{1}>1$ Ð$_{7}=(S_{7},L_{7},I_{7},E_{7},Y_{7},V_{7},C_{7}^{I},C_{7}^{Y})$ $\Re_{7}>1$ and $\Re_{8}>1$
 Equilibrium point Global stability conditions Ð$_{0}=(S_{0},0,0,0,0,0,0,0)$ $\Re_{1}\leq1$ and $\Re_{2}\leq1$ Ð$_{1}=(S_{1},L_{1},I_{1},0,0,V_{1},0,0)$ $\Re_{1}>1$, $\Re_{2}/\Re _{1}\leq1$ and $\Re_{3}\leq1$ Ð$_{2}=(S_{2},0,0,E_{2},Y_{2},0,0,0)$ $\Re_{2}>1$, $\Re_{1}/\Re_{2}\leq1$ and $\Re_{4}\leq1$ Ð$_{3}=(S_{3},L_{3},I_{3},0,0,V_{3},C_{3}^{I},0)$ $\Re_{3}>1$ and $\Re_{5}\leq1$ Ð$_{4}=(S_{4},0,0,E_{4},Y_{4},0,0,C_{4}^{Y})$ $\Re_{4}>1$ and $\Re _{6}\leq1$ Ð$_{5}=(S_{5},L_{5},I_{5},E_{5},Y_{5},V_{5},C_{5}^{I},0)$ $\Re_{5}^{\ast},$ $\Re_{5}>1$, $\Re_{8}\leq1$ and $\Re_{1}/\Re_{2}>1$ Ð$_{6}=(S_{6},L_{6},I_{6},E_{6},Y_{6},V_{6},0,C_{6}^{Y})$ $\Re_{6}>1$, $\Re_{7}\leq1$ and $\Re_{2}/\Re_{1}>1$ Ð$_{7}=(S_{7},L_{7},I_{7},E_{7},Y_{7},V_{7},C_{7}^{I},C_{7}^{Y})$ $\Re_{7}>1$ and $\Re_{8}>1$
The data of model (4)
 Parameter Value Parameter Value Parameter Value $\rho$ $10$ $\varphi$ $0.2$ $\varepsilon$ $2$ $\alpha$ $0.01$ $\delta$ $0.2$ $\gamma$ $0.02$ $\eta_{1}$ Varied $b$ $5$ $\sigma_{1}$ Varied $\eta_{2}$ Varied $\pi_{1}$ $0.1$ $\sigma_{2}$ Varied $\eta_{3}$ Varied $\pi_{2}$ $0.1$ $\lambda$ $0.2$ $\eta_{4}$ Varied $\mu_{1}$ $0.2$ $\omega$ $0.01$ $a$ $0.5$ $\mu_{2}$ $0.2$ $\psi$ $0.003$
 Parameter Value Parameter Value Parameter Value $\rho$ $10$ $\varphi$ $0.2$ $\varepsilon$ $2$ $\alpha$ $0.01$ $\delta$ $0.2$ $\gamma$ $0.02$ $\eta_{1}$ Varied $b$ $5$ $\sigma_{1}$ Varied $\eta_{2}$ Varied $\pi_{1}$ $0.1$ $\sigma_{2}$ Varied $\eta_{3}$ Varied $\pi_{2}$ $0.1$ $\lambda$ $0.2$ $\eta_{4}$ Varied $\mu_{1}$ $0.2$ $\omega$ $0.01$ $a$ $0.5$ $\mu_{2}$ $0.2$ $\psi$ $0.003$
Local stability of positive equilibria Ð$_{i}$, $i = 0,1,...,7$
 Scenario The equilibria $(\operatorname{Re}(\lambda_{i}),$ $i=1,2,...,6)$ Stability 1 $\begin{array} [c]{c} Ð_{0}=(1000,0,0,0,0,0,0,0) \end{array}$ $\begin{array} [c]{c} \left( -1.98,-0.63,-0.2,-0.1,-0.1,-0.07,-0.01,-0.01\right) \end{array}$ $\begin{array} [c]{l} \rm{stable} \end{array}$ 2 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 523.81,21.65,8.66,0,0,21.65,0,0\right) \end{array}$ $\begin{array} [c]{l} \left( -1.96,-0.7,-0.2,0.15,-0.1,-0.1,-0.01,-0.01\right) \\ \left( -1.98,-0.64,-0.2,-0.1,-0.07,-0.01,-0.01,-0.01\right) \end{array}$ $\begin{array} [c]{l} \rm{unstable} \\ \rm{stable} \end{array}$ 3 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{2}=\left( 722.22,0,0,42.74,0.64,0,0,0\right) \end{array}$ $\begin{array} [c]{l} \left( -1.96,-0.68,-0.22,-0.1,-0.1,-0.03,-0.01,0.004\right) \\ \left( -1.97,-0.64,-0.21,-0.1,-0.07,-0.07,-0.01,-0.01\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ {\rm{stable}} \end{array}$ 4 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 180.33,37.26,14.9,0,0,37.26,0,0\right) \\ Ð_{3}=\left( 569.59,19.56,2,0,0,5,7.28,0\right) \end{array}$ $\begin{array} [c]{l} \left( -1.51,-1.51,0.41,-0.2,-0.1,-0.1,-0.01,-0.01\right) \\ \left( -1.92,-0.79,0.65,-0.2,-0.1,-0.02,-0.02,-0.01\right) \\ \left( -2.03,-2.03,-0.2,-0.03,-0.03,-0.1,-0.02,-0.01\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array}$ 5 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 785.71,9.74,3.9,0,0,9.74,0,0\right) \\ Ð_{2}=\left( 123.81,0,0,134.8,2.02,0,0,0\right) \\ Ð_{3}=\left( 870.39,5.89,2,0,0,5,0.45,0\right) \\ Ð_{4}=\left( 533.33,0,0,71.79,0.25,0,0,3.31\right) \end{array}$ $\begin{array} [c]{l} \left( -1.96,-0.7,-0.28,-0.1,-0.1,0.07,0.04,-0.01\right) \\ \left( -1.97,-0.67,-0.27,-0.1,0.09,0.05,-0.01,-0.01\right) \\ \left( -2,0.71,-0.54,-0.22,-0.17,-0.1,-0.06,-0.01\right) \\ \left( -1.97,-0.75,-0.27,-0.1,0.06,-0.01,-0.01,-0.01\right) \\ \left( -1.98,-0.79,-0.63,-0.1,-0.06,-0.06,-0.03,-0.01\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array}$ 6 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 186.44,36.98,14.79,0,0,36.98,0,0\right) \\ Ð_{2}=\left( 393.94,0,0,93.24,1.4,0,0,0\right) \\ Ð_{3}=\left( 714.37,12.98,1,0,0,2.5,10.48,0\right) \\ Ð_{5}=\left( 393.94,5.89,1,73.31,1.1,2.5,3.39,0\right) \end{array}$ $\begin{array} [c]{l} \left( -1.51,-1.51,0.39,-0.23,-0.1,-0.1,0.02,-0.01\right) \\ \left( -1.92,1.38,-0.79,-0.21,-0.1,-0.02,-0.02,-0.01\right) \\ \left( -1.82,-1,-0.21,0.13,-0.1,-0.09,-0.01,-0.01\right) \\ \left( -2.34,-2.34,-0.22,-0.03,-0.03,-0.1,-0.01,0.01\right) \\ \left( -1.66,-1.66,-0.21,-0.02,-0.02,-0.09,-0.01,-0.01\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array}$ 7 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 282.05,32.63,13.05,0,0,32.63,0,0\right) \\ Ð_{2}=\left( 108.33,0,0,137.18,2.06,0,0,0\right) \\ Ð_{4}=\left( 636.36,0,0,55.94,0.14,0,0,4.87\right) \\ Ð_{6}=\left( 282.05,25.31,10.12,24.8,0.14,25.31,0,1.6\right) \end{array}$ $\begin{array} [c]{l} \left( -1.65,-1.24,-0.29,0.27,-0.1,-0.1,0.07,-0.01\right) \\ \left( -1.93,-0.77,-0.23,-0.1,-0.09,-0.01,-0.01,0.02\right) \\ \left( -1.98,1.34,-0.62,-0.22,-0.11,-0.1,-0.07,-0.01\right) \\ \left( -1.83,-1.1,-0.98,0.15,-0.1,-0.07,-0.02,-0.01\right) \\ \left( -1.93,-0.77,-0.47,-0.09,-0.02,-0.02,-0.05,-0.02\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array}$ 8 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 282.05,32.63,13.05,0,0,32.63,0,0\right) \\ Ð_{2}=\left( 144.44,0,0,131.62,1.97,0,0,0\right) \\ Ð_{3}=\left( 627.17,16.95,2.5,0,0,6.25,4.28,0\right) \\ Ð_{4}=\left( 625,0,0,57.69,0.2,0,0,3.33\right) \\ Ð_{6}=\left( 282.05,24.94,9.98,26.04,0.2,24.94,0,0.95\right) \\ Ð_{7}=\left( 467.37,11.46,2.5,43.14,0.2,6.25,2.09,2.24\right) \end{array}$ $\begin{array} [c]{l} \left( -1.65,-1.24,0.27,-0.27,-0.1,-0.1,0.06,-0.01\right) \\ \left( -1.93,-0.77,0.42,-0.22,-0.1,-0.01,-0.01,0.01\right) \\ \left( -1.97,0.89,-0.66,-0.22,-0.1,-0.08,-0.05,-0.01\right) \\ \left( -1.73,-1.73,-0.25,-0.1,-0.02,-0.02,0.04,-0.02\right) \\ \left( -1.83,-0.98,-0.8,0.15,-0.1,-0.06,-0.02,-0.01\right) \\ \left( -1.93,-0.77,-0.35,0.3,-0.02,-0.02,-0.03,-0.03\right) \\ \left( -1.81,-1.25,-0.59,-0.02,-0.02,-0.05,-0.03,-0.01\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array}$
 Scenario The equilibria $(\operatorname{Re}(\lambda_{i}),$ $i=1,2,...,6)$ Stability 1 $\begin{array} [c]{c} Ð_{0}=(1000,0,0,0,0,0,0,0) \end{array}$ $\begin{array} [c]{c} \left( -1.98,-0.63,-0.2,-0.1,-0.1,-0.07,-0.01,-0.01\right) \end{array}$ $\begin{array} [c]{l} \rm{stable} \end{array}$ 2 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 523.81,21.65,8.66,0,0,21.65,0,0\right) \end{array}$ $\begin{array} [c]{l} \left( -1.96,-0.7,-0.2,0.15,-0.1,-0.1,-0.01,-0.01\right) \\ \left( -1.98,-0.64,-0.2,-0.1,-0.07,-0.01,-0.01,-0.01\right) \end{array}$ $\begin{array} [c]{l} \rm{unstable} \\ \rm{stable} \end{array}$ 3 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{2}=\left( 722.22,0,0,42.74,0.64,0,0,0\right) \end{array}$ $\begin{array} [c]{l} \left( -1.96,-0.68,-0.22,-0.1,-0.1,-0.03,-0.01,0.004\right) \\ \left( -1.97,-0.64,-0.21,-0.1,-0.07,-0.07,-0.01,-0.01\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ {\rm{stable}} \end{array}$ 4 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 180.33,37.26,14.9,0,0,37.26,0,0\right) \\ Ð_{3}=\left( 569.59,19.56,2,0,0,5,7.28,0\right) \end{array}$ $\begin{array} [c]{l} \left( -1.51,-1.51,0.41,-0.2,-0.1,-0.1,-0.01,-0.01\right) \\ \left( -1.92,-0.79,0.65,-0.2,-0.1,-0.02,-0.02,-0.01\right) \\ \left( -2.03,-2.03,-0.2,-0.03,-0.03,-0.1,-0.02,-0.01\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array}$ 5 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 785.71,9.74,3.9,0,0,9.74,0,0\right) \\ Ð_{2}=\left( 123.81,0,0,134.8,2.02,0,0,0\right) \\ Ð_{3}=\left( 870.39,5.89,2,0,0,5,0.45,0\right) \\ Ð_{4}=\left( 533.33,0,0,71.79,0.25,0,0,3.31\right) \end{array}$ $\begin{array} [c]{l} \left( -1.96,-0.7,-0.28,-0.1,-0.1,0.07,0.04,-0.01\right) \\ \left( -1.97,-0.67,-0.27,-0.1,0.09,0.05,-0.01,-0.01\right) \\ \left( -2,0.71,-0.54,-0.22,-0.17,-0.1,-0.06,-0.01\right) \\ \left( -1.97,-0.75,-0.27,-0.1,0.06,-0.01,-0.01,-0.01\right) \\ \left( -1.98,-0.79,-0.63,-0.1,-0.06,-0.06,-0.03,-0.01\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array}$ 6 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 186.44,36.98,14.79,0,0,36.98,0,0\right) \\ Ð_{2}=\left( 393.94,0,0,93.24,1.4,0,0,0\right) \\ Ð_{3}=\left( 714.37,12.98,1,0,0,2.5,10.48,0\right) \\ Ð_{5}=\left( 393.94,5.89,1,73.31,1.1,2.5,3.39,0\right) \end{array}$ $\begin{array} [c]{l} \left( -1.51,-1.51,0.39,-0.23,-0.1,-0.1,0.02,-0.01\right) \\ \left( -1.92,1.38,-0.79,-0.21,-0.1,-0.02,-0.02,-0.01\right) \\ \left( -1.82,-1,-0.21,0.13,-0.1,-0.09,-0.01,-0.01\right) \\ \left( -2.34,-2.34,-0.22,-0.03,-0.03,-0.1,-0.01,0.01\right) \\ \left( -1.66,-1.66,-0.21,-0.02,-0.02,-0.09,-0.01,-0.01\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array}$ 7 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 282.05,32.63,13.05,0,0,32.63,0,0\right) \\ Ð_{2}=\left( 108.33,0,0,137.18,2.06,0,0,0\right) \\ Ð_{4}=\left( 636.36,0,0,55.94,0.14,0,0,4.87\right) \\ Ð_{6}=\left( 282.05,25.31,10.12,24.8,0.14,25.31,0,1.6\right) \end{array}$ $\begin{array} [c]{l} \left( -1.65,-1.24,-0.29,0.27,-0.1,-0.1,0.07,-0.01\right) \\ \left( -1.93,-0.77,-0.23,-0.1,-0.09,-0.01,-0.01,0.02\right) \\ \left( -1.98,1.34,-0.62,-0.22,-0.11,-0.1,-0.07,-0.01\right) \\ \left( -1.83,-1.1,-0.98,0.15,-0.1,-0.07,-0.02,-0.01\right) \\ \left( -1.93,-0.77,-0.47,-0.09,-0.02,-0.02,-0.05,-0.02\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array}$ 8 $\begin{array} [c]{l} Ð_{0}=(1000,0,0,0,0,0,0,0) \\ Ð_{1}=\left( 282.05,32.63,13.05,0,0,32.63,0,0\right) \\ Ð_{2}=\left( 144.44,0,0,131.62,1.97,0,0,0\right) \\ Ð_{3}=\left( 627.17,16.95,2.5,0,0,6.25,4.28,0\right) \\ Ð_{4}=\left( 625,0,0,57.69,0.2,0,0,3.33\right) \\ Ð_{6}=\left( 282.05,24.94,9.98,26.04,0.2,24.94,0,0.95\right) \\ Ð_{7}=\left( 467.37,11.46,2.5,43.14,0.2,6.25,2.09,2.24\right) \end{array}$ $\begin{array} [c]{l} \left( -1.65,-1.24,0.27,-0.27,-0.1,-0.1,0.06,-0.01\right) \\ \left( -1.93,-0.77,0.42,-0.22,-0.1,-0.01,-0.01,0.01\right) \\ \left( -1.97,0.89,-0.66,-0.22,-0.1,-0.08,-0.05,-0.01\right) \\ \left( -1.73,-1.73,-0.25,-0.1,-0.02,-0.02,0.04,-0.02\right) \\ \left( -1.83,-0.98,-0.8,0.15,-0.1,-0.06,-0.02,-0.01\right) \\ \left( -1.93,-0.77,-0.35,0.3,-0.02,-0.02,-0.03,-0.03\right) \\ \left( -1.81,-1.25,-0.59,-0.02,-0.02,-0.05,-0.03,-0.01\right) \end{array}$ $\begin{array} [c]{c} \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ \rm{unstable} \\ {\rm{stable}} \end{array}$
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