# American Institute of Mathematical Sciences

October  2021, 14(10): 3541-3556. doi: 10.3934/dcdss.2020441

## Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection

 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 4 Department of Physics, College of Sciences, University of Bisha, Bisha 61922, P.O. Box 344, Saudi Arabia 5 Department of Mathematics, Faculty of Science, Gauhati University, Guwahati 781014, India

* Corresponding author: A. M. Elaiw

Received  November 2019 Revised  January 2020 Published  October 2021 Early access  November 2020

This paper studies an $(n+2)$-dimensional nonlinear HIV dynamics model that characterizes the interactions of HIV particles, susceptible CD4$^{+}$ T cells and $n$-stages of infected CD4$^{+}$ T cells. Both virus-to-cell and cell-to-cell infection modes have been incorporated into the model. The incidence rates of viral and cellular infection as well as the production and death rates of all compartments are modeled by general nonlinear functions. We have revealed that the solutions of the system are nonnegative and bounded, which ensures the well-posedness of the proposed model. The basic reproduction number $\Re_{0}$ is determined which insures the existence of the two equilibria of the model under consideration. A set of conditions on the general functions has been established which is sufficient to investigate the global stability of the model's equilibria. The global asymptotic stability of the two equilibria is proven by utilizing Lyapunov function and LaSalle's invariance principle. We have proven that if $\Re_{0}\leq1$, then the infection-free equilibrium is globally asymptotically stable, and if $\Re _{0}>1$, then the chronic-infection equilibrium is globally asymptotically stable. The theoretical results are illustrated by numerical simulations of the model with specific forms of the general functions.

Citation: A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3541-3556. doi: 10.3934/dcdss.2020441
##### References:
 [1] S.-S. Chen, C.-Y. Cheng and Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, Journal of Mathematical Analysis and Applications, 442 (2016), 642-672.  doi: 10.1016/j.jmaa.2016.05.003. [2] R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, Journal of Mathematical Biology, 46 (2003), 425-444.  doi: 10.1007/s00285-002-0191-5. [3] A. M. Elaiw and E. Kh. Elnahary, Analysis of general humoral immunity HIV dynamics model with HAART and distributed delays, Mathematics, 7 (2019), Article Number 157. doi: 10.3390/math7020157. [4] A. M. Elaiw, E. Kh. Elnahary and A. A. Raezah, Effect of cellular reservoirs and delays on the global dynamics of HIV, Advances in Difference Equations, (2018) Paper No. 85, 36 pp. doi: 10.1186/s13662-018-1523-0. [5] 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) Paper No. 414, 25 pp. doi: 10.1186/s13662-018-1869-3. [6] A. M. Elaiw and N. H. AlShamrani, Stability of an adaptive immunity pathogen dynamics model with latency and multiple delays, Mathematical Methods in the Applied Sciences, 41 (2018), 6645-6672.  doi: 10.1002/mma.5182. [7] A. M. Elaiw and A. D. AlAgha, Global dynamics of reaction-diffusion oncolytic M1 virotherapy with immune response, Applied Mathematics and Computation, 367 (2020), Article No. 124758. doi: 10.1016/j.amc.2019.124758. [8] A. M. Elaiw and M. A. Alshaikh, Stability analysis of a general discrete-time pathogen infection model with humoral immunity, Journal of Difference Equations and Applications, 25 (2019), 1149-1172.  doi: 10.1080/10236198.2019.1662411. [9] L. Gibelli, A. Elaiw, M. A. Alghamdi and A. M. Althiabi, Heterogeneous population dynamics of active particles: Progression, mutations, and selection dynamics, Mathematical Models and Methods in Applied Sciences, 27 (2017), 617-640.  doi: 10.1142/S0218202517500117. [10] F. Graw and A. S. Perelson, Modeling viral spread, Annual Review of Virology, 3 (2016), 555-572.  doi: 10.1146/annurev-virology-110615-042249. [11] J. K. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM Journal of Applied Mathematics, 70 (2010), 2693-2708.  doi: 10.1137/090780821. [13] 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. doi: 10.7554/eLife.08150. [14] C. Jolly and Q. Sattentau, Retroviral spread by induction of virological synapses, Traffic, 5 (2004), 643-650.  doi: 10.1111/j.1600-0854.2004.00209.x. [15] 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. [16] P. D. Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM Journal of Applied Mathematics, 63 (2003), 1313-1327.  doi: 10.1137/S0036139902406905. [17] M. A. Nowak and C. R. M. Bangham., Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.  doi: 10.1126/science.272.5258.74. [18] K. M. Owolabi, Behavioural study of symbiosis dynamics via the Caputo and Atangana–Baleanu fractional derivatives, Chaos, Solitons & Fractals, 122 (2019), 89-101.  doi: 10.1016/j.chaos.2019.03.014. [19] 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.  doi: 10.1038/387188a0. [20] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41 (1999), 3-44.  doi: 10.1137/S0036144598335107. [21] 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.  doi: 10.1016/0042-6822(92)90038-Q. [22] H. Shu, Y. Chen and L. Wang, Impacts of the cell-free and cell-to-cell infection modes on viral dynamics, Journal of Dynamics and Differential Equations, 30 (2018), 1817-1836.  doi: 10.1007/s10884-017-9622-2. [23] 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.  doi: 10.1038/nature10347. [24] Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Mathematical Biosciences, 270 (2015), 183-191.  doi: 10.1016/j.mbs.2015.05.001. [25] J. Wang, C. Qin, Y. Chen and X. Wang, Hopf bifurcation in a CTL-inclusive HIV-1 infection model with two time delays, Mathematical Biosciences and Engineering, 16 (2019), 2587-2612.  doi: 10.3934/mbe.2019130.

show all references

##### References:
 [1] S.-S. Chen, C.-Y. Cheng and Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, Journal of Mathematical Analysis and Applications, 442 (2016), 642-672.  doi: 10.1016/j.jmaa.2016.05.003. [2] R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, Journal of Mathematical Biology, 46 (2003), 425-444.  doi: 10.1007/s00285-002-0191-5. [3] A. M. Elaiw and E. Kh. Elnahary, Analysis of general humoral immunity HIV dynamics model with HAART and distributed delays, Mathematics, 7 (2019), Article Number 157. doi: 10.3390/math7020157. [4] A. M. Elaiw, E. Kh. Elnahary and A. A. Raezah, Effect of cellular reservoirs and delays on the global dynamics of HIV, Advances in Difference Equations, (2018) Paper No. 85, 36 pp. doi: 10.1186/s13662-018-1523-0. [5] 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) Paper No. 414, 25 pp. doi: 10.1186/s13662-018-1869-3. [6] A. M. Elaiw and N. H. AlShamrani, Stability of an adaptive immunity pathogen dynamics model with latency and multiple delays, Mathematical Methods in the Applied Sciences, 41 (2018), 6645-6672.  doi: 10.1002/mma.5182. [7] A. M. Elaiw and A. D. AlAgha, Global dynamics of reaction-diffusion oncolytic M1 virotherapy with immune response, Applied Mathematics and Computation, 367 (2020), Article No. 124758. doi: 10.1016/j.amc.2019.124758. [8] A. M. Elaiw and M. A. Alshaikh, Stability analysis of a general discrete-time pathogen infection model with humoral immunity, Journal of Difference Equations and Applications, 25 (2019), 1149-1172.  doi: 10.1080/10236198.2019.1662411. [9] L. Gibelli, A. Elaiw, M. A. Alghamdi and A. M. Althiabi, Heterogeneous population dynamics of active particles: Progression, mutations, and selection dynamics, Mathematical Models and Methods in Applied Sciences, 27 (2017), 617-640.  doi: 10.1142/S0218202517500117. [10] F. Graw and A. S. Perelson, Modeling viral spread, Annual Review of Virology, 3 (2016), 555-572.  doi: 10.1146/annurev-virology-110615-042249. [11] J. K. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM Journal of Applied Mathematics, 70 (2010), 2693-2708.  doi: 10.1137/090780821. [13] 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. doi: 10.7554/eLife.08150. [14] C. Jolly and Q. Sattentau, Retroviral spread by induction of virological synapses, Traffic, 5 (2004), 643-650.  doi: 10.1111/j.1600-0854.2004.00209.x. [15] 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. [16] P. D. Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM Journal of Applied Mathematics, 63 (2003), 1313-1327.  doi: 10.1137/S0036139902406905. [17] M. A. Nowak and C. R. M. Bangham., Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.  doi: 10.1126/science.272.5258.74. [18] K. M. Owolabi, Behavioural study of symbiosis dynamics via the Caputo and Atangana–Baleanu fractional derivatives, Chaos, Solitons & Fractals, 122 (2019), 89-101.  doi: 10.1016/j.chaos.2019.03.014. [19] 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.  doi: 10.1038/387188a0. [20] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41 (1999), 3-44.  doi: 10.1137/S0036144598335107. [21] 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.  doi: 10.1016/0042-6822(92)90038-Q. [22] H. Shu, Y. Chen and L. Wang, Impacts of the cell-free and cell-to-cell infection modes on viral dynamics, Journal of Dynamics and Differential Equations, 30 (2018), 1817-1836.  doi: 10.1007/s10884-017-9622-2. [23] 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.  doi: 10.1038/nature10347. [24] Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Mathematical Biosciences, 270 (2015), 183-191.  doi: 10.1016/j.mbs.2015.05.001. [25] J. Wang, C. Qin, Y. Chen and X. Wang, Hopf bifurcation in a CTL-inclusive HIV-1 infection model with two time delays, Mathematical Biosciences and Engineering, 16 (2019), 2587-2612.  doi: 10.3934/mbe.2019130.
Solution trajectories of system (27)
Some values of the parameters of model (27)
 Parameter Value Parameter Value Parameter Value $\rho$ $10$ $\beta_{2}$ Varied $d_{2}$ $1$ $\alpha$ $0.01$ $b_{1}$ $0.6$ $d_{3}$ $5$ $\varsigma$ $0.005$ $b_{2}$ $0.7$ $\varepsilon$ $1.5$ $S_{\max}$ $1200$ $b_{3}$ $0.8$ $\beta_{1}$ Varied $d_{1}$ $0.2$
 Parameter Value Parameter Value Parameter Value $\rho$ $10$ $\beta_{2}$ Varied $d_{2}$ $1$ $\alpha$ $0.01$ $b_{1}$ $0.6$ $d_{3}$ $5$ $\varsigma$ $0.005$ $b_{2}$ $0.7$ $\varepsilon$ $1.5$ $S_{\max}$ $1200$ $b_{3}$ $0.8$ $\beta_{1}$ Varied $d_{1}$ $0.2$
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