# American Institute of Mathematical Sciences

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## 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  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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020441
##### References:

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##### References:
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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