Abstract
In this paper, we study the asymptotic and stability dynamics of a chemotaxis model in volume filling constraints on HIV-1-incorporating cytotoxic T lymphocytes (CTLs) cells in defense mechanism against the virus infection. The system of uninfected \(\hbox {CD}4^{+}\hbox {T}\)-cells, infected and CTL defense cells is globally well-defined in \(\Omega \times (0,\infty )\), with uninfected \(\hbox {CD}4^{+}\hbox {T}\) and CTL cells remaining bounded, while the HIV-1-activated cells decay to the null state at time \(t=\infty \). Routh–Hurwitz criteria yields asymptotical stability of the system, if the CTL threshold value is sufficiently large with CTL decay small, and instability otherwise. In control theory, it is implied that a bounded control yields the system not completely controllable, but bounded input-bounded output stable (b.i.b.o.-stable) with stabilizability and detectability not guaranteed. If guaranteed, the system is asymptotically stable if and only if it is b.i.b.o.-stable. In addition, numerical simulation results of the model are provided.
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Acknowledgements
The authors are sincerely grateful to the Editors in Chief of the Journal of Nonlinear Science and Reviewers’ suggestions that have furnished this revised version of the originally submitted manuscript. The first author is thankful to the Technical University of Madrid (UPM), South Campus Vallecas—Spain for their hospitality in hosting his Sabbatical Leave 2017–2018 and in particular, to Professor José Ignacio Tello del Castillo for the invitation.
Funding
Robert Willie was partially supported by the UKZN Sabbatical Leave Grant 2017–2018, and is partially supported by the UKZN Competitive Grant RC43-621994-2006-YY. Pan Zheng is partially supported by National Natural Science Foundation of China (Grant Nos: 11601053, 11526042), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No: KJ1500403), the Basic and Advanced Research Project of CQCSTC (Grant No: cstc2015jcyjA00008), the Doctor Start-up Funding and the Natural Science Foundation of Chongqing University of Posts and Telecommunications (Grant Nos: A2014-25 and A2014-106), and Natural Science Foundation of Chongqing(Grant No:cstc2019jcyj-msxmX0082). Nabendra Parumasur is partially supported by UKZN Competitive Research Grant 2019. Chunlai Mu is partially supported by National Natural Science Foundation of China (Grant Nos: 11771062, 11571062), the Basic and Advanced Research Project of CQCSTC (Grant No: cstc2015jcyjBX0007).
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Communicated by Ram Ramaswamy.
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Willie, R., Zheng, P., Parumasur, N. et al. Asymptotic and Stability Dynamics of an HIV-1-Cytotoxic T Lymphocytes (CTL) Chemotaxis Model. J Nonlinear Sci 30, 1055–1080 (2020). https://doi.org/10.1007/s00332-019-09601-z
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DOI: https://doi.org/10.1007/s00332-019-09601-z