Revisiting the classical target cell limited dynamical within-host HIV model - Basic mathematical properties and stability analysis

Benjamin Wacker; Benjamin Wacker

doi:10.3934/mbe.2024343

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 12: 7805-7829. doi: 10.3934/mbe.2024343

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Revisiting the classical target cell limited dynamical within-host HIV model - Basic mathematical properties and stability analysis

Benjamin Wacker ^,

Department of Engineering and Natural Sciences, University of Applied Sciences Merseburg, Eberhard-Leibnitz-Str. 2, D-06217 Merseburg, Germany

Academic Editor: Mingji Zhang

Received: 19 August 2024 Revised: 14 November 2024 Accepted: 03 December 2024 Published: 11 December 2024

In this article, we reconsider the classical target cell limited dynamical within-host HIV model, solely taking into account the interaction between $ {\rm{CD}}4^{+} $ T cells and virus particles. First, we summarize some analytical results regarding the corresponding dynamical system. For that purpose, we proved some analytical results regarding the system of differential equations as our first main contribution. Specifically, we showed non-negativity and boundedness of solutions, global existence in time and global uniqueness in time and examined stability properties of two possible equilibria. In particular, we demonstrated that the virus-free equilibrium and the plateau-phase equilibrium are locally asymptotically stable using the Routh–Hurwitz criterion under appropriate conditions. As our second main contribution, we underline our theoretical findings through some numerical experiments with standard Runge–Kutta time stepping schemes. We conclude this work with a summary of our main results and a suggestion of an extension for more complex dynamical systems with regard to HIV-infection.
- dynamical system,
- equilibria,
- existence,
- non-negativity,
- stability,
- uniqueness,
- within-host HIV model
Citation: Benjamin Wacker. Revisiting the classical target cell limited dynamical within-host HIV model - Basic mathematical properties and stability analysis[J]. Mathematical Biosciences and Engineering, 2024, 21(12): 7805-7829. doi: 10.3934/mbe.2024343

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Abstract

In this article, we reconsider the classical target cell limited dynamical within-host HIV model, solely taking into account the interaction between $ {\rm{CD}}4^{+} $ T cells and virus particles. First, we summarize some analytical results regarding the corresponding dynamical system. For that purpose, we proved some analytical results regarding the system of differential equations as our first main contribution. Specifically, we showed non-negativity and boundedness of solutions, global existence in time and global uniqueness in time and examined stability properties of two possible equilibria. In particular, we demonstrated that the virus-free equilibrium and the plateau-phase equilibrium are locally asymptotically stable using the Routh–Hurwitz criterion under appropriate conditions. As our second main contribution, we underline our theoretical findings through some numerical experiments with standard Runge–Kutta time stepping schemes. We conclude this work with a summary of our main results and a suggestion of an extension for more complex dynamical systems with regard to HIV-infection.

References

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