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Updated: Jul 31, 2025

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Sliding mode dynamics and optimal control for HIV model.

Dan Shi1, Shuo Ma1, Qimin Zhang1,2

  • 1School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China.

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|May 10, 2023
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Summary

This study introduces a Filippov control HIV model to explore drug treatment strategies. It analyzes threshold dynamics and optimal control for managing infected cells, offering insights into HIV therapy.

Keywords:
HIV modelcell-to-cell transmissionoptimal controlsliding mode dynamicsthreshold level

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Area of Science:

  • Mathematical modeling of infectious diseases
  • Control theory applications in virology
  • Computational biology and epidemiology

Background:

  • Human Immunodeficiency Virus (HIV) infection involves complex transmission dynamics, including virus-to-cell and cell-to-cell spread.
  • Effective drug treatment strategies are crucial for managing HIV, necessitating robust mathematical models to guide therapeutic interventions.
  • Understanding the threshold dynamics of infected cells is key to developing targeted treatment approaches.

Purpose of the Study:

  • To develop and analyze a novel HIV model incorporating Filippov control for drug treatment strategies.
  • To investigate the threshold dynamics of infected cells under different control scenarios.
  • To derive an optimal control strategy for managing HIV infection using dynamic programming.

Main Methods:

  • The study employs Filippov control theory to model HIV transmission dynamics.
  • Routh-Hurwitz Criterion is utilized to analyze system stability and threshold dynamics.
  • Utkin's equivalent control method is applied to derive sliding mode equations.
  • Dynamic programming and the Hamilton-Jacobi-Bellman (HJB) equation are used to determine the optimal control strategy.

Main Results:

  • The model exhibits distinct threshold dynamics based on the number of infected cells relative to a threshold level $N_t$.
  • Sliding mode control is achieved when infected cell counts equal $N_t$.
  • An optimal control strategy was derived for cases where infected cell counts exceed $N_t$, aiming to minimize infection levels.
  • Numerical simulations validated the theoretical findings and the effectiveness of the proposed control strategies.

Conclusions:

  • The Filippov control HIV model provides a framework for analyzing complex transmission dynamics and optimizing treatment.
  • Threshold-based control strategies, including sliding mode and optimal control, show promise in managing HIV infection.
  • The study highlights the importance of mathematical modeling and control theory in advancing HIV therapeutic interventions.