Sliding mode dynamics and optimal control for HIV model

Dan Shi; Shuo Ma; Qimin Zhang; Dan Shi; Shuo Ma; Qimin Zhang

doi:10.3934/mbe.2023315

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 4: 7273-7297. doi: 10.3934/mbe.2023315

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Research article

Sliding mode dynamics and optimal control for HIV model

Dan Shi ¹,
Shuo Ma ^{1
,
,},
Qimin Zhang ^{1,2
,
,}

1.
School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China
2.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China

Academic Editor: Hao Wang

Received: 07 November 2022 Revised: 12 January 2023 Accepted: 30 January 2023 Published: 13 February 2023

Considering the drug treatment strategy in both virus-to-cell and cell-to-cell transmissions, this paper presents an HIV model with Filippov control. Given the threshold level $ N_t $, when the total number of infected cells is less or greater than threshold level $ N_t $, the threshold dynamics of the model are studied by using the Routh-Hurwitz Criterion. When the total number of infected cells is equal to $ N_t $, the sliding mode equations are obtained by Utkin equivalent control method, and the dynamics are studied. In addition, the optimal control strategy is introduced for the case that the number of infected cells is greater than $ N_t $. By dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equation is constructed, and the optimal control is obtained. Numerical simulations are presented to illustrate the validity of our results.
- threshold level,
- HIV model,
- sliding mode dynamics,
- cell-to-cell transmission,
- optimal control
Citation: Dan Shi, Shuo Ma, Qimin Zhang. Sliding mode dynamics and optimal control for HIV model[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 7273-7297. doi: 10.3934/mbe.2023315

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Abstract

Considering the drug treatment strategy in both virus-to-cell and cell-to-cell transmissions, this paper presents an HIV model with Filippov control. Given the threshold level $ N_t $, when the total number of infected cells is less or greater than threshold level $ N_t $, the threshold dynamics of the model are studied by using the Routh-Hurwitz Criterion. When the total number of infected cells is equal to $ N_t $, the sliding mode equations are obtained by Utkin equivalent control method, and the dynamics are studied. In addition, the optimal control strategy is introduced for the case that the number of infected cells is greater than $ N_t $. By dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equation is constructed, and the optimal control is obtained. Numerical simulations are presented to illustrate the validity of our results.

References

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