Reaction-diffusion model of HIV infection of two target cells under optimal control strategy

Ziang Chen; Chunguang Dai; Lei Shi; Gaofang Chen; Peng Wu; Liping Wang; Ziang Chen; Chunguang Dai; Lei Shi; Gaofang Chen; Peng Wu; Liping Wang

doi:10.3934/era.2024186

Electronic Research Archive

2024, Volume 32, Issue 6: 4129-4163. doi: 10.3934/era.2024186

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Research article

Reaction-diffusion model of HIV infection of two target cells under optimal control strategy

1.
School of Mathematics and Statistics, Guilin University of Technology, Guilin 541004, China
2.
Intensive Care Unit, The Affiliated Hospital of Guilin Medical University, Guilin 541001, China
3.
Undergraduate School, Guilin University of Technology, Guilin 541004, China
4.
School of sciences, Hangzhou Dianzi University, Hangzhou 310018, China
5.
Department of Mathematics and Finance, Anhui Polytechnic University, Anhui 241000, China
† These authors contributed equally to the work

Received: 26 February 2024 Revised: 31 May 2024 Accepted: 12 June 2024 Published: 26 June 2024

In order to study the effects of reverse transcriptase inhibitors, protease inhibitors and flavonoids on two target cells infected by HIV in a heterogeneous environment, an HIV mathematical model at the cellular level was established. Research shows that infected cells can be categorized into immature infected cells, latent infected cells, and mature infected cells based on the infection process. The basic reproduction number $ R_{0} $ was established, and it is proved that $ R_{0} $ serves as a threshold parameter: When $ R_{0} < 1 $, the disease-free steady state is globally asymptotically stable, and the disease is extinct; when $ R_{0} > 1 $, the solution of the system is uniformly persistent, and the virus exists. Considering the huge advantages of drug intervention in controlling HIV infection, the optimal control problem was proposed under the condition that the constant diffusion coefficient is positive, so as to minimize the total number of HIV-infected cells and the cost of drug treatment. To illustrate our theoretical results, we performed numerical simulations in which the model parameters were obtained with reference to some medical studies. The results showed that: (1) as $ R_{0} $ increases, the risk of HIV transmission increases; (2) pharmacological interventions are important in early treatment of HIV spread and control of viral load in the body; (3) the treatment process must consider the heterogeneity of medication, otherwise it will not be conducive to suppressing the spread of the virus and will increase costs.
- HIV model,
- spatial heterogeneity,
- stability analysis,
- optimal control,
- numerical simulations
Citation: Ziang Chen, Chunguang Dai, Lei Shi, Gaofang Chen, Peng Wu, Liping Wang. Reaction-diffusion model of HIV infection of two target cells under optimal control strategy[J]. Electronic Research Archive, 2024, 32(6): 4129-4163. doi: 10.3934/era.2024186

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Abstract

In order to study the effects of reverse transcriptase inhibitors, protease inhibitors and flavonoids on two target cells infected by HIV in a heterogeneous environment, an HIV mathematical model at the cellular level was established. Research shows that infected cells can be categorized into immature infected cells, latent infected cells, and mature infected cells based on the infection process. The basic reproduction number $ R_{0} $ was established, and it is proved that $ R_{0} $ serves as a threshold parameter: When $ R_{0} < 1 $, the disease-free steady state is globally asymptotically stable, and the disease is extinct; when $ R_{0} > 1 $, the solution of the system is uniformly persistent, and the virus exists. Considering the huge advantages of drug intervention in controlling HIV infection, the optimal control problem was proposed under the condition that the constant diffusion coefficient is positive, so as to minimize the total number of HIV-infected cells and the cost of drug treatment. To illustrate our theoretical results, we performed numerical simulations in which the model parameters were obtained with reference to some medical studies. The results showed that: (1) as $ R_{0} $ increases, the risk of HIV transmission increases; (2) pharmacological interventions are important in early treatment of HIV spread and control of viral load in the body; (3) the treatment process must consider the heterogeneity of medication, otherwise it will not be conducive to suppressing the spread of the virus and will increase costs.

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