Analysis of drug-resistant tuberculosis in a two-patch environment using Caputo fractional-order modeling

Hongyan Wang; Shaoping Jiang; Yudie Hu; Supaporn Lonapalawong; Hongyan Wang; Shaoping Jiang; Yudie Hu; Supaporn Lonapalawong

doi:10.3934/math.20241565

AIMS Mathematics

2024, Volume 9, Issue 11: 32696-32733. doi: 10.3934/math.20241565

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Research article

Analysis of drug-resistant tuberculosis in a two-patch environment using Caputo fractional-order modeling

School of Mathematics and Computer Science, Yunnan Minzu University, Kunming 650500, China

Received: 24 September 2024 Revised: 31 October 2024 Accepted: 06 November 2024 Published: 19 November 2024
MSC : 34A08, 92B05, 92D30

In this study, a fractional-order mathematical model of the transmission dynamics of drug-resistant tuberculosis within a two-patch system incorporating population migration was proposed and analyzed using the Caputo operator. The positivity, boundedness, existence, and uniqueness of the solutions as well as the Ulam-Hyers stability of the model were guaranteed. Additionally, the basic reproduction numbers were derived and analyzed for sensitivity to identify the key parameters that affected the spread of drug-resistant tuberculosis. Moreover, the cure rates were used as control variables to formulate an optimal control problem, which examined the efficacy of the control measures and the influence of fractional order on the control values. The numerical results showed that controlling the cure rate can significantly reduce the number of drug-resistant tuberculosis infections, thus verifying the effectiveness of the proposed control strategy. As the fractional order decreased, the duration during which the maximum control intensity was applied in both patches increased.
- drug-resistant tuberculosis,
- individual migration,
- Caputo fractional-order derivative,
- Ulam-Hyers,
- sensitivity analysis,
- optimal control
Citation: Hongyan Wang, Shaoping Jiang, Yudie Hu, Supaporn Lonapalawong. Analysis of drug-resistant tuberculosis in a two-patch environment using Caputo fractional-order modeling[J]. AIMS Mathematics, 2024, 9(11): 32696-32733. doi: 10.3934/math.20241565

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Abstract

In this study, a fractional-order mathematical model of the transmission dynamics of drug-resistant tuberculosis within a two-patch system incorporating population migration was proposed and analyzed using the Caputo operator. The positivity, boundedness, existence, and uniqueness of the solutions as well as the Ulam-Hyers stability of the model were guaranteed. Additionally, the basic reproduction numbers were derived and analyzed for sensitivity to identify the key parameters that affected the spread of drug-resistant tuberculosis. Moreover, the cure rates were used as control variables to formulate an optimal control problem, which examined the efficacy of the control measures and the influence of fractional order on the control values. The numerical results showed that controlling the cure rate can significantly reduce the number of drug-resistant tuberculosis infections, thus verifying the effectiveness of the proposed control strategy. As the fractional order decreased, the duration during which the maximum control intensity was applied in both patches increased.

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