Dynamics and optimal control of tuberculosis model with the combined effects of vaccination, treatment and contaminated environments

Tao-Li Kang; Hai-Feng Huo; Hong Xiang; Tao-Li Kang; Hai-Feng Huo; Hong Xiang

doi:10.3934/mbe.2024234

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 4: 5308-5334. doi: 10.3934/mbe.2024234

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Dynamics and optimal control of tuberculosis model with the combined effects of vaccination, treatment and contaminated environments

1.
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China
2.
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China

Academic Editor: Sereno Denis

Received: 11 January 2024 Revised: 23 February 2024 Accepted: 28 February 2024 Published: 07 March 2024

Tuberculosis has affected human beings for thousands of years, and until today, tuberculosis still ranks third among 29 infectious diseases in China. However, most of the existing mathematical models consider a single factor, which is not conducive to the study of tuberculosis transmission dynamics. Therefore, this study considers the combined effects of vaccination, treatment, and contaminated environments on tuberculosis, and builds a new model with seven compartments of $ SVEITRW $ based on China's tuberculosis data. The study shows that when the basic reproduction number $ R_{0} $ is less than 1, the disease will eventually disappear, but when $ R_{0} $ is greater than 1, the disease may persist. In the numerical analysis part, we use Markov-chain Monte-Carlo method to obtain the optimal parameters of the model. Through the next generation matrix theory, we calculate that the $ R_{0} $ value of tuberculosis in China is $ 2.1102 $, that is, if not controlled, tuberculosis in China will not disappear over time. At the same time, through partial rank correlation coefficients, we find the most sensitive parameter to the basic reproduction number $ R_{0} $. On this basis, we combine the actual prevalence of tuberculosis in China, apply Pontryagin's maximum principle, and perform cost-effectiveness analysis to obtain the conditions required for optimal control. The analysis shows that four control strategies could effectively reduce the prevalence of TB, and simultaneously controlling $ u_{2}, u_{3}, u_{4} $ is the most cost-effective control strategy.
- tuberculosis,
- vaccination,
- contaminated environments,
- treatment,
- optimal control
Citation: Tao-Li Kang, Hai-Feng Huo, Hong Xiang. Dynamics and optimal control of tuberculosis model with the combined effects of vaccination, treatment and contaminated environments[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5308-5334. doi: 10.3934/mbe.2024234

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Abstract

Tuberculosis has affected human beings for thousands of years, and until today, tuberculosis still ranks third among 29 infectious diseases in China. However, most of the existing mathematical models consider a single factor, which is not conducive to the study of tuberculosis transmission dynamics. Therefore, this study considers the combined effects of vaccination, treatment, and contaminated environments on tuberculosis, and builds a new model with seven compartments of $ SVEITRW $ based on China's tuberculosis data. The study shows that when the basic reproduction number $ R_{0} $ is less than 1, the disease will eventually disappear, but when $ R_{0} $ is greater than 1, the disease may persist. In the numerical analysis part, we use Markov-chain Monte-Carlo method to obtain the optimal parameters of the model. Through the next generation matrix theory, we calculate that the $ R_{0} $ value of tuberculosis in China is $ 2.1102 $, that is, if not controlled, tuberculosis in China will not disappear over time. At the same time, through partial rank correlation coefficients, we find the most sensitive parameter to the basic reproduction number $ R_{0} $. On this basis, we combine the actual prevalence of tuberculosis in China, apply Pontryagin's maximum principle, and perform cost-effectiveness analysis to obtain the conditions required for optimal control. The analysis shows that four control strategies could effectively reduce the prevalence of TB, and simultaneously controlling $ u_{2}, u_{3}, u_{4} $ is the most cost-effective control strategy.

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