Global dynamics of a tuberculosis model with age-dependent latency and time delays in treatment

摘要

Abstract Since there exists heterogeneity in incubation periods of tuberculosis and a time lag between treatment and recovery. In this study, we develop a tuberculosis model that takes into account age-dependent latency and time delays in treatment to describe the transmission of tuberculosis. We first show that the solution semi-flow of the model is well-posed and has a global attractor Ddocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {D}}$$end{document} within an infinite dimensional space Γdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Gamma $$end{document}. Then we define the basic reproduction number R0documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {R}}_{0}$$end{document} and prove that it determines the global dynamics of the model. If R01documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {R}}_{0}>1$$end{document}, the semi-flow generated by the model is uniformly persistent, and there exists an interior global attractor D0documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {D}}_{0}$$end{document} for this uniformly persistent model. By constructing a suitable Lyapunov function and applying LaSalle’s Invariance Principle, we show that the global attractor D0documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {D}}_{0}$$end{document} is reduced to the endemic equilibrium state, which means that the endemic equilibrium state is globally asymptotically stable. Based on the tuberculosis data in China from 2007 to 2020, we simulate the parameters and initial values of the proposed model. Furthermore, we calculate the sensitivity of R0documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {R}}_{0}$$end{document} to the parameters and find the most sensitive parameters to R0documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {R}}_{0}$$end{document}. Finally, we present an improved strategy to achieve the WHO’s goal of reducing the incidence of tuberculosis by 90 by 2035 compared to 2015.