<p>This study develops a nonlinear mathematical model to describe the transmission dynamics of tuberculosis (TB) disease. The model incorporates endogenous reinfection and treatment of latently tuberculosis infected (LTBI) individuals through a non-monotonic saturated incidence and a saturation effect in the treatment function. A rigorous stability analysis of the equilibrium states is performed. The results show that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R_0 < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. When <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R_0 > 1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the endemic equilibrium point (EEP) achieves global asymptotic stability under certain conditions. Sensitivity analysis using Partial Rank Correlation Coefficient for both state variables and the reproduction number <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> with respect to model parameters identifies the key factors influencing disease dynamics. Interestingly, the model exhibits multiple endemic equilibria when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$R_0 < 1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> indicating the occurrence of a backward bifurcation. The existence and direction of Hopf bifurcations are examined, with analytical results validated through numerical simulations under different parameter choices. In particular, Hopf bifurcations are investigated with respect to parameters such as <i>m</i> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\omega .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> A key observation is that increasing <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> causes the unique EEP to lose stability via a supercritical Hopf bifurcation. Further increases in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> restore the stability of the EEP, producing a bifurcation diagram that exhibits the characteristic structure of an endemic bubble.</p>

Impact of media awareness, treatment saturation, and LTBI control on tuberculosis transmission: a mathematical modeling approach

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发表日期：

2025年12月15日

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Pooja Khoda,
Tapan Sarkar,
Sada Nand Prasad,
Om Prakash Meena

摘要

This study develops a nonlinear mathematical model to describe the transmission dynamics of tuberculosis (TB) disease. The model incorporates endogenous reinfection and treatment of latently tuberculosis infected (LTBI) individuals through a non-monotonic saturated incidence and a saturation effect in the treatment function. A rigorous stability analysis of the equilibrium states is performed. The results show that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number $R_0 < 1$ $R_{0} < 1$ . When $R_0 > 1,$ $R_{0} > 1,$ the endemic equilibrium point (EEP) achieves global asymptotic stability under certain conditions. Sensitivity analysis using Partial Rank Correlation Coefficient for both state variables and the reproduction number $R_0$ $R_{0}$ with respect to model parameters identifies the key factors influencing disease dynamics. Interestingly, the model exhibits multiple endemic equilibria when $R_0 < 1,$ $R_{0} < 1,$ indicating the occurrence of a backward bifurcation. The existence and direction of Hopf bifurcations are examined, with analytical results validated through numerical simulations under different parameter choices. In particular, Hopf bifurcations are investigated with respect to parameters such as m and $\omega .$ $ω .$ A key observation is that increasing $\omega $ $ω$ causes the unique EEP to lose stability via a supercritical Hopf bifurcation. Further increases in $\omega $ $ω$ restore the stability of the EEP, producing a bifurcation diagram that exhibits the characteristic structure of an endemic bubble.