<p>Transmission of tuberculosis (TB) among human population depends on an individual’s infectiousness, which is further determined by the concentration of Mycobacterium tuberculosis (Mtb) in the body. Additionally, Mtb is resistant to dryness, cold, acidic, and alkaline environments and can survive in acidic and alkaline environments for 4–5 years. Mtb in the environment plays a significant role in TB transmission and should not be overlooked. To investigate the epidemiologic relationships among pathogens, hosts, and the environment, we first develop a multiscale TB model that includes multiple transmission routes (human–to–human and environment–to–human) and links Mtb–immune response interactions to TB transmission in population. We comprehensively analyze the dynamic properties of the fast system, slow system, and full system. Analysis results reveal that coupling bacterial processes within-host with transmission mechanisms between-host can trigger diverse complex behaviors, including both forward and backward bifurcation phenomena. This implies that thresholds routinely used to control TB infection or eliminate Mtb from an epidemiological or immunological perspective may fail under specific conditions; that is, even if the basic reproduction number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is less than 1, endemic equilibria may still exist in the system. Second, from a microtherapeutic point of view, we establish an impulsive time–delayed differential equation to characterize the actual medication regimen for TB. The basic reproduction number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R_{0}^{\prime }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mrow> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> </math></EquationSource> </InlineEquation> is defined as the spectral radius of a linear integral operator. Then, we show that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R_{0}^{\prime }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mrow> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> </math></EquationSource> </InlineEquation> is a critical parameter that determines the persistence of the model. More precisely, if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$R_{0}^{\prime }<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>R</mi> <mrow> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the disease–free periodic solution is globally attractive; if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$R_{0}^{\prime }>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>R</mi> <mrow> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the disease is uniformly persistent. Finally, we employ numerical methods to elucidate the interactions between population transmission dynamics and pathogen dynamics. Specifically, the basic reproduction number <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$R_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> of the full system increases rapidly with the rise in Mtb release rate, while its change is relatively slower with an increase in the immune rate. These results highlight the dominant role of chemotherapy, with immunotherapy playing only a supporting role.</p>

The role of multiscale and delayed dynamics in tuberculosis Transmission and control: a mathematical approach

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发表日期：

2026年4月7日

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Wei Li,
Yi Wang,
Zhen Jin

摘要

Transmission of tuberculosis (TB) among human population depends on an individual’s infectiousness, which is further determined by the concentration of Mycobacterium tuberculosis (Mtb) in the body. Additionally, Mtb is resistant to dryness, cold, acidic, and alkaline environments and can survive in acidic and alkaline environments for 4–5 years. Mtb in the environment plays a significant role in TB transmission and should not be overlooked. To investigate the epidemiologic relationships among pathogens, hosts, and the environment, we first develop a multiscale TB model that includes multiple transmission routes (human–to–human and environment–to–human) and links Mtb–immune response interactions to TB transmission in population. We comprehensively analyze the dynamic properties of the fast system, slow system, and full system. Analysis results reveal that coupling bacterial processes within-host with transmission mechanisms between-host can trigger diverse complex behaviors, including both forward and backward bifurcation phenomena. This implies that thresholds routinely used to control TB infection or eliminate Mtb from an epidemiological or immunological perspective may fail under specific conditions; that is, even if the basic reproduction number $R_{0}$ $R_{0}$ is less than 1, endemic equilibria may still exist in the system. Second, from a microtherapeutic point of view, we establish an impulsive time–delayed differential equation to characterize the actual medication regimen for TB. The basic reproduction number $R_{0}^{\prime }$ $R_{0}^{'}$ is defined as the spectral radius of a linear integral operator. Then, we show that $R_{0}^{\prime }$ $R_{0}^{'}$ is a critical parameter that determines the persistence of the model. More precisely, if $R_{0}^{\prime }<1$ $R_{0}^{'} < 1$ , the disease–free periodic solution is globally attractive; if $R_{0}^{\prime }>1$ $R_{0}^{'} > 1$ , the disease is uniformly persistent. Finally, we employ numerical methods to elucidate the interactions between population transmission dynamics and pathogen dynamics. Specifically, the basic reproduction number $R_{0}$ $R_{0}$ of the full system increases rapidly with the rise in Mtb release rate, while its change is relatively slower with an increase in the immune rate. These results highlight the dominant role of chemotherapy, with immunotherapy playing only a supporting role.