<p>The co-infection of COVID-19 and tuberculosis (TB) currently poses a dual threat to healthcare systems in developing countries. Data from the world health organization (WHO) shows that the growth rate of co-infection cases in regions such as India reached 18.7% between 2020 and 2025, with overlapping epidemics exacerbating medical resource competition by 42%. Therefore, we have established a mathematical model for COVID-19 and TB co-infections. Considering the different epidemiological characteristics of the two diseases, we establish a deterministic infectious disease model with 13 compartments. For the convenience, we divide the model into COVID-19 only sub-model, TB only sub-model and co-infection model, and discuss them respectively to analyze their dynamic behavior and optimal control problem of co-infection model. Through the analysis of the model, it has been proven that the solution of the model is positive and bounded. Using the effective reproduction numbers corresponding to each disease, we analyze the existence and stability of its disease-free equilibrium point. When the effective reproduction number <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1895_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{c}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation> of the COVID-19 sub-model, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1895_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{t}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> of the tuberculosis sub-model and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1895_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{ct}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mrow> <mi mathvariant="italic">ct</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of the co-infection model are less than 1, the disease-free equilibrium point exists and is locally asymptotically stable. On this basis, the existence and bifurcation analysis of the model’s endemic equilibrium point are further analyzed, and the criterion for the backward branching phenomenon in the COVID-19 sub-model is obtained. We select the effective transmission rate <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1895_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$a_{c}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation> as the branch parameter for analysis and obtain the critical value of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1895_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$a_{c}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>. When <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1895_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{ct}>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mrow> <mi mathvariant="italic">ct</mi> </mrow> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, there are no branches in the co-infection model. Sensitivity analysis of effective reproduction number in co-infection models using LHS-PRCC method is carried out, the numerical simulation results indicate that the effective regeneration number of the system is most affected by the effective transmission rate and treatment rate. Finally, we study separately the effects of single control measures, combination control measures, and no control measures of vaccination or drug therapy on the dynamics of infectious disease transmission. Numerical simulation results demonstrate that the simultaneous implementation of vaccination and drug therapy is the optimal control strategy for preventing and controlling infectious diseases, and this strategy can reduce the effective reproduction number in the co-infection model by 37%.</p>

Modelling and Analysis on Co-infection of COVID-19 and TB and Optimal Control Strategy

期刊论文
发表日期：

2025年4月15日

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Xunyang Wang,
Fan Chen,
Di Wu

摘要

The co-infection of COVID-19 and tuberculosis (TB) currently poses a dual threat to healthcare systems in developing countries. Data from the world health organization (WHO) shows that the growth rate of co-infection cases in regions such as India reached 18.7% between 2020 and 2025, with overlapping epidemics exacerbating medical resource competition by 42%. Therefore, we have established a mathematical model for COVID-19 and TB co-infections. Considering the different epidemiological characteristics of the two diseases, we establish a deterministic infectious disease model with 13 compartments. For the convenience, we divide the model into COVID-19 only sub-model, TB only sub-model and co-infection model, and discuss them respectively to analyze their dynamic behavior and optimal control problem of co-infection model. Through the analysis of the model, it has been proven that the solution of the model is positive and bounded. Using the effective reproduction numbers corresponding to each disease, we analyze the existence and stability of its disease-free equilibrium point. When the effective reproduction number $R_{c}$ $R_{c}$ of the COVID-19 sub-model, $R_{t}$ $R_{t}$ of the tuberculosis sub-model and $R_{ct}$ $R_{ct}$ of the co-infection model are less than 1, the disease-free equilibrium point exists and is locally asymptotically stable. On this basis, the existence and bifurcation analysis of the model’s endemic equilibrium point are further analyzed, and the criterion for the backward branching phenomenon in the COVID-19 sub-model is obtained. We select the effective transmission rate $a_{c}$ $a_{c}$ as the branch parameter for analysis and obtain the critical value of $a_{c}$ $a_{c}$ . When $R_{ct}>1$ $R_{ct} > 1$ , there are no branches in the co-infection model. Sensitivity analysis of effective reproduction number in co-infection models using LHS-PRCC method is carried out, the numerical simulation results indicate that the effective regeneration number of the system is most affected by the effective transmission rate and treatment rate. Finally, we study separately the effects of single control measures, combination control measures, and no control measures of vaccination or drug therapy on the dynamics of infectious disease transmission. Numerical simulation results demonstrate that the simultaneous implementation of vaccination and drug therapy is the optimal control strategy for preventing and controlling infectious diseases, and this strategy can reduce the effective reproduction number in the co-infection model by 37%.