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A tuberculosis model with the impact of sputum smear microscopy

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Abstract

This paper proposes and analyzes a five-dimensional tuberculosis model incorporating slow-fast progression, endogenous reactivation, exogenous reinfection, and the assumption that infectives pass the smear microscopy test. This study disseminates information regarding how the presence and infectiousness of smear-negative patients in a tuberculosis epidemic significantly affect the threshold epidemic quantity \({\mathcal {R}_0}\) (basic reproduction number). Stability and bifurcation analysis is carried out, and we find that the exogenous reinfection causes backward bifurcation and multiple endemic steady states in the system. We analytically and numerically explored the case of the periodic oscillations in the population via Hopf bifurcation for \({\mathcal {R}_0<1}\) as well as \({\mathcal {R}_0>1}.\)

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References

  1. WHO.: The End TB Strategy. Aug 16, 2015. https://www.who.int/publications/i/item/WHO-HTM-TB-2015.19 (2015)

  2. WHO.: Global tuberculosis report. 2021. https://apps.who.int/iris/handle/10665/346387 (2021)

  3. Chadha, V.K.: Tuberculosis epidemiology in India: a review. Int. J. Tuberc. Lung Dis. 9(10), 1072–1082 (2005)

    CAS  PubMed  Google Scholar 

  4. Lillebaek, T., Dirksen, A., Baess, I., Strunge, B., Thomsen, V.Ø., Andersen, Å.B.: Molecular evidence of endogenous reactivation of mycobacterium tuberculosis after 33 years of latent infection. J. Infect. Dis. 185(3), 401–404 (2002)

    Article  CAS  PubMed  Google Scholar 

  5. Blower, S.M., Mclean, A.R., Porco, T.C., Small, P.M., Hopewell, P.C., Sanchez, M.A., Moss, A.R.: The intrinsic transmission dynamics of tuberculosis epidemics. Nat. Med. 1(8), 815–821 (1995)

    Article  CAS  PubMed  Google Scholar 

  6. Das, D.K., Khajanchi, S., Kar, T.: Transmission dynamics of tuberculosis with multiple re-infections. Chaos Solitons Fractals 130, 109450 (2020)

    Article  MathSciNet  Google Scholar 

  7. Das, D.K., Kar, T.: Global dynamics of a tuberculosis model with sensitivity of the smear microscopy. Chaos Solitons Fractals 146, 110879 (2021)

    Article  MathSciNet  Google Scholar 

  8. Feng, Z., Castillo-Chavez, C., Capurro, A.F.: A model for tuberculosis with exogenous reinfection. Theor. Popul. Biol. 57(3), 235–247 (2000)

    Article  CAS  PubMed  Google Scholar 

  9. Lipsitch, M., Murray, M.B.: Multiple equilibria: tuberculosis transmission require unrealistic assumptions. Theor. Popul. Biol. 2(63), 169–170 (2003)

    Article  Google Scholar 

  10. Sharomi, O.Y., Safi, M.A., Gumel, A.B., Gerberry, D.J.: Exogenous re-infection does not always cause backward bifurcation in TB transmission dynamics. Appl. Math. Comput. 298, 322–335 (2017)

    MathSciNet  Google Scholar 

  11. Gerberry, D.J.: Practical aspects of backward bifurcation in a mathematical model for tuberculosis. J. Theor. Biol. 388, 15–36 (2016)

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  12. Athithan, S., Ghosh, M.: Mathematical modelling of TB with the effects of case detection and treatment. Int. J. Dyn. Control 1, 223–230 (2013)

    Article  Google Scholar 

  13. Huo, H.-F., Zou, M.-X.: Modelling effects of treatment at home on tuberculosis transmission dynamics. Appl. Math. Model. 40(21–22), 9474–9484 (2016)

    Article  MathSciNet  Google Scholar 

  14. Mushayabasa, S., Bhunu, C.: Modeling the impact of early therapy for latent tuberculosis patients and its optimal control analysis. J. Biol. Phys. 39, 723–747 (2013)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  15. Sun, C., Yang, W.: Global results for an sirs model with vaccination and isolation. Nonlinear Anal. Real World Appl. 11(5), 4223–4237 (2010)

    Article  MathSciNet  Google Scholar 

  16. Egonmwan, A., Okuonghae, D.: Analysis of a mathematical model for tuberculosis with diagnosis. J. Appl. Math. Comput. 59, 129–162 (2019)

    Article  MathSciNet  Google Scholar 

  17. Okuonghae, D., Korobeinikov, A.: Dynamics of tuberculosis: the effect of direct observation therapy strategy (dots) in Nigeria. Math. Model. Nat. Phenomena 2(1), 113–128 (2007)

    Article  MathSciNet  Google Scholar 

  18. Moualeu, D., Yakam, A.N., Bowong, S., Temgoua, A.: Analysis of a tuberculosis model with undetected and lost-sight cases. Commun. Nonlinear Sci. Numer. Simul. 41, 48–63 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  19. Desikan, P.: Sputum smear microscopy in tuberculosis: is it still relevant? Indian J. Med. Res. 137(3), 442 (2013)

    PubMed  PubMed Central  Google Scholar 

  20. Linguissi, L.S.G., Vouvoungui, C.J., Poulain, P., Essassa, G.B., Kwedi, S., Ntoumi, F.: Diagnosis of smear-negative pulmonary tuberculosis based on clinical signs in the Republic of Congo. BMC Res. Notes 8, 1–7 (2015)

    Article  Google Scholar 

  21. Hernandez-Garduno, E., Cook, V., Kunimoto, D., Elwood, R., Black, W., FitzGerald, J.: Transmission of tuberculosis from smear negative patients: a molecular epidemiology study. Thorax 59(4), 286–290 (2004)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  22. Campos, L.C., Rocha, M.V.V., Willers, D.M.C., Silva, D.R.: Characteristics of patients with smear-negative pulmonary tuberculosis (TB) in a region with high TB and HIV prevalence. PLoS ONE 11(1), 0147933 (2016)

    Article  Google Scholar 

  23. Pandey, S., Chadha, V., Laxminarayan, R., Arinaminpathy, N.: Estimating tuberculosis incidence from primary survey data: a mathematical modeling approach. Int. J. Tuberc. Lung Dis. 21(4), 366–374 (2017)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  24. Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48 (2002)

    Article  MathSciNet  PubMed  Google Scholar 

  25. Chitnis, N., Hyman, J.M., Cushing, J.M.: Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol. 70(5), 1272–1296 (2008)

    Article  MathSciNet  PubMed  Google Scholar 

  26. Castillo, C., Feng, Z., Huang, W.: On the computation of R0 and its role on global stability. In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction (Minneapolis, MN, 1999), pp. 229–250 (2002)

  27. Li, M.Y., Muldowney, J.S.: A geometric approach to global-stability problems. SIAM J. Math. Anal. 27(4), 1070–1083 (1996)

    Article  MathSciNet  Google Scholar 

  28. Coppel, W.A.: Stability and Asymptotic Behavior of Differential Equations: Heath (1965)

  29. Muldowney, J.S.: Compound matrices and ordinary differential equations. Rocky Mt. J. Math. 20, 857–872 (1990)

    Article  MathSciNet  Google Scholar 

  30. Freedman, H.I., Ruan, S., Tang, M.: Uniform persistence and flows near a closed positively invariant set. J. Dyn. Differ. Equ. 6(4), 583–600 (1994)

    Article  MathSciNet  Google Scholar 

  31. Castillo-Chavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1(2), 361 (2004)

    Article  MathSciNet  PubMed  Google Scholar 

  32. Perko, L.: Differential Equations and Dynamical Systems. Springer, Berlin (2013)

    Google Scholar 

  33. Liu, W.-M.: Criterion of hopf bifurcations without using eigenvalues. J. Math. Anal. Appl. 182(1), 250–256 (1994)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

AS acknowledges financial support by the Indian Institute of Technology Patna. Authors are grateful to the anonymous referees for their constructive suggestions which has significantly improved the manuscript and its presentation.

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  1. Department of Mathematics, Indian Institute of Technology Patna, Patna, Bihar, 801103, India

    Akriti Srivastava & Prashant K. Srivastava

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  1. Akriti Srivastava
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  2. Prashant K. Srivastava
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Correspondence to Prashant K. Srivastava.

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Srivastava, A., Srivastava, P.K. A tuberculosis model with the impact of sputum smear microscopy. J. Appl. Math. Comput. 70, 711–740 (2024). https://doi.org/10.1007/s12190-023-01984-3

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