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2019 | OriginalPaper | Buchkapitel

A Tuberculosis Epidemic Model with Latent and Treatment Period Time Delays

verfasst von : Jay Michael R. Macalalag, Elvira P. De Lara-Tuprio, Timothy Robin Y. Teng

Erschienen in: Dynamical Systems, Bifurcation Analysis and Applications

Verlag: Springer Singapore

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Abstract

In this paper, a Susceptible-Exposed-Infectious-Treated (SEIT) epidemic model with two discrete time delays for the disease transmission of tuberculosis (TB) is proposed and analyzed. The first time delay \(\tau _1\) represents the time of progression of an individual from the latent TB infection to the active TB disease, and the other delay \(\tau _2\) corresponds to the treatment period. We begin our mathematical analysis of the model by establishing the existence, uniqueness, nonnegativity and boundedness of the solutions. We derive the basic reproductive number \(R_0\) for the model. Using LaSalle’s Invariance Principle, we determine the stability of the equilibrium points when the treatment success rate is equal to zero. We prove that if \(R_0<1\), then the disease-free equilibrium is globally asymptotically stable. If \(R_0>1\), then the disease-free equilibrium is unstable and a unique endemic equilibrium exists which is globally asymptotically stable. Numerical simulations are presented to illustrate the theoretical results.

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Vorheriges Kapitel Analysis of a Discrete-Time Fractional Order SIR Epidemic Model for Childhood Diseases

Nächstes Kapitel Numerical Bifurcation and Stability Analyses of Partial Differential Equations with Applications to Competitive System in Ecology

Babiarz, K.S., Suen, S.C., Goldhaber-Fiebert, J.D.: Tuberculosis treatment discontinuation and symptom persistence: an observational study of Bihar, India’s public care system covering \({>}\)100,000,000 inhabitants. BMC Public Health 14(418) (2014). https://doi.org/10.1186/1471-2458-14-418

Centers for Disease Control and Prevention (2018). Tuberculosis. https://www.cdc.gov/tb/topic/treatment/tbdisease.htm

Colijn, C., Cohen, T., Murray, M.: Emergent heterogeneity in declining tuberculosis epidemics. J. Theor. Biol. 247, 765–774 (2007). https://doi.org/10.1016/j.jtbi.2007.04.015MathSciNetCrossRef

Connolly, L.E., Edelstein, P.H., Ramakrishnan, L.: Why is long-term therapy required to cure tuberculosis? PLoS Med. 4(3), e120 (2007). https://doi.org/10.1371/journal.pmed.0040120CrossRef

Driver, R.: Ordinary and Delay Differential Equations. Springer, New York (1977)CrossRef

Feng, Z., Castillo-Chavez, C., Capurro, A.F.: A model for Tuberculosis with Exogenous Reinfection. Theor. Popul. Biol. 57, 235–247 (2000). https://doi.org/10.1006/tpbi.2000.1451CrossRefMATH

Feng, Z., Iannelli, M., Milner, F.: A two-strain tuberculosis model with age of infection. SIAM J. Appl. Math. 62, 1634–1656 (2002). https://doi.org/10.1137/S003613990038205XMathSciNetCrossRefMATH

Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press Inc., California (1993)MATH

Li, J., Ma, M.: Impact of prevention in a tuberculosis model with latent delay. Adv. Diff. Equ. (2016). https://doi.org/10.1186/s13662-016-0934-zMathSciNetCrossRefMATH

10.

Liu, L.J., Zhang, T.L.: Global stability for a tuberculosis model. Math. Comput. Model. 54, 836–845 (2011). https://doi.org/10.1016/j.mcm.2011.03.033MathSciNetCrossRefMATH

11.

Madkour, M.M.: Tuberculosis. Springer, Berlin, Heidelberg (2004)CrossRef

12.

Martcheva, M.: An Introduction to Mathematical Epidemiology. Springer, US (2015)CrossRef

13.

Mittal, H., Das, S., Faridi, M.M.: Management of newborn infant born to mother suffering from tuberculosis: current recommendations and gaps in knowledge. Indian J. Med. Res. 140(1), 32–9 (2014)

14.

Mittal, C.C., Gupta, S.: Noncompliance to DOTS: how it can be decreased. Indian J. Commun. Med. 36, 27–30 (2011). https://doi.org/10.4103/0970-0218.80789CrossRef

15.

Neyrolles, O., Hernandez-Pando, R., Pietri-Rouxel, F., et al.: Is Adipose Tissue a Place for Mycobacterium tuberculosis Persistence? PLOS One 1(1), e43 (2016). https://doi.org/10.1371/journal.pone.0000043CrossRef

16.

NIH/National Institute of Allergy and Infectious Diseases: TB can persist in lungs despite treatment, researchers find. Science Daily. http://www.sciencedaily.com/releases/2016/09/160906131454.htm

17.

Public Library of Science: Tuberculosis Bacillus hides from immune system in host’s fat cells. ScienceDaily. ScienceDaily, 24 December (2006). https://www.sciencedaily.com/releases/2006/12/061221074735.htm

18.

Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer (2011)

19.

Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002). https://doi.org/10.1016/S0025-5564(02)00108-6MathSciNetCrossRefMATH

20.

World Health Organization: Global Tuberculosis Report 2018. World Health Organization, Geneva (2018)

21.

Xu, Y., Huang, G., Zhao, Z.: Dynamics of a tuberculosis model with treatment and self-cure. In: Tan, H. (Eds.) Informatics in Control, Automation and Robotics, Lecture Notes in Electrical Engineering, Springer, Berlin, Heidelberg 132 (2015) https://doi.org/10.1007/978-3-642-25899-2_66.CrossRef

22.

Yang, Y.L., Li, J.Q., Zhou, Y.C.: Global stability of two tuberculosis models with treatment and self-cure. Rocky Mt. J. Math. 42(4), 1367–1386 (2012). https://doi.org/10.1216/RMJ-2012-42-4-1367MathSciNetCrossRefMATH

23.

Zhang, J., Feng, G.: Global stability for a tuberculosis model with isolation and incomplete treatment. Comput. Appl. Math. 34, 1237–1249 (2015). https://doi.org/10.1007/s40314-014-0177-0MathSciNetCrossRefMATH

Titel: A Tuberculosis Epidemic Model with Latent and Treatment Period Time Delays
verfasst von: Jay Michael R. Macalalag
Elvira P. De Lara-Tuprio
Timothy Robin Y. Teng
Verlag: Springer Singapore
Buch: Dynamical Systems, Bifurcation Analysis and Applications
Print ISBN: 978-981-329-831-6

Electronic ISBN: 978-981-329-832-3

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-981-32-9832-3_6

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