A synthesized model of tuberculosis transmission featuring treatment abandonment

Edwin Barrios-Rivera; Hanner E. Bastidas-Santacruz; Carmen A. Ramirez-Bernate; Olga Vasilieva; Edwin Barrios-Rivera; Hanner E. Bastidas-Santacruz; Carmen A. Ramirez-Bernate; Olga Vasilieva

doi:10.3934/mbe.2022509

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 11: 10882-10914. doi: 10.3934/mbe.2022509

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A synthesized model of tuberculosis transmission featuring treatment abandonment

Department of Mathematics, Universidad del Valle, Calle 13 No. 100-00, Cali 760032, Colombia

Academic Editor: Sheldon Wang

Received: 31 March 2022 Revised: 02 June 2022 Accepted: 07 June 2022 Published: 01 August 2022

In this paper, we propose and justify a synthesized version of the tuberculosis transmission model featuring treatment abandonment. In contrast to other models that account for the treatment abandonment, our model has only four state variables or classes (susceptible, latent, infectious, and treated), while people abandoning treatment are not gathered into an additional class. The proposed model retains the core properties that are highly desirable in epidemiological modeling. Namely, the disease transmission dynamics is characterized by the basic reproduction number $ \mathscr{R}_0 $, a threshold value that determines the number of possible steady states and their stability properties. It is shown that the disease-free equilibrium is globally asymptotically stable (GAS) only if $ \mathscr{R}_0 < 1 $, while a strictly positive endemic equilibrium exists and is GAS only if $ \mathscr{R}_0 > 1. $ Analysis of the dependence of $ \mathscr{R}_0 $ on the treatment abandonment rate shows that a reduction of the treatment abandonment rate has a positive effect on the disease incidence and results in avoiding disease-related fatalities.
- tuberculosis transmission model,
- treatment abandonment,
- non-sterilizing treatment,
- Lyapunov function,
- global stability
Citation: Edwin Barrios-Rivera, Hanner E. Bastidas-Santacruz, Carmen A. Ramirez-Bernate, Olga Vasilieva. A synthesized model of tuberculosis transmission featuring treatment abandonment[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 10882-10914. doi: 10.3934/mbe.2022509

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Abstract

In this paper, we propose and justify a synthesized version of the tuberculosis transmission model featuring treatment abandonment. In contrast to other models that account for the treatment abandonment, our model has only four state variables or classes (susceptible, latent, infectious, and treated), while people abandoning treatment are not gathered into an additional class. The proposed model retains the core properties that are highly desirable in epidemiological modeling. Namely, the disease transmission dynamics is characterized by the basic reproduction number $ \mathscr{R}_0 $, a threshold value that determines the number of possible steady states and their stability properties. It is shown that the disease-free equilibrium is globally asymptotically stable (GAS) only if $ \mathscr{R}_0 < 1 $, while a strictly positive endemic equilibrium exists and is GAS only if $ \mathscr{R}_0 > 1. $ Analysis of the dependence of $ \mathscr{R}_0 $ on the treatment abandonment rate shows that a reduction of the treatment abandonment rate has a positive effect on the disease incidence and results in avoiding disease-related fatalities.

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