Backward bifurcation and periodic dynamics in a tuberculosis model with integrated control strategies

Dipo Aldila; Chidozie Williams Chukwu; Eka D. A. Ginting; F. Fatmawati; Faishal Farrel Herdicho; Mohammad Ivan Azis; S. Sutrisno; Dipo Aldila; Chidozie Williams Chukwu; Eka D. A. Ginting; F. Fatmawati; Faishal Farrel Herdicho; Mohammad Ivan Azis; S. Sutrisno

doi:10.3934/mbe.2025100

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 10: 2720-2760. doi: 10.3934/mbe.2025100

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Backward bifurcation and periodic dynamics in a tuberculosis model with integrated control strategies

1.
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Indonesia, Depok 16424, Indonesia
2.
Innovative Mathematics and Predictive Analytics for Complex System and Technology Laboratory (IMPACT Lab), Universitas Indonesia, Depok 16424, Indonesia
3.
Department of Mathematical Sciences, Georgia Southern University, 65 Georgia Ave., P.O. Box 8093, Statesboro, Georgia 30460, USA
4.
Department of Mathematics, Faculty of Science and Technology, Universitas Airlangga, Surabaya, Indonesia
5.
Department of Mathematics, Hasanuddin University, Makassar, Indonesia
6.
Department of Mathematics, Diponegoro University, Tembalang, Semarang 50275, Indonesia

Academic Editor: Maia Martcheva

Received: 05 May 2025 Revised: 30 June 2025 Accepted: 15 July 2025 Published: 27 August 2025

In this study, we present a unified mathematical model for tuberculosis (TB) that integrates key interventions: Mask use and media campaigns to raise community awareness and promote vaccine booster uptake. The model also incorporates slow–fast disease progression and limited treatment capacity. A mathematical analysis was conducted to determine the existence and stability of equilibrium points. From the mathematical analysis on the stability criteria of the TB-free equilibrium point, we show that TB can be eradicated if the basic reproduction number is below one. However, due to insufficient treatment capacity, a backward bifurcation may occur when the reproduction number equals one, enabling the coexistence of endemic and disease-free equilibria even when the reproduction number is below one. The parameter estimation is based on TB incidence data per 100,000 individuals in Indonesia. Sensitivity analysis reveald that although both interventions are effective, media campaigns combined with vaccine boosters are more impactful in reducing TB transmission than the use of masks. Numerical simulations further suggest the possibility of periodic outbreaks, indicating potential seasonal TB patterns. To explore adaptive intervention strategies, we extended the model using an optimal control framework. Our findings suggested that combined implementation of face masks and media campaigns is more effective than using either alone, particularly when the likelihood of rapid disease progression increases.
- tuberculosis,
- media campaign,
- face mask,
- parameter estimation,
- global sensitivity analysis,
- periodic solution,
- optimal control model
Citation: Dipo Aldila, Chidozie Williams Chukwu, Eka D. A. Ginting, F. Fatmawati, Faishal Farrel Herdicho, Mohammad Ivan Azis, S. Sutrisno. Backward bifurcation and periodic dynamics in a tuberculosis model with integrated control strategies[J]. Mathematical Biosciences and Engineering, 2025, 22(10): 2720-2760. doi: 10.3934/mbe.2025100

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Abstract

In this study, we present a unified mathematical model for tuberculosis (TB) that integrates key interventions: Mask use and media campaigns to raise community awareness and promote vaccine booster uptake. The model also incorporates slow–fast disease progression and limited treatment capacity. A mathematical analysis was conducted to determine the existence and stability of equilibrium points. From the mathematical analysis on the stability criteria of the TB-free equilibrium point, we show that TB can be eradicated if the basic reproduction number is below one. However, due to insufficient treatment capacity, a backward bifurcation may occur when the reproduction number equals one, enabling the coexistence of endemic and disease-free equilibria even when the reproduction number is below one. The parameter estimation is based on TB incidence data per 100,000 individuals in Indonesia. Sensitivity analysis reveald that although both interventions are effective, media campaigns combined with vaccine boosters are more impactful in reducing TB transmission than the use of masks. Numerical simulations further suggest the possibility of periodic outbreaks, indicating potential seasonal TB patterns. To explore adaptive intervention strategies, we extended the model using an optimal control framework. Our findings suggested that combined implementation of face masks and media campaigns is more effective than using either alone, particularly when the likelihood of rapid disease progression increases.

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