Dynamical System for Tuberculosis Outbreak with Vaccination Treatment and Different Interventions on the Burden of Drug Resistance

Ratnah Kurniati MA, Sigit Sugiarto, Sugian Nurwijaya

Abstract


Tuberculosis, a highly contagious and lethal infectious disease, remains a global health concern with challenging treatment options. To combat its widespread impact, prevention strategies, such as vaccination, are imperative. This research focuses on developing a mathematical model with the addition of a vaccination compartment to understand the dynamics of tuberculosis transmission with vaccination. Subsequently, the study proceeds to identify the equilibrium points and calculate the basic reproduction number (ℜ0). Following this, a comprehensive stability analysis is conducted, and a numerical simulation is executed to observe the population dynamics. Furthermore, parameter sensitivity analysis is undertaken to assess the extent to which these parameters impact ℜ0. Preliminary analysis shows that the modified model has a solution that remains in the non-negative and bounded region. Furthermore, model analysis reveals two equilibrium points, namely the disease-free equilibrium and the endemic equilibrium. It is established that the disease-free equilibrium exhibits local asymptotic stability when ℜ0 < 1. Remarkably, the numerical simulation aligns with the analytical findings, reinforcing the robustness of the results. Analysis of the sensitivity of the parameter to ℜ0 shows that the parameter of the proportion of susceptible population entering the vaccination class has a significant effect on the value of ℜ0. The parameter of proportion of susceptible population entering the vaccination class has a negative effect on the number of populations with infection.


Keywords


Mathematical modeling; Drug-resistant Tuberculosis; Equilibrium Point; Stability Analysis; Sensitivity Analysis

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References


A. C. Bohrer et al., “Rapid GPR183-mediated recruitment of eosinophils to the lung after Mycobacterium tuberculosis infection,” Cell Reports, vol. 40, no. 4, pp. 1–23, 2022. DOI:10.1016/j.celrep.2022.111144

Y. A. Melsew et al., “Heterogeneous infectiousness in mathematical models of tuberculosis: A systematic review,” Epidemics, vol. 30, no. 100374, pp. 1–10, 2020. DOI:10.1016/j.epidem.2019.100374

M. Kubjane et al., “Heterogeneous infectiousness in mathematical models of tuberculosis: A systematic review,” International Journal of Infectious Diseases, vol. 122, pp. 811–819, 2022.

D. Iskandar et al., “Articles Clinical and economic burden of drug-susceptible tuberculosis in Indonesia : national trends 2017-19,” The Lancet Global Health, vol. 11, no. 1, pp. e117–e125, 2023. DOI:10.1016/S2214-109X(22)00455-7

M. L. Olaosebikan, M. K. Kolawole, and K. A. Bashiru, “Transmission Dynamics of Tuberculosis Model with Control Strategies,” Jambura Journal of Biomathematics (JJBM), vol. 4, no. 2, pp. 110–118, 2023. DOI:10.37905/jjbm.v4i2.21043

E. M. D. Moya, A. Pietrus, and S. M. Oliva, “Mathematical model with fractional order derivatives for Tuberculosis taking into account its relationship with HIV/AIDS and Diabetes,” Jambura Journal of Biomathematics (JJBM), vol. 2, no. 2, pp. 80–95, 2021. DOI:10.34312/jjbm.v2i2.11553

K. Das et al., “Mathematical transmission analysis of SEIR tuberculosis disease model,” Sensors International, vol. 2, pp. 100120, 2021. DOI:10.1016/j.sintl.2021.100120

K. G. Mekonen and L. L. Obsu, “Mathematical modeling and analysis for the co-infection of COVID-19 and tuberculosis,” Heliyon, vol. 8, no. 10, pp. 1–11, 2022. DOI:10.1016/j.heliyon.2022.e11195

Z. Zhang et al., “Dynamical aspects of a tuberculosis transmission model incorporating vaccination and time delay,” Alexandria Engineering Journal, vol. 66, pp. 287–300, 2023. DOI:10.1016/j.aej.2022.11.010

K. K. Avilov et al., “Mathematical modelling of the progression of active tuberculosis: Insights from fluorography data,” Infectious Disease Modelling, vol. 7, no. 3, pp. 374–386, 2022. DOI:10.1016/j.idm.2022.06.007

G. Günther et al., “Availability and costs of medicines for the treatment of tuberculosis in Europe,” Clinical Microbiology and Infection, vol. 29, no. 1, pp. 77–84, 2023. DOI:10.1016/j.cmi.2022.07.026

J. A. Gullón-Blanco et al., “[Translated article] Tuberculosis contacts tracing in Spain: Cost analysis,” Archivos de Bronconeumología, vol. 58, no. 5, pp. T448–T450, 2022. DOI:10.1016/j.arbres.2021.09.021

W. Huang et al., “The effect of BCG vaccination and risk factors for latent tuberculosis infection among college freshmen in China, ”International Journal of Infectious Diseases, vol. 122, pp. 321–326, 2022. DOI:10.1016/j.ijid.2022.06.010

Q. Liao et al., “Effectiveness of Bacillus Calmette-Guérin vaccination against severe childhood tuberculosis in China: a case-based, multicenter retrospective study,” International Journal of Infectious Diseases, vol. 121, pp. 113–119, 2022. DOI:10.1016/j.ijid.2022.04.023

L. Martinez et al., “Infant BCG vaccination and risk of pulmonary and extrapulmonary tuberculosis throughout the life course: a systematic review and individual participant data meta-analysis,” The Lancet Global Health, vol. 10, no. 9, pp. E1307-E1316, 2022. DOI:10.1016/S2214-109X(22)00283-2

A. Xu et al., “Prediction of different interventions on the burden of drug-resistant tuberculosis in China: a dynamic modelling study,” Journal of Global Antimicrobial Resistance, vol. 29, pp. 323–330, 2022. DOI:10.1016/j.jgar.2022.03.018

B. K. Mishra and J. Srivastava, “Mathematical model on pulmonary and multidrug-resistant tuberculosis patients with vaccination,” Journal of the Egyptian Mathematical Society, vol. 22, no. 2, pp. 311–316, 2014. DOI:10.1016/j.joems.2013.07.006

W. H. Organization, “Global tuberculosis report 2019,” Institutional Repository for Information SHaring, pp. 1–297, 2019.

P. Samui, J. Mondal, and S. Khajanchi, “A mathematical model for covid-19 transmission dynamics with a case study of india,” Chaos, Solitons & Fractals, vol. 140, p. 110173, 2020. DOI:10.1016/j.chaos.2020.110173

M. M. Ojo and E. F. D. Goufo, “Modeling, analyzing and simulating the dynamics of Lassa fever in Nigeria,” Journal of the Egyptian Mathematical Society, vol. 30, no. 1, pp. 1–31, 2022. DOI:10.1186/s42787-022-00138-x

S. Sugiarto, R. MA, and S. Nurwijaya, “Dynamical system for covid-19 outbreak within vaccination treatment,” BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 17, no. 2, pp. 0919–0930, 2023. DOI:10.30598/barekengvol17iss2pp0919-0930

S. Sugiarto and A. Alam, “Analisis Kestabilan Model Penyebaran Penyakit Antraks Tipe SVEIQR pada Ternak,” Jurnal Sains Matematika dan Statistika, vol. 9, no. 2, pp. 41–52, 2023. DOI:10.24014/jsms.v9i2.21529

P. Van Den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, no. 1–2, pp. 29–48, 2002. DOI:10.1016/S0025-5564(02)00108-6

S. Nurwijaya, R. MA, and S. Sugiarto, “Dynamical system for ebola outbreak within quarantine and vaccination treatments,” BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 17, no. 2, pp. 0615–0624, 2023. DOI:10.30598/barekengvol17iss2pp0615-0624

A. Alam and S. Sugiarto, “Analisis Sensitivitas Model Matematika Penyebaran Penyakit Antraks pada Ternak dengan Vaksinasi, Karantina dan Pengobatan,” Jurnal Ilmiah Matematika Dan Terapan, vol. 19, no. 2, pp. 180–191, 2022. DOI:10.22487/2540766X.2022.v19.i2.16017

S. F. AL-Azzawi, “Stability and bifurcation of pan chaotic system by using Routh-Hurwitz and gardan methods,” Applied Mathematics and Computation, vol. 219, no. 3, pp. 1144–1152, 2012. DOI:10.1016/j.amc.2012.07.022

S. R. Bandekar and M. Ghosh, “A co-infection model on tb - covid-19 with optimal control and sensitivity analysis,” Mathematics and Computers in Simulation, vol. 200, pp. 1–31, 2022. DOI:10.1016/j.matcom.2022.04.001

A. Malik et al., “Sensitivity analysis of covid-19 with quarantine and vaccination: A fractal-fractional model,” Alexandria Engineering Journal, vol. 61, no. 11, pp. 8859–8874, 2022. DOI:10.1016/j.aej.2022.02.024




DOI: https://doi.org/10.37905/jjbm.v5i1.21903

Copyright (c) 2024 Ratnah Kurniati MA, Sigit Sugiarto, Sugian Nurwijaya

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