Analysis of Fractal-Fractional -Order  Model of Tuberculosis with Vaccination and Treatment Interventions

Celestine Agbata Benedict; Fabiana Çullhaj; Emmanuel Abah; Raimonda Dervishi; William Obeng-Denteh; Homan Emadifar; Aseel Smerat; Mbah Godwin Christopher Ezike

doi:10.56345/ijrdv13n237

Authors

Celestine Agbata Benedict Department of Mathematics and Statistics, Faculty of Science, Confluence University ofScience and Technology, Osara, Nigeria
Fabiana Çullhaj Department of Mathematics, Faculty of Information Technology, Aleksandër Moisiu University of Durrës, Durrës, Albania
Emmanuel Abah Mathematical Sciences Department, Prince Abubakar Audu (formerly, Kogi State) University, Anyigba, Nigeria
Raimonda Dervishi Department of Mathematical Engineering, Mathematical and Physical Engineering Faculty, Polytechnic University of Tirana, Tirana, Albania
William Obeng-Denteh Department of Mathematics, College of Science, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana
Homan Emadifar Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai-602 105, Tamil Nadu, India; Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran
Aseel Smerat Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman 19328, Jordan; Department of Biosciences, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai, 602105, India
Mbah Godwin Christopher Ezike Department of Mathematics, Faculty of Physical Sciences, University of Nigeria, Nsukka, Nigeria

DOI:

https://doi.org/10.56345/ijrdv13n237

Keywords:

Tuberculosis (TB), Fractal–fractional derivatives, Data fitting, Sensitivity analysis, Numerical simulation

Abstract

Tuberculosis (TB) remains a major global health challenge, particularly in regions with limited healthcare resources and high transmission environments. This study develops and analyzes a deterministic compartmental model of TB using fractal–fractional derivatives with Mittag-Leffler kernels, incorporating memory and hereditary effects that better reflect the disease’s latent nature and long-term treatment outcomes. The total human population is stratified into seven classes: susceptible, vaccinated, exposed, acutely infected, chronically infected, treated, and recovered. The model explicitly integrates the influence of vaccination, treatment rates, immunity waning, and environmental interventions such as ventilation and sanitation improvements. Using fixed point theorems, we established the existence and uniqueness of model solutions, while positivity and boundedness analysis confirmed biological feasibility. Threshold conditions, including the basic reproduction number, were derived, and stability analysis for both disease-free and endemic equilibria was conducted. Sensitivity analysis revealed that parameters such as contact rate and exposure progression act as key drivers of disease spread, while vaccination, treatment, and environmental interventions effectively reduce the reproduction number. Numerical simulations using the Adams–Bashforth scheme demonstrated the dynamic effects of vaccination and treatment policies, showing significant reductions in susceptible and exposed populations as well as lower cumulative new cases under stronger intervention strategies. Data fitting with WHO TB incidence reports confirmed that the model aligns well with real-world observations.

Received: 20 February 2026 │ Accepted: 10 June 2026 │ Published: 23 July 2026