article · Decision Analytics Journal
A new six-dimensional mathematical model uses fractional calculus to capture the transmission dynamics of tuberculosis, incorporating memory effects through the Caputo derivative operator. The framework accounts for vaccination, the forces of tuberculosis infection, and exogenous re-infection driven by two actively infectious groups. Mathematical analysis establishes the existence and uniqueness of solutions and determines the effective reproduction number. Notably, the system exhibits backward bifurcation, meaning that stable disease-free and disease-endemic states can co-exist even when the reproduction number falls below one. After identifying sensitive parameters influencing disease spread, the study integrates four time-dependent interventions: public advocacy, vaccination, prophylaxis for latent cases, and treatment. Using Pontryagin's maximum principle, these measures are evaluated through efficiency and cost-effectiveness analyses to pinpoint the most effective strategies for curbing tuberculosis spread across varying fractional derivative orders.
Tuberculosis remains a major global public health challenge. Because the disease exhibits complex transmission dynamics, including re-infection and lingering biological memory, conventional models can miscalculate the effort needed for eradication. Demonstrating that the disease can persist even when the reproduction number is below unity highlights the necessity of combining multiple targeted interventions, such as advocacy, prophylaxis, and treatment, to design realistic and affordable disease control strategies.
This work provides an early-stage decision-support framework that could assist public health planners, epidemiologists, and healthcare economists in designing cost-effective intervention programmes for tuberculosis. Because the findings are purely theoretical and computational, real-world deployment would require integration into operational planning tools and validation with field-level epidemiological and economic data.
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This study is concerned with applying fractional calculus in modelling tuberculosis (TB) transmission dynamics. A new six-dimensional fractional-order mathematical model in the sense of the Caputo derivative operator is formulated to capture memory effects in the spread process of tuberculosis. Some key features such as vaccination, forces of TB infection, and exogenous re-infection by two actively infectious classes are considered. Banach fixed point theory is employed to prove the existence and uniqueness of a solution for the fractional-order model. Equilibrium points and effective reproduction number of the model are determined. Analysis reveals that the model exhibits backward bifurcation where stable TB-free and TB-present equilibrium points co-exist when the effective reproduction number is below unity. Some sensitive parameters of the model are identified to help design intervention measures against the disease. Hence, four time-dependent controls, namely advocacy effort against infection and re-infection, vaccination control, prophylaxis against latent infection, and treatment control, are considered to form a fractional-order optimal control model. The controls are characterized using Pontryagin’s maximum principle. Efficiency and cost-effectiveness assessments are conducted to quantify the most efficient and cost-effective intervention plan to minimize TB spread in the population. Behaviours of the fractional-order TB model at varying values of the order of the Caputo derivative are extensively explored to show memory effects in the transmission dynamics of the disease.
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DOI: 10.1016/j.dajour.2023.100324
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