Impact of Waning Immunity and Relapse on the Effective Reproduction Number in Tuberculosis Transmission: A Sensitivity Analysis
DOI:
https://doi.org/10.51459/jostir.2025.2.1.0100Keywords:
Disease Relapse, Effective Reproduction Number, Mathematical Model, Sensitivity Analysis, Tuberculosis, Waning ImmunityAbstract
It is estimated that one-third of the world’s population are infected with latent Tuberculosis (TB). Moreover, the emergent prevalence of Covid-19 resulted in a drastic decline of the diagnosis and treatment of TB. This research presents an exhaustive deterministic model for the transmission, prevention and control dynamics of TB. The model incorporates significant epidemiological factors including disease relapse, progression rate from latent to active TB, contact rate, treatment rate, vaccination and immunity wane. The model was shown to possess a positive and bounded solution region. Furthermore, by employing the next generation matrix approach and constructing an appropriate Lyapunov function respectively, it was obtained that there exists a locally and globally stable disease-free equilibrium point (DFE) for the model whenever the effective reproduction number, Re, is less than unity. Similarly, the model possess a unique globally asymptotically stable endemic equilibrium point (EEP) whenever Re >1. Sensitivity analysis of Re was performed using the forward index sensitivity approach. It was established that the recruitment rate into the susceptible population and the disease transmission rate have unit sensitivity indices. Rates of immunity wane after vaccination and progression from latent to active TB exhibit a direct variation while vaccination, treatment, natural and disease induced death rates exhibit an inverse variation with Re. Numerical simulation was performed on the model by implementing the fourth order Runge Kutta numerical computation method on MATLAB subroutine. The values of the model’s sensitive parameters were varied and their effects on the spread and control of TB discussed extensively.
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