A nonlinear time-delay modeling case study of Tuberculosis transmission in Algeria: 1990-2024.

This study develops a nonlinear time-delay mathematical model to investigate the transmission dynamics of tuberculosis in Algeria over the period 1990-2024. The model explicitly incorporates a delay representing the incubation period between infection and the onset of infectiousness, which significantly affects system behavior by potentially inducing instability and oscillatory dynamics. In particular, increasing the delay can destabilize the endemic equilibrium and lead to periodic outbreaks through Hopf bifurcation. The basic reproduction number $ \mathcal{R}_0 $ is derived, and rigorous analysis establishes the local and global stability conditions for both disease-free and endemic equilibria. Sensitivity analysis identifies key epidemiological parameters influencing $ \mathcal{R}_0 $, providing insight into effective intervention strategies. From a public health perspective, the results highlight that reducing diagnostic delays and improving timely treatment can stabilize disease dynamics and prevent recurrent outbreaks. Numerical simulations, calibrated with Algerian tuberculosis data from 1990 to 2024, support the theoretical findings and demonstrate the critical role of delay effects in shaping tuberculosis transmission and control.