Dynamics of Time-Delayed Epidemic Model for Noise-Induced Hearing Loss

A mathematical model with time delay is considered to describe the dynamics of hearing loss resulting from noise exposure. The analysis illustrates that the solutions consistently maintain positivity, and the model has at most two steady states: noise-free steady-state and noise-endemic. The stability of the noise-free steady-state and noise-endemic is shown analytically, demonstrating that the noise-endemic steady-state is stable below a critical value of time delay; beyond this threshold, it becomes unstable and undergoes a Hopf bifurcation when the time delay equals the critical value. The paper also examines the bifurcation and estimates the delay needed to keep the system stable. Furthermore, an assessment of parameter elasticity is performed to measure its influence on hearing loss control. Numerical bifurcation is conducted using DDE-BIFTOOL and traceDDE to explore various dynamical behaviors of the model via numerical continuation for varying values of system parameters. Analytical findings are supported by numerical simulations of the model to show different types of dynamical behavior.