An Ordinary Differential Equation Model for Hearing Threshold Dynamics Under Aging and Noise Exposure
DOI:
https://doi.org/10.54361/LJMR.20.3.87Keywords:
hearing loss, odinary differential equations, presbycusis, occupational noise, Runge–Kutta RK4, sensitivity analysis, parameter estimation, mathematical modeling; audiometryAbstract
Background: Hearing loss is a major and growing public-health challenge, driven by an aging population and by noise exposure that is, in principle, preventable. Longitudinal audiometric studies show that hearing thresholds deteriorate progressively across adulthood, at rates that depend on age, frequency, sex, and exposure history. Methods: We develop a continuous-time ordinary differential equation (ODE) framework for hearing-threshold progression that incorporates age-related degeneration, dynamic noise exposure, exposure clearance, threshold saturation, and protective intervention, with hearing threshold H(t), effective noise burden N(t), and chronological age a(t) as state variables. A closed-form solution is derived for a simplified linear regime, and the full coupled system is integrated numerically using fourth-order Runge–Kutta (RK4) over a 30-year horizon. Five exposure/intervention scenarios are evaluated, followed by stability, sensitivity, and parameter-estimation analyses. Results: During revision, independent recomputation uncovered and corrected inconsistencies between the originally reported hearing-threshold values and the stated differential equation; the corrected values are used throughout. Under the specified parameters, the 30-year threshold reaches 27.98 dB HL for low exposure and 51.43 dB HL for sustained high exposure, whereas 50% protection introduced at year 10 limits the year-30 threshold to 42.74 dB HL. The RK4 solution agrees with the matching analytical solution to within a maximum absolute error of roughly 1.03 × 10⁻¹⁰ dB HL.Conclusion: We present the framework as a transparent tool for hypothesis generation, trajectory simulation, and future calibration against longitudinal audiometric datasets, rather than as a clinically validated predictive tool pending external validation.
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20. Corrected main-scenario trajectories: the noise-burden trajectories reproduce the specified N*=E₀/δ values and the intervention path. The hearing-threshold trajectories were recomputed directly from the stated ODE and parameter table; the corrected values are used in the main text.
21. The recomputed year-30 thresholds are H_A(30)=27.98 dB HL and H_C(30)=51.43 dB HL, while Scenario D gives H_D(30)=42.74 dB HL. These values should be treated as the authoritative outputs of the model as currently specified.
22. Multi-frequency trajectories: the frequency-specific results are retained as model simulations. Because no external audiometric cohort is used for calibration, these trajectories demonstrate model behavior rather than validated clinical prognosis.
23. Analytical-versus-numerical verification: the comparison is performed with β=0 in both the analytical and numerical solutions. The maximum absolute difference is approximately 1.03×10⁻¹⁰ dB HL and R²=1.00000.
24. Sensitivity analysis: signs and qualitative trends were independently checked by finite differences. Exact magnitudes remain scenario-dependent and should be regenerated with a fully specified baseline trajectory before publication as definitive sensitivity coefficients.
25. Reference verification: the reference list has been expanded and bibliographic details were checked against PubMed, WHO, journal, or publisher records during revision. DOI information is provided where available.
26. Summary recommendation: the principal mathematical issue identified in the previous draft—the mismatch between the stated hearing-threshold equation and the reported trajectory values—has been corrected in the revised results. The manuscript should still undergo final equation/table/figure consistency checking and, ideally, external validation against longitudinal audiometric data before submission to a clinical or biomedical journal.
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