A Precision–Dispersion and Optimal Intervention Framework for Trachoma Transmission Dynamics under Random Environmental Fluctuations

Trachoma remains one of the leading infectious causes of preventable blindness in many developing regions, where environmental conditions, poor sanitation, and limited access to healthcare contribute significantly to disease persistence. This study develops a novel mathematical framework for analysing trachoma transmission dynamics through the integration of precision--dispersion measures, stochastic environmental fluctuations, and optimal intervention strategies. The total population is partitioned into susceptible, exposed, infectious, visually impaired, recovered, and protected classes, and a nonlinear system of differential equations is formulated to describe disease progression. Environmental uncertainty is incorporated through a fluctuating transmission parameter, while intervention effectiveness is evaluated using newly introduced precision and dispersion functionals. Fundamental properties of the model, including positivity, boundedness, existence of equilibria, and local stability conditions, are established. An effective reproduction number is derived using the next-generation matrix approach and sensitivity analysis is performed to identify the most influential epidemiological parameters. To quantify intervention reliability, a precision functional and dispersion index are combined into an intervention reliability measure that evaluates the consistency of control outcomes under uncertainty. Optimal control strategies involving hygiene promotion, antibiotic treatment, and surgical intervention are investigated using Pontryagin's Maximum Principle. Numerical simulations demonstrate that integrated intervention programmes substantially reduce disease prevalence while maintaining high reliability under fluctuating environmental conditions. Cost-effectiveness analysis further reveals that combined hygiene and treatment programmes provide the most favourable balance between implementation cost, disease reduction, and intervention stability. The proposed framework extends classical trachoma models by incorporating uncertainty quantification and intervention reliability analysis, providing a broader mathematical basis for evaluating infectious disease control programmes and supporting evidence-based public health decision-making.