Modeling Air Pollution–Mortality Interactions Using Fractional Calculus and Stochastic Processes

Air pollution is one of the most significant threats to environmental sustainability and public health, with particulate matter PM2.5PM_{2.5}PM2.5​ and PM10PM_{10}PM10​ recognized as among the most hazardous pollutants. Elevated concentrations of these fine particles are strongly associated with cardiovascular and respiratory diseases, leading to increased mortality worldwide. Understanding the dynamics of particulate matter and its health impacts is therefore essential for effective mitigation strategies and policy development. In this study, we propose a novel mathematical model to investigate air pollution dynamics by incorporating population, PM2.5, PM10, and pollution-induced mortality as key variables. The model is analyzed under deterministic, fractional-order, and stochastic frameworks. Fractional-order modeling is formulated using the Caputo derivative to capture memory and hereditary effects, while stochastic differential equations are employed to account for environmental randomness and uncertainty. The well-posedness of both the fractional and stochastic models is rigorously established. Stability and boundedness properties of the fractional-order system are examined, and the feasibility of equilibrium points is analyzed using isoclines and asymptotic behavior. For numerical simulations, the Adams–Bashforth–Moulton method is applied to the fractional-order model, whereas Milstein’s scheme is utilized for the stochastic system. Sensitivity analysis is conducted to evaluate the influence of key parameters on system dynamics. To enhance predictive performance, machine learning techniques are integrated with the mathematical framework. Data-driven forecasting methods, including the ARIMA model and random forest regression, are employed to capture both short-term fluctuations and long-term trends in pollutant levels. By combining analytical modeling with data-driven approaches, the proposed framework improves forecasting accuracy. It provides deeper insights into the complex interactions among air quality, particulate matter, and associated health risks.