Mathematical Modeling and Optimal Control of Asthma Dynamics Under Desert Dust Storm Exposure

Asthma is one of the most prevalent chronic respiratory diseases worldwide, and environmental pollutants such as desert dust play a significant role in triggering respiratory sensitization and aggravating asthma symptoms. In this study, a mathematical model is proposed to investigate the dynamics of dust-induced asthma progression. The population is divided into unaware susceptible individuals, aware susceptible individuals, dust-sensitized individuals, and asthmatic individuals, while two additional variables describe dust concentration in the human population and the environment. Fundamental qualitative properties of the model, including positivity, boundedness, existence of equilibria, and stability conditions, are established. It is shown that the asthma-free equilibrium is locally and globally asymptotically stable whenever the dust-induced asthma threshold is below unity, whereas a unique asthma-persistent equilibrium exists when the dust-induced asthma threshold exceeds unity. To reduce the burden of asthma, an optimal control problem is formulated by incorporating three time-dependent interventions: awareness campaigns, medical intervention for dust-sensitized individuals, and environmental dust reduction. Pontryagin’s Maximum Principle is employed to characterize the optimal controls and derive the corresponding adjoint system. Numerical simulations, performed using the forward–backward sweep method, demonstrate that the proposed interventions significantly reduce the number of asthmatic individuals compared with the uncontrolled case. Furthermore, a cost-effectiveness analysis based on the Average Cost-Effectiveness Ratio (ACER) and Incremental Cost-Effectiveness Ratio (ICER) is conducted to compare seven intervention strategies. The results indicate that the medical intervention strategy is the most economically attractive option, while the combined strategy involving all controls achieves the largest reduction in asthma burden.