In this paper, we study an SIR model with a nonlinear incidence rate \(\frac{\beta _0 e^{-\mu } SI}{1+\alpha I}\), which incorporates both a saturated incidence term and the exponential factor \(e^{-\mu }\), where \(\mu\) denotes the mask-wearing rate. We derive the equilibria of the system and compute the basic reproduction number \(R_0\). We also investigate the local and global stability of the model equilibria. To account for environmental and behavioral variability, bounded multiplicative random perturbations are introduced into the transmission parameter \(\beta _0\), and the resulting model is examined numerically under different perturbation intensities. In addition, we analyze the sensitivity of the basic reproduction number \(R_0\). Furthermore, Pontryagin’s maximum principle is employed to determine the optimal levels of two control strategies, namely vaccination and treatment. Finally, numerical simulations are performed to illustrate the theoretical findings and support the conclusions.
