Sliding Mode Control for Stochastic SIR Models with Telegraph and Lévy Noise: Theory and Applications

This paper establishes a new stochastic SIR epidemic model that incorporates telegraph noise and Lévy noise to simulate the complex environmental disturbances affecting disease transmission. Given the susceptibility of epidemic spread to environmental noise and its intricate dynamics, an adaptive sliding mode controller based on an integral sliding surface and an adaptive control law is proposed. This controller is capable of stabilizing the constructed model and effectively suppressing the spread of the disease. The main contributions of this paper include the following: establishing a comprehensive and realistic stochastic SIR model that accounts for the complex impacts of telegraph noise (symbolizing periodic environmental changes) and Lévy noise (representing sudden environmental shocks) on the dynamics of disease transmission; employing T-S fuzzy modeling, which considers the design of fuzzy rules and the symmetry of membership functions, to ensure linearization of the model; constructing an integral sliding surface and designing an adaptive sliding mode controller for the fuzzy-processed model. Finally, the effectiveness of the proposed control method is validated through numerical simulations.