Dynamical systems framework for modeling COVID-19 with Lévy noise

Stochastic fluctuations play a crucial role in shaping the dynamics of infectious diseases such as those caused by coronaviruses. To understand these dynamics, it is essential to examine the effects of random perturbations on epidemic models. In this study, we formulate a stochastic Susceptible–Infectious–Recovered (SIR) model for coronavirus transmission, where the contact rate is subject to Lévy noise. This approach captures discontinuous, high-intensity disturbances in transmission behavior. We first analyze the corresponding deterministic system. For the stochastic model driven by Lévy noise, we establish the existence, uniqueness, and global positivity of solutions-fundamental prerequisites for meaningful biological analysis. We then derive a stochastic threshold parameter that governs the long-term fate of the infection. Using Lyapunov analysis and martingale techniques, we provide rigorous criteria for almost sure exponential extinction when this parameter falls below one, and for persistence in mean when it exceeds one. These results demonstrate how Lévy-driven noise can fundamentally alter classical deterministic thresholds. Numerical simulations corroborate our theoretical findings: when the stochastic reproduction number is less than one, the disease dies out despite substantial stochastic perturbations; conversely, when it exceeds one, sustained fluctuations render control substantially more difficult. Collectively, this study offers fundamental theoretical insights into understanding and managing epidemic trajectories under the combined influence of continuous variability and discontinuous shocks.