Dynamic Systems Framework for Modeling COVID-19 with L\'evy Noise

Natural fluctuations have played a crucial role in affecting the dynamics of pervasive diseases such as the coronavirus. Examining the effects of irregular unsettling disturbances on epidemic models is important for understanding these dynamics. In this study, we introduce a mathematical model for the SIR (Susceptible-Infectious-Recovered) dynamics of the coronavirus, incorporating perturbations in the contact rate through Levy noise. The utilization of the Levy process is essential for the protection and control of diseases. We delve into the dynamics of both the deterministic model and the global positive solution of the stochastic model, establishing their existence and uniqueness. Additionally, we explore conditions for the termination and persistence of the infection. Furthermore, we derive the basic reproduction number, a critical determinant of disease extinction or persistence. Numerical results indicate that COVID-19 dissipates from the population when the reproduction number is less than one in the presence of significant or minor noise. Conversely, controlling epidemic diseases becomes challenging when the reproduction number exceeds one. To illustrate this phenomenon, we provide numerical simulations, offering insights into the dynamics of the disease and the efficacy of control measures