The generalized tempered stable (GTS) distribution is an optimal choice for modeling disease propagation, as it effectively captures the heavy-tailed nature of such events. This attribute is crucial for evaluating the impact of large-scale outbreaks and formulating effective public health interventions. In our study, we introduce a comprehensive stochastic epidemic model that incorporates various intervention strategies and utilizes Lévy jumps characterized by the GTS distribution. Notably, our proposed stochastic system does not exhibit endemic or disease-free states, challenging the conventional approach of assessing disease persistence or extinction based on asymptotic behavior. To address this, we employed a novel stochastic analysis approach to demonstrate the potential for disease eradication or continuation. We provide numerical examples to highlight the importance of incorporating the GTS distribution in epidemiological modeling. These examples validate the accuracy of our results and compare our model's outcomes with those of a standard system using basic Lévy jumps. The purposeful use of the GTS distribution accounts for the heavy-tailed nature of disease incidence or vector abundance, enhancing the precision of models and predictions in epidemiology.

The stochastic nature of ecological systems is fundamental to their modeling and understanding. In this paper, we introduce a comprehensive algae-zooplankton model that incorporates general interaction rate and second-order independent stochastic components. Our model's perturbation component encompasses both white noise and jump processes, enabling us to account for various sources of variability and capture a wide range of potential fluctuations in the system. By utilizing an auxiliary equation, we establish a global threshold for the stochastic system, distinguishing between scenarios of extinction and ergodicity. This threshold serves as a critical determinant of the system's long-term behavior and sheds light on the delicate balance between population persistence and decline in ecological communities. To elucidate the impact of noise on the dynamics of algae and zooplankton, we present a series of numerical illustrations. Through these simulations, we highlight how noise influences not only the extinction time but also the shape of the stationary distribution. Our findings underscore the significant role of stochasticity in shaping ecological dynamics and emphasize the importance of considering noise effects in ecological modeling and management practices.