Dynamics of a stochastic SIS model driven by Markovian switching and Lévy jump with heavy tailed increments

Empirical evidence shows that abrupt environmental shocks induce power-law jump behavior in epidemic dynamics, beyond the capacity of models restricted to finite Lévy measures. To more realistically characterize the transmission dynamics of dual SIS epidemics under abrupt environmental shocks and regime switching, this work develops a stochastic SIS model driven by Markov switching and heavy-tailed Lévy jumps, formulated within a general framework encompassing both finite and infinite Lévy measures. Rigorous analysis establishes explicit conditions for extinction and coexistence, showing that Lévy noise can suppress outbreaks, whereas moderate fluctuations permit the long-term coexistence of two diseases. Numerical experiments based on tempered stable processes confirm the theoretical predictions and illustrate how heavy-tailed environmental perturbations reshape epidemic dynamics. These findings provide new insight into how sudden shocks and regime transitions jointly govern multi-disease persistence in stochastic environments.