Dynamical analysis of SIR model with Gamma distribution delay driven by Lévy noise and switching

Considering the distributed time delay and the stochastic change of environment, this research primarily studied the dynamical behavior of the gamma-distributed delay SIR model driven by Lévy noise and Markov chain. First, it is proved that there is a unique global positive solution for stochastic model. Second, by constructing appropriate stochastic Lyapunov functions that account for regime switching, we derive a sufficient condition, $ {R_s} > 1 $, indicating the existence of a stationary distribution. This suggests the disease will persist over a long period. Finally, numerical simulations confirm the theoretical findings.