| Abstract: |
| We study a stochastic SIR model driven by multivariate tempered stable (TS) to capture heavy-tailed, cross-compartment shocks. The noise has a tail index $\alpha \in(0,1)$ with exponential tempering, accommodating rare, large jumps while preserving finite second moments and flexible dependence. Within a spectral-radial specification, finite spectral measure on the unit sphere coupled with a prescribed radial density, we characterize the infinite L\`evy measure and the resulting dynamics. We establish a sharp threshold separating extinction from persistence: below the threshold, the epidemic dies out; above it, the process admits a unique ergodic stationary distribution. Monte Carlo experiments reproduce jump-triggered flare-ups and abrupt regime shifts. Overall, our results show that epidemic persistence is governed not only by variance, but critically by tail shape and the magnitude of large jumps, underscoring the relevance of TS noise for data with sudden outbreaks and correlated shocks. |
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