We investigate the asymptotic behavior of nearly unstable Hawkes processes whose regression kernel has $L^1$ norm strictly greater than 1 and close to 1 as time goes to infinity. We find that the scaling size determines the scaling behavior of the processes as in Jaisson and Rosenbaum (2015). Specifically, after a suitable rescale of (Formula presented), the limit of the sequence of Hawkes processes is deterministic. Also, with another appropriate rescaling of 1/T² , the sequence converges in law to an integrated Cox–Ingersoll–Ross-like process. This theoretical result may apply to model the recent COVID-19 outbreak in epidemiology and phenomena in social networks.

Deep geothermal energy (~3-6 km depth) is a candidate for sustainable and carbon-free energy supply. One of the main concerns of deep geothermal systems is induced seismicity that may produce earthquakes of economic concerns, challenging the development of this form of alternative energy. So far, cold water injection has been overlooked but may contribute to induced seismicity due to fault reactivation through thermal stresses also beyond the cooling region. This can be of importance, in particular, in fractured and faulted geothermal reservoirs. In this study, we first compare different approaches to estimate induced seismic risk from slip-tendency analysis, rate-and-state friction theory and modified Gutenberg-Richter statistics based on frictional Coulomb-stress perturbations. Then, we systematically investigate effects of both, intrinsic geological parameters (e.g., fault-, host rock properties and in-situ stress), and operational parameters (e.g., well geometry and placement, injection schemes, induced pressure perturbation) on induced seismicity.