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Random walkers on a two-dimensional square lattice are used to explore the spatio-temporal growth of an epidemic. We have found that a simple random-walk system generates non-trivial dynamics compared with traditional well-mixed models. Phase diagrams characterizing the long-term behaviors of the epidemics are calculated numerically. The functional dependence of the basic reproductive number $$R_{0}$$ R 0 on the model’s defining parameters reveals the role of spatial fluctuations and leads to a novel expression for $$R_{0}$$ R 0 . Special attention is given to simulations of inter-regional transmission of the contagion. The scaling of the epidemic with respect to space and time scales is studied in detail in the critical region, which is shown to be compatible with the directed-percolation universality class.

Carbon capture and storage (CCS) is an essential technology for achieving carbon neutrality. Depositional environments with sandstone and interbedded shale layers are promising for CO2 storage because they can retain CO2 beneath continuous and discontinuous shale layers. However, conventional numerical simulation of shale–sandstone systems is computationally challenging due to the large contrast in properties between the shale and sandstone layers and significant impact of thin shale layers on CO2 migration. Extending recent advancements in Fourier neural operators (FNOs), we propose a new deep learning architecture, the RU-FNO, to predict CO2 migration in complex shale–sandstone reservoirs under various reservoir conditions, injection designs, and rock properties. The gas saturation plume and pressure buildup predictions of the RU-FNO model are 8000-times faster than traditional numerical models and exhibit remarkable accuracy. We utilize the model’s fast prediction to investigate the impact of shale layer characteristics on plume migration and pressure buildup. These case studies show that shale–sandstone reservoirs with moderate heterogeneity and spatial continuity can minimize the plume footprint and maximize storage efficiency.

Random walkers on a two-dimensional square lattice are used to explore the spatio-temporal growth of an epidemic. We have found that a simple random-walk system generates nontrivial dynamics compared with traditional well-mixed models. Phase diagrams characterizing the long-term behaviors of the epidemics are calculated numerically. The phase boundary separating those sets of parameters leading to outbreaks dying out and those leading to indefinite growth is mapped out in detail. The functional dependence of the basic reproductive number R0 on the model's defining parameters reveals the role of spatial fluctuations and leads to a novel expression for R0. Special attention is given to simulations of inter-regional transmission of the contagion. The attack rate and the (growing) radius of gyration of the affected zones are used as measures of the severity of the outbreaks, in cases where R0 is not sufficiently prescriptive to chart the epidemic dynamics.