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The fundamental models of epidemiology describe the progression of an infectious disease through a population using compartmentalized differential equations, but typically do not incorporate population-level heterogeneity in infection susceptibility. Here we combine a generalized analytical framework of contagion with computational models of epidemic dynamics to show that variation strongly influences the rate of infection, while the infection process simultaneously sculpts the susceptibility distribution. These joint dynamics influence the force of infection and are, in turn, influenced by the shape of the initial variability. We find that certain susceptibility distributions (the exponential and the gamma) are unchanged through the course of the outbreak, and lead naturally to power-law behavior in the force of infection; other distributions are often sculpted towards these "eigen-distributions" through the process of contagion. The power-law behavior fundamentally alters predictions of the long-term infection rate, and suggests that first-order epidemic models that are parameterized in the exponential-like phase may systematically and significantly over-estimate the final severity of the outbreak. In summary, our study suggests the need to examine the shape of susceptibility in natural populations as part of efforts to improve prediction models and to prioritize interventions that leverage heterogeneity to mitigate against spread.

Over the last 20 to 25 years theoretical chemistry (particularly theoretical chemical kinetics) has played an increasingly important role in developing chemical kinetics models for combustion. Theoretical methods of obtaining rate parameters are now competitive in accuracy with experiment, particularly for small molecules. Moreover, theoretical methods can deal with conditions that experiments frequently cannot. In addition to increased accuracy, theory has rejuvenated methods and discovered phenomena that were completely unappreciated, or at least underappreciated, in the 20th century. Our primary interest here is in molecular-level issues, i.e. in calculating rate and transport parameters. However, dealing with kinetics models that involve thousands of reactions and hundreds of species is important for practical applications and is relatively new to the 21st century. Theory, in a general sense, and theoretical methods development have a role to play here too. We discuss in this review all these topics in some detail with an emphasis on issues and methods that have emerged in the last 20 years or so. Even so, our review is selective, rather than comprehensive, out of necessity.

Optimisation of waste-dump planning at the nickel/cobalt open-pit, truck-and-shovel mines in New Caledonia, south Pacific Ocean by backfilling the waste material is described. Extraction of the laterite and saprolite ores, recovery of nickel and cobalt by high-pressure acid leaching, uses of the metals in applications such as batteries, and data on cobalt usage by year (2006, 2016 and 2020) are discussed. The design of mine pushbacks and operational and final pit slopes, and the backfilling and dump sequencing are reviewed. The importance of achieving the correct balance between finding the appropriate blend and, at the same time, removal of material from the previous dump site for relocation to the newest in-pit site is discussed., Optimisation of waste-dump planning at the nickel/cobalt open-pit, truck-and-shovel mines in New Caledonia, south Pacific Ocean by backfilling the waste material is described. Extraction of the laterite and saprolite ores, recovery of nickel and cobalt by high-pressure acid leaching, uses of the metals in applications such as batteries, and data on cobalt usage by year (2006, 2016 and 2020) are discussed. The design of mine pushbacks and operational and final pit slopes, and the backfilling and dump sequencing are reviewed. The importance of achieving the correct balance between finding the appropriate blend and, at the same time, removal of material from the previous dump site for relocation to the newest in-pit site is discussed.