Hyper-exponential growth in an epidemic model: some explicit solutions and their significance

The COVID-19 pandemic highlighted the ability of epidemics to evolve through the emergence of successive strains of greater infectiousness, which prompted the insight that hyper-exponential growth (HEG) can arise in the development of an epidemic. The phenomenon of HEG has intrigued many researchers, because of some radical differences from exponential growth such as the finite-time singularity. However, the actual mechanism of HEG was usually hidden or needed to be added phenomenologically. Thus, there was little insight into how constraints would be triggered. In this study, we explore an SIR model with evolving parameters leading to a discrete sequence of variants. This allows us to consider the HEG phenomenon in greater depth and with greater mathematical rigour. The model yields a mechanistic description of what happens at the collapse of HEG. The model also yields closed expressions for important features, such as the critical time to the singularity, in terms of basic demographic parameters. Our analysis flags a few important issues needing further research, such as the stochastic character of the emergence of variants. Greater understanding of the HEG process will yield dividends in other fields, since modern societies exhibit HEG at several levels, such as human population growth, economic indicators and technological innovation. For this reason, more research on HEG remains imperative, especially on HEG in the presence of limited resources.