On the Reproduction Number of Epidemics with Sub-exponential Growth

Abstract

Mathematical models for infectious disease epidemics often assume, explicitly or implicitly, an initial exponential growth for the number of new infections. Recent studies have highlighted that some historical epidemics actually grew sub-exponentially. Using models that presume exponential growth for such epidemics may not faithfully characterize the epidemiological parameters, especially the reproduction number. Here, using a well-established “generalized-growth” model, we derive analytical expressions of the time-dependent reproduction number and show that this quantity for epidemics with sub-exponential growth decreases and approaches unity over disease generation intervals. We use this theoretical framework to estimate the reproduction number for synthetic and real epidemics. Our findings suggest that estimates of the reproduction number during the early stages of disease outset are subject to substantial uncertainty regardless of the underlying assumptions for the epidemic growth.